+++ /dev/null
-/*
- * FILE: d0.h
- * AUTHOR: Rudolf Polzer - divVerent@xonotic.org
- *
- * Copyright (c) 2010, Rudolf Polzer
- * All rights reserved.
- *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions
- * are met:
- * 1. Redistributions of source code must retain the above copyright
- * notice, this list of conditions and the following disclaimer.
- * 2. Redistributions in binary form must reproduce the above copyright
- * notice, this list of conditions and the following disclaimer in the
- * documentation and/or other materials provided with the distribution.
- * 3. Neither the name of the copyright holder nor the names of contributors
- * may be used to endorse or promote products derived from this software
- * without specific prior written permission.
- *
- * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTOR(S) ``AS IS'' AND
- * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
- * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
- * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTOR(S) BE LIABLE
- * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
- * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
- * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
- * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
- * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
- * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
- * SUCH DAMAGE.
- *
- * $Format:commit %H$
- * $Id: 6c55afeb50f24bd316079ae46582e65f8020b19b $
- */
-
-#ifndef __D0_H__
-#define __D0_H__
-
-#include <unistd.h> // size_t
-
-#define D0_EXPORT __attribute__((__visibility__("default")))
-#define D0_USED __attribute__((used))
-#define D0_WARN_UNUSED_RESULT __attribute__((warn_unused_result))
-#define D0_BOOL int
-
-typedef void *(d0_malloc_t)(size_t len);
-typedef void (d0_free_t)(void *p);
-typedef void *(d0_createmutex_t)(void);
-typedef void (d0_destroymutex_t)(void *);
-typedef int (d0_lockmutex_t)(void *); // zero on success
-typedef int (d0_unlockmutex_t)(void *); // zero on success
-
-extern d0_malloc_t *d0_malloc;
-extern d0_free_t *d0_free;
-extern d0_createmutex_t *d0_createmutex;
-extern d0_destroymutex_t *d0_destroymutex;
-extern d0_lockmutex_t *d0_lockmutex;
-extern d0_unlockmutex_t *d0_unlockmutex;
-
-void d0_setmallocfuncs(d0_malloc_t *m, d0_free_t *f);
-void d0_setmutexfuncs(d0_createmutex_t *c, d0_destroymutex_t *d, d0_lockmutex_t *l, d0_unlockmutex_t *u);
-void d0_initfuncs(void); // initializes them, this needs to be only called internally once
-
-extern const char *d0_bsd_license_notice;
-
-#endif
+++ /dev/null
-/*
- * FILE: d0_blind_id.h
- * AUTHOR: Rudolf Polzer - divVerent@xonotic.org
- *
- * Copyright (c) 2010, Rudolf Polzer
- * All rights reserved.
- *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions
- * are met:
- * 1. Redistributions of source code must retain the above copyright
- * notice, this list of conditions and the following disclaimer.
- * 2. Redistributions in binary form must reproduce the above copyright
- * notice, this list of conditions and the following disclaimer in the
- * documentation and/or other materials provided with the distribution.
- * 3. Neither the name of the copyright holder nor the names of contributors
- * may be used to endorse or promote products derived from this software
- * without specific prior written permission.
- *
- * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTOR(S) ``AS IS'' AND
- * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
- * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
- * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTOR(S) BE LIABLE
- * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
- * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
- * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
- * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
- * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
- * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
- * SUCH DAMAGE.
- *
- * $Format:commit %H$
- * $Id: bf838f43093aceadcd2d20071684f1e7148a4332 $
- */
-
-#ifndef __D0_BLIND_ID_H__
-#define __D0_BLIND_ID_H__
-
-#include "d0.h"
-
-typedef struct d0_blind_id_s d0_blind_id_t;
-typedef D0_BOOL (*d0_fastreject_function) (const d0_blind_id_t *ctx, void *pass);
-
-D0_EXPORT D0_WARN_UNUSED_RESULT d0_blind_id_t *d0_blind_id_new(void);
-D0_EXPORT void d0_blind_id_free(d0_blind_id_t *a);
-D0_EXPORT void d0_blind_id_clear(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_copy(d0_blind_id_t *ctx, const d0_blind_id_t *src);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_key(d0_blind_id_t *ctx, int k);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_key_fastreject(d0_blind_id_t *ctx, int k, d0_fastreject_function reject, void *pass);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_key(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_public_key(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_key(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_public_key(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_fingerprint64_public_key(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_id_modulus(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_id_modulus(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_id_modulus(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_id_start(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_id_request(d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_answer_private_id_request(const d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_finish_private_id_request(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_id_request_camouflage(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_id_request_camouflage(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_id(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_public_id(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_public_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_start(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL send_modulus, const char *message, size_t msglen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_challenge(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL recv_modulus, const char *inbuf, size_t inbuflen, char *outbuf, size_t *outbuflen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_response(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_verify(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen, char *msg, size_t *msglen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_generate_missing_signature(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_sign(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL send_modulus, const char *message, size_t msglen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_sign_detached(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL send_modulus, const char *message, size_t msglen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_verify(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL recv_modulus, const char *inbuf, size_t inbuflen, char *msg, size_t *msglen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_verify_detached(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL recv_modulus, const char *inbuf, size_t inbuflen, const char *msg, size_t msglen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_fingerprint64_public_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_verify_public_id(const d0_blind_id_t *ctx, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_verify_private_id(const d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sessionkey_public_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen); // can only be done after successful key exchange, this performs a modpow; key length is limited by SHA_DIGESTSIZE for now; also ONLY valid after successful d0_blind_id_authenticate_with_private_id_verify/d0_blind_id_fingerprint64_public_id
-
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_INITIALIZE(void);
-D0_EXPORT void d0_blind_id_SHUTDOWN(void);
-
-D0_EXPORT void d0_blind_id_util_sha256(char *out, const char *in, size_t n);
-
-// for exporting
-D0_EXPORT void d0_blind_id_setmallocfuncs(d0_malloc_t *m, d0_free_t *f);
-D0_EXPORT void d0_blind_id_setmutexfuncs(d0_createmutex_t *c, d0_destroymutex_t *d, d0_lockmutex_t *l, d0_unlockmutex_t *u);
-
-#endif
+++ /dev/null
-// from http://www.efgh.com/software/rijndael.htm (public domain)
-
-#ifndef H__RIJNDAEL
-#define H__RIJNDAEL
-
-#include "d0.h"
-
-D0_EXPORT int d0_rijndael_setup_encrypt(unsigned long *rk, const unsigned char *key,
- int keybits);
-D0_EXPORT int d0_rijndael_setup_decrypt(unsigned long *rk, const unsigned char *key,
- int keybits);
-D0_EXPORT void d0_rijndael_encrypt(const unsigned long *rk, int nrounds,
- const unsigned char plaintext[16], unsigned char ciphertext[16]);
-D0_EXPORT void d0_rijndael_decrypt(const unsigned long *rk, int nrounds,
- const unsigned char ciphertext[16], unsigned char plaintext[16]);
-
-#define D0_RIJNDAEL_KEYLENGTH(keybits) ((keybits)/8)
-#define D0_RIJNDAEL_RKLENGTH(keybits) ((keybits)/8+28)
-#define D0_RIJNDAEL_NROUNDS(keybits) ((keybits)/32+6)
-
-#endif
+++ /dev/null
-# libd0_blind_id.la - a libtool library file
-# Generated by ltmain.sh (GNU libtool) 2.2.6b Debian-2.2.6b-2ubuntu1
-#
-# Please DO NOT delete this file!
-# It is necessary for linking the library.
-
-# The name that we can dlopen(3).
-dlname=''
-
-# Names of this library.
-library_names=''
-
-# The name of the static archive.
-old_library='libd0_blind_id.a'
-
-# Linker flags that can not go in dependency_libs.
-inherited_linker_flags=''
-
-# Libraries that this one depends upon.
-dependency_libs=' -L/tmp/d0_blind_id.deps/lib/ /tmp/gg/lib/libgmp.la'
-
-# Names of additional weak libraries provided by this library
-weak_library_names=''
-
-# Version information for libd0_blind_id.
-current=0
-age=0
-revision=0
-
-# Is this an already installed library?
-installed=yes
-
-# Should we warn about portability when linking against -modules?
-shouldnotlink=no
-
-# Files to dlopen/dlpreopen
-dlopen=''
-dlpreopen=''
-
-# Directory that this library needs to be installed in:
-libdir='/usr/local/lib'
+++ /dev/null
-libd0_blind_id.so.0.0.0
\ No newline at end of file
+++ /dev/null
-libd0_blind_id.so.0.0.0
\ No newline at end of file
+++ /dev/null
-# libd0_rijndael.la - a libtool library file
-# Generated by ltmain.sh (GNU libtool) 2.2.6b Debian-2.2.6b-2ubuntu1
-#
-# Please DO NOT delete this file!
-# It is necessary for linking the library.
-
-# The name that we can dlopen(3).
-dlname=''
-
-# Names of this library.
-library_names=''
-
-# The name of the static archive.
-old_library='libd0_rijndael.a'
-
-# Linker flags that can not go in dependency_libs.
-inherited_linker_flags=''
-
-# Libraries that this one depends upon.
-dependency_libs=' -L/tmp/d0_blind_id.deps/lib/ /tmp/gg/lib/libgmp.la'
-
-# Names of additional weak libraries provided by this library
-weak_library_names=''
-
-# Version information for libd0_rijndael.
-current=0
-age=0
-revision=0
-
-# Is this an already installed library?
-installed=yes
-
-# Should we warn about portability when linking against -modules?
-shouldnotlink=no
-
-# Files to dlopen/dlpreopen
-dlopen=''
-dlpreopen=''
-
-# Directory that this library needs to be installed in:
-libdir='/usr/local/lib'
+++ /dev/null
-libd0_rijndael.so.0.0.0
\ No newline at end of file
+++ /dev/null
-libd0_rijndael.so.0.0.0
\ No newline at end of file
+++ /dev/null
-prefix=/usr/local
-exec_prefix=${prefix}
-libdir=${exec_prefix}/lib
-includedir=${prefix}/include
-
-Name: Blind-ID
-Description: Library for user identification using RSA blind signatures
-Requires:
-Version: 0.5
-Libs: -L${libdir} -ld0_blind_id
-Cflags: -I${includedir}/d0_blind_id
+++ /dev/null
-prefix=/usr/local
-exec_prefix=${prefix}
-libdir=${exec_prefix}/lib
-includedir=${prefix}/include
-
-Name: Rijndael
-Description: Library for Rijndael encryption
-Requires:
-Version: 0.5
-Libs: -L${libdir} -ld0_rijndael
-Cflags: -I${includedir}/d0_blind_id
+++ /dev/null
-/* Definitions for GNU multiple precision functions. -*- mode: c -*-
-
-Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1999, 2000, 2001, 2002, 2003,
-2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
-
-This file is part of the GNU MP Library.
-
-The GNU MP Library is free software; you can redistribute it and/or modify
-it under the terms of the GNU Lesser General Public License as published by
-the Free Software Foundation; either version 3 of the License, or (at your
-option) any later version.
-
-The GNU MP Library is distributed in the hope that it will be useful, but
-WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
-or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
-License for more details.
-
-You should have received a copy of the GNU Lesser General Public License
-along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */
-
-#ifndef __GMP_H__
-
-#if defined (__cplusplus)
-#include <iosfwd> /* for std::istream, std::ostream, std::string */
-#include <cstdio>
-#endif
-
-
-/* Instantiated by configure. */
-#if ! defined (__GMP_WITHIN_CONFIGURE)
-#define __GMP_HAVE_HOST_CPU_FAMILY_power 0
-#define __GMP_HAVE_HOST_CPU_FAMILY_powerpc 0
-#define GMP_LIMB_BITS 32
-#define GMP_NAIL_BITS 0
-#endif
-#define GMP_NUMB_BITS (GMP_LIMB_BITS - GMP_NAIL_BITS)
-#define GMP_NUMB_MASK ((~ __GMP_CAST (mp_limb_t, 0)) >> GMP_NAIL_BITS)
-#define GMP_NUMB_MAX GMP_NUMB_MASK
-#define GMP_NAIL_MASK (~ GMP_NUMB_MASK)
-
-
-/* The following (everything under ifndef __GNU_MP__) must be identical in
- gmp.h and mp.h to allow both to be included in an application or during
- the library build. */
-#ifndef __GNU_MP__
-#define __GNU_MP__ 5
-
-#define __need_size_t /* tell gcc stddef.h we only want size_t */
-#if defined (__cplusplus)
-#include <cstddef> /* for size_t */
-#else
-#include <stddef.h> /* for size_t */
-#endif
-#undef __need_size_t
-
-/* Instantiated by configure. */
-#if ! defined (__GMP_WITHIN_CONFIGURE)
-/* #undef _LONG_LONG_LIMB */
-#define __GMP_LIBGMP_DLL 0
-#endif
-
-
-/* __STDC__ - some ANSI compilers define this only to 0, hence the use of
- "defined" and not "__STDC__-0". In particular Sun workshop C 5.0
- sets __STDC__ to 0, but requires "##" for token pasting.
-
- _AIX - gnu ansidecl.h asserts that all known AIX compilers are ANSI but
- don't always define __STDC__.
-
- __DECC - current versions of DEC C (5.9 for instance) for alpha are ANSI,
- but don't define __STDC__ in their default mode. Don't know if old
- versions might have been K&R, but let's not worry about that unless
- someone is still using one.
-
- _mips - gnu ansidecl.h says the RISC/OS MIPS compiler is ANSI in SVR4
- mode, but doesn't define __STDC__.
-
- _MSC_VER - Microsoft C is ANSI, but __STDC__ is undefined unless the /Za
- option is given (in which case it's 1).
-
- _WIN32 - tested for by gnu ansidecl.h, no doubt on the assumption that
- all w32 compilers are ansi.
-
- Note: This same set of tests is used by gen-psqr.c and
- demos/expr/expr-impl.h, so if anything needs adding, then be sure to
- update those too. */
-
-#if defined (__STDC__) \
- || defined (__cplusplus) \
- || defined (_AIX) \
- || defined (__DECC) \
- || (defined (__mips) && defined (_SYSTYPE_SVR4)) \
- || defined (_MSC_VER) \
- || defined (_WIN32)
-#define __GMP_HAVE_CONST 1
-#define __GMP_HAVE_PROTOTYPES 1
-#define __GMP_HAVE_TOKEN_PASTE 1
-#else
-#define __GMP_HAVE_CONST 0
-#define __GMP_HAVE_PROTOTYPES 0
-#define __GMP_HAVE_TOKEN_PASTE 0
-#endif
-
-
-#if __GMP_HAVE_CONST
-#define __gmp_const const
-#define __gmp_signed signed
-#else
-#define __gmp_const
-#define __gmp_signed
-#endif
-
-
-/* __GMP_DECLSPEC supports Windows DLL versions of libgmp, and is empty in
- all other circumstances.
-
- When compiling objects for libgmp, __GMP_DECLSPEC is an export directive,
- or when compiling for an application it's an import directive. The two
- cases are differentiated by __GMP_WITHIN_GMP defined by the GMP Makefiles
- (and not defined from an application).
-
- __GMP_DECLSPEC_XX is similarly used for libgmpxx. __GMP_WITHIN_GMPXX
- indicates when building libgmpxx, and in that case libgmpxx functions are
- exports, but libgmp functions which might get called are imports.
-
- libmp.la uses __GMP_DECLSPEC, just as if it were libgmp.la. libgmp and
- libmp don't call each other, so there's no conflict or confusion.
-
- Libtool DLL_EXPORT define is not used.
-
- There's no attempt to support GMP built both static and DLL. Doing so
- would mean applications would have to tell us which of the two is going
- to be used when linking, and that seems very tedious and error prone if
- using GMP by hand, and equally tedious from a package since autoconf and
- automake don't give much help.
-
- __GMP_DECLSPEC is required on all documented global functions and
- variables, the various internals in gmp-impl.h etc can be left unadorned.
- But internals used by the test programs or speed measuring programs
- should have __GMP_DECLSPEC, and certainly constants or variables must
- have it or the wrong address will be resolved.
-
- In gcc __declspec can go at either the start or end of a prototype.
-
- In Microsoft C __declspec must go at the start, or after the type like
- void __declspec(...) *foo()". There's no __dllexport or anything to
- guard against someone foolish #defining dllexport. _export used to be
- available, but no longer.
-
- In Borland C _export still exists, but needs to go after the type, like
- "void _export foo();". Would have to change the __GMP_DECLSPEC syntax to
- make use of that. Probably more trouble than it's worth. */
-
-#if defined (__GNUC__)
-#define __GMP_DECLSPEC_EXPORT __declspec(__dllexport__)
-#define __GMP_DECLSPEC_IMPORT __declspec(__dllimport__)
-#endif
-#if defined (_MSC_VER) || defined (__BORLANDC__)
-#define __GMP_DECLSPEC_EXPORT __declspec(dllexport)
-#define __GMP_DECLSPEC_IMPORT __declspec(dllimport)
-#endif
-#ifdef __WATCOMC__
-#define __GMP_DECLSPEC_EXPORT __export
-#define __GMP_DECLSPEC_IMPORT __import
-#endif
-#ifdef __IBMC__
-#define __GMP_DECLSPEC_EXPORT _Export
-#define __GMP_DECLSPEC_IMPORT _Import
-#endif
-
-#if __GMP_LIBGMP_DLL
-#if __GMP_WITHIN_GMP
-/* compiling to go into a DLL libgmp */
-#define __GMP_DECLSPEC __GMP_DECLSPEC_EXPORT
-#else
-/* compiling to go into an application which will link to a DLL libgmp */
-#define __GMP_DECLSPEC __GMP_DECLSPEC_IMPORT
-#endif
-#else
-/* all other cases */
-#define __GMP_DECLSPEC
-#endif
-
-
-#ifdef __GMP_SHORT_LIMB
-typedef unsigned int mp_limb_t;
-typedef int mp_limb_signed_t;
-#else
-#ifdef _LONG_LONG_LIMB
-typedef unsigned long long int mp_limb_t;
-typedef long long int mp_limb_signed_t;
-#else
-typedef unsigned long int mp_limb_t;
-typedef long int mp_limb_signed_t;
-#endif
-#endif
-typedef unsigned long int mp_bitcnt_t;
-
-/* For reference, note that the name __mpz_struct gets into C++ mangled
- function names, which means although the "__" suggests an internal, we
- must leave this name for binary compatibility. */
-typedef struct
-{
- int _mp_alloc; /* Number of *limbs* allocated and pointed
- to by the _mp_d field. */
- int _mp_size; /* abs(_mp_size) is the number of limbs the
- last field points to. If _mp_size is
- negative this is a negative number. */
- mp_limb_t *_mp_d; /* Pointer to the limbs. */
-} __mpz_struct;
-
-#endif /* __GNU_MP__ */
-
-
-typedef __mpz_struct MP_INT; /* gmp 1 source compatibility */
-typedef __mpz_struct mpz_t[1];
-
-typedef mp_limb_t * mp_ptr;
-typedef __gmp_const mp_limb_t * mp_srcptr;
-#if defined (_CRAY) && ! defined (_CRAYMPP)
-/* plain `int' is much faster (48 bits) */
-#define __GMP_MP_SIZE_T_INT 1
-typedef int mp_size_t;
-typedef int mp_exp_t;
-#else
-#define __GMP_MP_SIZE_T_INT 0
-typedef long int mp_size_t;
-typedef long int mp_exp_t;
-#endif
-
-typedef struct
-{
- __mpz_struct _mp_num;
- __mpz_struct _mp_den;
-} __mpq_struct;
-
-typedef __mpq_struct MP_RAT; /* gmp 1 source compatibility */
-typedef __mpq_struct mpq_t[1];
-
-typedef struct
-{
- int _mp_prec; /* Max precision, in number of `mp_limb_t's.
- Set by mpf_init and modified by
- mpf_set_prec. The area pointed to by the
- _mp_d field contains `prec' + 1 limbs. */
- int _mp_size; /* abs(_mp_size) is the number of limbs the
- last field points to. If _mp_size is
- negative this is a negative number. */
- mp_exp_t _mp_exp; /* Exponent, in the base of `mp_limb_t'. */
- mp_limb_t *_mp_d; /* Pointer to the limbs. */
-} __mpf_struct;
-
-/* typedef __mpf_struct MP_FLOAT; */
-typedef __mpf_struct mpf_t[1];
-
-/* Available random number generation algorithms. */
-typedef enum
-{
- GMP_RAND_ALG_DEFAULT = 0,
- GMP_RAND_ALG_LC = GMP_RAND_ALG_DEFAULT /* Linear congruential. */
-} gmp_randalg_t;
-
-/* Random state struct. */
-typedef struct
-{
- mpz_t _mp_seed; /* _mp_d member points to state of the generator. */
- gmp_randalg_t _mp_alg; /* Currently unused. */
- union {
- void *_mp_lc; /* Pointer to function pointers structure. */
- } _mp_algdata;
-} __gmp_randstate_struct;
-typedef __gmp_randstate_struct gmp_randstate_t[1];
-
-/* Types for function declarations in gmp files. */
-/* ??? Should not pollute user name space with these ??? */
-typedef __gmp_const __mpz_struct *mpz_srcptr;
-typedef __mpz_struct *mpz_ptr;
-typedef __gmp_const __mpf_struct *mpf_srcptr;
-typedef __mpf_struct *mpf_ptr;
-typedef __gmp_const __mpq_struct *mpq_srcptr;
-typedef __mpq_struct *mpq_ptr;
-
-
-/* This is not wanted in mp.h, so put it outside the __GNU_MP__ common
- section. */
-#if __GMP_LIBGMP_DLL
-#if __GMP_WITHIN_GMPXX
-/* compiling to go into a DLL libgmpxx */
-#define __GMP_DECLSPEC_XX __GMP_DECLSPEC_EXPORT
-#else
-/* compiling to go into a application which will link to a DLL libgmpxx */
-#define __GMP_DECLSPEC_XX __GMP_DECLSPEC_IMPORT
-#endif
-#else
-/* all other cases */
-#define __GMP_DECLSPEC_XX
-#endif
-
-
-#if __GMP_HAVE_PROTOTYPES
-#define __GMP_PROTO(x) x
-#else
-#define __GMP_PROTO(x) ()
-#endif
-
-#ifndef __MPN
-#if __GMP_HAVE_TOKEN_PASTE
-#define __MPN(x) __gmpn_##x
-#else
-#define __MPN(x) __gmpn_/**/x
-#endif
-#endif
-
-/* For reference, "defined(EOF)" cannot be used here. In g++ 2.95.4,
- <iostream> defines EOF but not FILE. */
-#if defined (FILE) \
- || defined (H_STDIO) \
- || defined (_H_STDIO) /* AIX */ \
- || defined (_STDIO_H) /* glibc, Sun, SCO */ \
- || defined (_STDIO_H_) /* BSD, OSF */ \
- || defined (__STDIO_H) /* Borland */ \
- || defined (__STDIO_H__) /* IRIX */ \
- || defined (_STDIO_INCLUDED) /* HPUX */ \
- || defined (__dj_include_stdio_h_) /* DJGPP */ \
- || defined (_FILE_DEFINED) /* Microsoft */ \
- || defined (__STDIO__) /* Apple MPW MrC */ \
- || defined (_MSL_STDIO_H) /* Metrowerks */ \
- || defined (_STDIO_H_INCLUDED) /* QNX4 */ \
- || defined (_ISO_STDIO_ISO_H) /* Sun C++ */
-#define _GMP_H_HAVE_FILE 1
-#endif
-
-/* In ISO C, if a prototype involving "struct obstack *" is given without
- that structure defined, then the struct is scoped down to just the
- prototype, causing a conflict if it's subsequently defined for real. So
- only give prototypes if we've got obstack.h. */
-#if defined (_OBSTACK_H) /* glibc <obstack.h> */
-#define _GMP_H_HAVE_OBSTACK 1
-#endif
-
-/* The prototypes for gmp_vprintf etc are provided only if va_list is
- available, via an application having included <stdarg.h> or <varargs.h>.
- Usually va_list is a typedef so can't be tested directly, but C99
- specifies that va_start is a macro (and it was normally a macro on past
- systems too), so look for that.
-
- <stdio.h> will define some sort of va_list for vprintf and vfprintf, but
- let's not bother trying to use that since it's not standard and since
- application uses for gmp_vprintf etc will almost certainly require the
- whole <stdarg.h> or <varargs.h> anyway. */
-
-#ifdef va_start
-#define _GMP_H_HAVE_VA_LIST 1
-#endif
-
-/* Test for gcc >= maj.min, as per __GNUC_PREREQ in glibc */
-#if defined (__GNUC__) && defined (__GNUC_MINOR__)
-#define __GMP_GNUC_PREREQ(maj, min) \
- ((__GNUC__ << 16) + __GNUC_MINOR__ >= ((maj) << 16) + (min))
-#else
-#define __GMP_GNUC_PREREQ(maj, min) 0
-#endif
-
-/* "pure" is in gcc 2.96 and up, see "(gcc)Function Attributes". Basically
- it means a function does nothing but examine its arguments and memory
- (global or via arguments) to generate a return value, but changes nothing
- and has no side-effects. __GMP_NO_ATTRIBUTE_CONST_PURE lets
- tune/common.c etc turn this off when trying to write timing loops. */
-#if __GMP_GNUC_PREREQ (2,96) && ! defined (__GMP_NO_ATTRIBUTE_CONST_PURE)
-#define __GMP_ATTRIBUTE_PURE __attribute__ ((__pure__))
-#else
-#define __GMP_ATTRIBUTE_PURE
-#endif
-
-
-/* __GMP_CAST allows us to use static_cast in C++, so our macros are clean
- to "g++ -Wold-style-cast".
-
- Casts in "extern inline" code within an extern "C" block don't induce
- these warnings, so __GMP_CAST only needs to be used on documented
- macros. */
-
-#ifdef __cplusplus
-#define __GMP_CAST(type, expr) (static_cast<type> (expr))
-#else
-#define __GMP_CAST(type, expr) ((type) (expr))
-#endif
-
-
-/* An empty "throw ()" means the function doesn't throw any C++ exceptions,
- this can save some stack frame info in applications.
-
- Currently it's given only on functions which never divide-by-zero etc,
- don't allocate memory, and are expected to never need to allocate memory.
- This leaves open the possibility of a C++ throw from a future GMP
- exceptions scheme.
-
- mpz_set_ui etc are omitted to leave open the lazy allocation scheme
- described in doc/tasks.html. mpz_get_d etc are omitted to leave open
- exceptions for float overflows.
-
- Note that __GMP_NOTHROW must be given on any inlines the same as on their
- prototypes (for g++ at least, where they're used together). Note also
- that g++ 3.0 demands that __GMP_NOTHROW is before other attributes like
- __GMP_ATTRIBUTE_PURE. */
-
-#if defined (__cplusplus)
-#define __GMP_NOTHROW throw ()
-#else
-#define __GMP_NOTHROW
-#endif
-
-
-/* PORTME: What other compilers have a useful "extern inline"? "static
- inline" would be an acceptable substitute if the compiler (or linker)
- discards unused statics. */
-
- /* gcc has __inline__ in all modes, including strict ansi. Give a prototype
- for an inline too, so as to correctly specify "dllimport" on windows, in
- case the function is called rather than inlined.
- GCC 4.3 and above with -std=c99 or -std=gnu99 implements ISO C99
- inline semantics, unless -fgnu89-inline is used. */
-#ifdef __GNUC__
-#if (defined __GNUC_STDC_INLINE__) || (__GNUC__ == 4 && __GNUC_MINOR__ == 2)
-#define __GMP_EXTERN_INLINE extern __inline__ __attribute__ ((__gnu_inline__))
-#else
-#define __GMP_EXTERN_INLINE extern __inline__
-#endif
-#define __GMP_INLINE_PROTOTYPES 1
-#endif
-
-/* DEC C (eg. version 5.9) supports "static __inline foo()", even in -std1
- strict ANSI mode. Inlining is done even when not optimizing (ie. -O0
- mode, which is the default), but an unnecessary local copy of foo is
- emitted unless -O is used. "extern __inline" is accepted, but the
- "extern" appears to be ignored, ie. it becomes a plain global function
- but which is inlined within its file. Don't know if all old versions of
- DEC C supported __inline, but as a start let's do the right thing for
- current versions. */
-#ifdef __DECC
-#define __GMP_EXTERN_INLINE static __inline
-#endif
-
-/* SCO OpenUNIX 8 cc supports "static inline foo()" but not in -Xc strict
- ANSI mode (__STDC__ is 1 in that mode). Inlining only actually takes
- place under -O. Without -O "foo" seems to be emitted whether it's used
- or not, which is wasteful. "extern inline foo()" isn't useful, the
- "extern" is apparently ignored, so foo is inlined if possible but also
- emitted as a global, which causes multiple definition errors when
- building a shared libgmp. */
-#ifdef __SCO_VERSION__
-#if __SCO_VERSION__ > 400000000 && __STDC__ != 1 \
- && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE static inline
-#endif
-#endif
-
-/* Microsoft's C compiler accepts __inline */
-#ifdef _MSC_VER
-#define __GMP_EXTERN_INLINE __inline
-#endif
-
-/* Recent enough Sun C compilers want "inline" */
-#if defined (__SUNPRO_C) && __SUNPRO_C >= 0x560 \
- && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE inline
-#endif
-
-/* Somewhat older Sun C compilers want "static inline" */
-#if defined (__SUNPRO_C) && __SUNPRO_C >= 0x540 \
- && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE static inline
-#endif
-
-
-/* C++ always has "inline" and since it's a normal feature the linker should
- discard duplicate non-inlined copies, or if it doesn't then that's a
- problem for everyone, not just GMP. */
-#if defined (__cplusplus) && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE inline
-#endif
-
-/* Don't do any inlining within a configure run, since if the compiler ends
- up emitting copies of the code into the object file it can end up
- demanding the various support routines (like mpn_popcount) for linking,
- making the "alloca" test and perhaps others fail. And on hppa ia64 a
- pre-release gcc 3.2 was seen not respecting the "extern" in "extern
- __inline__", triggering this problem too. */
-#if defined (__GMP_WITHIN_CONFIGURE) && ! __GMP_WITHIN_CONFIGURE_INLINE
-#undef __GMP_EXTERN_INLINE
-#endif
-
-/* By default, don't give a prototype when there's going to be an inline
- version. Note in particular that Cray C++ objects to the combination of
- prototype and inline. */
-#ifdef __GMP_EXTERN_INLINE
-#ifndef __GMP_INLINE_PROTOTYPES
-#define __GMP_INLINE_PROTOTYPES 0
-#endif
-#else
-#define __GMP_INLINE_PROTOTYPES 1
-#endif
-
-
-#define __GMP_ABS(x) ((x) >= 0 ? (x) : -(x))
-#define __GMP_MAX(h,i) ((h) > (i) ? (h) : (i))
-
-/* __GMP_USHRT_MAX is not "~ (unsigned short) 0" because short is promoted
- to int by "~". */
-#define __GMP_UINT_MAX (~ (unsigned) 0)
-#define __GMP_ULONG_MAX (~ (unsigned long) 0)
-#define __GMP_USHRT_MAX ((unsigned short) ~0)
-
-
-/* __builtin_expect is in gcc 3.0, and not in 2.95. */
-#if __GMP_GNUC_PREREQ (3,0)
-#define __GMP_LIKELY(cond) __builtin_expect ((cond) != 0, 1)
-#define __GMP_UNLIKELY(cond) __builtin_expect ((cond) != 0, 0)
-#else
-#define __GMP_LIKELY(cond) (cond)
-#define __GMP_UNLIKELY(cond) (cond)
-#endif
-
-#ifdef _CRAY
-#define __GMP_CRAY_Pragma(str) _Pragma (str)
-#else
-#define __GMP_CRAY_Pragma(str)
-#endif
-
-
-/* Allow direct user access to numerator and denominator of a mpq_t object. */
-#define mpq_numref(Q) (&((Q)->_mp_num))
-#define mpq_denref(Q) (&((Q)->_mp_den))
-
-
-#if defined (__cplusplus)
-extern "C" {
-using std::FILE;
-#endif
-
-#define mp_set_memory_functions __gmp_set_memory_functions
-__GMP_DECLSPEC void mp_set_memory_functions __GMP_PROTO ((void *(*) (size_t),
- void *(*) (void *, size_t, size_t),
- void (*) (void *, size_t))) __GMP_NOTHROW;
-
-#define mp_get_memory_functions __gmp_get_memory_functions
-__GMP_DECLSPEC void mp_get_memory_functions __GMP_PROTO ((void *(**) (size_t),
- void *(**) (void *, size_t, size_t),
- void (**) (void *, size_t))) __GMP_NOTHROW;
-
-#define mp_bits_per_limb __gmp_bits_per_limb
-__GMP_DECLSPEC extern __gmp_const int mp_bits_per_limb;
-
-#define gmp_errno __gmp_errno
-__GMP_DECLSPEC extern int gmp_errno;
-
-#define gmp_version __gmp_version
-__GMP_DECLSPEC extern __gmp_const char * __gmp_const gmp_version;
-
-
-/**************** Random number routines. ****************/
-
-/* obsolete */
-#define gmp_randinit __gmp_randinit
-__GMP_DECLSPEC void gmp_randinit __GMP_PROTO ((gmp_randstate_t, gmp_randalg_t, ...));
-
-#define gmp_randinit_default __gmp_randinit_default
-__GMP_DECLSPEC void gmp_randinit_default __GMP_PROTO ((gmp_randstate_t));
-
-#define gmp_randinit_lc_2exp __gmp_randinit_lc_2exp
-__GMP_DECLSPEC void gmp_randinit_lc_2exp __GMP_PROTO ((gmp_randstate_t,
- mpz_srcptr, unsigned long int,
- mp_bitcnt_t));
-
-#define gmp_randinit_lc_2exp_size __gmp_randinit_lc_2exp_size
-__GMP_DECLSPEC int gmp_randinit_lc_2exp_size __GMP_PROTO ((gmp_randstate_t, mp_bitcnt_t));
-
-#define gmp_randinit_mt __gmp_randinit_mt
-__GMP_DECLSPEC void gmp_randinit_mt __GMP_PROTO ((gmp_randstate_t));
-
-#define gmp_randinit_set __gmp_randinit_set
-__GMP_DECLSPEC void gmp_randinit_set __GMP_PROTO ((gmp_randstate_t, __gmp_const __gmp_randstate_struct *));
-
-#define gmp_randseed __gmp_randseed
-__GMP_DECLSPEC void gmp_randseed __GMP_PROTO ((gmp_randstate_t, mpz_srcptr));
-
-#define gmp_randseed_ui __gmp_randseed_ui
-__GMP_DECLSPEC void gmp_randseed_ui __GMP_PROTO ((gmp_randstate_t, unsigned long int));
-
-#define gmp_randclear __gmp_randclear
-__GMP_DECLSPEC void gmp_randclear __GMP_PROTO ((gmp_randstate_t));
-
-#define gmp_urandomb_ui __gmp_urandomb_ui
-__GMP_DECLSPEC unsigned long gmp_urandomb_ui __GMP_PROTO ((gmp_randstate_t, unsigned long));
-
-#define gmp_urandomm_ui __gmp_urandomm_ui
-__GMP_DECLSPEC unsigned long gmp_urandomm_ui __GMP_PROTO ((gmp_randstate_t, unsigned long));
-
-
-/**************** Formatted output routines. ****************/
-
-#define gmp_asprintf __gmp_asprintf
-__GMP_DECLSPEC int gmp_asprintf __GMP_PROTO ((char **, __gmp_const char *, ...));
-
-#define gmp_fprintf __gmp_fprintf
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC int gmp_fprintf __GMP_PROTO ((FILE *, __gmp_const char *, ...));
-#endif
-
-#define gmp_obstack_printf __gmp_obstack_printf
-#if defined (_GMP_H_HAVE_OBSTACK)
-__GMP_DECLSPEC int gmp_obstack_printf __GMP_PROTO ((struct obstack *, __gmp_const char *, ...));
-#endif
-
-#define gmp_obstack_vprintf __gmp_obstack_vprintf
-#if defined (_GMP_H_HAVE_OBSTACK) && defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_obstack_vprintf __GMP_PROTO ((struct obstack *, __gmp_const char *, va_list));
-#endif
-
-#define gmp_printf __gmp_printf
-__GMP_DECLSPEC int gmp_printf __GMP_PROTO ((__gmp_const char *, ...));
-
-#define gmp_snprintf __gmp_snprintf
-__GMP_DECLSPEC int gmp_snprintf __GMP_PROTO ((char *, size_t, __gmp_const char *, ...));
-
-#define gmp_sprintf __gmp_sprintf
-__GMP_DECLSPEC int gmp_sprintf __GMP_PROTO ((char *, __gmp_const char *, ...));
-
-#define gmp_vasprintf __gmp_vasprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vasprintf __GMP_PROTO ((char **, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vfprintf __gmp_vfprintf
-#if defined (_GMP_H_HAVE_FILE) && defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vfprintf __GMP_PROTO ((FILE *, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vprintf __gmp_vprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vprintf __GMP_PROTO ((__gmp_const char *, va_list));
-#endif
-
-#define gmp_vsnprintf __gmp_vsnprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vsnprintf __GMP_PROTO ((char *, size_t, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vsprintf __gmp_vsprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vsprintf __GMP_PROTO ((char *, __gmp_const char *, va_list));
-#endif
-
-
-/**************** Formatted input routines. ****************/
-
-#define gmp_fscanf __gmp_fscanf
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC int gmp_fscanf __GMP_PROTO ((FILE *, __gmp_const char *, ...));
-#endif
-
-#define gmp_scanf __gmp_scanf
-__GMP_DECLSPEC int gmp_scanf __GMP_PROTO ((__gmp_const char *, ...));
-
-#define gmp_sscanf __gmp_sscanf
-__GMP_DECLSPEC int gmp_sscanf __GMP_PROTO ((__gmp_const char *, __gmp_const char *, ...));
-
-#define gmp_vfscanf __gmp_vfscanf
-#if defined (_GMP_H_HAVE_FILE) && defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vfscanf __GMP_PROTO ((FILE *, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vscanf __gmp_vscanf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vscanf __GMP_PROTO ((__gmp_const char *, va_list));
-#endif
-
-#define gmp_vsscanf __gmp_vsscanf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vsscanf __GMP_PROTO ((__gmp_const char *, __gmp_const char *, va_list));
-#endif
-
-
-/**************** Integer (i.e. Z) routines. ****************/
-
-#define _mpz_realloc __gmpz_realloc
-#define mpz_realloc __gmpz_realloc
-__GMP_DECLSPEC void *_mpz_realloc __GMP_PROTO ((mpz_ptr, mp_size_t));
-
-#define mpz_abs __gmpz_abs
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_abs)
-__GMP_DECLSPEC void mpz_abs __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-#endif
-
-#define mpz_add __gmpz_add
-__GMP_DECLSPEC void mpz_add __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_add_ui __gmpz_add_ui
-__GMP_DECLSPEC void mpz_add_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_addmul __gmpz_addmul
-__GMP_DECLSPEC void mpz_addmul __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_addmul_ui __gmpz_addmul_ui
-__GMP_DECLSPEC void mpz_addmul_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_and __gmpz_and
-__GMP_DECLSPEC void mpz_and __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_array_init __gmpz_array_init
-__GMP_DECLSPEC void mpz_array_init __GMP_PROTO ((mpz_ptr, mp_size_t, mp_size_t));
-
-#define mpz_bin_ui __gmpz_bin_ui
-__GMP_DECLSPEC void mpz_bin_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_bin_uiui __gmpz_bin_uiui
-__GMP_DECLSPEC void mpz_bin_uiui __GMP_PROTO ((mpz_ptr, unsigned long int, unsigned long int));
-
-#define mpz_cdiv_q __gmpz_cdiv_q
-__GMP_DECLSPEC void mpz_cdiv_q __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_cdiv_q_2exp __gmpz_cdiv_q_2exp
-__GMP_DECLSPEC void mpz_cdiv_q_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long));
-
-#define mpz_cdiv_q_ui __gmpz_cdiv_q_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_q_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_cdiv_qr __gmpz_cdiv_qr
-__GMP_DECLSPEC void mpz_cdiv_qr __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_cdiv_qr_ui __gmpz_cdiv_qr_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_qr_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_cdiv_r __gmpz_cdiv_r
-__GMP_DECLSPEC void mpz_cdiv_r __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_cdiv_r_2exp __gmpz_cdiv_r_2exp
-__GMP_DECLSPEC void mpz_cdiv_r_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_cdiv_r_ui __gmpz_cdiv_r_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_r_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_cdiv_ui __gmpz_cdiv_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_clear __gmpz_clear
-__GMP_DECLSPEC void mpz_clear __GMP_PROTO ((mpz_ptr));
-
-#define mpz_clears __gmpz_clears
-__GMP_DECLSPEC void mpz_clears __GMP_PROTO ((mpz_ptr, ...));
-
-#define mpz_clrbit __gmpz_clrbit
-__GMP_DECLSPEC void mpz_clrbit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_cmp __gmpz_cmp
-__GMP_DECLSPEC int mpz_cmp __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmp_d __gmpz_cmp_d
-__GMP_DECLSPEC int mpz_cmp_d __GMP_PROTO ((mpz_srcptr, double)) __GMP_ATTRIBUTE_PURE;
-
-#define _mpz_cmp_si __gmpz_cmp_si
-__GMP_DECLSPEC int _mpz_cmp_si __GMP_PROTO ((mpz_srcptr, signed long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define _mpz_cmp_ui __gmpz_cmp_ui
-__GMP_DECLSPEC int _mpz_cmp_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmpabs __gmpz_cmpabs
-__GMP_DECLSPEC int mpz_cmpabs __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmpabs_d __gmpz_cmpabs_d
-__GMP_DECLSPEC int mpz_cmpabs_d __GMP_PROTO ((mpz_srcptr, double)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmpabs_ui __gmpz_cmpabs_ui
-__GMP_DECLSPEC int mpz_cmpabs_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_com __gmpz_com
-__GMP_DECLSPEC void mpz_com __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_combit __gmpz_combit
-__GMP_DECLSPEC void mpz_combit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_congruent_p __gmpz_congruent_p
-__GMP_DECLSPEC int mpz_congruent_p __GMP_PROTO ((mpz_srcptr, mpz_srcptr, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_congruent_2exp_p __gmpz_congruent_2exp_p
-__GMP_DECLSPEC int mpz_congruent_2exp_p __GMP_PROTO ((mpz_srcptr, mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_congruent_ui_p __gmpz_congruent_ui_p
-__GMP_DECLSPEC int mpz_congruent_ui_p __GMP_PROTO ((mpz_srcptr, unsigned long, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_divexact __gmpz_divexact
-__GMP_DECLSPEC void mpz_divexact __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_divexact_ui __gmpz_divexact_ui
-__GMP_DECLSPEC void mpz_divexact_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long));
-
-#define mpz_divisible_p __gmpz_divisible_p
-__GMP_DECLSPEC int mpz_divisible_p __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_divisible_ui_p __gmpz_divisible_ui_p
-__GMP_DECLSPEC int mpz_divisible_ui_p __GMP_PROTO ((mpz_srcptr, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_divisible_2exp_p __gmpz_divisible_2exp_p
-__GMP_DECLSPEC int mpz_divisible_2exp_p __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_dump __gmpz_dump
-__GMP_DECLSPEC void mpz_dump __GMP_PROTO ((mpz_srcptr));
-
-#define mpz_export __gmpz_export
-__GMP_DECLSPEC void *mpz_export __GMP_PROTO ((void *, size_t *, int, size_t, int, size_t, mpz_srcptr));
-
-#define mpz_fac_ui __gmpz_fac_ui
-__GMP_DECLSPEC void mpz_fac_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_fdiv_q __gmpz_fdiv_q
-__GMP_DECLSPEC void mpz_fdiv_q __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_fdiv_q_2exp __gmpz_fdiv_q_2exp
-__GMP_DECLSPEC void mpz_fdiv_q_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_fdiv_q_ui __gmpz_fdiv_q_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_q_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_fdiv_qr __gmpz_fdiv_qr
-__GMP_DECLSPEC void mpz_fdiv_qr __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_fdiv_qr_ui __gmpz_fdiv_qr_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_qr_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_fdiv_r __gmpz_fdiv_r
-__GMP_DECLSPEC void mpz_fdiv_r __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_fdiv_r_2exp __gmpz_fdiv_r_2exp
-__GMP_DECLSPEC void mpz_fdiv_r_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_fdiv_r_ui __gmpz_fdiv_r_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_r_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_fdiv_ui __gmpz_fdiv_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fib_ui __gmpz_fib_ui
-__GMP_DECLSPEC void mpz_fib_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_fib2_ui __gmpz_fib2_ui
-__GMP_DECLSPEC void mpz_fib2_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, unsigned long int));
-
-#define mpz_fits_sint_p __gmpz_fits_sint_p
-__GMP_DECLSPEC int mpz_fits_sint_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fits_slong_p __gmpz_fits_slong_p
-__GMP_DECLSPEC int mpz_fits_slong_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fits_sshort_p __gmpz_fits_sshort_p
-__GMP_DECLSPEC int mpz_fits_sshort_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fits_uint_p __gmpz_fits_uint_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_fits_uint_p)
-__GMP_DECLSPEC int mpz_fits_uint_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_fits_ulong_p __gmpz_fits_ulong_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_fits_ulong_p)
-__GMP_DECLSPEC int mpz_fits_ulong_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_fits_ushort_p __gmpz_fits_ushort_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_fits_ushort_p)
-__GMP_DECLSPEC int mpz_fits_ushort_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_gcd __gmpz_gcd
-__GMP_DECLSPEC void mpz_gcd __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_gcd_ui __gmpz_gcd_ui
-__GMP_DECLSPEC unsigned long int mpz_gcd_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_gcdext __gmpz_gcdext
-__GMP_DECLSPEC void mpz_gcdext __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_get_d __gmpz_get_d
-__GMP_DECLSPEC double mpz_get_d __GMP_PROTO ((mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_get_d_2exp __gmpz_get_d_2exp
-__GMP_DECLSPEC double mpz_get_d_2exp __GMP_PROTO ((signed long int *, mpz_srcptr));
-
-#define mpz_get_si __gmpz_get_si
-__GMP_DECLSPEC /* signed */ long int mpz_get_si __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_get_str __gmpz_get_str
-__GMP_DECLSPEC char *mpz_get_str __GMP_PROTO ((char *, int, mpz_srcptr));
-
-#define mpz_get_ui __gmpz_get_ui
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_get_ui)
-__GMP_DECLSPEC unsigned long int mpz_get_ui __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_getlimbn __gmpz_getlimbn
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_getlimbn)
-__GMP_DECLSPEC mp_limb_t mpz_getlimbn __GMP_PROTO ((mpz_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_hamdist __gmpz_hamdist
-__GMP_DECLSPEC mp_bitcnt_t mpz_hamdist __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_import __gmpz_import
-__GMP_DECLSPEC void mpz_import __GMP_PROTO ((mpz_ptr, size_t, int, size_t, int, size_t, __gmp_const void *));
-
-#define mpz_init __gmpz_init
-__GMP_DECLSPEC void mpz_init __GMP_PROTO ((mpz_ptr));
-
-#define mpz_init2 __gmpz_init2
-__GMP_DECLSPEC void mpz_init2 __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_inits __gmpz_inits
-__GMP_DECLSPEC void mpz_inits __GMP_PROTO ((mpz_ptr, ...));
-
-#define mpz_init_set __gmpz_init_set
-__GMP_DECLSPEC void mpz_init_set __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_init_set_d __gmpz_init_set_d
-__GMP_DECLSPEC void mpz_init_set_d __GMP_PROTO ((mpz_ptr, double));
-
-#define mpz_init_set_si __gmpz_init_set_si
-__GMP_DECLSPEC void mpz_init_set_si __GMP_PROTO ((mpz_ptr, signed long int));
-
-#define mpz_init_set_str __gmpz_init_set_str
-__GMP_DECLSPEC int mpz_init_set_str __GMP_PROTO ((mpz_ptr, __gmp_const char *, int));
-
-#define mpz_init_set_ui __gmpz_init_set_ui
-__GMP_DECLSPEC void mpz_init_set_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_inp_raw __gmpz_inp_raw
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_inp_raw __GMP_PROTO ((mpz_ptr, FILE *));
-#endif
-
-#define mpz_inp_str __gmpz_inp_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_inp_str __GMP_PROTO ((mpz_ptr, FILE *, int));
-#endif
-
-#define mpz_invert __gmpz_invert
-__GMP_DECLSPEC int mpz_invert __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_ior __gmpz_ior
-__GMP_DECLSPEC void mpz_ior __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_jacobi __gmpz_jacobi
-__GMP_DECLSPEC int mpz_jacobi __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_kronecker mpz_jacobi /* alias */
-
-#define mpz_kronecker_si __gmpz_kronecker_si
-__GMP_DECLSPEC int mpz_kronecker_si __GMP_PROTO ((mpz_srcptr, long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_kronecker_ui __gmpz_kronecker_ui
-__GMP_DECLSPEC int mpz_kronecker_ui __GMP_PROTO ((mpz_srcptr, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_si_kronecker __gmpz_si_kronecker
-__GMP_DECLSPEC int mpz_si_kronecker __GMP_PROTO ((long, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_ui_kronecker __gmpz_ui_kronecker
-__GMP_DECLSPEC int mpz_ui_kronecker __GMP_PROTO ((unsigned long, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_lcm __gmpz_lcm
-__GMP_DECLSPEC void mpz_lcm __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_lcm_ui __gmpz_lcm_ui
-__GMP_DECLSPEC void mpz_lcm_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long));
-
-#define mpz_legendre mpz_jacobi /* alias */
-
-#define mpz_lucnum_ui __gmpz_lucnum_ui
-__GMP_DECLSPEC void mpz_lucnum_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_lucnum2_ui __gmpz_lucnum2_ui
-__GMP_DECLSPEC void mpz_lucnum2_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, unsigned long int));
-
-#define mpz_millerrabin __gmpz_millerrabin
-__GMP_DECLSPEC int mpz_millerrabin __GMP_PROTO ((mpz_srcptr, int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_mod __gmpz_mod
-__GMP_DECLSPEC void mpz_mod __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_mod_ui mpz_fdiv_r_ui /* same as fdiv_r because divisor unsigned */
-
-#define mpz_mul __gmpz_mul
-__GMP_DECLSPEC void mpz_mul __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_mul_2exp __gmpz_mul_2exp
-__GMP_DECLSPEC void mpz_mul_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_mul_si __gmpz_mul_si
-__GMP_DECLSPEC void mpz_mul_si __GMP_PROTO ((mpz_ptr, mpz_srcptr, long int));
-
-#define mpz_mul_ui __gmpz_mul_ui
-__GMP_DECLSPEC void mpz_mul_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_neg __gmpz_neg
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_neg)
-__GMP_DECLSPEC void mpz_neg __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-#endif
-
-#define mpz_nextprime __gmpz_nextprime
-__GMP_DECLSPEC void mpz_nextprime __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_out_raw __gmpz_out_raw
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_out_raw __GMP_PROTO ((FILE *, mpz_srcptr));
-#endif
-
-#define mpz_out_str __gmpz_out_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_out_str __GMP_PROTO ((FILE *, int, mpz_srcptr));
-#endif
-
-#define mpz_perfect_power_p __gmpz_perfect_power_p
-__GMP_DECLSPEC int mpz_perfect_power_p __GMP_PROTO ((mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_perfect_square_p __gmpz_perfect_square_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_perfect_square_p)
-__GMP_DECLSPEC int mpz_perfect_square_p __GMP_PROTO ((mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_popcount __gmpz_popcount
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_popcount)
-__GMP_DECLSPEC mp_bitcnt_t mpz_popcount __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_pow_ui __gmpz_pow_ui
-__GMP_DECLSPEC void mpz_pow_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_powm __gmpz_powm
-__GMP_DECLSPEC void mpz_powm __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_powm_sec __gmpz_powm_sec
-__GMP_DECLSPEC void mpz_powm_sec __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_powm_ui __gmpz_powm_ui
-__GMP_DECLSPEC void mpz_powm_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int, mpz_srcptr));
-
-#define mpz_probab_prime_p __gmpz_probab_prime_p
-__GMP_DECLSPEC int mpz_probab_prime_p __GMP_PROTO ((mpz_srcptr, int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_random __gmpz_random
-__GMP_DECLSPEC void mpz_random __GMP_PROTO ((mpz_ptr, mp_size_t));
-
-#define mpz_random2 __gmpz_random2
-__GMP_DECLSPEC void mpz_random2 __GMP_PROTO ((mpz_ptr, mp_size_t));
-
-#define mpz_realloc2 __gmpz_realloc2
-__GMP_DECLSPEC void mpz_realloc2 __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_remove __gmpz_remove
-__GMP_DECLSPEC unsigned long int mpz_remove __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_root __gmpz_root
-__GMP_DECLSPEC int mpz_root __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_rootrem __gmpz_rootrem
-__GMP_DECLSPEC void mpz_rootrem __GMP_PROTO ((mpz_ptr,mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_rrandomb __gmpz_rrandomb
-__GMP_DECLSPEC void mpz_rrandomb __GMP_PROTO ((mpz_ptr, gmp_randstate_t, mp_bitcnt_t));
-
-#define mpz_scan0 __gmpz_scan0
-__GMP_DECLSPEC mp_bitcnt_t mpz_scan0 __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_scan1 __gmpz_scan1
-__GMP_DECLSPEC mp_bitcnt_t mpz_scan1 __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_set __gmpz_set
-__GMP_DECLSPEC void mpz_set __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_set_d __gmpz_set_d
-__GMP_DECLSPEC void mpz_set_d __GMP_PROTO ((mpz_ptr, double));
-
-#define mpz_set_f __gmpz_set_f
-__GMP_DECLSPEC void mpz_set_f __GMP_PROTO ((mpz_ptr, mpf_srcptr));
-
-#define mpz_set_q __gmpz_set_q
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_set_q)
-__GMP_DECLSPEC void mpz_set_q __GMP_PROTO ((mpz_ptr, mpq_srcptr));
-#endif
-
-#define mpz_set_si __gmpz_set_si
-__GMP_DECLSPEC void mpz_set_si __GMP_PROTO ((mpz_ptr, signed long int));
-
-#define mpz_set_str __gmpz_set_str
-__GMP_DECLSPEC int mpz_set_str __GMP_PROTO ((mpz_ptr, __gmp_const char *, int));
-
-#define mpz_set_ui __gmpz_set_ui
-__GMP_DECLSPEC void mpz_set_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_setbit __gmpz_setbit
-__GMP_DECLSPEC void mpz_setbit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_size __gmpz_size
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_size)
-__GMP_DECLSPEC size_t mpz_size __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_sizeinbase __gmpz_sizeinbase
-__GMP_DECLSPEC size_t mpz_sizeinbase __GMP_PROTO ((mpz_srcptr, int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_sqrt __gmpz_sqrt
-__GMP_DECLSPEC void mpz_sqrt __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_sqrtrem __gmpz_sqrtrem
-__GMP_DECLSPEC void mpz_sqrtrem __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr));
-
-#define mpz_sub __gmpz_sub
-__GMP_DECLSPEC void mpz_sub __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_sub_ui __gmpz_sub_ui
-__GMP_DECLSPEC void mpz_sub_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_ui_sub __gmpz_ui_sub
-__GMP_DECLSPEC void mpz_ui_sub __GMP_PROTO ((mpz_ptr, unsigned long int, mpz_srcptr));
-
-#define mpz_submul __gmpz_submul
-__GMP_DECLSPEC void mpz_submul __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_submul_ui __gmpz_submul_ui
-__GMP_DECLSPEC void mpz_submul_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_swap __gmpz_swap
-__GMP_DECLSPEC void mpz_swap __GMP_PROTO ((mpz_ptr, mpz_ptr)) __GMP_NOTHROW;
-
-#define mpz_tdiv_ui __gmpz_tdiv_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_tdiv_q __gmpz_tdiv_q
-__GMP_DECLSPEC void mpz_tdiv_q __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_tdiv_q_2exp __gmpz_tdiv_q_2exp
-__GMP_DECLSPEC void mpz_tdiv_q_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_tdiv_q_ui __gmpz_tdiv_q_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_q_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_tdiv_qr __gmpz_tdiv_qr
-__GMP_DECLSPEC void mpz_tdiv_qr __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_tdiv_qr_ui __gmpz_tdiv_qr_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_qr_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_tdiv_r __gmpz_tdiv_r
-__GMP_DECLSPEC void mpz_tdiv_r __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_tdiv_r_2exp __gmpz_tdiv_r_2exp
-__GMP_DECLSPEC void mpz_tdiv_r_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_tdiv_r_ui __gmpz_tdiv_r_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_r_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_tstbit __gmpz_tstbit
-__GMP_DECLSPEC int mpz_tstbit __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_ui_pow_ui __gmpz_ui_pow_ui
-__GMP_DECLSPEC void mpz_ui_pow_ui __GMP_PROTO ((mpz_ptr, unsigned long int, unsigned long int));
-
-#define mpz_urandomb __gmpz_urandomb
-__GMP_DECLSPEC void mpz_urandomb __GMP_PROTO ((mpz_ptr, gmp_randstate_t, mp_bitcnt_t));
-
-#define mpz_urandomm __gmpz_urandomm
-__GMP_DECLSPEC void mpz_urandomm __GMP_PROTO ((mpz_ptr, gmp_randstate_t, mpz_srcptr));
-
-#define mpz_xor __gmpz_xor
-#define mpz_eor __gmpz_xor
-__GMP_DECLSPEC void mpz_xor __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-
-/**************** Rational (i.e. Q) routines. ****************/
-
-#define mpq_abs __gmpq_abs
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpq_abs)
-__GMP_DECLSPEC void mpq_abs __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-#endif
-
-#define mpq_add __gmpq_add
-__GMP_DECLSPEC void mpq_add __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_canonicalize __gmpq_canonicalize
-__GMP_DECLSPEC void mpq_canonicalize __GMP_PROTO ((mpq_ptr));
-
-#define mpq_clear __gmpq_clear
-__GMP_DECLSPEC void mpq_clear __GMP_PROTO ((mpq_ptr));
-
-#define mpq_clears __gmpq_clears
-__GMP_DECLSPEC void mpq_clears __GMP_PROTO ((mpq_ptr, ...));
-
-#define mpq_cmp __gmpq_cmp
-__GMP_DECLSPEC int mpq_cmp __GMP_PROTO ((mpq_srcptr, mpq_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define _mpq_cmp_si __gmpq_cmp_si
-__GMP_DECLSPEC int _mpq_cmp_si __GMP_PROTO ((mpq_srcptr, long, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define _mpq_cmp_ui __gmpq_cmp_ui
-__GMP_DECLSPEC int _mpq_cmp_ui __GMP_PROTO ((mpq_srcptr, unsigned long int, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpq_div __gmpq_div
-__GMP_DECLSPEC void mpq_div __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_div_2exp __gmpq_div_2exp
-__GMP_DECLSPEC void mpq_div_2exp __GMP_PROTO ((mpq_ptr, mpq_srcptr, mp_bitcnt_t));
-
-#define mpq_equal __gmpq_equal
-__GMP_DECLSPEC int mpq_equal __GMP_PROTO ((mpq_srcptr, mpq_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpq_get_num __gmpq_get_num
-__GMP_DECLSPEC void mpq_get_num __GMP_PROTO ((mpz_ptr, mpq_srcptr));
-
-#define mpq_get_den __gmpq_get_den
-__GMP_DECLSPEC void mpq_get_den __GMP_PROTO ((mpz_ptr, mpq_srcptr));
-
-#define mpq_get_d __gmpq_get_d
-__GMP_DECLSPEC double mpq_get_d __GMP_PROTO ((mpq_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpq_get_str __gmpq_get_str
-__GMP_DECLSPEC char *mpq_get_str __GMP_PROTO ((char *, int, mpq_srcptr));
-
-#define mpq_init __gmpq_init
-__GMP_DECLSPEC void mpq_init __GMP_PROTO ((mpq_ptr));
-
-#define mpq_inits __gmpq_inits
-__GMP_DECLSPEC void mpq_inits __GMP_PROTO ((mpq_ptr, ...));
-
-#define mpq_inp_str __gmpq_inp_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpq_inp_str __GMP_PROTO ((mpq_ptr, FILE *, int));
-#endif
-
-#define mpq_inv __gmpq_inv
-__GMP_DECLSPEC void mpq_inv __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-
-#define mpq_mul __gmpq_mul
-__GMP_DECLSPEC void mpq_mul __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_mul_2exp __gmpq_mul_2exp
-__GMP_DECLSPEC void mpq_mul_2exp __GMP_PROTO ((mpq_ptr, mpq_srcptr, mp_bitcnt_t));
-
-#define mpq_neg __gmpq_neg
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpq_neg)
-__GMP_DECLSPEC void mpq_neg __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-#endif
-
-#define mpq_out_str __gmpq_out_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpq_out_str __GMP_PROTO ((FILE *, int, mpq_srcptr));
-#endif
-
-#define mpq_set __gmpq_set
-__GMP_DECLSPEC void mpq_set __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-
-#define mpq_set_d __gmpq_set_d
-__GMP_DECLSPEC void mpq_set_d __GMP_PROTO ((mpq_ptr, double));
-
-#define mpq_set_den __gmpq_set_den
-__GMP_DECLSPEC void mpq_set_den __GMP_PROTO ((mpq_ptr, mpz_srcptr));
-
-#define mpq_set_f __gmpq_set_f
-__GMP_DECLSPEC void mpq_set_f __GMP_PROTO ((mpq_ptr, mpf_srcptr));
-
-#define mpq_set_num __gmpq_set_num
-__GMP_DECLSPEC void mpq_set_num __GMP_PROTO ((mpq_ptr, mpz_srcptr));
-
-#define mpq_set_si __gmpq_set_si
-__GMP_DECLSPEC void mpq_set_si __GMP_PROTO ((mpq_ptr, signed long int, unsigned long int));
-
-#define mpq_set_str __gmpq_set_str
-__GMP_DECLSPEC int mpq_set_str __GMP_PROTO ((mpq_ptr, __gmp_const char *, int));
-
-#define mpq_set_ui __gmpq_set_ui
-__GMP_DECLSPEC void mpq_set_ui __GMP_PROTO ((mpq_ptr, unsigned long int, unsigned long int));
-
-#define mpq_set_z __gmpq_set_z
-__GMP_DECLSPEC void mpq_set_z __GMP_PROTO ((mpq_ptr, mpz_srcptr));
-
-#define mpq_sub __gmpq_sub
-__GMP_DECLSPEC void mpq_sub __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_swap __gmpq_swap
-__GMP_DECLSPEC void mpq_swap __GMP_PROTO ((mpq_ptr, mpq_ptr)) __GMP_NOTHROW;
-
-
-/**************** Float (i.e. F) routines. ****************/
-
-#define mpf_abs __gmpf_abs
-__GMP_DECLSPEC void mpf_abs __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_add __gmpf_add
-__GMP_DECLSPEC void mpf_add __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_add_ui __gmpf_add_ui
-__GMP_DECLSPEC void mpf_add_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-#define mpf_ceil __gmpf_ceil
-__GMP_DECLSPEC void mpf_ceil __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_clear __gmpf_clear
-__GMP_DECLSPEC void mpf_clear __GMP_PROTO ((mpf_ptr));
-
-#define mpf_clears __gmpf_clears
-__GMP_DECLSPEC void mpf_clears __GMP_PROTO ((mpf_ptr, ...));
-
-#define mpf_cmp __gmpf_cmp
-__GMP_DECLSPEC int mpf_cmp __GMP_PROTO ((mpf_srcptr, mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_cmp_d __gmpf_cmp_d
-__GMP_DECLSPEC int mpf_cmp_d __GMP_PROTO ((mpf_srcptr, double)) __GMP_ATTRIBUTE_PURE;
-
-#define mpf_cmp_si __gmpf_cmp_si
-__GMP_DECLSPEC int mpf_cmp_si __GMP_PROTO ((mpf_srcptr, signed long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_cmp_ui __gmpf_cmp_ui
-__GMP_DECLSPEC int mpf_cmp_ui __GMP_PROTO ((mpf_srcptr, unsigned long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_div __gmpf_div
-__GMP_DECLSPEC void mpf_div __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_div_2exp __gmpf_div_2exp
-__GMP_DECLSPEC void mpf_div_2exp __GMP_PROTO ((mpf_ptr, mpf_srcptr, mp_bitcnt_t));
-
-#define mpf_div_ui __gmpf_div_ui
-__GMP_DECLSPEC void mpf_div_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_dump __gmpf_dump
-__GMP_DECLSPEC void mpf_dump __GMP_PROTO ((mpf_srcptr));
-
-#define mpf_eq __gmpf_eq
-__GMP_DECLSPEC int mpf_eq __GMP_PROTO ((mpf_srcptr, mpf_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_sint_p __gmpf_fits_sint_p
-__GMP_DECLSPEC int mpf_fits_sint_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_slong_p __gmpf_fits_slong_p
-__GMP_DECLSPEC int mpf_fits_slong_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_sshort_p __gmpf_fits_sshort_p
-__GMP_DECLSPEC int mpf_fits_sshort_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_uint_p __gmpf_fits_uint_p
-__GMP_DECLSPEC int mpf_fits_uint_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_ulong_p __gmpf_fits_ulong_p
-__GMP_DECLSPEC int mpf_fits_ulong_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_ushort_p __gmpf_fits_ushort_p
-__GMP_DECLSPEC int mpf_fits_ushort_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_floor __gmpf_floor
-__GMP_DECLSPEC void mpf_floor __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_get_d __gmpf_get_d
-__GMP_DECLSPEC double mpf_get_d __GMP_PROTO ((mpf_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_d_2exp __gmpf_get_d_2exp
-__GMP_DECLSPEC double mpf_get_d_2exp __GMP_PROTO ((signed long int *, mpf_srcptr));
-
-#define mpf_get_default_prec __gmpf_get_default_prec
-__GMP_DECLSPEC mp_bitcnt_t mpf_get_default_prec __GMP_PROTO ((void)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_prec __gmpf_get_prec
-__GMP_DECLSPEC mp_bitcnt_t mpf_get_prec __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_si __gmpf_get_si
-__GMP_DECLSPEC long mpf_get_si __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_str __gmpf_get_str
-__GMP_DECLSPEC char *mpf_get_str __GMP_PROTO ((char *, mp_exp_t *, int, size_t, mpf_srcptr));
-
-#define mpf_get_ui __gmpf_get_ui
-__GMP_DECLSPEC unsigned long mpf_get_ui __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_init __gmpf_init
-__GMP_DECLSPEC void mpf_init __GMP_PROTO ((mpf_ptr));
-
-#define mpf_init2 __gmpf_init2
-__GMP_DECLSPEC void mpf_init2 __GMP_PROTO ((mpf_ptr, mp_bitcnt_t));
-
-#define mpf_inits __gmpf_inits
-__GMP_DECLSPEC void mpf_inits __GMP_PROTO ((mpf_ptr, ...));
-
-#define mpf_init_set __gmpf_init_set
-__GMP_DECLSPEC void mpf_init_set __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_init_set_d __gmpf_init_set_d
-__GMP_DECLSPEC void mpf_init_set_d __GMP_PROTO ((mpf_ptr, double));
-
-#define mpf_init_set_si __gmpf_init_set_si
-__GMP_DECLSPEC void mpf_init_set_si __GMP_PROTO ((mpf_ptr, signed long int));
-
-#define mpf_init_set_str __gmpf_init_set_str
-__GMP_DECLSPEC int mpf_init_set_str __GMP_PROTO ((mpf_ptr, __gmp_const char *, int));
-
-#define mpf_init_set_ui __gmpf_init_set_ui
-__GMP_DECLSPEC void mpf_init_set_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
-
-#define mpf_inp_str __gmpf_inp_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpf_inp_str __GMP_PROTO ((mpf_ptr, FILE *, int));
-#endif
-
-#define mpf_integer_p __gmpf_integer_p
-__GMP_DECLSPEC int mpf_integer_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_mul __gmpf_mul
-__GMP_DECLSPEC void mpf_mul __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_mul_2exp __gmpf_mul_2exp
-__GMP_DECLSPEC void mpf_mul_2exp __GMP_PROTO ((mpf_ptr, mpf_srcptr, mp_bitcnt_t));
-
-#define mpf_mul_ui __gmpf_mul_ui
-__GMP_DECLSPEC void mpf_mul_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_neg __gmpf_neg
-__GMP_DECLSPEC void mpf_neg __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_out_str __gmpf_out_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpf_out_str __GMP_PROTO ((FILE *, int, size_t, mpf_srcptr));
-#endif
-
-#define mpf_pow_ui __gmpf_pow_ui
-__GMP_DECLSPEC void mpf_pow_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_random2 __gmpf_random2
-__GMP_DECLSPEC void mpf_random2 __GMP_PROTO ((mpf_ptr, mp_size_t, mp_exp_t));
-
-#define mpf_reldiff __gmpf_reldiff
-__GMP_DECLSPEC void mpf_reldiff __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_set __gmpf_set
-__GMP_DECLSPEC void mpf_set __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_set_d __gmpf_set_d
-__GMP_DECLSPEC void mpf_set_d __GMP_PROTO ((mpf_ptr, double));
-
-#define mpf_set_default_prec __gmpf_set_default_prec
-__GMP_DECLSPEC void mpf_set_default_prec __GMP_PROTO ((mp_bitcnt_t)) __GMP_NOTHROW;
-
-#define mpf_set_prec __gmpf_set_prec
-__GMP_DECLSPEC void mpf_set_prec __GMP_PROTO ((mpf_ptr, mp_bitcnt_t));
-
-#define mpf_set_prec_raw __gmpf_set_prec_raw
-__GMP_DECLSPEC void mpf_set_prec_raw __GMP_PROTO ((mpf_ptr, mp_bitcnt_t)) __GMP_NOTHROW;
-
-#define mpf_set_q __gmpf_set_q
-__GMP_DECLSPEC void mpf_set_q __GMP_PROTO ((mpf_ptr, mpq_srcptr));
-
-#define mpf_set_si __gmpf_set_si
-__GMP_DECLSPEC void mpf_set_si __GMP_PROTO ((mpf_ptr, signed long int));
-
-#define mpf_set_str __gmpf_set_str
-__GMP_DECLSPEC int mpf_set_str __GMP_PROTO ((mpf_ptr, __gmp_const char *, int));
-
-#define mpf_set_ui __gmpf_set_ui
-__GMP_DECLSPEC void mpf_set_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
-
-#define mpf_set_z __gmpf_set_z
-__GMP_DECLSPEC void mpf_set_z __GMP_PROTO ((mpf_ptr, mpz_srcptr));
-
-#define mpf_size __gmpf_size
-__GMP_DECLSPEC size_t mpf_size __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_sqrt __gmpf_sqrt
-__GMP_DECLSPEC void mpf_sqrt __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_sqrt_ui __gmpf_sqrt_ui
-__GMP_DECLSPEC void mpf_sqrt_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
-
-#define mpf_sub __gmpf_sub
-__GMP_DECLSPEC void mpf_sub __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_sub_ui __gmpf_sub_ui
-__GMP_DECLSPEC void mpf_sub_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_swap __gmpf_swap
-__GMP_DECLSPEC void mpf_swap __GMP_PROTO ((mpf_ptr, mpf_ptr)) __GMP_NOTHROW;
-
-#define mpf_trunc __gmpf_trunc
-__GMP_DECLSPEC void mpf_trunc __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_ui_div __gmpf_ui_div
-__GMP_DECLSPEC void mpf_ui_div __GMP_PROTO ((mpf_ptr, unsigned long int, mpf_srcptr));
-
-#define mpf_ui_sub __gmpf_ui_sub
-__GMP_DECLSPEC void mpf_ui_sub __GMP_PROTO ((mpf_ptr, unsigned long int, mpf_srcptr));
-
-#define mpf_urandomb __gmpf_urandomb
-__GMP_DECLSPEC void mpf_urandomb __GMP_PROTO ((mpf_t, gmp_randstate_t, mp_bitcnt_t));
-
-
-/************ Low level positive-integer (i.e. N) routines. ************/
-
-/* This is ugly, but we need to make user calls reach the prefixed function. */
-
-#define mpn_add __MPN(add)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_add)
-__GMP_DECLSPEC mp_limb_t mpn_add __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_srcptr,mp_size_t));
-#endif
-
-#define mpn_add_1 __MPN(add_1)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_add_1)
-__GMP_DECLSPEC mp_limb_t mpn_add_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t)) __GMP_NOTHROW;
-#endif
-
-#define mpn_add_n __MPN(add_n)
-__GMP_DECLSPEC mp_limb_t mpn_add_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_addmul_1 __MPN(addmul_1)
-__GMP_DECLSPEC mp_limb_t mpn_addmul_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_cmp __MPN(cmp)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_cmp)
-__GMP_DECLSPEC int mpn_cmp __GMP_PROTO ((mp_srcptr, mp_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpn_divexact_by3(dst,src,size) \
- mpn_divexact_by3c (dst, src, size, __GMP_CAST (mp_limb_t, 0))
-
-#define mpn_divexact_by3c __MPN(divexact_by3c)
-__GMP_DECLSPEC mp_limb_t mpn_divexact_by3c __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_divmod_1(qp,np,nsize,dlimb) \
- mpn_divrem_1 (qp, __GMP_CAST (mp_size_t, 0), np, nsize, dlimb)
-
-#define mpn_divrem __MPN(divrem)
-__GMP_DECLSPEC mp_limb_t mpn_divrem __GMP_PROTO ((mp_ptr, mp_size_t, mp_ptr, mp_size_t, mp_srcptr, mp_size_t));
-
-#define mpn_divrem_1 __MPN(divrem_1)
-__GMP_DECLSPEC mp_limb_t mpn_divrem_1 __GMP_PROTO ((mp_ptr, mp_size_t, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_divrem_2 __MPN(divrem_2)
-__GMP_DECLSPEC mp_limb_t mpn_divrem_2 __GMP_PROTO ((mp_ptr, mp_size_t, mp_ptr, mp_size_t, mp_srcptr));
-
-#define mpn_gcd __MPN(gcd)
-__GMP_DECLSPEC mp_size_t mpn_gcd __GMP_PROTO ((mp_ptr, mp_ptr, mp_size_t, mp_ptr, mp_size_t));
-
-#define mpn_gcd_1 __MPN(gcd_1)
-__GMP_DECLSPEC mp_limb_t mpn_gcd_1 __GMP_PROTO ((mp_srcptr, mp_size_t, mp_limb_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_gcdext_1 __MPN(gcdext_1)
-__GMP_DECLSPEC mp_limb_t mpn_gcdext_1 __GMP_PROTO ((mp_limb_signed_t *, mp_limb_signed_t *, mp_limb_t, mp_limb_t));
-
-#define mpn_gcdext __MPN(gcdext)
-__GMP_DECLSPEC mp_size_t mpn_gcdext __GMP_PROTO ((mp_ptr, mp_ptr, mp_size_t *, mp_ptr, mp_size_t, mp_ptr, mp_size_t));
-
-#define mpn_get_str __MPN(get_str)
-__GMP_DECLSPEC size_t mpn_get_str __GMP_PROTO ((unsigned char *, int, mp_ptr, mp_size_t));
-
-#define mpn_hamdist __MPN(hamdist)
-__GMP_DECLSPEC mp_bitcnt_t mpn_hamdist __GMP_PROTO ((mp_srcptr, mp_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpn_lshift __MPN(lshift)
-__GMP_DECLSPEC mp_limb_t mpn_lshift __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, unsigned int));
-
-#define mpn_mod_1 __MPN(mod_1)
-__GMP_DECLSPEC mp_limb_t mpn_mod_1 __GMP_PROTO ((mp_srcptr, mp_size_t, mp_limb_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_mul __MPN(mul)
-__GMP_DECLSPEC mp_limb_t mpn_mul __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t));
-
-#define mpn_mul_1 __MPN(mul_1)
-__GMP_DECLSPEC mp_limb_t mpn_mul_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_mul_n __MPN(mul_n)
-__GMP_DECLSPEC void mpn_mul_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_sqr __MPN(sqr)
-__GMP_DECLSPEC void mpn_sqr __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-
-#define mpn_neg __MPN(neg)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_neg)
-__GMP_DECLSPEC mp_limb_t mpn_neg __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#endif
-
-#define mpn_com __MPN(com)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_com)
-__GMP_DECLSPEC void mpn_com __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#endif
-
-#define mpn_perfect_square_p __MPN(perfect_square_p)
-__GMP_DECLSPEC int mpn_perfect_square_p __GMP_PROTO ((mp_srcptr, mp_size_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_perfect_power_p __MPN(perfect_power_p)
-__GMP_DECLSPEC int mpn_perfect_power_p __GMP_PROTO ((mp_srcptr, mp_size_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_popcount __MPN(popcount)
-__GMP_DECLSPEC mp_bitcnt_t mpn_popcount __GMP_PROTO ((mp_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpn_pow_1 __MPN(pow_1)
-__GMP_DECLSPEC mp_size_t mpn_pow_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t, mp_ptr));
-
-/* undocumented now, but retained here for upward compatibility */
-#define mpn_preinv_mod_1 __MPN(preinv_mod_1)
-__GMP_DECLSPEC mp_limb_t mpn_preinv_mod_1 __GMP_PROTO ((mp_srcptr, mp_size_t, mp_limb_t, mp_limb_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_random __MPN(random)
-__GMP_DECLSPEC void mpn_random __GMP_PROTO ((mp_ptr, mp_size_t));
-
-#define mpn_random2 __MPN(random2)
-__GMP_DECLSPEC void mpn_random2 __GMP_PROTO ((mp_ptr, mp_size_t));
-
-#define mpn_rshift __MPN(rshift)
-__GMP_DECLSPEC mp_limb_t mpn_rshift __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, unsigned int));
-
-#define mpn_scan0 __MPN(scan0)
-__GMP_DECLSPEC mp_bitcnt_t mpn_scan0 __GMP_PROTO ((mp_srcptr, mp_bitcnt_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_scan1 __MPN(scan1)
-__GMP_DECLSPEC mp_bitcnt_t mpn_scan1 __GMP_PROTO ((mp_srcptr, mp_bitcnt_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_set_str __MPN(set_str)
-__GMP_DECLSPEC mp_size_t mpn_set_str __GMP_PROTO ((mp_ptr, __gmp_const unsigned char *, size_t, int));
-
-#define mpn_sqrtrem __MPN(sqrtrem)
-__GMP_DECLSPEC mp_size_t mpn_sqrtrem __GMP_PROTO ((mp_ptr, mp_ptr, mp_srcptr, mp_size_t));
-
-#define mpn_sub __MPN(sub)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_sub)
-__GMP_DECLSPEC mp_limb_t mpn_sub __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_srcptr,mp_size_t));
-#endif
-
-#define mpn_sub_1 __MPN(sub_1)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_sub_1)
-__GMP_DECLSPEC mp_limb_t mpn_sub_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t)) __GMP_NOTHROW;
-#endif
-
-#define mpn_sub_n __MPN(sub_n)
-__GMP_DECLSPEC mp_limb_t mpn_sub_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_submul_1 __MPN(submul_1)
-__GMP_DECLSPEC mp_limb_t mpn_submul_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_tdiv_qr __MPN(tdiv_qr)
-__GMP_DECLSPEC void mpn_tdiv_qr __GMP_PROTO ((mp_ptr, mp_ptr, mp_size_t, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t));
-
-#define mpn_and_n __MPN(and_n)
-__GMP_DECLSPEC void mpn_and_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_andn_n __MPN(andn_n)
-__GMP_DECLSPEC void mpn_andn_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_nand_n __MPN(nand_n)
-__GMP_DECLSPEC void mpn_nand_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_ior_n __MPN(ior_n)
-__GMP_DECLSPEC void mpn_ior_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_iorn_n __MPN(iorn_n)
-__GMP_DECLSPEC void mpn_iorn_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_nior_n __MPN(nior_n)
-__GMP_DECLSPEC void mpn_nior_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_xor_n __MPN(xor_n)
-__GMP_DECLSPEC void mpn_xor_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_xnor_n __MPN(xnor_n)
-__GMP_DECLSPEC void mpn_xnor_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_copyi __MPN(copyi)
-__GMP_DECLSPEC void mpn_copyi __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#define mpn_copyd __MPN(copyd)
-__GMP_DECLSPEC void mpn_copyd __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#define mpn_zero __MPN(zero)
-__GMP_DECLSPEC void mpn_zero __GMP_PROTO ((mp_ptr, mp_size_t));
-
-/**************** mpz inlines ****************/
-
-/* The following are provided as inlines where possible, but always exist as
- library functions too, for binary compatibility.
-
- Within gmp itself this inlining generally isn't relied on, since it
- doesn't get done for all compilers, whereas if something is worth
- inlining then it's worth arranging always.
-
- There are two styles of inlining here. When the same bit of code is
- wanted for the inline as for the library version, then __GMP_FORCE_foo
- arranges for that code to be emitted and the __GMP_EXTERN_INLINE
- directive suppressed, eg. mpz_fits_uint_p. When a different bit of code
- is wanted for the inline than for the library version, then
- __GMP_FORCE_foo arranges the inline to be suppressed, eg. mpz_abs. */
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpz_abs)
-__GMP_EXTERN_INLINE void
-mpz_abs (mpz_ptr __gmp_w, mpz_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpz_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_size = __GMP_ABS (__gmp_w->_mp_size);
-}
-#endif
-
-#if GMP_NAIL_BITS == 0
-#define __GMPZ_FITS_UTYPE_P(z,maxval) \
- mp_size_t __gmp_n = z->_mp_size; \
- mp_ptr __gmp_p = z->_mp_d; \
- return (__gmp_n == 0 || (__gmp_n == 1 && __gmp_p[0] <= maxval));
-#else
-#define __GMPZ_FITS_UTYPE_P(z,maxval) \
- mp_size_t __gmp_n = z->_mp_size; \
- mp_ptr __gmp_p = z->_mp_d; \
- return (__gmp_n == 0 || (__gmp_n == 1 && __gmp_p[0] <= maxval) \
- || (__gmp_n == 2 && __gmp_p[1] <= ((mp_limb_t) maxval >> GMP_NUMB_BITS)));
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_fits_uint_p)
-#if ! defined (__GMP_FORCE_mpz_fits_uint_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_fits_uint_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_UINT_MAX);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_fits_ulong_p)
-#if ! defined (__GMP_FORCE_mpz_fits_ulong_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_fits_ulong_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_ULONG_MAX);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_fits_ushort_p)
-#if ! defined (__GMP_FORCE_mpz_fits_ushort_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_fits_ushort_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_USHRT_MAX);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_get_ui)
-#if ! defined (__GMP_FORCE_mpz_get_ui)
-__GMP_EXTERN_INLINE
-#endif
-unsigned long
-mpz_get_ui (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- mp_ptr __gmp_p = __gmp_z->_mp_d;
- mp_size_t __gmp_n = __gmp_z->_mp_size;
- mp_limb_t __gmp_l = __gmp_p[0];
- /* This is a "#if" rather than a plain "if" so as to avoid gcc warnings
- about "<< GMP_NUMB_BITS" exceeding the type size, and to avoid Borland
- C++ 6.0 warnings about condition always true for something like
- "__GMP_ULONG_MAX < GMP_NUMB_MASK". */
-#if GMP_NAIL_BITS == 0 || defined (_LONG_LONG_LIMB)
- /* limb==long and no nails, or limb==longlong, one limb is enough */
- return (__gmp_n != 0 ? __gmp_l : 0);
-#else
- /* limb==long and nails, need two limbs when available */
- __gmp_n = __GMP_ABS (__gmp_n);
- if (__gmp_n <= 1)
- return (__gmp_n != 0 ? __gmp_l : 0);
- else
- return __gmp_l + (__gmp_p[1] << GMP_NUMB_BITS);
-#endif
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_getlimbn)
-#if ! defined (__GMP_FORCE_mpz_getlimbn)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpz_getlimbn (mpz_srcptr __gmp_z, mp_size_t __gmp_n) __GMP_NOTHROW
-{
- mp_limb_t __gmp_result = 0;
- if (__GMP_LIKELY (__gmp_n >= 0 && __gmp_n < __GMP_ABS (__gmp_z->_mp_size)))
- __gmp_result = __gmp_z->_mp_d[__gmp_n];
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpz_neg)
-__GMP_EXTERN_INLINE void
-mpz_neg (mpz_ptr __gmp_w, mpz_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpz_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_size = - __gmp_w->_mp_size;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_perfect_square_p)
-#if ! defined (__GMP_FORCE_mpz_perfect_square_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_perfect_square_p (mpz_srcptr __gmp_a)
-{
- mp_size_t __gmp_asize;
- int __gmp_result;
-
- __gmp_asize = __gmp_a->_mp_size;
- __gmp_result = (__gmp_asize >= 0); /* zero is a square, negatives are not */
- if (__GMP_LIKELY (__gmp_asize > 0))
- __gmp_result = mpn_perfect_square_p (__gmp_a->_mp_d, __gmp_asize);
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_popcount)
-#if ! defined (__GMP_FORCE_mpz_popcount)
-__GMP_EXTERN_INLINE
-#endif
-mp_bitcnt_t
-mpz_popcount (mpz_srcptr __gmp_u) __GMP_NOTHROW
-{
- mp_size_t __gmp_usize;
- mp_bitcnt_t __gmp_result;
-
- __gmp_usize = __gmp_u->_mp_size;
- __gmp_result = (__gmp_usize < 0 ? __GMP_ULONG_MAX : 0);
- if (__GMP_LIKELY (__gmp_usize > 0))
- __gmp_result = mpn_popcount (__gmp_u->_mp_d, __gmp_usize);
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_set_q)
-#if ! defined (__GMP_FORCE_mpz_set_q)
-__GMP_EXTERN_INLINE
-#endif
-void
-mpz_set_q (mpz_ptr __gmp_w, mpq_srcptr __gmp_u)
-{
- mpz_tdiv_q (__gmp_w, mpq_numref (__gmp_u), mpq_denref (__gmp_u));
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_size)
-#if ! defined (__GMP_FORCE_mpz_size)
-__GMP_EXTERN_INLINE
-#endif
-size_t
-mpz_size (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- return __GMP_ABS (__gmp_z->_mp_size);
-}
-#endif
-
-
-/**************** mpq inlines ****************/
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpq_abs)
-__GMP_EXTERN_INLINE void
-mpq_abs (mpq_ptr __gmp_w, mpq_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpq_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_num._mp_size = __GMP_ABS (__gmp_w->_mp_num._mp_size);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpq_neg)
-__GMP_EXTERN_INLINE void
-mpq_neg (mpq_ptr __gmp_w, mpq_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpq_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_num._mp_size = - __gmp_w->_mp_num._mp_size;
-}
-#endif
-
-
-/**************** mpn inlines ****************/
-
-/* The comments with __GMPN_ADD_1 below apply here too.
-
- The test for FUNCTION returning 0 should predict well. If it's assumed
- {yp,ysize} will usually have a random number of bits then the high limb
- won't be full and a carry out will occur a good deal less than 50% of the
- time.
-
- ysize==0 isn't a documented feature, but is used internally in a few
- places.
-
- Producing cout last stops it using up a register during the main part of
- the calculation, though gcc (as of 3.0) on an "if (mpn_add (...))"
- doesn't seem able to move the true and false legs of the conditional up
- to the two places cout is generated. */
-
-#define __GMPN_AORS(cout, wp, xp, xsize, yp, ysize, FUNCTION, TEST) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x; \
- \
- /* ASSERT ((ysize) >= 0); */ \
- /* ASSERT ((xsize) >= (ysize)); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE2_P (wp, xsize, xp, xsize)); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE2_P (wp, xsize, yp, ysize)); */ \
- \
- __gmp_i = (ysize); \
- if (__gmp_i != 0) \
- { \
- if (FUNCTION (wp, xp, yp, __gmp_i)) \
- { \
- do \
- { \
- if (__gmp_i >= (xsize)) \
- { \
- (cout) = 1; \
- goto __gmp_done; \
- } \
- __gmp_x = (xp)[__gmp_i]; \
- } \
- while (TEST); \
- } \
- } \
- if ((wp) != (xp)) \
- __GMPN_COPY_REST (wp, xp, xsize, __gmp_i); \
- (cout) = 0; \
- __gmp_done: \
- ; \
- } while (0)
-
-#define __GMPN_ADD(cout, wp, xp, xsize, yp, ysize) \
- __GMPN_AORS (cout, wp, xp, xsize, yp, ysize, mpn_add_n, \
- (((wp)[__gmp_i++] = (__gmp_x + 1) & GMP_NUMB_MASK) == 0))
-#define __GMPN_SUB(cout, wp, xp, xsize, yp, ysize) \
- __GMPN_AORS (cout, wp, xp, xsize, yp, ysize, mpn_sub_n, \
- (((wp)[__gmp_i++] = (__gmp_x - 1) & GMP_NUMB_MASK), __gmp_x == 0))
-
-
-/* The use of __gmp_i indexing is designed to ensure a compile time src==dst
- remains nice and clear to the compiler, so that __GMPN_COPY_REST can
- disappear, and the load/add/store gets a chance to become a
- read-modify-write on CISC CPUs.
-
- Alternatives:
-
- Using a pair of pointers instead of indexing would be possible, but gcc
- isn't able to recognise compile-time src==dst in that case, even when the
- pointers are incremented more or less together. Other compilers would
- very likely have similar difficulty.
-
- gcc could use "if (__builtin_constant_p(src==dst) && src==dst)" or
- similar to detect a compile-time src==dst. This works nicely on gcc
- 2.95.x, it's not good on gcc 3.0 where __builtin_constant_p(p==p) seems
- to be always false, for a pointer p. But the current code form seems
- good enough for src==dst anyway.
-
- gcc on x86 as usual doesn't give particularly good flags handling for the
- carry/borrow detection. It's tempting to want some multi instruction asm
- blocks to help it, and this was tried, but in truth there's only a few
- instructions to save and any gain is all too easily lost by register
- juggling setting up for the asm. */
-
-#if GMP_NAIL_BITS == 0
-#define __GMPN_AORS_1(cout, dst, src, n, v, OP, CB) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x, __gmp_r; \
- \
- /* ASSERT ((n) >= 1); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE_P (dst, src, n)); */ \
- \
- __gmp_x = (src)[0]; \
- __gmp_r = __gmp_x OP (v); \
- (dst)[0] = __gmp_r; \
- if (CB (__gmp_r, __gmp_x, (v))) \
- { \
- (cout) = 1; \
- for (__gmp_i = 1; __gmp_i < (n);) \
- { \
- __gmp_x = (src)[__gmp_i]; \
- __gmp_r = __gmp_x OP 1; \
- (dst)[__gmp_i] = __gmp_r; \
- ++__gmp_i; \
- if (!CB (__gmp_r, __gmp_x, 1)) \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, __gmp_i); \
- (cout) = 0; \
- break; \
- } \
- } \
- } \
- else \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, 1); \
- (cout) = 0; \
- } \
- } while (0)
-#endif
-
-#if GMP_NAIL_BITS >= 1
-#define __GMPN_AORS_1(cout, dst, src, n, v, OP, CB) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x, __gmp_r; \
- \
- /* ASSERT ((n) >= 1); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE_P (dst, src, n)); */ \
- \
- __gmp_x = (src)[0]; \
- __gmp_r = __gmp_x OP (v); \
- (dst)[0] = __gmp_r & GMP_NUMB_MASK; \
- if (__gmp_r >> GMP_NUMB_BITS != 0) \
- { \
- (cout) = 1; \
- for (__gmp_i = 1; __gmp_i < (n);) \
- { \
- __gmp_x = (src)[__gmp_i]; \
- __gmp_r = __gmp_x OP 1; \
- (dst)[__gmp_i] = __gmp_r & GMP_NUMB_MASK; \
- ++__gmp_i; \
- if (__gmp_r >> GMP_NUMB_BITS == 0) \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, __gmp_i); \
- (cout) = 0; \
- break; \
- } \
- } \
- } \
- else \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, 1); \
- (cout) = 0; \
- } \
- } while (0)
-#endif
-
-#define __GMPN_ADDCB(r,x,y) ((r) < (y))
-#define __GMPN_SUBCB(r,x,y) ((x) < (y))
-
-#define __GMPN_ADD_1(cout, dst, src, n, v) \
- __GMPN_AORS_1(cout, dst, src, n, v, +, __GMPN_ADDCB)
-#define __GMPN_SUB_1(cout, dst, src, n, v) \
- __GMPN_AORS_1(cout, dst, src, n, v, -, __GMPN_SUBCB)
-
-
-/* Compare {xp,size} and {yp,size}, setting "result" to positive, zero or
- negative. size==0 is allowed. On random data usually only one limb will
- need to be examined to get a result, so it's worth having it inline. */
-#define __GMPN_CMP(result, xp, yp, size) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x, __gmp_y; \
- \
- /* ASSERT ((size) >= 0); */ \
- \
- (result) = 0; \
- __gmp_i = (size); \
- while (--__gmp_i >= 0) \
- { \
- __gmp_x = (xp)[__gmp_i]; \
- __gmp_y = (yp)[__gmp_i]; \
- if (__gmp_x != __gmp_y) \
- { \
- /* Cannot use __gmp_x - __gmp_y, may overflow an "int" */ \
- (result) = (__gmp_x > __gmp_y ? 1 : -1); \
- break; \
- } \
- } \
- } while (0)
-
-
-#if defined (__GMPN_COPY) && ! defined (__GMPN_COPY_REST)
-#define __GMPN_COPY_REST(dst, src, size, start) \
- do { \
- /* ASSERT ((start) >= 0); */ \
- /* ASSERT ((start) <= (size)); */ \
- __GMPN_COPY ((dst)+(start), (src)+(start), (size)-(start)); \
- } while (0)
-#endif
-
-/* Copy {src,size} to {dst,size}, starting at "start". This is designed to
- keep the indexing dst[j] and src[j] nice and simple for __GMPN_ADD_1,
- __GMPN_ADD, etc. */
-#if ! defined (__GMPN_COPY_REST)
-#define __GMPN_COPY_REST(dst, src, size, start) \
- do { \
- mp_size_t __gmp_j; \
- /* ASSERT ((size) >= 0); */ \
- /* ASSERT ((start) >= 0); */ \
- /* ASSERT ((start) <= (size)); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE_P (dst, src, size)); */ \
- __GMP_CRAY_Pragma ("_CRI ivdep"); \
- for (__gmp_j = (start); __gmp_j < (size); __gmp_j++) \
- (dst)[__gmp_j] = (src)[__gmp_j]; \
- } while (0)
-#endif
-
-/* Enhancement: Use some of the smarter code from gmp-impl.h. Maybe use
- mpn_copyi if there's a native version, and if we don't mind demanding
- binary compatibility for it (on targets which use it). */
-
-#if ! defined (__GMPN_COPY)
-#define __GMPN_COPY(dst, src, size) __GMPN_COPY_REST (dst, src, size, 0)
-#endif
-
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_add)
-#if ! defined (__GMP_FORCE_mpn_add)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_add (mp_ptr __gmp_wp, mp_srcptr __gmp_xp, mp_size_t __gmp_xsize, mp_srcptr __gmp_yp, mp_size_t __gmp_ysize)
-{
- mp_limb_t __gmp_c;
- __GMPN_ADD (__gmp_c, __gmp_wp, __gmp_xp, __gmp_xsize, __gmp_yp, __gmp_ysize);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_add_1)
-#if ! defined (__GMP_FORCE_mpn_add_1)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_add_1 (mp_ptr __gmp_dst, mp_srcptr __gmp_src, mp_size_t __gmp_size, mp_limb_t __gmp_n) __GMP_NOTHROW
-{
- mp_limb_t __gmp_c;
- __GMPN_ADD_1 (__gmp_c, __gmp_dst, __gmp_src, __gmp_size, __gmp_n);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_cmp)
-#if ! defined (__GMP_FORCE_mpn_cmp)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpn_cmp (mp_srcptr __gmp_xp, mp_srcptr __gmp_yp, mp_size_t __gmp_size) __GMP_NOTHROW
-{
- int __gmp_result;
- __GMPN_CMP (__gmp_result, __gmp_xp, __gmp_yp, __gmp_size);
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_sub)
-#if ! defined (__GMP_FORCE_mpn_sub)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_sub (mp_ptr __gmp_wp, mp_srcptr __gmp_xp, mp_size_t __gmp_xsize, mp_srcptr __gmp_yp, mp_size_t __gmp_ysize)
-{
- mp_limb_t __gmp_c;
- __GMPN_SUB (__gmp_c, __gmp_wp, __gmp_xp, __gmp_xsize, __gmp_yp, __gmp_ysize);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_sub_1)
-#if ! defined (__GMP_FORCE_mpn_sub_1)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_sub_1 (mp_ptr __gmp_dst, mp_srcptr __gmp_src, mp_size_t __gmp_size, mp_limb_t __gmp_n) __GMP_NOTHROW
-{
- mp_limb_t __gmp_c;
- __GMPN_SUB_1 (__gmp_c, __gmp_dst, __gmp_src, __gmp_size, __gmp_n);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_neg)
-#if ! defined (__GMP_FORCE_mpn_neg)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_neg (mp_ptr __gmp_rp, mp_srcptr __gmp_up, mp_size_t __gmp_n)
-{
- mp_limb_t __gmp_ul, __gmp_cy;
- __gmp_cy = 0;
- do {
- __gmp_ul = *__gmp_up++;
- *__gmp_rp++ = -__gmp_ul - __gmp_cy;
- __gmp_cy |= __gmp_ul != 0;
- } while (--__gmp_n != 0);
- return __gmp_cy;
-}
-#endif
-
-#if defined (__cplusplus)
-}
-#endif
-
-
-/* Allow faster testing for negative, zero, and positive. */
-#define mpz_sgn(Z) ((Z)->_mp_size < 0 ? -1 : (Z)->_mp_size > 0)
-#define mpf_sgn(F) ((F)->_mp_size < 0 ? -1 : (F)->_mp_size > 0)
-#define mpq_sgn(Q) ((Q)->_mp_num._mp_size < 0 ? -1 : (Q)->_mp_num._mp_size > 0)
-
-/* When using GCC, optimize certain common comparisons. */
-#if defined (__GNUC__) && __GNUC__ >= 2
-#define mpz_cmp_ui(Z,UI) \
- (__builtin_constant_p (UI) && (UI) == 0 \
- ? mpz_sgn (Z) : _mpz_cmp_ui (Z,UI))
-#define mpz_cmp_si(Z,SI) \
- (__builtin_constant_p (SI) && (SI) == 0 ? mpz_sgn (Z) \
- : __builtin_constant_p (SI) && (SI) > 0 \
- ? _mpz_cmp_ui (Z, __GMP_CAST (unsigned long int, SI)) \
- : _mpz_cmp_si (Z,SI))
-#define mpq_cmp_ui(Q,NUI,DUI) \
- (__builtin_constant_p (NUI) && (NUI) == 0 \
- ? mpq_sgn (Q) : _mpq_cmp_ui (Q,NUI,DUI))
-#define mpq_cmp_si(q,n,d) \
- (__builtin_constant_p ((n) >= 0) && (n) >= 0 \
- ? mpq_cmp_ui (q, __GMP_CAST (unsigned long, n), d) \
- : _mpq_cmp_si (q, n, d))
-#else
-#define mpz_cmp_ui(Z,UI) _mpz_cmp_ui (Z,UI)
-#define mpz_cmp_si(Z,UI) _mpz_cmp_si (Z,UI)
-#define mpq_cmp_ui(Q,NUI,DUI) _mpq_cmp_ui (Q,NUI,DUI)
-#define mpq_cmp_si(q,n,d) _mpq_cmp_si(q,n,d)
-#endif
-
-
-/* Using "&" rather than "&&" means these can come out branch-free. Every
- mpz_t has at least one limb allocated, so fetching the low limb is always
- allowed. */
-#define mpz_odd_p(z) (((z)->_mp_size != 0) & __GMP_CAST (int, (z)->_mp_d[0]))
-#define mpz_even_p(z) (! mpz_odd_p (z))
-
-
-/**************** C++ routines ****************/
-
-#ifdef __cplusplus
-__GMP_DECLSPEC_XX std::ostream& operator<< (std::ostream &, mpz_srcptr);
-__GMP_DECLSPEC_XX std::ostream& operator<< (std::ostream &, mpq_srcptr);
-__GMP_DECLSPEC_XX std::ostream& operator<< (std::ostream &, mpf_srcptr);
-__GMP_DECLSPEC_XX std::istream& operator>> (std::istream &, mpz_ptr);
-__GMP_DECLSPEC_XX std::istream& operator>> (std::istream &, mpq_ptr);
-__GMP_DECLSPEC_XX std::istream& operator>> (std::istream &, mpf_ptr);
-#endif
-
-
-/* Source-level compatibility with GMP 2 and earlier. */
-#define mpn_divmod(qp,np,nsize,dp,dsize) \
- mpn_divrem (qp, __GMP_CAST (mp_size_t, 0), np, nsize, dp, dsize)
-
-/* Source-level compatibility with GMP 1. */
-#define mpz_mdiv mpz_fdiv_q
-#define mpz_mdivmod mpz_fdiv_qr
-#define mpz_mmod mpz_fdiv_r
-#define mpz_mdiv_ui mpz_fdiv_q_ui
-#define mpz_mdivmod_ui(q,r,n,d) \
- (((r) == 0) ? mpz_fdiv_q_ui (q,n,d) : mpz_fdiv_qr_ui (q,r,n,d))
-#define mpz_mmod_ui(r,n,d) \
- (((r) == 0) ? mpz_fdiv_ui (n,d) : mpz_fdiv_r_ui (r,n,d))
-
-/* Useful synonyms, but not quite compatible with GMP 1. */
-#define mpz_div mpz_fdiv_q
-#define mpz_divmod mpz_fdiv_qr
-#define mpz_div_ui mpz_fdiv_q_ui
-#define mpz_divmod_ui mpz_fdiv_qr_ui
-#define mpz_div_2exp mpz_fdiv_q_2exp
-#define mpz_mod_2exp mpz_fdiv_r_2exp
-
-enum
-{
- GMP_ERROR_NONE = 0,
- GMP_ERROR_UNSUPPORTED_ARGUMENT = 1,
- GMP_ERROR_DIVISION_BY_ZERO = 2,
- GMP_ERROR_SQRT_OF_NEGATIVE = 4,
- GMP_ERROR_INVALID_ARGUMENT = 8
-};
-
-/* Define CC and CFLAGS which were used to build this version of GMP */
-#define __GMP_CC "gcc -std=gnu99"
-#define __GMP_CFLAGS "-m32 -O2 -pedantic -fomit-frame-pointer -mtune=pentiumpro -march=pentiumpro"
-
-/* Major version number is the value of __GNU_MP__ too, above and in mp.h. */
-#define __GNU_MP_VERSION 5
-#define __GNU_MP_VERSION_MINOR 0
-#define __GNU_MP_VERSION_PATCHLEVEL 1
-#define __GMP_MP_RELEASE (__GNU_MP_VERSION * 10000 + __GNU_MP_VERSION_MINOR * 100 + __GNU_MP_VERSION_PATCHLEVEL)
-
-#define __GMP_H__
-#endif /* __GMP_H__ */
+++ /dev/null
-# libgmp.la - a libtool library file
-# Generated by ltmain.sh (GNU libtool) 2.2.6b
-#
-# Please DO NOT delete this file!
-# It is necessary for linking the library.
-
-# The name that we can dlopen(3).
-dlname=''
-
-# Names of this library.
-library_names=''
-
-# The name of the static archive.
-old_library='libgmp.a'
-
-# Linker flags that can not go in dependency_libs.
-inherited_linker_flags=''
-
-# Libraries that this one depends upon.
-dependency_libs=''
-
-# Names of additional weak libraries provided by this library
-weak_library_names=''
-
-# Version information for libgmp.
-current=10
-age=0
-revision=1
-
-# Is this an already installed library?
-installed=yes
-
-# Should we warn about portability when linking against -modules?
-shouldnotlink=no
-
-# Files to dlopen/dlpreopen
-dlopen=''
-dlpreopen=''
-
-# Directory that this library needs to be installed in:
-libdir='/tmp/gg/lib'
+++ /dev/null
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-END-INFO-DIR-ENTRY
-
-\1f
-Indirect:
-gmp.info-1: 981
-gmp.info-2: 300864
-\1f
-Tag Table:
-(Indirect)
-Node: Top\7f981
-Node: Copying\7f3211
-Node: Introduction to GMP\7f5062
-Node: Installing GMP\7f7773
-Node: Build Options\7f8505
-Node: ABI and ISA\7f24573
-Node: Notes for Package Builds\7f34251
-Node: Notes for Particular Systems\7f37338
-Node: Known Build Problems\7f43895
-Node: Performance optimization\7f47429
-Node: GMP Basics\7f48563
-Node: Headers and Libraries\7f49211
-Node: Nomenclature and Types\7f50635
-Node: Function Classes\7f52632
-Node: Variable Conventions\7f54325
-Node: Parameter Conventions\7f55934
-Node: Memory Management\7f57990
-Node: Reentrancy\7f59118
-Node: Useful Macros and Constants\7f60991
-Node: Compatibility with older versions\7f61989
-Node: Demonstration Programs\7f62950
-Node: Efficiency\7f64815
-Node: Debugging\7f72439
-Node: Profiling\7f78997
-Node: Autoconf\7f82988
-Node: Emacs\7f84767
-Node: Reporting Bugs\7f85373
-Node: Integer Functions\7f87916
-Node: Initializing Integers\7f88692
-Node: Assigning Integers\7f90839
-Node: Simultaneous Integer Init & Assign\7f92426
-Node: Converting Integers\7f94051
-Node: Integer Arithmetic\7f96973
-Node: Integer Division\7f98559
-Node: Integer Exponentiation\7f104869
-Node: Integer Roots\7f106309
-Node: Number Theoretic Functions\7f107983
-Node: Integer Comparisons\7f114126
-Node: Integer Logic and Bit Fiddling\7f115504
-Node: I/O of Integers\7f118051
-Node: Integer Random Numbers\7f120935
-Node: Integer Import and Export\7f123546
-Node: Miscellaneous Integer Functions\7f127556
-Node: Integer Special Functions\7f129416
-Node: Rational Number Functions\7f132503
-Node: Initializing Rationals\7f133696
-Node: Rational Conversions\7f136157
-Node: Rational Arithmetic\7f137888
-Node: Comparing Rationals\7f139192
-Node: Applying Integer Functions\7f140559
-Node: I/O of Rationals\7f142042
-Node: Floating-point Functions\7f143902
-Node: Initializing Floats\7f146787
-Node: Assigning Floats\7f150874
-Node: Simultaneous Float Init & Assign\7f153441
-Node: Converting Floats\7f154969
-Node: Float Arithmetic\7f158217
-Node: Float Comparison\7f160230
-Node: I/O of Floats\7f161811
-Node: Miscellaneous Float Functions\7f164409
-Node: Low-level Functions\7f166303
-Node: Random Number Functions\7f190437
-Node: Random State Initialization\7f191505
-Node: Random State Seeding\7f194363
-Node: Random State Miscellaneous\7f195752
-Node: Formatted Output\7f196393
-Node: Formatted Output Strings\7f196638
-Node: Formatted Output Functions\7f201852
-Node: C++ Formatted Output\7f205927
-Node: Formatted Input\7f208609
-Node: Formatted Input Strings\7f208845
-Node: Formatted Input Functions\7f213497
-Node: C++ Formatted Input\7f216466
-Node: C++ Class Interface\7f218369
-Node: C++ Interface General\7f219370
-Node: C++ Interface Integers\7f222440
-Node: C++ Interface Rationals\7f225871
-Node: C++ Interface Floats\7f229548
-Node: C++ Interface Random Numbers\7f234830
-Node: C++ Interface Limitations\7f237236
-Node: BSD Compatible Functions\7f240056
-Node: Custom Allocation\7f244767
-Node: Language Bindings\7f249085
-Node: Algorithms\7f253038
-Node: Multiplication Algorithms\7f253738
-Node: Basecase Multiplication\7f254710
-Node: Karatsuba Multiplication\7f256618
-Node: Toom 3-Way Multiplication\7f260243
-Node: Toom 4-Way Multiplication\7f266657
-Node: FFT Multiplication\7f268029
-Node: Other Multiplication\7f273365
-Node: Unbalanced Multiplication\7f275839
-Node: Division Algorithms\7f276627
-Node: Single Limb Division\7f277006
-Node: Basecase Division\7f279897
-Node: Divide and Conquer Division\7f281100
-Node: Block-Wise Barrett Division\7f283169
-Node: Exact Division\7f283821
-Node: Exact Remainder\7f286986
-Node: Small Quotient Division\7f289213
-Node: Greatest Common Divisor Algorithms\7f290811
-Node: Binary GCD\7f291108
-Node: Lehmer's Algorithm\7f293957
-Node: Subquadratic GCD\7f296177
-Node: Extended GCD\7f298636
-Node: Jacobi Symbol\7f299948
-Node: Powering Algorithms\7f300864
-Node: Normal Powering Algorithm\7f301127
-Node: Modular Powering Algorithm\7f301655
-Node: Root Extraction Algorithms\7f302435
-Node: Square Root Algorithm\7f302750
-Node: Nth Root Algorithm\7f304891
-Node: Perfect Square Algorithm\7f305676
-Node: Perfect Power Algorithm\7f307762
-Node: Radix Conversion Algorithms\7f308383
-Node: Binary to Radix\7f308759
-Node: Radix to Binary\7f312688
-Node: Other Algorithms\7f314776
-Node: Prime Testing Algorithm\7f315128
-Node: Factorial Algorithm\7f316312
-Node: Binomial Coefficients Algorithm\7f317715
-Node: Fibonacci Numbers Algorithm\7f318609
-Node: Lucas Numbers Algorithm\7f321083
-Node: Random Number Algorithms\7f321804
-Node: Assembly Coding\7f323925
-Node: Assembly Code Organisation\7f324885
-Node: Assembly Basics\7f325852
-Node: Assembly Carry Propagation\7f327002
-Node: Assembly Cache Handling\7f328833
-Node: Assembly Functional Units\7f330994
-Node: Assembly Floating Point\7f332607
-Node: Assembly SIMD Instructions\7f336385
-Node: Assembly Software Pipelining\7f337367
-Node: Assembly Loop Unrolling\7f338429
-Node: Assembly Writing Guide\7f340644
-Node: Internals\7f343409
-Node: Integer Internals\7f343921
-Node: Rational Internals\7f346177
-Node: Float Internals\7f347415
-Node: Raw Output Internals\7f354829
-Node: C++ Interface Internals\7f356023
-Node: Contributors\7f359309
-Node: References\7f364267
-Node: GNU Free Documentation License\7f369925
-Node: Concept Index\7f395094
-Node: Function Index\7f441276
-\1f
-End Tag Table
+++ /dev/null
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-END-INFO-DIR-ENTRY
-
-\1f
-File: gmp.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
-
-GNU MP
-******
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-
-* Menu:
-
-* Copying:: GMP Copying Conditions (LGPL).
-* Introduction to GMP:: Brief introduction to GNU MP.
-* Installing GMP:: How to configure and compile the GMP library.
-* GMP Basics:: What every GMP user should know.
-* Reporting Bugs:: How to usefully report bugs.
-* Integer Functions:: Functions for arithmetic on signed integers.
-* Rational Number Functions:: Functions for arithmetic on rational numbers.
-* Floating-point Functions:: Functions for arithmetic on floats.
-* Low-level Functions:: Fast functions for natural numbers.
-* Random Number Functions:: Functions for generating random numbers.
-* Formatted Output:: `printf' style output.
-* Formatted Input:: `scanf' style input.
-* C++ Class Interface:: Class wrappers around GMP types.
-* BSD Compatible Functions:: All functions found in BSD MP.
-* Custom Allocation:: How to customize the internal allocation.
-* Language Bindings:: Using GMP from other languages.
-* Algorithms:: What happens behind the scenes.
-* Internals:: How values are represented behind the scenes.
-
-* Contributors:: Who brings you this library?
-* References:: Some useful papers and books to read.
-* GNU Free Documentation License::
-* Concept Index::
-* Function Index::
-
-\1f
-File: gmp.info, Node: Copying, Next: Introduction to GMP, Prev: Top, Up: Top
-
-GNU MP Copying Conditions
-*************************
-
-This library is "free"; this means that everyone is free to use it and
-free to redistribute it on a free basis. The library is not in the
-public domain; it is copyrighted and there are restrictions on its
-distribution, but these restrictions are designed to permit everything
-that a good cooperating citizen would want to do. What is not allowed
-is to try to prevent others from further sharing any version of this
-library that they might get from you.
-
- Specifically, we want to make sure that you have the right to give
-away copies of the library, that you receive source code or else can
-get it if you want it, that you can change this library or use pieces
-of it in new free programs, and that you know you can do these things.
-
- To make sure that everyone has such rights, we have to forbid you to
-deprive anyone else of these rights. For example, if you distribute
-copies of the GNU MP library, you must give the recipients all the
-rights that you have. You must make sure that they, too, receive or
-can get the source code. And you must tell them their rights.
-
- Also, for our own protection, we must make certain that everyone
-finds out that there is no warranty for the GNU MP library. If it is
-modified by someone else and passed on, we want their recipients to
-know that what they have is not what we distributed, so that any
-problems introduced by others will not reflect on our reputation.
-
- The precise conditions of the license for the GNU MP library are
-found in the Lesser General Public License version 3 that accompanies
-the source code, see `COPYING.LIB'. Certain demonstration programs are
-provided under the terms of the plain General Public License version 3,
-see `COPYING'.
-
-\1f
-File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top
-
-1 Introduction to GNU MP
-************************
-
-GNU MP is a portable library written in C for arbitrary precision
-arithmetic on integers, rational numbers, and floating-point numbers.
-It aims to provide the fastest possible arithmetic for all applications
-that need higher precision than is directly supported by the basic C
-types.
-
- Many applications use just a few hundred bits of precision; but some
-applications may need thousands or even millions of bits. GMP is
-designed to give good performance for both, by choosing algorithms
-based on the sizes of the operands, and by carefully keeping the
-overhead at a minimum.
-
- The speed of GMP is achieved by using fullwords as the basic
-arithmetic type, by using sophisticated algorithms, by including
-carefully optimized assembly code for the most common inner loops for
-many different CPUs, and by a general emphasis on speed (as opposed to
-simplicity or elegance).
-
- There is assembly code for these CPUs: ARM, DEC Alpha 21064, 21164,
-and 21264, AMD 29000, AMD K6, K6-2, Athlon, and Athlon64, Hitachi
-SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium
-Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola
-MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64,
-National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC,
-generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some
-optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and
-Pyramid AP/XP.
-
-For up-to-date information on GMP, please see the GMP web pages at
-
- `http://gmplib.org/'
-
-The latest version of the library is available at
-
- `ftp://ftp.gnu.org/gnu/gmp/'
-
- Many sites around the world mirror `ftp.gnu.org', please use a mirror
-near you, see `http://www.gnu.org/order/ftp.html' for a full list.
-
- There are three public mailing lists of interest. One for release
-announcements, one for general questions and discussions about usage of
-the GMP library and one for bug reports. For more information, see
-
- `http://gmplib.org/mailman/listinfo/'.
-
- The proper place for bug reports is <gmp-bugs@gmplib.org>. See
-*Note Reporting Bugs:: for information about reporting bugs.
-
-
-1.1 How to use this Manual
-==========================
-
-Everyone should read *Note GMP Basics::. If you need to install the
-library yourself, then read *Note Installing GMP::. If you have a
-system with multiple ABIs, then read *Note ABI and ISA::, for the
-compiler options that must be used on applications.
-
- The rest of the manual can be used for later reference, although it
-is probably a good idea to glance through it.
-
-\1f
-File: gmp.info, Node: Installing GMP, Next: GMP Basics, Prev: Introduction to GMP, Up: Top
-
-2 Installing GMP
-****************
-
-GMP has an autoconf/automake/libtool based configuration system. On a
-Unix-like system a basic build can be done with
-
- ./configure
- make
-
-Some self-tests can be run with
-
- make check
-
-And you can install (under `/usr/local' by default) with
-
- make install
-
- If you experience problems, please report them to
-<gmp-bugs@gmplib.org>. See *Note Reporting Bugs::, for information on
-what to include in useful bug reports.
-
-* Menu:
-
-* Build Options::
-* ABI and ISA::
-* Notes for Package Builds::
-* Notes for Particular Systems::
-* Known Build Problems::
-* Performance optimization::
-
-\1f
-File: gmp.info, Node: Build Options, Next: ABI and ISA, Prev: Installing GMP, Up: Installing GMP
-
-2.1 Build Options
-=================
-
-All the usual autoconf configure options are available, run `./configure
---help' for a summary. The file `INSTALL.autoconf' has some generic
-installation information too.
-
-Tools
- `configure' requires various Unix-like tools. See *Note Notes for
- Particular Systems::, for some options on non-Unix systems.
-
- It might be possible to build without the help of `configure',
- certainly all the code is there, but unfortunately you'll be on
- your own.
-
-Build Directory
- To compile in a separate build directory, `cd' to that directory,
- and prefix the configure command with the path to the GMP source
- directory. For example
-
- cd /my/build/dir
- /my/sources/gmp-5.0.1/configure
-
- Not all `make' programs have the necessary features (`VPATH') to
- support this. In particular, SunOS and Slowaris `make' have bugs
- that make them unable to build in a separate directory. Use GNU
- `make' instead.
-
-`--prefix' and `--exec-prefix'
- The `--prefix' option can be used in the normal way to direct GMP
- to install under a particular tree. The default is `/usr/local'.
-
- `--exec-prefix' can be used to direct architecture-dependent files
- like `libgmp.a' to a different location. This can be used to share
- architecture-independent parts like the documentation, but
- separate the dependent parts. Note however that `gmp.h' and
- `mp.h' are architecture-dependent since they encode certain
- aspects of `libgmp', so it will be necessary to ensure both
- `$prefix/include' and `$exec_prefix/include' are available to the
- compiler.
-
-`--disable-shared', `--disable-static'
- By default both shared and static libraries are built (where
- possible), but one or other can be disabled. Shared libraries
- result in smaller executables and permit code sharing between
- separate running processes, but on some CPUs are slightly slower,
- having a small cost on each function call.
-
-Native Compilation, `--build=CPU-VENDOR-OS'
- For normal native compilation, the system can be specified with
- `--build'. By default `./configure' uses the output from running
- `./config.guess'. On some systems `./config.guess' can determine
- the exact CPU type, on others it will be necessary to give it
- explicitly. For example,
-
- ./configure --build=ultrasparc-sun-solaris2.7
-
- In all cases the `OS' part is important, since it controls how
- libtool generates shared libraries. Running `./config.guess' is
- the simplest way to see what it should be, if you don't know
- already.
-
-Cross Compilation, `--host=CPU-VENDOR-OS'
- When cross-compiling, the system used for compiling is given by
- `--build' and the system where the library will run is given by
- `--host'. For example when using a FreeBSD Athlon system to build
- GNU/Linux m68k binaries,
-
- ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
-
- Compiler tools are sought first with the host system type as a
- prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
- plain `ranlib'. This makes it possible for a set of
- cross-compiling tools to co-exist with native tools. The prefix
- is the argument to `--host', and this can be an alias, such as
- `m68k-linux'. But note that tools don't have to be setup this
- way, it's enough to just have a `PATH' with a suitable
- cross-compiling `cc' etc.
-
- Compiling for a different CPU in the same family as the build
- system is a form of cross-compilation, though very possibly this
- would merely be special options on a native compiler. In any case
- `./configure' avoids depending on being able to run code on the
- build system, which is important when creating binaries for a
- newer CPU since they very possibly won't run on the build system.
-
- In all cases the compiler must be able to produce an executable
- (of whatever format) from a standard C `main'. Although only
- object files will go to make up `libgmp', `./configure' uses
- linking tests for various purposes, such as determining what
- functions are available on the host system.
-
- Currently a warning is given unless an explicit `--build' is used
- when cross-compiling, because it may not be possible to correctly
- guess the build system type if the `PATH' has only a
- cross-compiling `cc'.
-
- Note that the `--target' option is not appropriate for GMP. It's
- for use when building compiler tools, with `--host' being where
- they will run, and `--target' what they'll produce code for.
- Ordinary programs or libraries like GMP are only interested in the
- `--host' part, being where they'll run. (Some past versions of
- GMP used `--target' incorrectly.)
-
-CPU types
- In general, if you want a library that runs as fast as possible,
- you should configure GMP for the exact CPU type your system uses.
- However, this may mean the binaries won't run on older members of
- the family, and might run slower on other members, older or newer.
- The best idea is always to build GMP for the exact machine type
- you intend to run it on.
-
- The following CPUs have specific support. See `configure.in' for
- details of what code and compiler options they select.
-
- * Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
- alphaev6, alphaev67, alphaev68 alphaev7
-
- * Cray: c90, j90, t90, sv1
-
- * HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64
-
- * IA-64: ia64, itanium, itanium2
-
- * MIPS: mips, mips3, mips64
-
- * Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
- m68060, m68302, m68360, m88k, m88110
-
- * POWER: power, power1, power2, power2sc
-
- * PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
- powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
- powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
- powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
- powerpc821, powerpc823, powerpc860, powerpc970
-
- * SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
- ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
-
- * x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro,
- pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64,
- viac3, viac32
-
- * Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
- z8k
-
- CPUs not listed will use generic C code.
-
-Generic C Build
- If some of the assembly code causes problems, or if otherwise
- desired, the generic C code can be selected with CPU `none'. For
- example,
-
- ./configure --host=none-unknown-freebsd3.5
-
- Note that this will run quite slowly, but it should be portable
- and should at least make it possible to get something running if
- all else fails.
-
-Fat binary, `--enable-fat'
- Using `--enable-fat' selects a "fat binary" build on x86, where
- optimized low level subroutines are chosen at runtime according to
- the CPU detected. This means more code, but gives good
- performance on all x86 chips. (This option might become available
- for more architectures in the future.)
-
-`ABI'
- On some systems GMP supports multiple ABIs (application binary
- interfaces), meaning data type sizes and calling conventions. By
- default GMP chooses the best ABI available, but a particular ABI
- can be selected. For example
-
- ./configure --host=mips64-sgi-irix6 ABI=n32
-
- See *Note ABI and ISA::, for the available choices on relevant
- CPUs, and what applications need to do.
-
-`CC', `CFLAGS'
- By default the C compiler used is chosen from among some likely
- candidates, with `gcc' normally preferred if it's present. The
- usual `CC=whatever' can be passed to `./configure' to choose
- something different.
-
- For various systems, default compiler flags are set based on the
- CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to
- `./configure' to use something different or to set good flags for
- systems GMP doesn't otherwise know.
-
- The `CC' and `CFLAGS' used are printed during `./configure', and
- can be found in each generated `Makefile'. This is the easiest way
- to check the defaults when considering changing or adding
- something.
-
- Note that when `CC' and `CFLAGS' are specified on a system
- supporting multiple ABIs it's important to give an explicit
- `ABI=whatever', since GMP can't determine the ABI just from the
- flags and won't be able to select the correct assembly code.
-
- If just `CC' is selected then normal default `CFLAGS' for that
- compiler will be used (if GMP recognises it). For example
- `CC=gcc' can be used to force the use of GCC, with default flags
- (and default ABI).
-
-`CPPFLAGS'
- Any flags like `-D' defines or `-I' includes required by the
- preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
- Compiling is done with both `CPPFLAGS' and `CFLAGS', but
- preprocessing uses just `CPPFLAGS'. This distinction is because
- most preprocessors won't accept all the flags the compiler does.
- Preprocessing is done separately in some configure tests, and in
- the `ansi2knr' support for K&R compilers.
-
-`CC_FOR_BUILD'
- Some build-time programs are compiled and run to generate
- host-specific data tables. `CC_FOR_BUILD' is the compiler used
- for this. It doesn't need to be in any particular ABI or mode, it
- merely needs to generate executables that can run. The default is
- to try the selected `CC' and some likely candidates such as `cc'
- and `gcc', looking for something that works.
-
- No flags are used with `CC_FOR_BUILD' because a simple invocation
- like `cc foo.c' should be enough. If some particular options are
- required they can be included as for instance `CC_FOR_BUILD="cc
- -whatever"'.
-
-C++ Support, `--enable-cxx'
- C++ support in GMP can be enabled with `--enable-cxx', in which
- case a C++ compiler will be required. As a convenience
- `--enable-cxx=detect' can be used to enable C++ support only if a
- compiler can be found. The C++ support consists of a library
- `libgmpxx.la' and header file `gmpxx.h' (*note Headers and
- Libraries::).
-
- A separate `libgmpxx.la' has been adopted rather than having C++
- objects within `libgmp.la' in order to ensure dynamic linked C
- programs aren't bloated by a dependency on the C++ standard
- library, and to avoid any chance that the C++ compiler could be
- required when linking plain C programs.
-
- `libgmpxx.la' will use certain internals from `libgmp.la' and can
- only be expected to work with `libgmp.la' from the same GMP
- version. Future changes to the relevant internals will be
- accompanied by renaming, so a mismatch will cause unresolved
- symbols rather than perhaps mysterious misbehaviour.
-
- In general `libgmpxx.la' will be usable only with the C++ compiler
- that built it, since name mangling and runtime support are usually
- incompatible between different compilers.
-
-`CXX', `CXXFLAGS'
- When C++ support is enabled, the C++ compiler and its flags can be
- set with variables `CXX' and `CXXFLAGS' in the usual way. The
- default for `CXX' is the first compiler that works from a list of
- likely candidates, with `g++' normally preferred when available.
- The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
- `-g', then for `g++' either `-g -O2' or `-O2', or for other
- compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
- when using `gcc' and `g++' together, since the flags for `gcc' will
- usually suit `g++'.
-
- It's important that the C and C++ compilers match, meaning their
- startup and runtime support routines are compatible and that they
- generate code in the same ABI (if there's a choice of ABIs on the
- system). `./configure' isn't currently able to check these things
- very well itself, so for that reason `--disable-cxx' is the
- default, to avoid a build failure due to a compiler mismatch.
- Perhaps this will change in the future.
-
- Incidentally, it's normally not good enough to set `CXX' to the
- same as `CC'. Although `gcc' for instance recognises `foo.cc' as
- C++ code, only `g++' will invoke the linker the right way when
- building an executable or shared library from C++ object files.
-
-Temporary Memory, `--enable-alloca=<choice>'
- GMP allocates temporary workspace using one of the following three
- methods, which can be selected with for instance
- `--enable-alloca=malloc-reentrant'.
-
- * `alloca' - C library or compiler builtin.
-
- * `malloc-reentrant' - the heap, in a re-entrant fashion.
-
- * `malloc-notreentrant' - the heap, with global variables.
-
- For convenience, the following choices are also available.
- `--disable-alloca' is the same as `no'.
-
- * `yes' - a synonym for `alloca'.
-
- * `no' - a synonym for `malloc-reentrant'.
-
- * `reentrant' - `alloca' if available, otherwise
- `malloc-reentrant'. This is the default.
-
- * `notreentrant' - `alloca' if available, otherwise
- `malloc-notreentrant'.
-
- `alloca' is reentrant and fast, and is recommended. It actually
- allocates just small blocks on the stack; larger ones use
- malloc-reentrant.
-
- `malloc-reentrant' is, as the name suggests, reentrant and thread
- safe, but `malloc-notreentrant' is faster and should be used if
- reentrancy is not required.
-
- The two malloc methods in fact use the memory allocation functions
- selected by `mp_set_memory_functions', these being `malloc' and
- friends by default. *Note Custom Allocation::.
-
- An additional choice `--enable-alloca=debug' is available, to help
- when debugging memory related problems (*note Debugging::).
-
-FFT Multiplication, `--disable-fft'
- By default multiplications are done using Karatsuba, 3-way Toom,
- and Fermat FFT. The FFT is only used on large to very large
- operands and can be disabled to save code size if desired.
-
-Berkeley MP, `--enable-mpbsd'
- The Berkeley MP compatibility library (`libmp') and header file
- (`mp.h') are built and installed only if `--enable-mpbsd' is used.
- *Note BSD Compatible Functions::.
-
-Assertion Checking, `--enable-assert'
- This option enables some consistency checking within the library.
- This can be of use while debugging, *note Debugging::.
-
-Execution Profiling, `--enable-profiling=prof/gprof/instrument'
- Enable profiling support, in one of various styles, *note
- Profiling::.
-
-`MPN_PATH'
- Various assembly versions of each mpn subroutines are provided.
- For a given CPU, a search is made though a path to choose a
- version of each. For example `sparcv8' has
-
- MPN_PATH="sparc32/v8 sparc32 generic"
-
- which means look first for v8 code, then plain sparc32 (which is
- v7), and finally fall back on generic C. Knowledgeable users with
- special requirements can specify a different path. Normally this
- is completely unnecessary.
-
-Documentation
- The source for the document you're now reading is `doc/gmp.texi',
- in Texinfo format, see *Note Texinfo: (texinfo)Top.
-
- Info format `doc/gmp.info' is included in the distribution. The
- usual automake targets are available to make PostScript, DVI, PDF
- and HTML (these will require various TeX and Texinfo tools).
-
- DocBook and XML can be generated by the Texinfo `makeinfo' program
- too, see *Note Options for `makeinfo': (texinfo)makeinfo options.
-
- Some supplementary notes can also be found in the `doc'
- subdirectory.
-
-
-\1f
-File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP
-
-2.2 ABI and ISA
-===============
-
-ABI (Application Binary Interface) refers to the calling conventions
-between functions, meaning what registers are used and what sizes the
-various C data types are. ISA (Instruction Set Architecture) refers to
-the instructions and registers a CPU has available.
-
- Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
-defined, the latter for compatibility with older CPUs in the family.
-GMP supports some CPUs like this in both ABIs. In fact within GMP
-`ABI' means a combination of chip ABI, plus how GMP chooses to use it.
-For example in some 32-bit ABIs, GMP may support a limb as either a
-32-bit `long' or a 64-bit `long long'.
-
- By default GMP chooses the best ABI available for a given system,
-and this generally gives significantly greater speed. But an ABI can
-be chosen explicitly to make GMP compatible with other libraries, or
-particular application requirements. For example,
-
- ./configure ABI=32
-
- In all cases it's vital that all object code used in a given program
-is compiled for the same ABI.
-
- Usually a limb is implemented as a `long'. When a `long long' limb
-is used this is encoded in the generated `gmp.h'. This is convenient
-for applications, but it does mean that `gmp.h' will vary, and can't be
-just copied around. `gmp.h' remains compiler independent though, since
-all compilers for a particular ABI will be expected to use the same
-limb type.
-
- Currently no attempt is made to follow whatever conventions a system
-has for installing library or header files built for a particular ABI.
-This will probably only matter when installing multiple builds of GMP,
-and it might be as simple as configuring with a special `libdir', or it
-might require more than that. Note that builds for different ABIs need
-to done separately, with a fresh `./configure' and `make' each.
-
-
-AMD64 (`x86_64')
- On AMD64 systems supporting both 32-bit and 64-bit modes for
- applications, the following ABI choices are available.
-
- `ABI=64'
- The 64-bit ABI uses 64-bit limbs and pointers and makes full
- use of the chip architecture. This is the default.
- Applications will usually not need special compiler flags,
- but for reference the option is
-
- gcc -m64
-
- `ABI=32'
- The 32-bit ABI is the usual i386 conventions. This will be
- slower, and is not recommended except for inter-operating
- with other code not yet 64-bit capable. Applications must be
- compiled with
-
- gcc -m32
-
- (In GCC 2.95 and earlier there's no `-m32' option, it's the
- only mode.)
-
-
-HPPA 2.0 (`hppa2.0*', `hppa64')
-
- `ABI=2.0w'
- The 2.0w ABI uses 64-bit limbs and pointers and is available
- on HP-UX 11 or up. Applications must be compiled with
-
- gcc [built for 2.0w]
- cc +DD64
-
- `ABI=2.0n'
- The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal
- calling conventions, but with 64-bit instructions permitted
- within functions. GMP uses a 64-bit `long long' for a limb.
- This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or
- higher. Applications must be compiled with
-
- gcc [built for 2.0n]
- cc +DA2.0 +e
-
- Note that current versions of GCC (eg. 3.2) don't generate
- 64-bit instructions for `long long' operations and so may be
- slower than for 2.0w. (The GMP assembly code is the same
- though.)
-
- `ABI=1.0'
- HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit
- HPPA 1.0 ABI. No special compiler options are needed for
- applications.
-
- All three ABIs are available for CPU types `hppa2.0w', `hppa2.0'
- and `hppa64', but for CPU type `hppa2.0n' only 2.0n or 1.0 are
- considered.
-
- Note that GCC on HP-UX has no options to choose between 2.0n and
- 2.0w modes, unlike HP `cc'. Instead it must be built for one or
- the other ABI. GMP will detect how it was built, and skip to the
- corresponding `ABI'.
-
-
-IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
- HP-UX supports two ABIs for IA-64. GMP performance is the same in
- both.
-
- `ABI=32'
- In the 32-bit ABI, pointers, `int's and `long's are 32 bits
- and GMP uses a 64 bit `long long' for a limb. Applications
- can be compiled without any special flags since this ABI is
- the default in both HP C and GCC, but for reference the flags
- are
-
- gcc -milp32
- cc +DD32
-
- `ABI=64'
- In the 64-bit ABI, `long's and pointers are 64 bits and GMP
- uses a `long' for a limb. Applications must be compiled with
-
- gcc -mlp64
- cc +DD64
-
- On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the
- only choice.
-
-
-MIPS under IRIX 6 (`mips*-*-irix[6789]')
- IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs
- o32, n32, and 64. n32 or 64 are recommended, and GMP performance
- will be the same in each. The default is n32.
-
- `ABI=o32'
- The o32 ABI is 32-bit pointers and integers, and no 64-bit
- operations. GMP will be slower than in n32 or 64, this
- option only exists to support old compilers, eg. GCC 2.7.2.
- Applications can be compiled with no special flags on an old
- compiler, or on a newer compiler with
-
- gcc -mabi=32
- cc -32
-
- `ABI=n32'
- The n32 ABI is 32-bit pointers and integers, but with a
- 64-bit limb using a `long long'. Applications must be
- compiled with
-
- gcc -mabi=n32
- cc -n32
-
- `ABI=64'
- The 64-bit ABI is 64-bit pointers and integers. Applications
- must be compiled with
-
- gcc -mabi=64
- cc -64
-
- Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have
- the necessary support for n32 or 64 and so only gets a 32-bit limb
- and the MIPS 2 code.
-
-
-PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970', `power4', `power5')
-
- `ABI=aix64'
- The AIX 64 ABI uses 64-bit limbs and pointers and is the
- default on PowerPC 64 `*-*-aix*' systems. Applications must
- be compiled with
-
- gcc -maix64
- xlc -q64
-
- `ABI=mode64'
- The `mode64' ABI uses 64-bit limbs and pointers, and is the
- default on 64-bit GNU/Linux, BSD, and Mac OS X/Darwin
- systems. Applications must be compiled with
-
- gcc -m64
-
- `ABI=mode32'
- The `mode32' ABI uses a 64-bit `long long' limb but with the
- chip still in 32-bit mode and using 32-bit calling
- conventions. This is the default on for systems where the
- true 64-bit ABIs are unavailable. No special compiler
- options are needed for applications.
-
- `ABI=32'
- This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
- special compiler options are needed for applications.
-
- GMP speed is greatest in `aix64' and `mode32'. In `ABI=32' only
- the 32-bit ISA is used and this doesn't make full use of a 64-bit
- chip. On a suitable system we could perhaps use more of the ISA,
- but there are no plans to do so.
-
-
-Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
-
- `ABI=64'
- The 64-bit V9 ABI is available on the various BSD sparc64
- ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
- and up (when the kernel is in 64-bit mode). GCC 3.2 or
- higher, or Sun `cc' is required. On GNU/Linux, depending on
- the default `gcc' mode, applications must be compiled with
-
- gcc -m64
-
- On Solaris applications must be compiled with
-
- gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
- cc -xarch=v9
-
- On the BSD sparc64 systems no special options are required,
- since 64-bits is the only ABI available.
-
- `ABI=32'
- For the basic 32-bit ABI, GMP still uses as much of the V9
- ISA as it can. In the Sun documentation this combination is
- known as "v8plus". On GNU/Linux, depending on the default
- `gcc' mode, applications may need to be compiled with
-
- gcc -m32
-
- On Solaris, no special compiler options are required for
- applications, though using something like the following is
- recommended. (`gcc' 2.8 and earlier only support `-mv8'
- though.)
-
- gcc -mv8plus
- cc -xarch=v8plus
-
- GMP speed is greatest in `ABI=64', so it's the default where
- available. The speed is partly because there are extra registers
- available and partly because 64-bits is considered the more
- important case and has therefore had better code written for it.
-
- Don't be confused by the names of the `-m' and `-x' compiler
- options, they're called `arch' but effectively control both ABI
- and ISA.
-
- On Solaris 2.6 and earlier, only `ABI=32' is available since the
- kernel doesn't save all registers.
-
- On Solaris 2.7 with the kernel in 32-bit mode, a normal native
- build will reject `ABI=64' because the resulting executables won't
- run. `ABI=64' can still be built if desired by making it look
- like a cross-compile, for example
-
- ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
-
-\1f
-File: gmp.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing GMP
-
-2.3 Notes for Package Builds
-============================
-
-GMP should present no great difficulties for packaging in a binary
-distribution.
-
- Libtool is used to build the library and `-version-info' is set
-appropriately, having started from `3:0:0' in GMP 3.0 (*note Library
-interface versions: (libtool)Versioning.).
-
- The GMP 4 series will be upwardly binary compatible in each release
-and will be upwardly binary compatible with all of the GMP 3 series.
-Additional function interfaces may be added in each release, so on
-systems where libtool versioning is not fully checked by the loader an
-auxiliary mechanism may be needed to express that a dynamic linked
-application depends on a new enough GMP.
-
- An auxiliary mechanism may also be needed to express that
-`libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires
-`libgmp.la' from the same GMP version, since this is not done by the
-libtool versioning, nor otherwise. A mismatch will result in
-unresolved symbols from the linker, or perhaps the loader.
-
- When building a package for a CPU family, care should be taken to use
-`--host' (or `--build') to choose the least common denominator among
-the CPUs which might use the package. For example this might mean plain
-`sparc' (meaning V7) for SPARCs.
-
- For x86s, `--enable-fat' sets things up for a fat binary build,
-making a runtime selection of optimized low level routines. This is a
-good choice for packaging to run on a range of x86 chips.
-
- Users who care about speed will want GMP built for their exact CPU
-type, to make best use of the available optimizations. Providing a way
-to suitably rebuild a package may be useful. This could be as simple
-as making it possible for a user to omit `--build' (and `--host') so
-`./config.guess' will detect the CPU. But a way to manually specify a
-`--build' will be wanted for systems where `./config.guess' is inexact.
-
- On systems with multiple ABIs, a packaged build will need to decide
-which among the choices is to be provided, see *Note ABI and ISA::. A
-given run of `./configure' etc will only build one ABI. If a second
-ABI is also required then a second run of `./configure' etc must be
-made, starting from a clean directory tree (`make distclean').
-
- As noted under "ABI and ISA", currently no attempt is made to follow
-system conventions for install locations that vary with ABI, such as
-`/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'.
-A package build can override `libdir' and other standard variables as
-necessary.
-
- Note that `gmp.h' is a generated file, and will be architecture and
-ABI dependent. When attempting to install two ABIs simultaneously it
-will be important that an application compile gets the correct `gmp.h'
-for its desired ABI. If compiler include paths don't vary with ABI
-options then it might be necessary to create a `/usr/include/gmp.h'
-which tests preprocessor symbols and chooses the correct actual `gmp.h'.
-
-\1f
-File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP
-
-2.4 Notes for Particular Systems
-================================
-
-AIX 3 and 4
- On systems `*-*-aix[34]*' shared libraries are disabled by
- default, since some versions of the native `ar' fail on the
- convenience libraries used. A shared build can be attempted with
-
- ./configure --enable-shared --disable-static
-
- Note that the `--disable-static' is necessary because in a shared
- build libtool makes `libgmp.a' a symlink to `libgmp.so',
- apparently for the benefit of old versions of `ld' which only
- recognise `.a', but unfortunately this is done even if a fully
- functional `ld' is available.
-
-ARM
- On systems `arm*-*-*', versions of GCC up to and including 2.95.3
- have a bug in unsigned division, giving wrong results for some
- operands. GMP `./configure' will demand GCC 2.95.4 or later.
-
-Compaq C++
- Compaq C++ on OSF 5.1 has two flavours of `iostream', a standard
- one and an old pre-standard one (see `man iostream_intro'). GMP
- can only use the standard one, which unfortunately is not the
- default but must be selected by defining `__USE_STD_IOSTREAM'.
- Configure with for instance
-
- ./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
-
-Floating Point Mode
- On some systems, the hardware floating point has a control mode
- which can set all operations to be done in a particular precision,
- for instance single, double or extended on x86 systems (x87
- floating point). The GMP functions involving a `double' cannot be
- expected to operate to their full precision when the hardware is
- in single precision mode. Of course this affects all code,
- including application code, not just GMP.
-
-MS-DOS and MS Windows
- On an MS-DOS system DJGPP can be used to build GMP, and on an MS
- Windows system Cygwin, DJGPP and MINGW can be used. All three are
- excellent ports of GCC and the various GNU tools.
-
- `http://www.cygwin.com/'
- `http://www.delorie.com/djgpp/'
- `http://www.mingw.org/'
-
- Microsoft also publishes an Interix "Services for Unix" which can
- be used to build GMP on Windows (with a normal `./configure'), but
- it's not free software.
-
-MS Windows DLLs
- On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
- GMP builds only a static library, but a DLL can be built instead
- using
-
- ./configure --disable-static --enable-shared
-
- Static and DLL libraries can't both be built, since certain export
- directives in `gmp.h' must be different.
-
- A MINGW DLL build of GMP can be used with Microsoft C. Libtool
- doesn't install a `.lib' format import library, but it can be
- created with MS `lib' as follows, and copied to the install
- directory. Similarly for `libmp' and `libgmpxx'.
-
- cd .libs
- lib /def:libgmp-3.dll.def /out:libgmp-3.lib
-
- MINGW uses the C runtime library `msvcrt.dll' for I/O, so
- applications wanting to use the GMP I/O routines must be compiled
- with `cl /MD' to do the same. If one of the other C runtime
- library choices provided by MS C is desired then the suggestion is
- to use the GMP string functions and confine I/O to the application.
-
-Motorola 68k CPU Types
- `m68k' is taken to mean 68000. `m68020' or higher will give a
- performance boost on applicable CPUs. `m68360' can be used for
- CPU32 series chips. `m68302' can be used for "Dragonball" series
- chips, though this is merely a synonym for `m68000'.
-
-OpenBSD 2.6
- `m4' in this release of OpenBSD has a bug in `eval' that makes it
- unsuitable for `.asm' file processing. `./configure' will detect
- the problem and either abort or choose another m4 in the `PATH'.
- The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
-
-Power CPU Types
- In GMP, CPU types `power*' and `powerpc*' will each use
- instructions not available on the other, so it's important to
- choose the right one for the CPU that will be used. Currently GMP
- has no assembly code support for using just the common instruction
- subset. To get executables that run on both, the current
- suggestion is to use the generic C code (CPU `none'), possibly
- with appropriate compiler options (like `-mcpu=common' for `gcc').
- CPU `rs6000' (which is not a CPU but a family of workstations) is
- accepted by `config.sub', but is currently equivalent to `none'.
-
-Sparc CPU Types
- `sparcv8' or `supersparc' on relevant systems will give a
- significant performance increase over the V7 code selected by plain
- `sparc'.
-
-Sparc App Regs
- The GMP assembly code for both 32-bit and 64-bit Sparc clobbers the
- "application registers" `g2', `g3' and `g4', the same way that the
- GCC default `-mapp-regs' does (*note SPARC Options: (gcc)SPARC
- Options.).
-
- This makes that code unsuitable for use with the special V9
- `-mcmodel=embmedany' (which uses `g4' as a data segment pointer),
- and for applications wanting to use those registers for special
- purposes. In these cases the only suggestion currently is to
- build GMP with CPU `none' to avoid the assembly code.
-
-SunOS 4
- `/usr/bin/m4' lacks various features needed to process `.asm'
- files, and instead `./configure' will automatically use
- `/usr/5bin/m4', which we believe is always available (if not then
- use GNU m4).
-
-x86 CPU Types
- `i586', `pentium' or `pentiummmx' code is good for its intended P5
- Pentium chips, but quite slow when run on Intel P6 class chips
- (PPro, P-II, P-III). `i386' is a better choice when making
- binaries that must run on both.
-
-x86 MMX and SSE2 Code
- If the CPU selected has MMX code but the assembler doesn't support
- it, a warning is given and non-MMX code is used instead. This
- will be an inferior build, since the MMX code that's present is
- there because it's faster than the corresponding plain integer
- code. The same applies to SSE2.
-
- Old versions of `gas' don't support MMX instructions, in particular
- version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent
- OpenBSD 3.1 doesn't.
-
- Solaris 2.6 and 2.7 `as' generate incorrect object code for
- register to register `movq' instructions, and so can't be used for
- MMX code. Install a recent `gas' if MMX code is wanted on these
- systems.
-
-\1f
-File: gmp.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing GMP
-
-2.5 Known Build Problems
-========================
-
-You might find more up-to-date information at `http://gmplib.org/'.
-
-Compiler link options
- The version of libtool currently in use rather aggressively strips
- compiler options when linking a shared library. This will
- hopefully be relaxed in the future, but for now if this is a
- problem the suggestion is to create a little script to hide them,
- and for instance configure with
-
- ./configure CC=gcc-with-my-options
-
-DJGPP (`*-*-msdosdjgpp*')
- The DJGPP port of `bash' 2.03 is unable to run the `configure'
- script, it exits silently, having died writing a preamble to
- `config.log'. Use `bash' 2.04 or higher.
-
- `make all' was found to run out of memory during the final
- `libgmp.la' link on one system tested, despite having 64Mb
- available. Running `make libgmp.la' directly helped, perhaps
- recursing into the various subdirectories uses up memory.
-
-GNU binutils `strip' prior to 2.12
- `strip' from GNU binutils 2.11 and earlier should not be used on
- the static libraries `libgmp.a' and `libmp.a' since it will
- discard all but the last of multiple archive members with the same
- name, like the three versions of `init.o' in `libgmp.a'. Binutils
- 2.12 or higher can be used successfully.
-
- The shared libraries `libgmp.so' and `libmp.so' are not affected by
- this and any version of `strip' can be used on them.
-
-`make' syntax error
- On certain versions of SCO OpenServer 5 and IRIX 6.5 the native
- `make' is unable to handle the long dependencies list for
- `libgmp.la'. The symptom is a "syntax error" on the following
- line of the top-level `Makefile'.
-
- libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
-
- Either use GNU Make, or as a workaround remove
- `$(libgmp_la_DEPENDENCIES)' from that line (which will make the
- initial build work, but if any recompiling is done `libgmp.la'
- might not be rebuilt).
-
-MacOS X (`*-*-darwin*')
- Libtool currently only knows how to create shared libraries on
- MacOS X using the native `cc' (which is a modified GCC), not a
- plain GCC. A static-only build should work though
- (`--disable-shared').
-
-NeXT prior to 3.3
- The system compiler on old versions of NeXT was a massacred and
- old GCC, even if it called itself `cc'. This compiler cannot be
- used to build GMP, you need to get a real GCC, and install that.
- (NeXT may have fixed this in release 3.3 of their system.)
-
-POWER and PowerPC
- Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP
- on POWER or PowerPC. If you want to use GCC for these machines,
- get GCC 2.7.2.1 (or later).
-
-Sequent Symmetry
- Use the GNU assembler instead of the system assembler, since the
- latter has serious bugs.
-
-Solaris 2.6
- The system `sed' prints an error "Output line too long" when
- libtool builds `libgmp.la'. This doesn't seem to cause any
- obvious ill effects, but GNU `sed' is recommended, to avoid any
- doubt.
-
-Sparc Solaris 2.7 with gcc 2.95.2 in `ABI=32'
- A shared library build of GMP seems to fail in this combination,
- it builds but then fails the tests, apparently due to some
- incorrect data relocations within `gmp_randinit_lc_2exp_size'.
- The exact cause is unknown, `--disable-shared' is recommended.
-
-\1f
-File: gmp.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing GMP
-
-2.6 Performance optimization
-============================
-
-For optimal performance, build GMP for the exact CPU type of the target
-computer, see *Note Build Options::.
-
- Unlike what is the case for most other programs, the compiler
-typically doesn't matter much, since GMP uses assembly language for the
-most critical operation.
-
- In particular for long-running GMP applications, and applications
-demanding extremely large numbers, building and running the `tuneup'
-program in the `tune' subdirectory, can be important. For example,
-
- cd tune
- make tuneup
- ./tuneup
-
- will generate better contents for the `gmp-mparam.h' parameter file.
-
- To use the results, put the output in the file file indicated in the
-`Parameters for ...' header. Then recompile from scratch.
-
- The `tuneup' program takes one useful parameter, `-f NNN', which
-instructs the program how long to check FFT multiply parameters. If
-you're going to use GMP for extremely large numbers, you may want to
-run `tuneup' with a large NNN value.
-
-\1f
-File: gmp.info, Node: GMP Basics, Next: Reporting Bugs, Prev: Installing GMP, Up: Top
-
-3 GMP Basics
-************
-
-*Using functions, macros, data types, etc. not documented in this
-manual is strongly discouraged. If you do so your application is
-guaranteed to be incompatible with future versions of GMP.*
-
-* Menu:
-
-* Headers and Libraries::
-* Nomenclature and Types::
-* Function Classes::
-* Variable Conventions::
-* Parameter Conventions::
-* Memory Management::
-* Reentrancy::
-* Useful Macros and Constants::
-* Compatibility with older versions::
-* Demonstration Programs::
-* Efficiency::
-* Debugging::
-* Profiling::
-* Autoconf::
-* Emacs::
-
-\1f
-File: gmp.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: GMP Basics, Up: GMP Basics
-
-3.1 Headers and Libraries
-=========================
-
-All declarations needed to use GMP are collected in the include file
-`gmp.h'. It is designed to work with both C and C++ compilers.
-
- #include <gmp.h>
-
- Note however that prototypes for GMP functions with `FILE *'
-parameters are only provided if `<stdio.h>' is included too.
-
- #include <stdio.h>
- #include <gmp.h>
-
- Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes
-with `va_list' parameters, such as `gmp_vprintf'. And `<obstack.h>'
-for prototypes with `struct obstack' parameters, such as
-`gmp_obstack_printf', when available.
-
- All programs using GMP must link against the `libgmp' library. On a
-typical Unix-like system this can be done with `-lgmp', for example
-
- gcc myprogram.c -lgmp
-
- GMP C++ functions are in a separate `libgmpxx' library. This is
-built and installed if C++ support has been enabled (*note Build
-Options::). For example,
-
- g++ mycxxprog.cc -lgmpxx -lgmp
-
- GMP is built using Libtool and an application can use that to link
-if desired, *note GNU Libtool: (libtool)Top.
-
- If GMP has been installed to a non-standard location then it may be
-necessary to use `-I' and `-L' compiler options to point to the right
-directories, and some sort of run-time path for a shared library.
-
-\1f
-File: gmp.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: GMP Basics
-
-3.2 Nomenclature and Types
-==========================
-
-In this manual, "integer" usually means a multiple precision integer, as
-defined by the GMP library. The C data type for such integers is
-`mpz_t'. Here are some examples of how to declare such integers:
-
- mpz_t sum;
-
- struct foo { mpz_t x, y; };
-
- mpz_t vec[20];
-
- "Rational number" means a multiple precision fraction. The C data
-type for these fractions is `mpq_t'. For example:
-
- mpq_t quotient;
-
- "Floating point number" or "Float" for short, is an arbitrary
-precision mantissa with a limited precision exponent. The C data type
-for such objects is `mpf_t'. For example:
-
- mpf_t fp;
-
- The floating point functions accept and return exponents in the C
-type `mp_exp_t'. Currently this is usually a `long', but on some
-systems it's an `int' for efficiency.
-
- A "limb" means the part of a multi-precision number that fits in a
-single machine word. (We chose this word because a limb of the human
-body is analogous to a digit, only larger, and containing several
-digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
-is `mp_limb_t'.
-
- Counts of limbs of a multi-precision number represented in the C type
-`mp_size_t'. Currently this is normally a `long', but on some systems
-it's an `int' for efficiency, and on some systems it will be `long
-long' in the future.
-
- Counts of bits of a multi-precision number are represented in the C
-type `mp_bitcnt_t'. Currently this is always an `unsigned long', but on
-some systems it will be an `unsigned long long' in the future .
-
- "Random state" means an algorithm selection and current state data.
-The C data type for such objects is `gmp_randstate_t'. For example:
-
- gmp_randstate_t rstate;
-
- Also, in general `mp_bitcnt_t' is used for bit counts and ranges, and
-`size_t' is used for byte or character counts.
-
-\1f
-File: gmp.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics
-
-3.3 Function Classes
-====================
-
-There are six classes of functions in the GMP library:
-
- 1. Functions for signed integer arithmetic, with names beginning with
- `mpz_'. The associated type is `mpz_t'. There are about 150
- functions in this class. (*note Integer Functions::)
-
- 2. Functions for rational number arithmetic, with names beginning with
- `mpq_'. The associated type is `mpq_t'. There are about 40
- functions in this class, but the integer functions can be used for
- arithmetic on the numerator and denominator separately. (*note
- Rational Number Functions::)
-
- 3. Functions for floating-point arithmetic, with names beginning with
- `mpf_'. The associated type is `mpf_t'. There are about 60
- functions is this class. (*note Floating-point Functions::)
-
- 4. Functions compatible with Berkeley MP, such as `itom', `madd', and
- `mult'. The associated type is `MINT'. (*note BSD Compatible
- Functions::)
-
- 5. Fast low-level functions that operate on natural numbers. These
- are used by the functions in the preceding groups, and you can
- also call them directly from very time-critical user programs.
- These functions' names begin with `mpn_'. The associated type is
- array of `mp_limb_t'. There are about 30 (hard-to-use) functions
- in this class. (*note Low-level Functions::)
-
- 6. Miscellaneous functions. Functions for setting up custom
- allocation and functions for generating random numbers. (*note
- Custom Allocation::, and *note Random Number Functions::)
-
-\1f
-File: gmp.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: GMP Basics
-
-3.4 Variable Conventions
-========================
-
-GMP functions generally have output arguments before input arguments.
-This notation is by analogy with the assignment operator. The BSD MP
-compatibility functions are exceptions, having the output arguments
-last.
-
- GMP lets you use the same variable for both input and output in one
-call. For example, the main function for integer multiplication,
-`mpz_mul', can be used to square `x' and put the result back in `x' with
-
- mpz_mul (x, x, x);
-
- Before you can assign to a GMP variable, you need to initialize it
-by calling one of the special initialization functions. When you're
-done with a variable, you need to clear it out, using one of the
-functions for that purpose. Which function to use depends on the type
-of variable. See the chapters on integer functions, rational number
-functions, and floating-point functions for details.
-
- A variable should only be initialized once, or at least cleared
-between each initialization. After a variable has been initialized, it
-may be assigned to any number of times.
-
- For efficiency reasons, avoid excessive initializing and clearing.
-In general, initialize near the start of a function and clear near the
-end. For example,
-
- void
- foo (void)
- {
- mpz_t n;
- int i;
- mpz_init (n);
- for (i = 1; i < 100; i++)
- {
- mpz_mul (n, ...);
- mpz_fdiv_q (n, ...);
- ...
- }
- mpz_clear (n);
- }
-
-\1f
-File: gmp.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics
-
-3.5 Parameter Conventions
-=========================
-
-When a GMP variable is used as a function parameter, it's effectively a
-call-by-reference, meaning if the function stores a value there it will
-change the original in the caller. Parameters which are input-only can
-be designated `const' to provoke a compiler error or warning on
-attempting to modify them.
-
- When a function is going to return a GMP result, it should designate
-a parameter that it sets, like the library functions do. More than one
-value can be returned by having more than one output parameter, again
-like the library functions. A `return' of an `mpz_t' etc doesn't
-return the object, only a pointer, and this is almost certainly not
-what's wanted.
-
- Here's an example accepting an `mpz_t' parameter, doing a
-calculation, and storing the result to the indicated parameter.
-
- void
- foo (mpz_t result, const mpz_t param, unsigned long n)
- {
- unsigned long i;
- mpz_mul_ui (result, param, n);
- for (i = 1; i < n; i++)
- mpz_add_ui (result, result, i*7);
- }
-
- int
- main (void)
- {
- mpz_t r, n;
- mpz_init (r);
- mpz_init_set_str (n, "123456", 0);
- foo (r, n, 20L);
- gmp_printf ("%Zd\n", r);
- return 0;
- }
-
- `foo' works even if the mainline passes the same variable for
-`param' and `result', just like the library functions. But sometimes
-it's tricky to make that work, and an application might not want to
-bother supporting that sort of thing.
-
- For interest, the GMP types `mpz_t' etc are implemented as
-one-element arrays of certain structures. This is why declaring a
-variable creates an object with the fields GMP needs, but then using it
-as a parameter passes a pointer to the object. Note that the actual
-fields in each `mpz_t' etc are for internal use only and should not be
-accessed directly by code that expects to be compatible with future GMP
-releases.
-
-\1f
-File: gmp.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: GMP Basics
-
-3.6 Memory Management
-=====================
-
-The GMP types like `mpz_t' are small, containing only a couple of sizes,
-and pointers to allocated data. Once a variable is initialized, GMP
-takes care of all space allocation. Additional space is allocated
-whenever a variable doesn't have enough.
-
- `mpz_t' and `mpq_t' variables never reduce their allocated space.
-Normally this is the best policy, since it avoids frequent reallocation.
-Applications that need to return memory to the heap at some particular
-point can use `mpz_realloc2', or clear variables no longer needed.
-
- `mpf_t' variables, in the current implementation, use a fixed amount
-of space, determined by the chosen precision and allocated at
-initialization, so their size doesn't change.
-
- All memory is allocated using `malloc' and friends by default, but
-this can be changed, see *Note Custom Allocation::. Temporary memory
-on the stack is also used (via `alloca'), but this can be changed at
-build-time if desired, see *Note Build Options::.
-
-\1f
-File: gmp.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: GMP Basics
-
-3.7 Reentrancy
-==============
-
-GMP is reentrant and thread-safe, with some exceptions:
-
- * If configured with `--enable-alloca=malloc-notreentrant' (or with
- `--enable-alloca=notreentrant' when `alloca' is not available),
- then naturally GMP is not reentrant.
-
- * `mpf_set_default_prec' and `mpf_init' use a global variable for the
- selected precision. `mpf_init2' can be used instead, and in the
- C++ interface an explicit precision to the `mpf_class' constructor.
-
- * `mpz_random' and the other old random number functions use a global
- random state and are hence not reentrant. The newer random number
- functions that accept a `gmp_randstate_t' parameter can be used
- instead.
-
- * `gmp_randinit' (obsolete) returns an error indication through a
- global variable, which is not thread safe. Applications are
- advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp'
- instead.
-
- * `mp_set_memory_functions' uses global variables to store the
- selected memory allocation functions.
-
- * If the memory allocation functions set by a call to
- `mp_set_memory_functions' (or `malloc' and friends by default) are
- not reentrant, then GMP will not be reentrant either.
-
- * If the standard I/O functions such as `fwrite' are not reentrant
- then the GMP I/O functions using them will not be reentrant either.
-
- * It's safe for two threads to read from the same GMP variable
- simultaneously, but it's not safe for one to read while the
- another might be writing, nor for two threads to write
- simultaneously. It's not safe for two threads to generate a
- random number from the same `gmp_randstate_t' simultaneously,
- since this involves an update of that variable.
-
-\1f
-File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: GMP Basics
-
-3.8 Useful Macros and Constants
-===============================
-
- -- Global Constant: const int mp_bits_per_limb
- The number of bits per limb.
-
- -- Macro: __GNU_MP_VERSION
- -- Macro: __GNU_MP_VERSION_MINOR
- -- Macro: __GNU_MP_VERSION_PATCHLEVEL
- The major and minor GMP version, and patch level, respectively, as
- integers. For GMP i.j, these numbers will be i, j, and 0,
- respectively. For GMP i.j.k, these numbers will be i, j, and k,
- respectively.
-
- -- Global Constant: const char * const gmp_version
- The GMP version number, as a null-terminated string, in the form
- "i.j.k". This release is "5.0.1". Note that the format "i.j" was
- used when k was zero was used before version 4.3.0.
-
- -- Macro: __GMP_CC
- -- Macro: __GMP_CFLAGS
- The compiler and compiler flags, respectively, used when compiling
- GMP, as strings.
-
-\1f
-File: gmp.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: GMP Basics
-
-3.9 Compatibility with older versions
-=====================================
-
-This version of GMP is upwardly binary compatible with all 4.x and 3.x
-versions, and upwardly compatible at the source level with all 2.x
-versions, with the following exceptions.
-
- * `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
- consistency with other `mpn' functions.
-
- * `mpf_get_prec' counted precision slightly differently in GMP 3.0
- and 3.0.1, but in 3.1 reverted to the 2.x style.
-
- There are a number of compatibility issues between GMP 1 and GMP 2
-that of course also apply when porting applications from GMP 1 to GMP
-4. Please see the GMP 2 manual for details.
-
- The Berkeley MP compatibility library (*note BSD Compatible
-Functions::) is source and binary compatible with the standard `libmp'.
-
-\1f
-File: gmp.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: GMP Basics
-
-3.10 Demonstration programs
-===========================
-
-The `demos' subdirectory has some sample programs using GMP. These
-aren't built or installed, but there's a `Makefile' with rules for them.
-For instance,
-
- make pexpr
- ./pexpr 68^975+10
-
-The following programs are provided
-
- * `pexpr' is an expression evaluator, the program used on the GMP
- web page.
-
- * The `calc' subdirectory has a similar but simpler evaluator using
- `lex' and `yacc'.
-
- * The `expr' subdirectory is yet another expression evaluator, a
- library designed for ease of use within a C program. See
- `demos/expr/README' for more information.
-
- * `factorize' is a Pollard-Rho factorization program.
-
- * `isprime' is a command-line interface to the `mpz_probab_prime_p'
- function.
-
- * `primes' counts or lists primes in an interval, using a sieve.
-
- * `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic
- class numbers.
-
- * The `perl' subdirectory is a comprehensive perl interface to GMP.
- See `demos/perl/INSTALL' for more information. Documentation is
- in POD format in `demos/perl/GMP.pm'.
-
- As an aside, consideration has been given at various times to some
-sort of expression evaluation within the main GMP library. Going
-beyond something minimal quickly leads to matters like user-defined
-functions, looping, fixnums for control variables, etc, which are
-considered outside the scope of GMP (much closer to language
-interpreters or compilers, *Note Language Bindings::.) Something
-simple for program input convenience may yet be a possibility, a
-combination of the `expr' demo and the `pexpr' tree back-end perhaps.
-But for now the above evaluators are offered as illustrations.
-
-\1f
-File: gmp.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: GMP Basics
-
-3.11 Efficiency
-===============
-
-Small Operands
- On small operands, the time for function call overheads and memory
- allocation can be significant in comparison to actual calculation.
- This is unavoidable in a general purpose variable precision
- library, although GMP attempts to be as efficient as it can on
- both large and small operands.
-
-Static Linking
- On some CPUs, in particular the x86s, the static `libgmp.a' should
- be used for maximum speed, since the PIC code in the shared
- `libgmp.so' will have a small overhead on each function call and
- global data address. For many programs this will be
- insignificant, but for long calculations there's a gain to be had.
-
-Initializing and Clearing
- Avoid excessive initializing and clearing of variables, since this
- can be quite time consuming, especially in comparison to otherwise
- fast operations like addition.
-
- A language interpreter might want to keep a free list or stack of
- initialized variables ready for use. It should be possible to
- integrate something like that with a garbage collector too.
-
-Reallocations
- An `mpz_t' or `mpq_t' variable used to hold successively increasing
- values will have its memory repeatedly `realloc'ed, which could be
- quite slow or could fragment memory, depending on the C library.
- If an application can estimate the final size then `mpz_init2' or
- `mpz_realloc2' can be called to allocate the necessary space from
- the beginning (*note Initializing Integers::).
-
- It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
- is too small, since all functions will do a further reallocation
- if necessary. Badly overestimating memory required will waste
- space though.
-
-`2exp' Functions
- It's up to an application to call functions like `mpz_mul_2exp'
- when appropriate. General purpose functions like `mpz_mul' make
- no attempt to identify powers of two or other special forms,
- because such inputs will usually be very rare and testing every
- time would be wasteful.
-
-`ui' and `si' Functions
- The `ui' functions and the small number of `si' functions exist for
- convenience and should be used where applicable. But if for
- example an `mpz_t' contains a value that fits in an `unsigned
- long' there's no need extract it and call a `ui' function, just
- use the regular `mpz' function.
-
-In-Place Operations
- `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
- `mpf_neg' are fast when used for in-place operations like
- `mpz_abs(x,x)', since in the current implementation only a single
- field of `x' needs changing. On suitable compilers (GCC for
- instance) this is inlined too.
-
- `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
- from an in-place operation like `mpz_add_ui(x,x,y)', since usually
- only one or two limbs of `x' will need to be changed. The same
- applies to the full precision `mpz_add' etc if `y' is small. If
- `y' is big then cache locality may be helped, but that's all.
-
- `mpz_mul' is currently the opposite, a separate destination is
- slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
- only one limb, make a temporary copy of `x' before forming the
- result. Normally that copying will only be a tiny fraction of the
- time for the multiply, so this is not a particularly important
- consideration.
-
- `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
- attempt to recognise a copy of something to itself, so a call like
- `mpz_set(x,x)' will be wasteful. Naturally that would never be
- written deliberately, but if it might arise from two pointers to
- the same object then a test to avoid it might be desirable.
-
- if (x != y)
- mpz_set (x, y);
-
- Note that it's never worth introducing extra `mpz_set' calls just
- to get in-place operations. If a result should go to a particular
- variable then just direct it there and let GMP take care of data
- movement.
-
-Divisibility Testing (Small Integers)
- `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
- functions for testing whether an `mpz_t' is divisible by an
- individual small integer. They use an algorithm which is faster
- than `mpz_tdiv_ui', but which gives no useful information about
- the actual remainder, only whether it's zero (or a particular
- value).
-
- However when testing divisibility by several small integers, it's
- best to take a remainder modulo their product, to save
- multi-precision operations. For instance to test whether a number
- is divisible by any of 23, 29 or 31 take a remainder modulo
- 23*29*31 = 20677 and then test that.
-
- The division functions like `mpz_tdiv_q_ui' which give a quotient
- as well as a remainder are generally a little slower than the
- remainder-only functions like `mpz_tdiv_ui'. If the quotient is
- only rarely wanted then it's probably best to just take a
- remainder and then go back and calculate the quotient if and when
- it's wanted (`mpz_divexact_ui' can be used if the remainder is
- zero).
-
-Rational Arithmetic
- The `mpq' functions operate on `mpq_t' values with no common
- factors in the numerator and denominator. Common factors are
- checked-for and cast out as necessary. In general, cancelling
- factors every time is the best approach since it minimizes the
- sizes for subsequent operations.
-
- However, applications that know something about the factorization
- of the values they're working with might be able to avoid some of
- the GCDs used for canonicalization, or swap them for divisions.
- For example when multiplying by a prime it's enough to check for
- factors of it in the denominator instead of doing a full GCD. Or
- when forming a big product it might be known that very little
- cancellation will be possible, and so canonicalization can be left
- to the end.
-
- The `mpq_numref' and `mpq_denref' macros give access to the
- numerator and denominator to do things outside the scope of the
- supplied `mpq' functions. *Note Applying Integer Functions::.
-
- The canonical form for rationals allows mixed-type `mpq_t' and
- integer additions or subtractions to be done directly with
- multiples of the denominator. This will be somewhat faster than
- `mpq_add'. For example,
-
- /* mpq increment */
- mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
-
- /* mpq += unsigned long */
- mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
-
- /* mpq -= mpz */
- mpz_submul (mpq_numref(q), mpq_denref(q), z);
-
-Number Sequences
- Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
- designed for calculating isolated values. If a range of values is
- wanted it's probably best to call to get a starting point and
- iterate from there.
-
-Text Input/Output
- Hexadecimal or octal are suggested for input or output in text
- form. Power-of-2 bases like these can be converted much more
- efficiently than other bases, like decimal. For big numbers
- there's usually nothing of particular interest to be seen in the
- digits, so the base doesn't matter much.
-
- Maybe we can hope octal will one day become the normal base for
- everyday use, as proposed by King Charles XII of Sweden and later
- reformers.
-
-\1f
-File: gmp.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: GMP Basics
-
-3.12 Debugging
-==============
-
-Stack Overflow
- Depending on the system, a segmentation violation or bus error
- might be the only indication of stack overflow. See
- `--enable-alloca' choices in *Note Build Options::, for how to
- address this.
-
- In new enough versions of GCC, `-fstack-check' may be able to
- ensure an overflow is recognised by the system before too much
- damage is done, or `-fstack-limit-symbol' or
- `-fstack-limit-register' may be able to add checking if the system
- itself doesn't do any (*note Options for Code Generation:
- (gcc)Code Gen Options.). These options must be added to the
- `CFLAGS' used in the GMP build (*note Build Options::), adding
- them just to an application will have no effect. Note also
- they're a slowdown, adding overhead to each function call and each
- stack allocation.
-
-Heap Problems
- The most likely cause of application problems with GMP is heap
- corruption. Failing to `init' GMP variables will have
- unpredictable effects, and corruption arising elsewhere in a
- program may well affect GMP. Initializing GMP variables more than
- once or failing to clear them will cause memory leaks.
-
- In all such cases a `malloc' debugger is recommended. On a GNU or
- BSD system the standard C library `malloc' has some diagnostic
- facilities, see *Note Allocation Debugging: (libc)Allocation
- Debugging, or `man 3 malloc'. Other possibilities, in no
- particular order, include
-
- `http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/'
- `http://dmalloc.com/'
- `http://www.perens.com/FreeSoftware/' (electric fence)
- `http://packages.debian.org/stable/devel/fda'
- `http://www.gnupdate.org/components/leakbug/'
- `http://people.redhat.com/~otaylor/memprof/'
- `http://www.cbmamiga.demon.co.uk/mpatrol/'
-
- The GMP default allocation routines in `memory.c' also have a
- simple sentinel scheme which can be enabled with `#define DEBUG'
- in that file. This is mainly designed for detecting buffer
- overruns during GMP development, but might find other uses.
-
-Stack Backtraces
- On some systems the compiler options GMP uses by default can
- interfere with debugging. In particular on x86 and 68k systems
- `-fomit-frame-pointer' is used and this generally inhibits stack
- backtracing. Recompiling without such options may help while
- debugging, though the usual caveats about it potentially moving a
- memory problem or hiding a compiler bug will apply.
-
-GDB, the GNU Debugger
- A sample `.gdbinit' is included in the distribution, showing how
- to call some undocumented dump functions to print GMP variables
- from within GDB. Note that these functions shouldn't be used in
- final application code since they're undocumented and may be
- subject to incompatible changes in future versions of GMP.
-
-Source File Paths
- GMP has multiple source files with the same name, in different
- directories. For example `mpz', `mpq' and `mpf' each have an
- `init.c'. If the debugger can't already determine the right one
- it may help to build with absolute paths on each C file. One way
- to do that is to use a separate object directory with an absolute
- path to the source directory.
-
- cd /my/build/dir
- /my/source/dir/gmp-5.0.1/configure
-
- This works via `VPATH', and might require GNU `make'. Alternately
- it might be possible to change the `.c.lo' rules appropriately.
-
-Assertion Checking
- The build option `--enable-assert' is available to add some
- consistency checks to the library (see *Note Build Options::).
- These are likely to be of limited value to most applications.
- Assertion failures are just as likely to indicate memory
- corruption as a library or compiler bug.
-
- Applications using the low-level `mpn' functions, however, will
- benefit from `--enable-assert' since it adds checks on the
- parameters of most such functions, many of which have subtle
- restrictions on their usage. Note however that only the generic C
- code has checks, not the assembly code, so CPU `none' should be
- used for maximum checking.
-
-Temporary Memory Checking
- The build option `--enable-alloca=debug' arranges that each block
- of temporary memory in GMP is allocated with a separate call to
- `malloc' (or the allocation function set with
- `mp_set_memory_functions').
-
- This can help a malloc debugger detect accesses outside the
- intended bounds, or detect memory not released. In a normal
- build, on the other hand, temporary memory is allocated in blocks
- which GMP divides up for its own use, or may be allocated with a
- compiler builtin `alloca' which will go nowhere near any malloc
- debugger hooks.
-
-Maximum Debuggability
- To summarize the above, a GMP build for maximum debuggability
- would be
-
- ./configure --disable-shared --enable-assert \
- --enable-alloca=debug --host=none CFLAGS=-g
-
- For C++, add `--enable-cxx CXXFLAGS=-g'.
-
-Checker
- The GCC checker (`http://savannah.nongnu.org/projects/checker/')
- can be used with GMP. It contains a stub library which means GMP
- applications compiled with checker can use a normal GMP build.
-
- A build of GMP with checking within GMP itself can be made. This
- will run very very slowly. On GNU/Linux for example,
-
- ./configure --host=none-pc-linux-gnu CC=checkergcc
-
- `--host=none' must be used, since the GMP assembly code doesn't
- support the checking scheme. The GMP C++ features cannot be used,
- since current versions of checker (0.9.9.1) don't yet support the
- standard C++ library.
-
-Valgrind
- The valgrind program (`http://valgrind.org/') is a memory checker
- for x86s. It translates and emulates machine instructions to do
- strong checks for uninitialized data (at the level of individual
- bits), memory accesses through bad pointers, and memory leaks.
-
- Recent versions of Valgrind are getting support for MMX and
- SSE/SSE2 instructions, for past versions GMP will need to be
- configured not to use those, ie. for an x86 without them (for
- instance plain `i486').
-
-Other Problems
- Any suspected bug in GMP itself should be isolated to make sure
- it's not an application problem, see *Note Reporting Bugs::.
-
-\1f
-File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics
-
-3.13 Profiling
-==============
-
-Running a program under a profiler is a good way to find where it's
-spending most time and where improvements can be best sought. The
-profiling choices for a GMP build are as follows.
-
-`--disable-profiling'
- The default is to add nothing special for profiling.
-
- It should be possible to just compile the mainline of a program
- with `-p' and use `prof' to get a profile consisting of
- timer-based sampling of the program counter. Most of the GMP
- assembly code has the necessary symbol information.
-
- This approach has the advantage of minimizing interference with
- normal program operation, but on most systems the resolution of
- the sampling is quite low (10 milliseconds for instance),
- requiring long runs to get accurate information.
-
-`--enable-profiling=prof'
- Build with support for the system `prof', which means `-p' added
- to the `CFLAGS'.
-
- This provides call counting in addition to program counter
- sampling, which allows the most frequently called routines to be
- identified, and an average time spent in each routine to be
- determined.
-
- The x86 assembly code has support for this option, but on other
- processors the assembly routines will be as if compiled without
- `-p' and therefore won't appear in the call counts.
-
- On some systems, such as GNU/Linux, `-p' in fact means `-pg' and in
- this case `--enable-profiling=gprof' described below should be used
- instead.
-
-`--enable-profiling=gprof'
- Build with support for `gprof', which means `-pg' added to the
- `CFLAGS'.
-
- This provides call graph construction in addition to call counting
- and program counter sampling, which makes it possible to count
- calls coming from different locations. For example the number of
- calls to `mpn_mul' from `mpz_mul' versus the number from
- `mpf_mul'. The program counter sampling is still flat though, so
- only a total time in `mpn_mul' would be accumulated, not a
- separate amount for each call site.
-
- The x86 assembly code has support for this option, but on other
- processors the assembly routines will be as if compiled without
- `-pg' and therefore not be included in the call counts.
-
- On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
- incompatible, so the latter is omitted from the default flags in
- that case, which might result in poorer code generation.
-
- Incidentally, it should be possible to use the `gprof' program
- with a plain `--enable-profiling=prof' build. But in that case
- only the `gprof -p' flat profile and call counts can be expected
- to be valid, not the `gprof -q' call graph.
-
-`--enable-profiling=instrument'
- Build with the GCC option `-finstrument-functions' added to the
- `CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
- Options.).
-
- This inserts special instrumenting calls at the start and end of
- each function, allowing exact timing and full call graph
- construction.
-
- This instrumenting is not normally a standard system feature and
- will require support from an external library, such as
-
- `http://sourceforge.net/projects/fnccheck/'
-
- This should be included in `LIBS' during the GMP configure so that
- test programs will link. For example,
-
- ./configure --enable-profiling=instrument LIBS=-lfc
-
- On a GNU system the C library provides dummy instrumenting
- functions, so programs compiled with this option will link. In
- this case it's only necessary to ensure the correct library is
- added when linking an application.
-
- The x86 assembly code supports this option, but on other
- processors the assembly routines will be as if compiled without
- `-finstrument-functions' meaning time spent in them will
- effectively be attributed to their caller.
-
-\1f
-File: gmp.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: GMP Basics
-
-3.14 Autoconf
-=============
-
-Autoconf based applications can easily check whether GMP is installed.
-The only thing to be noted is that GMP library symbols from version 3
-onwards have prefixes like `__gmpz'. The following therefore would be
-a simple test,
-
- AC_CHECK_LIB(gmp, __gmpz_init)
-
- This just uses the default `AC_CHECK_LIB' actions for found or not
-found, but an application that must have GMP would want to generate an
-error if not found. For example,
-
- AC_CHECK_LIB(gmp, __gmpz_init, ,
- [AC_MSG_ERROR([GNU MP not found, see http://gmplib.org/])])
-
- If functions added in some particular version of GMP are required,
-then one of those can be used when checking. For example `mpz_mul_si'
-was added in GMP 3.1,
-
- AC_CHECK_LIB(gmp, __gmpz_mul_si, ,
- [AC_MSG_ERROR(
- [GNU MP not found, or not 3.1 or up, see http://gmplib.org/])])
-
- An alternative would be to test the version number in `gmp.h' using
-say `AC_EGREP_CPP'. That would make it possible to test the exact
-version, if some particular sub-minor release is known to be necessary.
-
- In general it's recommended that applications should simply demand a
-new enough GMP rather than trying to provide supplements for features
-not available in past versions.
-
- Occasionally an application will need or want to know the size of a
-type at configuration or preprocessing time, not just with `sizeof' in
-the code. This can be done in the normal way with `mp_limb_t' etc, but
-GMP 4.0 or up is best for this, since prior versions needed certain
-`-D' defines on systems using a `long long' limb. The following would
-suit Autoconf 2.50 or up,
-
- AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
-
-\1f
-File: gmp.info, Node: Emacs, Prev: Autoconf, Up: GMP Basics
-
-3.15 Emacs
-==========
-
-<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
-C functions while editing (*note Info Documentation Lookup: (emacs)Info
-Lookup.).
-
- The GMP manual can be included in such lookups by putting the
-following in your `.emacs',
-
- (eval-after-load "info-look"
- '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
- (setcar (nthcdr 3 mode-value)
- (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
- (nth 3 mode-value)))))
-
-\1f
-File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: GMP Basics, Up: Top
-
-4 Reporting Bugs
-****************
-
-If you think you have found a bug in the GMP library, please
-investigate it and report it. We have made this library available to
-you, and it is not too much to ask you to report the bugs you find.
-
- Before you report a bug, check it's not already addressed in *Note
-Known Build Problems::, or perhaps *Note Notes for Particular
-Systems::. You may also want to check `http://gmplib.org/' for patches
-for this release.
-
- Please include the following in any report,
-
- * The GMP version number, and if pre-packaged or patched then say so.
-
- * A test program that makes it possible for us to reproduce the bug.
- Include instructions on how to run the program.
-
- * A description of what is wrong. If the results are incorrect, in
- what way. If you get a crash, say so.
-
- * If you get a crash, include a stack backtrace from the debugger if
- it's informative (`where' in `gdb', or `$C' in `adb').
-
- * Please do not send core dumps, executables or `strace's.
-
- * The configuration options you used when building GMP, if any.
-
- * The name of the compiler and its version. For `gcc', get the
- version with `gcc -v', otherwise perhaps `what `which cc`', or
- similar.
-
- * The output from running `uname -a'.
-
- * The output from running `./config.guess', and from running
- `./configfsf.guess' (might be the same).
-
- * If the bug is related to `configure', then the compressed contents
- of `config.log'.
-
- * If the bug is related to an `asm' file not assembling, then the
- contents of `config.m4' and the offending line or lines from the
- temporary `mpn/tmp-<file>.s'.
-
- Please make an effort to produce a self-contained report, with
-something definite that can be tested or debugged. Vague queries or
-piecemeal messages are difficult to act on and don't help the
-development effort.
-
- It is not uncommon that an observed problem is actually due to a bug
-in the compiler; the GMP code tends to explore interesting corners in
-compilers.
-
- If your bug report is good, we will do our best to help you get a
-corrected version of the library; if the bug report is poor, we won't
-do anything about it (except maybe ask you to send a better report).
-
- Send your report to: <gmp-bugs@gmplib.org>.
-
- If you think something in this manual is unclear, or downright
-incorrect, or if the language needs to be improved, please send a note
-to the same address.
-
-\1f
-File: gmp.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
-
-5 Integer Functions
-*******************
-
-This chapter describes the GMP functions for performing integer
-arithmetic. These functions start with the prefix `mpz_'.
-
- GMP integers are stored in objects of type `mpz_t'.
-
-* Menu:
-
-* Initializing Integers::
-* Assigning Integers::
-* Simultaneous Integer Init & Assign::
-* Converting Integers::
-* Integer Arithmetic::
-* Integer Division::
-* Integer Exponentiation::
-* Integer Roots::
-* Number Theoretic Functions::
-* Integer Comparisons::
-* Integer Logic and Bit Fiddling::
-* I/O of Integers::
-* Integer Random Numbers::
-* Integer Import and Export::
-* Miscellaneous Integer Functions::
-* Integer Special Functions::
-
-\1f
-File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions
-
-5.1 Initialization Functions
-============================
-
-The functions for integer arithmetic assume that all integer objects are
-initialized. You do that by calling the function `mpz_init'. For
-example,
-
- {
- mpz_t integ;
- mpz_init (integ);
- ...
- mpz_add (integ, ...);
- ...
- mpz_sub (integ, ...);
-
- /* Unless the program is about to exit, do ... */
- mpz_clear (integ);
- }
-
- As you can see, you can store new values any number of times, once an
-object is initialized.
-
- -- Function: void mpz_init (mpz_t X)
- Initialize X, and set its value to 0.
-
- -- Function: void mpz_inits (mpz_t X, ...)
- Initialize a NULL-terminated list of `mpz_t' variables, and set
- their values to 0.
-
- -- Function: void mpz_init2 (mpz_t X, mp_bitcnt_t N)
- Initialize X, with space for N-bit numbers, and set its value to 0.
- Calling this function instead of `mpz_init' or `mpz_inits' is never
- necessary; reallocation is handled automatically by GMP when
- needed.
-
- N is only the initial space, X will grow automatically in the
- normal way, if necessary, for subsequent values stored.
- `mpz_init2' makes it possible to avoid such reallocations if a
- maximum size is known in advance.
-
- -- Function: void mpz_clear (mpz_t X)
- Free the space occupied by X. Call this function for all `mpz_t'
- variables when you are done with them.
-
- -- Function: void mpz_clears (mpz_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpz_t'
- variables.
-
- -- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N)
- Change the space allocated for X to N bits. The value in X is
- preserved if it fits, or is set to 0 if not.
-
- Calling this function is never necessary; reallocation is handled
- automatically by GMP when needed. But this function can be used
- to increase the space for a variable in order to avoid repeated
- automatic reallocations, or to decrease it to give memory back to
- the heap.
-
-\1f
-File: gmp.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions
-
-5.2 Assignment Functions
-========================
-
-These functions assign new values to already initialized integers
-(*note Initializing Integers::).
-
- -- Function: void mpz_set (mpz_t ROP, mpz_t OP)
- -- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
- -- Function: void mpz_set_si (mpz_t ROP, signed long int OP)
- -- Function: void mpz_set_d (mpz_t ROP, double OP)
- -- Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
- -- Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
- Set the value of ROP from OP.
-
- `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
- integer.
-
- -- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
- Set the value of ROP from STR, a null-terminated C string in base
- BASE. White space is allowed in the string, and is simply ignored.
-
- The BASE may vary from 2 to 62, or if BASE is 0, then the leading
- characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
- for binary, `0' for octal, or decimal otherwise.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value. For bases 37 to 62, upper-case
- letter represent the usual 10..35 while lower-case letter
- represent 36..61.
-
- This function returns 0 if the entire string is a valid number in
- base BASE. Otherwise it returns -1.
-
- -- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
- Swap the values ROP1 and ROP2 efficiently.
-
-\1f
-File: gmp.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions
-
-5.3 Combined Initialization and Assignment Functions
-====================================================
-
-For convenience, GMP provides a parallel series of initialize-and-set
-functions which initialize the output and then store the value there.
-These functions' names have the form `mpz_init_set...'
-
- Here is an example of using one:
-
- {
- mpz_t pie;
- mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
- ...
- mpz_sub (pie, ...);
- ...
- mpz_clear (pie);
- }
-
-Once the integer has been initialized by any of the `mpz_init_set...'
-functions, it can be used as the source or destination operand for the
-ordinary integer functions. Don't use an initialize-and-set function
-on a variable already initialized!
-
- -- Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
- -- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
- -- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
- -- Function: void mpz_init_set_d (mpz_t ROP, double OP)
- Initialize ROP with limb space and set the initial numeric value
- from OP.
-
- -- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
- Initialize ROP and set its value like `mpz_set_str' (see its
- documentation above for details).
-
- If the string is a correct base BASE number, the function returns
- 0; if an error occurs it returns -1. ROP is initialized even if
- an error occurs. (I.e., you have to call `mpz_clear' for it.)
-
-\1f
-File: gmp.info, Node: Converting Integers, Next: Integer Arithmetic, Prev: Simultaneous Integer Init & Assign, Up: Integer Functions
-
-5.4 Conversion Functions
-========================
-
-This section describes functions for converting GMP integers to
-standard C types. Functions for converting _to_ GMP integers are
-described in *Note Assigning Integers:: and *Note I/O of Integers::.
-
- -- Function: unsigned long int mpz_get_ui (mpz_t OP)
- Return the value of OP as an `unsigned long'.
-
- If OP is too big to fit an `unsigned long' then just the least
- significant bits that do fit are returned. The sign of OP is
- ignored, only the absolute value is used.
-
- -- Function: signed long int mpz_get_si (mpz_t OP)
- If OP fits into a `signed long int' return the value of OP.
- Otherwise return the least significant part of OP, with the same
- sign as OP.
-
- If OP is too big to fit in a `signed long int', the returned
- result is probably not very useful. To find out if the value will
- fit, use the function `mpz_fits_slong_p'.
-
- -- Function: double mpz_get_d (mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent from the conversion is too big, the result is
- system dependent. An infinity is returned where available. A
- hardware overflow trap may or may not occur.
-
- -- Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero), and returning the exponent separately.
-
- The return value is in the range 0.5<=abs(D)<1 and the exponent is
- stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
- is zero, the return is 0.0 and 0 is stored to `*EXP'.
-
- This is similar to the standard C `frexp' function (*note
- Normalization Functions: (libc)Normalization Functions.).
-
- -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
- Convert OP to a string of digits in base BASE. The base argument
- may vary from 2 to 62 or from -2 to -36.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of storage large
- enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
- The two extra bytes are for a possible minus sign, and the
- null-terminator.
-
- A pointer to the result string is returned, being either the
- allocated block, or the given STR.
-
-\1f
-File: gmp.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
-
-5.5 Arithmetic Functions
-========================
-
- -- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 + OP2.
-
- -- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
- -- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
- OP2)
- Set ROP to OP1 - OP2.
-
- -- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2)
- -- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 times OP2.
-
- -- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long
- int OP2)
- Set ROP to ROP + OP1 times OP2.
-
- -- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long
- int OP2)
- Set ROP to ROP - OP1 times OP2.
-
- -- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2. This operation can also be
- defined as a left shift by OP2 bits.
-
- -- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
- Set ROP to -OP.
-
- -- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
- Set ROP to the absolute value of OP.
-
-\1f
-File: gmp.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
-
-5.6 Division Functions
-======================
-
-Division is undefined if the divisor is zero. Passing a zero divisor
-to the division or modulo functions (including the modular powering
-functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
-division by zero. This lets a program handle arithmetic exceptions in
-these functions the same way as for normal C `int' arithmetic.
-
- -- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_ui (mpz_t N,
- unsigned long int D)
- -- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- -- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_ui (mpz_t N,
- unsigned long int D)
- -- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- -- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_ui (mpz_t N,
- unsigned long int D)
- -- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- Divide N by D, forming a quotient Q and/or remainder R. For the
- `2exp' functions, D=2^B. The rounding is in three styles, each
- suiting different applications.
-
- * `cdiv' rounds Q up towards +infinity, and R will have the
- opposite sign to D. The `c' stands for "ceil".
-
- * `fdiv' rounds Q down towards -infinity, and R will have the
- same sign as D. The `f' stands for "floor".
-
- * `tdiv' rounds Q towards zero, and R will have the same sign
- as N. The `t' stands for "truncate".
-
- In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
- 0<=abs(R)<abs(D).
-
- The `q' functions calculate only the quotient, the `r' functions
- only the remainder, and the `qr' functions calculate both. Note
- that for `qr' the same variable cannot be passed for both Q and R,
- or results will be unpredictable.
-
- For the `ui' variants the return value is the remainder, and in
- fact returning the remainder is all the `div_ui' functions do. For
- `tdiv' and `cdiv' the remainder can be negative, so for those the
- return value is the absolute value of the remainder.
-
- For the `2exp' variants the divisor is 2^B. These functions are
- implemented as right shifts and bit masks, but of course they
- round the same as the other functions.
-
- For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
- simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
- effectively an arithmetic right shift treating N as twos complement
- the same as the bitwise logical functions do, whereas
- `mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
-
- -- Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- Set R to N `mod' D. The sign of the divisor is ignored; the
- result is always non-negative.
-
- `mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
- remainder as well as setting R. See `mpz_fdiv_ui' above if only
- the return value is wanted.
-
- -- Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_divexact_ui (mpz_t Q, mpz_t N, unsigned long D)
- Set Q to N/D. These functions produce correct results only when
- it is known in advance that D divides N.
-
- These routines are much faster than the other division functions,
- and are the best choice when exact division is known to occur, for
- example reducing a rational to lowest terms.
-
- -- Function: int mpz_divisible_p (mpz_t N, mpz_t D)
- -- Function: int mpz_divisible_ui_p (mpz_t N, unsigned long int D)
- -- Function: int mpz_divisible_2exp_p (mpz_t N, mp_bitcnt_t B)
- Return non-zero if N is exactly divisible by D, or in the case of
- `mpz_divisible_2exp_p' by 2^B.
-
- N is divisible by D if there exists an integer Q satisfying N =
- Q*D. Unlike the other division functions, D=0 is accepted and
- following the rule it can be seen that only 0 is considered
- divisible by 0.
-
- -- Function: int mpz_congruent_p (mpz_t N, mpz_t C, mpz_t D)
- -- Function: int mpz_congruent_ui_p (mpz_t N, unsigned long int C,
- unsigned long int D)
- -- Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, mp_bitcnt_t B)
- Return non-zero if N is congruent to C modulo D, or in the case of
- `mpz_congruent_2exp_p' modulo 2^B.
-
- N is congruent to C mod D if there exists an integer Q satisfying
- N = C + Q*D. Unlike the other division functions, D=0 is accepted
- and following the rule it can be seen that N and C are considered
- congruent mod 0 only when exactly equal.
-
-\1f
-File: gmp.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
-
-5.7 Exponentiation Functions
-============================
-
- -- Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t
- MOD)
- -- Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long
- int EXP, mpz_t MOD)
- Set ROP to (BASE raised to EXP) modulo MOD.
-
- Negative EXP is supported if an inverse BASE^-1 mod MOD exists
- (see `mpz_invert' in *Note Number Theoretic Functions::). If an
- inverse doesn't exist then a divide by zero is raised.
-
- -- Function: void mpz_powm_sec (mpz_t ROP, mpz_t BASE, mpz_t EXP,
- mpz_t MOD)
- Set ROP to (BASE raised to EXP) modulo MOD.
-
- It is required that EXP > 0 and that MOD is odd.
-
- This function is designed to take the same time and have the same
- cache access patterns for any two same-size arguments, assuming
- that function arguments are placed at the same position and that
- the machine state is identical upon function entry. This function
- is intended for cryptographic purposes, where resilience to
- side-channel attacks is desired.
-
- -- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
- EXP)
- -- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
- unsigned long int EXP)
- Set ROP to BASE raised to EXP. The case 0^0 yields 1.
-
-\1f
-File: gmp.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions
-
-5.8 Root Extraction Functions
-=============================
-
- -- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
- Set ROP to the truncated integer part of the Nth root of OP.
- Return non-zero if the computation was exact, i.e., if OP is ROP
- to the Nth power.
-
- -- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, mpz_t U,
- unsigned long int N)
- Set ROOT to the truncated integer part of the Nth root of U. Set
- REM to the remainder, U-ROOT**N.
-
- -- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
- Set ROP to the truncated integer part of the square root of OP.
-
- -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
- Set ROP1 to the truncated integer part of the square root of OP,
- like `mpz_sqrt'. Set ROP2 to the remainder OP-ROP1*ROP1, which
- will be zero if OP is a perfect square.
-
- If ROP1 and ROP2 are the same variable, the results are undefined.
-
- -- Function: int mpz_perfect_power_p (mpz_t OP)
- Return non-zero if OP is a perfect power, i.e., if there exist
- integers A and B, with B>1, such that OP equals A raised to the
- power B.
-
- Under this definition both 0 and 1 are considered to be perfect
- powers. Negative values of OP are accepted, but of course can
- only be odd perfect powers.
-
- -- Function: int mpz_perfect_square_p (mpz_t OP)
- Return non-zero if OP is a perfect square, i.e., if the square
- root of OP is an integer. Under this definition both 0 and 1 are
- considered to be perfect squares.
-
-\1f
-File: gmp.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
-
-5.9 Number Theoretic Functions
-==============================
-
- -- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
- Determine whether N is prime. Return 2 if N is definitely prime,
- return 1 if N is probably prime (without being certain), or return
- 0 if N is definitely composite.
-
- This function does some trial divisions, then some Miller-Rabin
- probabilistic primality tests. REPS controls how many such tests
- are done, 5 to 10 is a reasonable number, more will reduce the
- chances of a composite being returned as "probably prime".
-
- Miller-Rabin and similar tests can be more properly called
- compositeness tests. Numbers which fail are known to be composite
- but those which pass might be prime or might be composite. Only a
- few composites pass, hence those which pass are considered
- probably prime.
-
- -- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
- Set ROP to the next prime greater than OP.
-
- This function uses a probabilistic algorithm to identify primes.
- For practical purposes it's adequate, the chance of a composite
- passing will be extremely small.
-
- -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to the greatest common divisor of OP1 and OP2. The result
- is always positive even if one or both input operands are negative.
-
- -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
- unsigned long int OP2)
- Compute the greatest common divisor of OP1 and OP2. If ROP is not
- `NULL', store the result there.
-
- If the result is small enough to fit in an `unsigned long int', it
- is returned. If the result does not fit, 0 is returned, and the
- result is equal to the argument OP1. Note that the result will
- always fit if OP2 is non-zero.
-
- -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A,
- mpz_t B)
- Set G to the greatest common divisor of A and B, and in addition
- set S and T to coefficients satisfying A*S + B*T = G. The value
- in G is always positive, even if one or both of A and B are
- negative. The values in S and T are chosen such that abs(S) <=
- abs(B) and abs(T) <= abs(A).
-
- If T is `NULL' then that value is not computed.
-
- -- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2)
- Set ROP to the least common multiple of OP1 and OP2. ROP is
- always positive, irrespective of the signs of OP1 and OP2. ROP
- will be zero if either OP1 or OP2 is zero.
-
- -- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Compute the inverse of OP1 modulo OP2 and put the result in ROP.
- If the inverse exists, the return value is non-zero and ROP will
- satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return
- value is zero and ROP is undefined.
-
- -- Function: int mpz_jacobi (mpz_t A, mpz_t B)
- Calculate the Jacobi symbol (A/B). This is defined only for B odd.
-
- -- Function: int mpz_legendre (mpz_t A, mpz_t P)
- Calculate the Legendre symbol (A/P). This is defined only for P
- an odd positive prime, and for such P it's identical to the Jacobi
- symbol.
-
- -- Function: int mpz_kronecker (mpz_t A, mpz_t B)
- -- Function: int mpz_kronecker_si (mpz_t A, long B)
- -- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
- -- Function: int mpz_si_kronecker (long A, mpz_t B)
- -- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
- Calculate the Jacobi symbol (A/B) with the Kronecker extension
- (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
-
- When B is odd the Jacobi symbol and Kronecker symbol are
- identical, so `mpz_kronecker_ui' etc can be used for mixed
- precision Jacobi symbols too.
-
- For more information see Henri Cohen section 1.4.2 (*note
- References::), or any number theory textbook. See also the
- example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
-
- -- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
- Remove all occurrences of the factor F from OP and store the
- result in ROP. The return value is how many such occurrences were
- removed.
-
- -- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
- Set ROP to OP!, the factorial of OP.
-
- -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
- -- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
- unsigned long int K)
- Compute the binomial coefficient N over K and store the result in
- ROP. Negative values of N are supported by `mpz_bin_ui', using
- the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
- section 1.2.6 part G.
-
- -- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
- -- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
- int N)
- `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
- `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
-
- These functions are designed for calculating isolated Fibonacci
- numbers. When a sequence of values is wanted it's best to start
- with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
- similar.
-
- -- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
- -- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
- long int N)
- `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
- `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
-
- These functions are designed for calculating isolated Lucas
- numbers. When a sequence of values is wanted it's best to start
- with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
- or similar.
-
- The Fibonacci numbers and Lucas numbers are related sequences, so
- it's never necessary to call both `mpz_fib2_ui' and
- `mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
- can be found in *Note Lucas Numbers Algorithm::, the reverse is
- straightforward too.
-
-\1f
-File: gmp.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
-
-5.10 Comparison Functions
-=========================
-
- -- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
- -- Function: int mpz_cmp_d (mpz_t OP1, double OP2)
- -- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
- -- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, or a negative value if OP1 < OP2.
-
- `mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their
- arguments more than once. `mpz_cmp_d' can be called with an
- infinity, but results are undefined for a NaN.
-
- -- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
- -- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
- -- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
- Compare the absolute values of OP1 and OP2. Return a positive
- value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
- negative value if abs(OP1) < abs(OP2).
-
- `mpz_cmpabs_d' can be called with an infinity, but results are
- undefined for a NaN.
-
- -- Macro: int mpz_sgn (mpz_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates
- its argument multiple times.
-
-\1f
-File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
-
-5.11 Logical and Bit Manipulation Functions
-===========================================
-
-These functions behave as if twos complement arithmetic were used
-(although sign-magnitude is the actual implementation). The least
-significant bit is number 0.
-
- -- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise-and OP2.
-
- -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise inclusive-or OP2.
-
- -- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise exclusive-or OP2.
-
- -- Function: void mpz_com (mpz_t ROP, mpz_t OP)
- Set ROP to the one's complement of OP.
-
- -- Function: mp_bitcnt_t mpz_popcount (mpz_t OP)
- If OP>=0, return the population count of OP, which is the number
- of 1 bits in the binary representation. If OP<0, the number of 1s
- is infinite, and the return value is the largest possible
- `mp_bitcnt_t'.
-
- -- Function: mp_bitcnt_t mpz_hamdist (mpz_t OP1, mpz_t OP2)
- If OP1 and OP2 are both >=0 or both <0, return the hamming
- distance between the two operands, which is the number of bit
- positions where OP1 and OP2 have different bit values. If one
- operand is >=0 and the other <0 then the number of bits different
- is infinite, and the return value is the largest possible
- `mp_bitcnt_t'.
-
- -- Function: mp_bitcnt_t mpz_scan0 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
- -- Function: mp_bitcnt_t mpz_scan1 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
- Scan OP, starting from bit STARTING_BIT, towards more significant
- bits, until the first 0 or 1 bit (respectively) is found. Return
- the index of the found bit.
-
- If the bit at STARTING_BIT is already what's sought, then
- STARTING_BIT is returned.
-
- If there's no bit found, then the largest possible `mp_bitcnt_t' is
- returned. This will happen in `mpz_scan0' past the end of a
- negative number, or `mpz_scan1' past the end of a nonnegative
- number.
-
- -- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Set bit BIT_INDEX in ROP.
-
- -- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Clear bit BIT_INDEX in ROP.
-
- -- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Complement bit BIT_INDEX in ROP.
-
- -- Function: int mpz_tstbit (mpz_t OP, mp_bitcnt_t BIT_INDEX)
- Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
-
-\1f
-File: gmp.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
-
-5.12 Input and Output Functions
-===============================
-
-Functions that perform input from a stdio stream, and functions that
-output to a stdio stream. Passing a `NULL' pointer for a STREAM
-argument to any of these functions will make them read from `stdin' and
-write to `stdout', respectively.
-
- When using any of these functions, it is a good idea to include
-`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
-prototypes for these functions.
-
- -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
- Output OP on stdio stream STREAM, as a string of digits in base
- BASE. The base argument may vary from 2 to 62 or from -2 to -36.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- -- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
- Input a possibly white-space preceded string in base BASE from
- stdio stream STREAM, and put the read integer in ROP.
-
- The BASE may vary from 2 to 62, or if BASE is 0, then the leading
- characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
- for binary, `0' for octal, or decimal otherwise.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value. For bases 37 to 62, upper-case
- letter represent the usual 10..35 while lower-case letter
- represent 36..61.
-
- Return the number of bytes read, or if an error occurred, return 0.
-
- -- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
- Output OP on stdio stream STREAM, in raw binary format. The
- integer is written in a portable format, with 4 bytes of size
- information, and that many bytes of limbs. Both the size and the
- limbs are written in decreasing significance order (i.e., in
- big-endian).
-
- The output can be read with `mpz_inp_raw'.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- The output of this can not be read by `mpz_inp_raw' from GMP 1,
- because of changes necessary for compatibility between 32-bit and
- 64-bit machines.
-
- -- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
- Input from stdio stream STREAM in the format written by
- `mpz_out_raw', and put the result in ROP. Return the number of
- bytes read, or if an error occurred, return 0.
-
- This routine can read the output from `mpz_out_raw' also from GMP
- 1, in spite of changes necessary for compatibility between 32-bit
- and 64-bit machines.
-
-\1f
-File: gmp.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
-
-5.13 Random Number Functions
-============================
-
-The random number functions of GMP come in two groups; older function
-that rely on a global state, and newer functions that accept a state
-parameter that is read and modified. Please see the *Note Random
-Number Functions:: for more information on how to use and not to use
-random number functions.
-
- -- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t N)
- Generate a uniformly distributed random integer in the range 0 to
- 2^N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
- mpz_t N)
- Generate a uniform random integer in the range 0 to N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t N)
- Generate a random integer with long strings of zeros and ones in
- the binary representation. Useful for testing functions and
- algorithms, since this kind of random numbers have proven to be
- more likely to trigger corner-case bugs. The random number will
- be in the range 0 to 2^N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
- Generate a random integer of at most MAX_SIZE limbs. The generated
- random number doesn't satisfy any particular requirements of
- randomness. Negative random numbers are generated when MAX_SIZE
- is negative.
-
- This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
- instead.
-
- -- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
- Generate a random integer of at most MAX_SIZE limbs, with long
- strings of zeros and ones in the binary representation. Useful
- for testing functions and algorithms, since this kind of random
- numbers have proven to be more likely to trigger corner-case bugs.
- Negative random numbers are generated when MAX_SIZE is negative.
-
- This function is obsolete. Use `mpz_rrandomb' instead.
-
-\1f
-File: gmp.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions
-
-5.14 Integer Import and Export
-==============================
-
-`mpz_t' variables can be converted to and from arbitrary words of binary
-data with the following functions.
-
- -- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
- size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
- Set ROP from an array of word data at OP.
-
- The parameters specify the format of the data. COUNT many words
- are read, each SIZE bytes. ORDER can be 1 for most significant
- word first or -1 for least significant first. Within each word
- ENDIAN can be 1 for most significant byte first, -1 for least
- significant first, or 0 for the native endianness of the host CPU.
- The most significant NAILS bits of each word are skipped, this
- can be 0 to use the full words.
-
- There is no sign taken from the data, ROP will simply be a positive
- integer. An application can handle any sign itself, and apply it
- for instance with `mpz_neg'.
-
- There are no data alignment restrictions on OP, any address is
- allowed.
-
- Here's an example converting an array of `unsigned long' data, most
- significant element first, and host byte order within each value.
-
- unsigned long a[20];
- /* Initialize Z and A */
- mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
-
- This example assumes the full `sizeof' bytes are used for data in
- the given type, which is usually true, and certainly true for
- `unsigned long' everywhere we know of. However on Cray vector
- systems it may be noted that `short' and `int' are always stored
- in 8 bytes (and with `sizeof' indicating that) but use only 32 or
- 46 bits. The NAILS feature can account for this, by passing for
- instance `8*sizeof(int)-INT_BIT'.
-
- -- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
- size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
- Fill ROP with word data from OP.
-
- The parameters specify the format of the data produced. Each word
- will be SIZE bytes and ORDER can be 1 for most significant word
- first or -1 for least significant first. Within each word ENDIAN
- can be 1 for most significant byte first, -1 for least significant
- first, or 0 for the native endianness of the host CPU. The most
- significant NAILS bits of each word are unused and set to zero,
- this can be 0 to produce full words.
-
- The number of words produced is written to `*COUNTP', or COUNTP
- can be `NULL' to discard the count. ROP must have enough space
- for the data, or if ROP is `NULL' then a result array of the
- necessary size is allocated using the current GMP allocation
- function (*note Custom Allocation::). In either case the return
- value is the destination used, either ROP or the allocated block.
-
- If OP is non-zero then the most significant word produced will be
- non-zero. If OP is zero then the count returned will be zero and
- nothing written to ROP. If ROP is `NULL' in this case, no block
- is allocated, just `NULL' is returned.
-
- The sign of OP is ignored, just the absolute value is exported. An
- application can use `mpz_sgn' to get the sign and handle it as
- desired. (*note Integer Comparisons::)
-
- There are no data alignment restrictions on ROP, any address is
- allowed.
-
- When an application is allocating space itself the required size
- can be determined with a calculation like the following. Since
- `mpz_sizeinbase' always returns at least 1, `count' here will be
- at least one, which avoids any portability problems with
- `malloc(0)', though if `z' is zero no space at all is actually
- needed (or written).
-
- numb = 8*size - nail;
- count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
- p = malloc (count * size);
-
-\1f
-File: gmp.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
-
-5.15 Miscellaneous Functions
-============================
-
- -- Function: int mpz_fits_ulong_p (mpz_t OP)
- -- Function: int mpz_fits_slong_p (mpz_t OP)
- -- Function: int mpz_fits_uint_p (mpz_t OP)
- -- Function: int mpz_fits_sint_p (mpz_t OP)
- -- Function: int mpz_fits_ushort_p (mpz_t OP)
- -- Function: int mpz_fits_sshort_p (mpz_t OP)
- Return non-zero iff the value of OP fits in an `unsigned long int',
- `signed long int', `unsigned int', `signed int', `unsigned short
- int', or `signed short int', respectively. Otherwise, return zero.
-
- -- Macro: int mpz_odd_p (mpz_t OP)
- -- Macro: int mpz_even_p (mpz_t OP)
- Determine whether OP is odd or even, respectively. Return
- non-zero if yes, zero if no. These macros evaluate their argument
- more than once.
-
- -- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
- Return the size of OP measured in number of digits in the given
- BASE. BASE can vary from 2 to 62. The sign of OP is ignored,
- just the absolute value is used. The result will be either exact
- or 1 too big. If BASE is a power of 2, the result is always
- exact. If OP is zero the return value is always 1.
-
- This function can be used to determine the space required when
- converting OP to a string. The right amount of allocation is
- normally two more than the value returned by `mpz_sizeinbase', one
- extra for a minus sign and one for the null-terminator.
-
- It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
- the most significant 1 bit in OP, counting from 1. (Unlike the
- bitwise functions which start from 0, *Note Logical and Bit
- Manipulation Functions: Integer Logic and Bit Fiddling.)
-
-\1f
-File: gmp.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
-
-5.16 Special Functions
-======================
-
-The functions in this section are for various special purposes. Most
-applications will not need them.
-
- -- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_size_t
- ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
- This is a special type of initialization. *Fixed* space of
- FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in
- INTEGER_ARRAY. There is no way to free the storage allocated by
- this function. Don't call `mpz_clear'!
-
- The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For
- example,
-
- mpz_t arr[20000];
- mpz_array_init (arr[0], 20000, 512);
-
- This function is only intended for programs that create a large
- number of integers and need to reduce memory usage by avoiding the
- overheads of allocating and reallocating lots of small blocks. In
- normal programs this function is not recommended.
-
- The space allocated to each integer by this function will not be
- automatically increased, unlike the normal `mpz_init', so an
- application must ensure it is sufficient for any value stored.
- The following space requirements apply to various routines,
-
- * `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
- `mpz_set_ui' need room for the value they store.
-
- * `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
- for the larger of the two operands, plus an extra
- `mp_bits_per_limb'.
-
- * `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
- of the number of bits in their operands, but each rounded up
- to a multiple of `mp_bits_per_limb'.
-
- * `mpz_swap' can be used between two array variables, but not
- between an array and a normal variable.
-
- For other functions, or if in doubt, the suggestion is to
- calculate in a regular `mpz_init' variable and copy the result to
- an array variable with `mpz_set'.
-
- -- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
- Change the space for INTEGER to NEW_ALLOC limbs. The value in
- INTEGER is preserved if it fits, or is set to 0 if not. The return
- value is not useful to applications and should be ignored.
-
- `mpz_realloc2' is the preferred way to accomplish allocation
- changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
- except that `_mpz_realloc' takes its size in limbs.
-
- -- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
- Return limb number N from OP. The sign of OP is ignored, just the
- absolute value is used. The least significant limb is number 0.
-
- `mpz_size' can be used to find how many limbs make up OP.
- `mpz_getlimbn' returns zero if N is outside the range 0 to
- `mpz_size(OP)-1'.
-
- -- Function: size_t mpz_size (mpz_t OP)
- Return the size of OP measured in number of limbs. If OP is zero,
- the returned value will be zero.
-
-\1f
-File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
-
-6 Rational Number Functions
-***************************
-
-This chapter describes the GMP functions for performing arithmetic on
-rational numbers. These functions start with the prefix `mpq_'.
-
- Rational numbers are stored in objects of type `mpq_t'.
-
- All rational arithmetic functions assume operands have a canonical
-form, and canonicalize their result. The canonical from means that the
-denominator and the numerator have no common factors, and that the
-denominator is positive. Zero has the unique representation 0/1.
-
- Pure assignment functions do not canonicalize the assigned variable.
-It is the responsibility of the user to canonicalize the assigned
-variable before any arithmetic operations are performed on that
-variable.
-
- -- Function: void mpq_canonicalize (mpq_t OP)
- Remove any factors that are common to the numerator and
- denominator of OP, and make the denominator positive.
-
-* Menu:
-
-* Initializing Rationals::
-* Rational Conversions::
-* Rational Arithmetic::
-* Comparing Rationals::
-* Applying Integer Functions::
-* I/O of Rationals::
-
-\1f
-File: gmp.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
-
-6.1 Initialization and Assignment Functions
-===========================================
-
- -- Function: void mpq_init (mpq_t X)
- Initialize X and set it to 0/1. Each variable should normally
- only be initialized once, or at least cleared out (using the
- function `mpq_clear') between each initialization.
-
- -- Function: void mpq_inits (mpq_t X, ...)
- Initialize a NULL-terminated list of `mpq_t' variables, and set
- their values to 0/1.
-
- -- Function: void mpq_clear (mpq_t X)
- Free the space occupied by X. Make sure to call this function for
- all `mpq_t' variables when you are done with them.
-
- -- Function: void mpq_clears (mpq_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpq_t'
- variables.
-
- -- Function: void mpq_set (mpq_t ROP, mpq_t OP)
- -- Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
- Assign ROP from OP.
-
- -- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
- unsigned long int OP2)
- -- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
- long int OP2)
- Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
- common factors, ROP has to be passed to `mpq_canonicalize' before
- any operations are performed on ROP.
-
- -- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
- Set ROP from a null-terminated string STR in the given BASE.
-
- The string can be an integer like "41" or a fraction like
- "41/152". The fraction must be in canonical form (*note Rational
- Number Functions::), or if not then `mpq_canonicalize' must be
- called.
-
- The numerator and optional denominator are parsed the same as in
- `mpz_set_str' (*note Assigning Integers::). White space is
- allowed in the string, and is simply ignored. The BASE can vary
- from 2 to 62, or if BASE is 0 then the leading characters are
- used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
- octal, or decimal otherwise. Note that this is done separately
- for the numerator and denominator, so for instance `0xEF/100' is
- 239/100, whereas `0xEF/0x100' is 239/256.
-
- The return value is 0 if the entire string is a valid number, or
- -1 if not.
-
- -- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
- Swap the values ROP1 and ROP2 efficiently.
-
-\1f
-File: gmp.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
-
-6.2 Conversion Functions
-========================
-
- -- Function: double mpq_get_d (mpq_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent from the conversion is too big or too small to fit
- a `double' then the result is system dependent. For too big an
- infinity is returned when available. For too small 0.0 is
- normally returned. Hardware overflow, underflow and denorm traps
- may or may not occur.
-
- -- Function: void mpq_set_d (mpq_t ROP, double OP)
- -- Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
- Set ROP to the value of OP. There is no rounding, this conversion
- is exact.
-
- -- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
- Convert OP to a string of digits in base BASE. The base may vary
- from 2 to 36. The string will be of the form `num/den', or if the
- denominator is 1 then just `num'.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of storage large
- enough for the result, that being
-
- mpz_sizeinbase (mpq_numref(OP), BASE)
- + mpz_sizeinbase (mpq_denref(OP), BASE) + 3
-
- The three extra bytes are for a possible minus sign, possible
- slash, and the null-terminator.
-
- A pointer to the result string is returned, being either the
- allocated block, or the given STR.
-
-\1f
-File: gmp.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
-
-6.3 Arithmetic Functions
-========================
-
- -- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
- Set SUM to ADDEND1 + ADDEND2.
-
- -- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
- SUBTRAHEND)
- Set DIFFERENCE to MINUEND - SUBTRAHEND.
-
- -- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
- MULTIPLICAND)
- Set PRODUCT to MULTIPLIER times MULTIPLICAND.
-
- -- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2.
-
- -- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
- DIVISOR)
- Set QUOTIENT to DIVIDEND/DIVISOR.
-
- -- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 divided by 2 raised to OP2.
-
- -- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
- Set NEGATED_OPERAND to -OPERAND.
-
- -- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
- Set ROP to the absolute value of OP.
-
- -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
- Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
- this routine will divide by zero.
-
-\1f
-File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
-
-6.4 Comparison Functions
-========================
-
- -- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- To determine if two rationals are equal, `mpq_equal' is faster than
- `mpq_cmp'.
-
- -- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
- long int DEN2)
- -- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
- DEN2)
- Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
- NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
- NUM2/DEN2.
-
- NUM2 and DEN2 are allowed to have common factors.
-
- These functions are implemented as a macros and evaluate their
- arguments multiple times.
-
- -- Macro: int mpq_sgn (mpq_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates its
- arguments multiple times.
-
- -- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
- Return non-zero if OP1 and OP2 are equal, zero if they are
- non-equal. Although `mpq_cmp' can be used for the same purpose,
- this function is much faster.
-
-\1f
-File: gmp.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
-
-6.5 Applying Integer Functions to Rationals
-===========================================
-
-The set of `mpq' functions is quite small. In particular, there are few
-functions for either input or output. The following functions give
-direct access to the numerator and denominator of an `mpq_t'.
-
- Note that if an assignment to the numerator and/or denominator could
-take an `mpq_t' out of the canonical form described at the start of
-this chapter (*note Rational Number Functions::) then
-`mpq_canonicalize' must be called before any other `mpq' functions are
-applied to that `mpq_t'.
-
- -- Macro: mpz_t mpq_numref (mpq_t OP)
- -- Macro: mpz_t mpq_denref (mpq_t OP)
- Return a reference to the numerator and denominator of OP,
- respectively. The `mpz' functions can be used on the result of
- these macros.
-
- -- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
- -- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
- -- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
- -- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
- Get or set the numerator or denominator of a rational. These
- functions are equivalent to calling `mpz_set' with an appropriate
- `mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
- `mpq_denref' is recommended instead of these functions.
-
-\1f
-File: gmp.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
-
-6.6 Input and Output Functions
-==============================
-
-When using any of these functions, it's a good idea to include `stdio.h'
-before `gmp.h', since that will allow `gmp.h' to define prototypes for
-these functions.
-
- Passing a `NULL' pointer for a STREAM argument to any of these
-functions will make them read from `stdin' and write to `stdout',
-respectively.
-
- -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
- Output OP on stdio stream STREAM, as a string of digits in base
- BASE. The base may vary from 2 to 36. Output is in the form
- `num/den' or if the denominator is 1 then just `num'.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- -- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
- Read a string of digits from STREAM and convert them to a rational
- in ROP. Any initial white-space characters are read and
- discarded. Return the number of characters read (including white
- space), or 0 if a rational could not be read.
-
- The input can be a fraction like `17/63' or just an integer like
- `123'. Reading stops at the first character not in this form, and
- white space is not permitted within the string. If the input
- might not be in canonical form, then `mpq_canonicalize' must be
- called (*note Rational Number Functions::).
-
- The BASE can be between 2 and 36, or can be 0 in which case the
- leading characters of the string determine the base, `0x' or `0X'
- for hexadecimal, `0' for octal, or decimal otherwise. The leading
- characters are examined separately for the numerator and
- denominator of a fraction, so for instance `0x10/11' is 16/11,
- whereas `0x10/0x11' is 16/17.
-
-\1f
-File: gmp.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
-
-7 Floating-point Functions
-**************************
-
-GMP floating point numbers are stored in objects of type `mpf_t' and
-functions operating on them have an `mpf_' prefix.
-
- The mantissa of each float has a user-selectable precision, limited
-only by available memory. Each variable has its own precision, and
-that can be increased or decreased at any time.
-
- The exponent of each float is a fixed precision, one machine word on
-most systems. In the current implementation the exponent is a count of
-limbs, so for example on a 32-bit system this means a range of roughly
-2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
-greater. Note however `mpf_get_str' can only return an exponent which
-fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
-bigger than a `long'.
-
- Each variable keeps a size for the mantissa data actually in use.
-This means that if a float is exactly represented in only a few bits
-then only those bits will be used in a calculation, even if the
-selected precision is high.
-
- All calculations are performed to the precision of the destination
-variable. Each function is defined to calculate with "infinite
-precision" followed by a truncation to the destination precision, but
-of course the work done is only what's needed to determine a result
-under that definition.
-
- The precision selected for a variable is a minimum value, GMP may
-increase it a little to facilitate efficient calculation. Currently
-this means rounding up to a whole limb, and then sometimes having a
-further partial limb, depending on the high limb of the mantissa. But
-applications shouldn't be concerned by such details.
-
- The mantissa in stored in binary, as might be imagined from the fact
-precisions are expressed in bits. One consequence of this is that
-decimal fractions like 0.1 cannot be represented exactly. The same is
-true of plain IEEE `double' floats. This makes both highly unsuitable
-for calculations involving money or other values that should be exact
-decimal fractions. (Suitably scaled integers, or perhaps rationals,
-are better choices.)
-
- `mpf' functions and variables have no special notion of infinity or
-not-a-number, and applications must take care not to overflow the
-exponent or results will be unpredictable. This might change in a
-future release.
-
- Note that the `mpf' functions are _not_ intended as a smooth
-extension to IEEE P754 arithmetic. In particular results obtained on
-one computer often differ from the results on a computer with a
-different word size.
-
-* Menu:
-
-* Initializing Floats::
-* Assigning Floats::
-* Simultaneous Float Init & Assign::
-* Converting Floats::
-* Float Arithmetic::
-* Float Comparison::
-* I/O of Floats::
-* Miscellaneous Float Functions::
-
-\1f
-File: gmp.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
-
-7.1 Initialization Functions
-============================
-
- -- Function: void mpf_set_default_prec (mp_bitcnt_t PREC)
- Set the default precision to be *at least* PREC bits. All
- subsequent calls to `mpf_init' will use this precision, but
- previously initialized variables are unaffected.
-
- -- Function: mp_bitcnt_t mpf_get_default_prec (void)
- Return the default precision actually used.
-
- An `mpf_t' object must be initialized before storing the first value
-in it. The functions `mpf_init' and `mpf_init2' are used for that
-purpose.
-
- -- Function: void mpf_init (mpf_t X)
- Initialize X to 0. Normally, a variable should be initialized
- once only or at least be cleared, using `mpf_clear', between
- initializations. The precision of X is undefined unless a default
- precision has already been established by a call to
- `mpf_set_default_prec'.
-
- -- Function: void mpf_init2 (mpf_t X, mp_bitcnt_t PREC)
- Initialize X to 0 and set its precision to be *at least* PREC
- bits. Normally, a variable should be initialized once only or at
- least be cleared, using `mpf_clear', between initializations.
-
- -- Function: void mpf_inits (mpf_t X, ...)
- Initialize a NULL-terminated list of `mpf_t' variables, and set
- their values to 0. The precision of the initialized variables is
- undefined unless a default precision has already been established
- by a call to `mpf_set_default_prec'.
-
- -- Function: void mpf_clear (mpf_t X)
- Free the space occupied by X. Make sure to call this function for
- all `mpf_t' variables when you are done with them.
-
- -- Function: void mpf_clears (mpf_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpf_t'
- variables.
-
- Here is an example on how to initialize floating-point variables:
- {
- mpf_t x, y;
- mpf_init (x); /* use default precision */
- mpf_init2 (y, 256); /* precision _at least_ 256 bits */
- ...
- /* Unless the program is about to exit, do ... */
- mpf_clear (x);
- mpf_clear (y);
- }
-
- The following three functions are useful for changing the precision
-during a calculation. A typical use would be for adjusting the
-precision gradually in iterative algorithms like Newton-Raphson, making
-the computation precision closely match the actual accurate part of the
-numbers.
-
- -- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP)
- Return the current precision of OP, in bits.
-
- -- Function: void mpf_set_prec (mpf_t ROP, mp_bitcnt_t PREC)
- Set the precision of ROP to be *at least* PREC bits. The value in
- ROP will be truncated to the new precision.
-
- This function requires a call to `realloc', and so should not be
- used in a tight loop.
-
- -- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC)
- Set the precision of ROP to be *at least* PREC bits, without
- changing the memory allocated.
-
- PREC must be no more than the allocated precision for ROP, that
- being the precision when ROP was initialized, or in the most recent
- `mpf_set_prec'.
-
- The value in ROP is unchanged, and in particular if it had a higher
- precision than PREC it will retain that higher precision. New
- values written to ROP will use the new PREC.
-
- Before calling `mpf_clear' or the full `mpf_set_prec', another
- `mpf_set_prec_raw' call must be made to restore ROP to its original
- allocated precision. Failing to do so will have unpredictable
- results.
-
- `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
- original allocated precision. After `mpf_set_prec_raw' it
- reflects the PREC value set.
-
- `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
- at different precisions during a calculation, perhaps to gradually
- increase precision in an iteration, or just to use various
- different precisions for different purposes during a calculation.
-
-\1f
-File: gmp.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
-
-7.2 Assignment Functions
-========================
-
-These functions assign new values to already initialized floats (*note
-Initializing Floats::).
-
- -- Function: void mpf_set (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
- -- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
- -- Function: void mpf_set_d (mpf_t ROP, double OP)
- -- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
- -- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
- Set the value of ROP from OP.
-
- -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
- Set the value of ROP from the string in STR. The string is of the
- form `M@N' or, if the base is 10 or less, alternatively `MeN'.
- `M' is the mantissa and `N' is the exponent. The mantissa is
- always in the specified base. The exponent is either in the
- specified base or, if BASE is negative, in decimal. The decimal
- point expected is taken from the current locale, on systems
- providing `localeconv'.
-
- The argument BASE may be in the ranges 2 to 62, or -62 to -2.
- Negative values are used to specify that the exponent is in
- decimal.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value; for bases 37 to 62, upper-case letter
- represent the usual 10..35 while lower-case letter represent
- 36..61.
-
- Unlike the corresponding `mpz' function, the base will not be
- determined from the leading characters of the string if BASE is 0.
- This is so that numbers like `0.23' are not interpreted as octal.
-
- White space is allowed in the string, and is simply ignored.
- [This is not really true; white-space is ignored in the beginning
- of the string and within the mantissa, but not in other places,
- such as after a minus sign or in the exponent. We are considering
- changing the definition of this function, making it fail when
- there is any white-space in the input, since that makes a lot of
- sense. Please tell us your opinion about this change. Do you
- really want it to accept "3 14" as meaning 314 as it does now?]
-
- This function returns 0 if the entire string is a valid number in
- base BASE. Otherwise it returns -1.
-
- -- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
- Swap ROP1 and ROP2 efficiently. Both the values and the
- precisions of the two variables are swapped.
-
-\1f
-File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
-
-7.3 Combined Initialization and Assignment Functions
-====================================================
-
-For convenience, GMP provides a parallel series of initialize-and-set
-functions which initialize the output and then store the value there.
-These functions' names have the form `mpf_init_set...'
-
- Once the float has been initialized by any of the `mpf_init_set...'
-functions, it can be used as the source or destination operand for the
-ordinary float functions. Don't use an initialize-and-set function on
-a variable already initialized!
-
- -- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
- -- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
- -- Function: void mpf_init_set_d (mpf_t ROP, double OP)
- Initialize ROP and set its value from OP.
-
- The precision of ROP will be taken from the active default
- precision, as set by `mpf_set_default_prec'.
-
- -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
- Initialize ROP and set its value from the string in STR. See
- `mpf_set_str' above for details on the assignment operation.
-
- Note that ROP is initialized even if an error occurs. (I.e., you
- have to call `mpf_clear' for it.)
-
- The precision of ROP will be taken from the active default
- precision, as set by `mpf_set_default_prec'.
-
-\1f
-File: gmp.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
-
-7.4 Conversion Functions
-========================
-
- -- Function: double mpf_get_d (mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent in OP is too big or too small to fit a `double'
- then the result is system dependent. For too big an infinity is
- returned when available. For too small 0.0 is normally returned.
- Hardware overflow, underflow and denorm traps may or may not occur.
-
- -- Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero), and with an exponent returned separately.
-
- The return value is in the range 0.5<=abs(D)<1 and the exponent is
- stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
- is zero, the return is 0.0 and 0 is stored to `*EXP'.
-
- This is similar to the standard C `frexp' function (*note
- Normalization Functions: (libc)Normalization Functions.).
-
- -- Function: long mpf_get_si (mpf_t OP)
- -- Function: unsigned long mpf_get_ui (mpf_t OP)
- Convert OP to a `long' or `unsigned long', truncating any fraction
- part. If OP is too big for the return type, the result is
- undefined.
-
- See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
- Miscellaneous Float Functions::).
-
- -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
- BASE, size_t N_DIGITS, mpf_t OP)
- Convert OP to a string of digits in base BASE. The base argument
- may vary from 2 to 62 or from -2 to -36. Up to N_DIGITS digits
- will be generated. Trailing zeros are not returned. No more
- digits than can be accurately represented by OP are ever
- generated. If N_DIGITS is 0 then that accurate maximum number of
- digits are generated.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of N_DIGITS + 2
- bytes, that being enough for the mantissa, a possible minus sign,
- and a null-terminator. When N_DIGITS is 0 to get all significant
- digits, an application won't be able to know the space required,
- and STR should be `NULL' in that case.
-
- The generated string is a fraction, with an implicit radix point
- immediately to the left of the first digit. The applicable
- exponent is written through the EXPPTR pointer. For example, the
- number 3.1416 would be returned as string "31416" and exponent 1.
-
- When OP is zero, an empty string is produced and the exponent
- returned is 0.
-
- A pointer to the result string is returned, being either the
- allocated block or the given STR.
-
-\1f
-File: gmp.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
-
-7.5 Arithmetic Functions
-========================
-
- -- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 + OP2.
-
- -- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
- OP2)
- -- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 - OP2.
-
- -- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 times OP2.
-
- Division is undefined if the divisor is zero, and passing a zero
-divisor to the divide functions will make these functions intentionally
-divide by zero. This lets the user handle arithmetic exceptions in
-these functions in the same manner as other arithmetic exceptions.
-
- -- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
- OP2)
- -- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1/OP2.
-
- -- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
- Set ROP to the square root of OP.
-
- -- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 raised to the power OP2.
-
- -- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
- Set ROP to -OP.
-
- -- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
- Set ROP to the absolute value of OP.
-
- -- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2.
-
- -- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 divided by 2 raised to OP2.
-
-\1f
-File: gmp.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
-
-7.6 Comparison Functions
-========================
-
- -- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
- -- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
- -- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
- -- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- `mpf_cmp_d' can be called with an infinity, but results are
- undefined for a NaN.
-
- -- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, mp_bitcnt_t op3)
- Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
- zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
-
- Caution 1: All version of GMP up to version 4.2.4 compared just
- whole limbs, meaning sometimes more than OP3 bits, sometimes fewer.
-
- Caution 2: This function will consider XXX11...111 and XX100...000
- different, even if ... is replaced by a semi-infinite number of
- bits. Such numbers are really just one ulp off, and should be
- considered equal.
-
- -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Compute the relative difference between OP1 and OP2 and store the
- result in ROP. This is abs(OP1-OP2)/OP1.
-
- -- Macro: int mpf_sgn (mpf_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates
- its arguments multiple times.
-
-\1f
-File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
-
-7.7 Input and Output Functions
-==============================
-
-Functions that perform input from a stdio stream, and functions that
-output to a stdio stream. Passing a `NULL' pointer for a STREAM
-argument to any of these functions will make them read from `stdin' and
-write to `stdout', respectively.
-
- When using any of these functions, it is a good idea to include
-`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
-prototypes for these functions.
-
- -- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
- N_DIGITS, mpf_t OP)
- Print OP to STREAM, as a string of digits. Return the number of
- bytes written, or if an error occurred, return 0.
-
- The mantissa is prefixed with an `0.' and is in the given BASE,
- which may vary from 2 to 62 or from -2 to -36. An exponent is
- then printed, separated by an `e', or if the base is greater than
- 10 then by an `@'. The exponent is always in decimal. The
- decimal point follows the current locale, on systems providing
- `localeconv'.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- Up to N_DIGITS will be printed from the mantissa, except that no
- more digits than are accurately representable by OP will be
- printed. N_DIGITS can be 0 to select that accurate maximum.
-
- -- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
- Read a string in base BASE from STREAM, and put the read float in
- ROP. The string is of the form `M@N' or, if the base is 10 or
- less, alternatively `MeN'. `M' is the mantissa and `N' is the
- exponent. The mantissa is always in the specified base. The
- exponent is either in the specified base or, if BASE is negative,
- in decimal. The decimal point expected is taken from the current
- locale, on systems providing `localeconv'.
-
- The argument BASE may be in the ranges 2 to 36, or -36 to -2.
- Negative values are used to specify that the exponent is in
- decimal.
-
- Unlike the corresponding `mpz' function, the base will not be
- determined from the leading characters of the string if BASE is 0.
- This is so that numbers like `0.23' are not interpreted as octal.
-
- Return the number of bytes read, or if an error occurred, return 0.
-
-\1f
-File: gmp.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
-
-7.8 Miscellaneous Functions
-===========================
-
- -- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
- Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
- next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
- to the integer towards zero.
-
- -- Function: int mpf_integer_p (mpf_t OP)
- Return non-zero if OP is an integer.
-
- -- Function: int mpf_fits_ulong_p (mpf_t OP)
- -- Function: int mpf_fits_slong_p (mpf_t OP)
- -- Function: int mpf_fits_uint_p (mpf_t OP)
- -- Function: int mpf_fits_sint_p (mpf_t OP)
- -- Function: int mpf_fits_ushort_p (mpf_t OP)
- -- Function: int mpf_fits_sshort_p (mpf_t OP)
- Return non-zero if OP would fit in the respective C data type, when
- truncated to an integer.
-
- -- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t NBITS)
- Generate a uniformly distributed random float in ROP, such that 0
- <= ROP < 1, with NBITS significant bits in the mantissa.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
- EXP)
- Generate a random float of at most MAX_SIZE limbs, with long
- strings of zeros and ones in the binary representation. The
- exponent of the number is in the interval -EXP to EXP (in limbs).
- This function is useful for testing functions and algorithms,
- since these kind of random numbers have proven to be more likely
- to trigger corner-case bugs. Negative random numbers are
- generated when MAX_SIZE is negative.
-
-\1f
-File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
-
-8 Low-level Functions
-*********************
-
-This chapter describes low-level GMP functions, used to implement the
-high-level GMP functions, but also intended for time-critical user code.
-
- These functions start with the prefix `mpn_'.
-
- The `mpn' functions are designed to be as fast as possible, *not* to
-provide a coherent calling interface. The different functions have
-somewhat similar interfaces, but there are variations that make them
-hard to use. These functions do as little as possible apart from the
-real multiple precision computation, so that no time is spent on things
-that not all callers need.
-
- A source operand is specified by a pointer to the least significant
-limb and a limb count. A destination operand is specified by just a
-pointer. It is the responsibility of the caller to ensure that the
-destination has enough space for storing the result.
-
- With this way of specifying operands, it is possible to perform
-computations on subranges of an argument, and store the result into a
-subrange of a destination.
-
- A common requirement for all functions is that each source area
-needs at least one limb. No size argument may be zero. Unless
-otherwise stated, in-place operations are allowed where source and
-destination are the same, but not where they only partly overlap.
-
- The `mpn' functions are the base for the implementation of the
-`mpz_', `mpf_', and `mpq_' functions.
-
- This example adds the number beginning at S1P and the number
-beginning at S2P and writes the sum at DESTP. All areas have N limbs.
-
- cy = mpn_add_n (destp, s1p, s2p, n)
-
- It should be noted that the `mpn' functions make no attempt to
-identify high or low zero limbs on their operands, or other special
-forms. On random data such cases will be unlikely and it'd be wasteful
-for every function to check every time. An application knowing
-something about its data can take steps to trim or perhaps split its
-calculations.
-
-
-In the notation used below, a source operand is identified by the
-pointer to the least significant limb, and the limb count in braces.
-For example, {S1P, S1N}.
-
- -- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Add {S1P, N} and {S2P, N}, and write the N least significant limbs
- of the result to RP. Return carry, either 0 or 1.
-
- This is the lowest-level function for addition. It is the
- preferred function for addition, since it is written in assembly
- for most CPUs. For addition of a variable to itself (i.e., S1P
- equals S2P) use `mpn_lshift' with a count of 1 for optimal speed.
-
- -- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Add {S1P, N} and S2LIMB, and write the N least significant limbs
- of the result to RP. Return carry, either 0 or 1.
-
- -- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
- limbs of the result to RP. Return carry, either 0 or 1.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Subtract {S2P, N} from {S1P, N}, and write the N least significant
- limbs of the result to RP. Return borrow, either 0 or 1.
-
- This is the lowest-level function for subtraction. It is the
- preferred function for subtraction, since it is written in
- assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Subtract S2LIMB from {S1P, N}, and write the N least significant
- limbs of the result to RP. Return borrow, either 0 or 1.
-
- -- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
- significant limbs of the result to RP. Return borrow, either 0 or
- 1.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: void mpn_neg (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N)
- Perform the negation of {SP, N}, and write the result to {RP, N}.
- Return carry-out.
-
- -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
- RP.
-
- The destination has to have space for 2*N limbs, even if the
- product's most significant limb is zero. No overlap is permitted
- between the destination and either source.
-
- If the two input operands are the same, use `mpn_sqr'.
-
- -- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Multiply {S1P, S1N} and {S2P, S2N}, and write the (S1N+S2N)-limb
- result to RP. Return the most significant limb of the result.
-
- The destination has to have space for S1N + S2N limbs, even if the
- product's most significant limb is zero. No overlap is permitted
- between the destination and either source.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Compute the square of {S1P, N} and write the 2*N-limb result to RP.
-
- The destination has to have space for 2*N limbs, even if the
- result's most significant limb is zero. No overlap is permitted
- between the destination and the source.
-
- -- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} by S2LIMB, and write the N least significant
- limbs of the product to RP. Return the most significant limb of
- the product. {S1P, N} and {RP, N} are allowed to overlap provided
- RP <= S1P.
-
- This is a low-level function that is a building block for general
- multiplication as well as other operations in GMP. It is written
- in assembly for most CPUs.
-
- Don't call this function if S2LIMB is a power of 2; use
- `mpn_lshift' with a count equal to the logarithm of S2LIMB
- instead, for optimal speed.
-
- -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
- *S1P, mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} and S2LIMB, and add the N least significant
- limbs of the product to {RP, N} and write the result to RP.
- Return the most significant limb of the product, plus carry-out
- from the addition.
-
- This is a low-level function that is a building block for general
- multiplication as well as other operations in GMP. It is written
- in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
- *S1P, mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} and S2LIMB, and subtract the N least significant
- limbs of the product from {RP, N} and write the result to RP.
- Return the most significant limb of the product, plus borrow-out
- from the subtraction.
-
- This is a low-level function that is a building block for general
- multiplication and division as well as other operations in GMP.
- It is written in assembly for most CPUs.
-
- -- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
- QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
- mp_size_t DN)
- Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
- and the remainder at {RP, DN}. The quotient is rounded towards 0.
-
- No overlap is permitted between arguments, except that NP might
- equal RP. The dividend size NN must be greater than or equal to
- divisor size DN. The most significant limb of the divisor must be
- non-zero. The QXN operand must be zero.
-
- -- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
- mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
- mp_size_t S3N)
- [This function is obsolete. Please call `mpn_tdiv_qr' instead for
- best performance.]
-
- Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
- with the exception of the most significant limb, which is
- returned. The remainder replaces the dividend at RS2P; it will be
- S3N limbs long (i.e., as many limbs as the divisor).
-
- In addition to an integer quotient, QXN fraction limbs are
- developed, and stored after the integral limbs. For most usages,
- QXN will be zero.
-
- It is required that RS2N is greater than or equal to S3N. It is
- required that the most significant bit of the divisor is set.
-
- If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
- that special case, no overlap between arguments is permitted.
-
- Return the most significant limb of the quotient, either 0 or 1.
-
- The area at R1P needs to be RS2N - S3N + QXN limbs large.
-
- -- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
- mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
- -- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
- mp_size_t S2N, mp_limb_t S3LIMB)
- Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
- Return the remainder.
-
- The integer quotient is written to {R1P+QXN, S2N} and in addition
- QXN fraction limbs are developed and written to {R1P, QXN}.
- Either or both S2N and QXN can be zero. For most usages, QXN will
- be zero.
-
- `mpn_divmod_1' exists for upward source compatibility and is
- simply a macro calling `mpn_divrem_1' with a QXN of 0.
-
- The areas at R1P and S2P have to be identical or completely
- separate, not partially overlapping.
-
- -- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
- mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
- [This function is obsolete. Please call `mpn_tdiv_qr' instead for
- best performance.]
-
- -- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
- mp_size_t N)
- -- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t
- *SP, mp_size_t N, mp_limb_t CARRY)
- Divide {SP, N} by 3, expecting it to divide exactly, and writing
- the result to {RP, N}. If 3 divides exactly, the return value is
- zero and the result is the quotient. If not, the return value is
- non-zero and the result won't be anything useful.
-
- `mpn_divexact_by3c' takes an initial carry parameter, which can be
- the return value from a previous call, so a large calculation can
- be done piece by piece from low to high. `mpn_divexact_by3' is
- simply a macro calling `mpn_divexact_by3c' with a 0 carry
- parameter.
-
- These routines use a multiply-by-inverse and will be faster than
- `mpn_divrem_1' on CPUs with fast multiplication but slow division.
-
- The source a, result q, size n, initial carry i, and return value
- c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
- c is always 0, 1 or 2, and the initial carry i must also be 0, 1
- or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
- When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b
- == 1 mod 3 (when `mp_bits_per_limb' is even, which is always so
- currently).
-
- -- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
- mp_limb_t S2LIMB)
- Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
- zero.
-
- -- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N, unsigned int COUNT)
- Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
- The bits shifted out at the left are returned in the least
- significant COUNT bits of the return value (the rest of the return
- value is zero).
-
- COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
- {SP, N} and {RP, N} may overlap, provided RP >= SP.
-
- This function is written in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N, unsigned int COUNT)
- Shift {SP, N} right by COUNT bits, and write the result to {RP,
- N}. The bits shifted out at the right are returned in the most
- significant COUNT bits of the return value (the rest of the return
- value is zero).
-
- COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
- {SP, N} and {RP, N} may overlap, provided RP <= SP.
-
- This function is written in assembly for most CPUs.
-
- -- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
- mp_size_t N)
- Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
- S2, 0 if they are equal, or a negative value if S1 < S2.
-
- -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *XP,
- mp_size_t XN, mp_limb_t *YP, mp_size_t YN)
- Set {RP, RETVAL} to the greatest common divisor of {XP, XN} and
- {YP, YN}. The result can be up to YN limbs, the return value is
- the actual number produced. Both source operands are destroyed.
-
- {XP, XN} must have at least as many bits as {YP, YN}. {YP, YN}
- must be odd. Both operands must have non-zero most significant
- limbs. No overlap is permitted between {XP, XN} and {YP, YN}.
-
- -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *XP, mp_size_t XN,
- mp_limb_t YLIMB)
- Return the greatest common divisor of {XP, XN} and YLIMB. Both
- operands must be non-zero.
-
- -- Function: mp_size_t mpn_gcdext (mp_limb_t *GP, mp_limb_t *SP,
- mp_size_t *SN, mp_limb_t *XP, mp_size_t XN, mp_limb_t *YP,
- mp_size_t YN)
- Let U be defined by {XP, XN} and let V be defined by {YP, YN}.
-
- Compute the greatest common divisor G of U and V. Compute a
- cofactor S such that G = US + VT. The second cofactor T is not
- computed but can easily be obtained from (G - U*S) / V (the
- division will be exact). It is required that U >= V > 0.
-
- S satisfies S = 1 or abs(S) < V / (2 G). S = 0 if and only if V
- divides U (i.e., G = V).
-
- Store G at GP and let the return value define its limb count.
- Store S at SP and let |*SN| define its limb count. S can be
- negative; when this happens *SN will be negative. The areas at GP
- and SP should each have room for XN+1 limbs.
-
- The areas {XP, XN+1} and {YP, YN+1} are destroyed (i.e. the input
- operands plus an extra limb past the end of each).
-
- Compatibility note: GMP 4.3.0 and 4.3.1 defined S less strictly.
- Earlier as well as later GMP releases define S as described here.
-
- -- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
- const mp_limb_t *SP, mp_size_t N)
- Compute the square root of {SP, N} and put the result at {R1P,
- ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space
- for N limbs, but the return value indicates how many are produced.
-
- The most significant limb of {SP, N} must be non-zero. The areas
- {R1P, ceil(N/2)} and {SP, N} must be completely separate. The
- areas {R2P, N} and {SP, N} must be either identical or completely
- separate.
-
- If the remainder is not wanted then R2P can be `NULL', and in this
- case the return value is zero or non-zero according to whether the
- remainder would have been zero or non-zero.
-
- A return value of zero indicates a perfect square. See also
- `mpz_perfect_square_p'.
-
- -- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
- mp_limb_t *S1P, mp_size_t S1N)
- Convert {S1P, S1N} to a raw unsigned char array at STR in base
- BASE, and return the number of characters produced. There may be
- leading zeros in the string. The string is not in ASCII; to
- convert it to printable format, add the ASCII codes for `0' or
- `A', depending on the base and range. BASE can vary from 2 to 256.
-
- The most significant limb of the input {S1P, S1N} must be
- non-zero. The input {S1P, S1N} is clobbered, except when BASE is
- a power of 2, in which case it's unchanged.
-
- The area at STR has to have space for the largest possible number
- represented by a S1N long limb array, plus one extra character.
-
- -- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
- *STR, size_t STRSIZE, int BASE)
- Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
-
- STR[0] is the most significant byte and STR[STRSIZE-1] is the
- least significant. Each byte should be a value in the range 0 to
- BASE-1, not an ASCII character. BASE can vary from 2 to 256.
-
- The return value is the number of limbs written to RP. If the most
- significant input byte is non-zero then the high limb at RP will be
- non-zero, and only that exact number of limbs will be required
- there.
-
- If the most significant input byte is zero then there may be high
- zero limbs written to RP and included in the return value.
-
- STRSIZE must be at least 1, and no overlap is permitted between
- {STR,STRSIZE} and the result at RP.
-
- -- Function: mp_bitcnt_t mpn_scan0 (const mp_limb_t *S1P, mp_bitcnt_t
- BIT)
- Scan S1P from bit position BIT for the next clear bit.
-
- It is required that there be a clear bit within the area at S1P at
- or beyond bit position BIT, so that the function has something to
- return.
-
- -- Function: mp_bitcnt_t mpn_scan1 (const mp_limb_t *S1P, mp_bitcnt_t
- BIT)
- Scan S1P from bit position BIT for the next set bit.
-
- It is required that there be a set bit within the area at S1P at or
- beyond bit position BIT, so that the function has something to
- return.
-
- -- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
- -- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
- Generate a random number of length R1N and store it at R1P. The
- most significant limb is always non-zero. `mpn_random' generates
- uniformly distributed limb data, `mpn_random2' generates long
- strings of zeros and ones in the binary representation.
-
- `mpn_random2' is intended for testing the correctness of the `mpn'
- routines.
-
- -- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t
- N)
- Count the number of set bits in {S1P, N}.
-
- -- Function: mp_bitcnt_t mpn_hamdist (const mp_limb_t *S1P, const
- mp_limb_t *S2P, mp_size_t N)
- Compute the hamming distance between {S1P, N} and {S2P, N}, which
- is the number of bit positions where the two operands have
- different bit values.
-
- -- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
- N)
- Return non-zero iff {S1P, N} is a perfect square.
-
- -- Function: void mpn_and_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
- write the result to {RP, N}.
-
- -- Function: void mpn_ior_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
- and write the result to {RP, N}.
-
- -- Function: void mpn_xor_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
- and write the result to {RP, N}.
-
- -- Function: void mpn_andn_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and the bitwise
- complement of {S2P, N}, and write the result to {RP, N}.
-
- -- Function: void mpn_iorn_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and the
- bitwise complement of {S2P, N}, and write the result to {RP, N}.
-
- -- Function: void mpn_nand_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
- write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_nior_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
- and write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_xnor_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
- and write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_com (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N)
- Perform the bitwise complement of {SP, N}, and write the result to
- {RP, N}.
-
- -- Function: void mpn_copyi (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Copy from {S1P, N} to {RP, N}, increasingly.
-
- -- Function: void mpn_copyd (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Copy from {S1P, N} to {RP, N}, decreasingly.
-
- -- Function: void mpn_zero (mp_limb_t *RP, mp_size_t N)
- Zero {RP, N}.
-
-
-8.1 Nails
-=========
-
-*Everything in this section is highly experimental and may disappear or
-be subject to incompatible changes in a future version of GMP.*
-
- Nails are an experimental feature whereby a few bits are left unused
-at the top of each `mp_limb_t'. This can significantly improve carry
-handling on some processors.
-
- All the `mpn' functions accepting limb data will expect the nail
-bits to be zero on entry, and will return data with the nails similarly
-all zero. This applies both to limb vectors and to single limb
-arguments.
-
- Nails can be enabled by configuring with `--enable-nails'. By
-default the number of bits will be chosen according to what suits the
-host processor, but a particular number can be selected with
-`--enable-nails=N'.
-
- At the mpn level, a nail build is neither source nor binary
-compatible with a non-nail build, strictly speaking. But programs
-acting on limbs only through the mpn functions are likely to work
-equally well with either build, and judicious use of the definitions
-below should make any program compatible with either build, at the
-source level.
-
- For the higher level routines, meaning `mpz' etc, a nail build
-should be fully source and binary compatible with a non-nail build.
-
- -- Macro: GMP_NAIL_BITS
- -- Macro: GMP_NUMB_BITS
- -- Macro: GMP_LIMB_BITS
- `GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
- not in use. `GMP_NUMB_BITS' is the number of data bits in a limb.
- `GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In
- all cases
-
- GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
-
- -- Macro: GMP_NAIL_MASK
- -- Macro: GMP_NUMB_MASK
- Bit masks for the nail and number parts of a limb.
- `GMP_NAIL_MASK' is 0 when nails are not in use.
-
- `GMP_NAIL_MASK' is not often needed, since the nail part can be
- obtained with `x >> GMP_NUMB_BITS', and that means one less large
- constant, which can help various RISC chips.
-
- -- Macro: GMP_NUMB_MAX
- The maximum value that can be stored in the number part of a limb.
- This is the same as `GMP_NUMB_MASK', but can be used for clarity
- when doing comparisons rather than bit-wise operations.
-
- The term "nails" comes from finger or toe nails, which are at the
-ends of a limb (arm or leg). "numb" is short for number, but is also
-how the developers felt after trying for a long time to come up with
-sensible names for these things.
-
- In the future (the distant future most likely) a non-zero nail might
-be permitted, giving non-unique representations for numbers in a limb
-vector. This would help vector processors since carries would only
-ever need to propagate one or two limbs.
-
-\1f
-File: gmp.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
-
-9 Random Number Functions
-*************************
-
-Sequences of pseudo-random numbers in GMP are generated using a
-variable of type `gmp_randstate_t', which holds an algorithm selection
-and a current state. Such a variable must be initialized by a call to
-one of the `gmp_randinit' functions, and can be seeded with one of the
-`gmp_randseed' functions.
-
- The functions actually generating random numbers are described in
-*Note Integer Random Numbers::, and *Note Miscellaneous Float
-Functions::.
-
- The older style random number functions don't accept a
-`gmp_randstate_t' parameter but instead share a global variable of that
-type. They use a default algorithm and are currently not seeded
-(though perhaps that will change in the future). The new functions
-accepting a `gmp_randstate_t' are recommended for applications that
-care about randomness.
-
-* Menu:
-
-* Random State Initialization::
-* Random State Seeding::
-* Random State Miscellaneous::
-
-\1f
-File: gmp.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
-
-9.1 Random State Initialization
-===============================
-
- -- Function: void gmp_randinit_default (gmp_randstate_t STATE)
- Initialize STATE with a default algorithm. This will be a
- compromise between speed and randomness, and is recommended for
- applications with no special requirements. Currently this is
- `gmp_randinit_mt'.
-
- -- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
- Initialize STATE for a Mersenne Twister algorithm. This algorithm
- is fast and has good randomness properties.
-
- -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t
- A, unsigned long C, mp_bitcnt_t M2EXP)
- Initialize STATE with a linear congruential algorithm X = (A*X +
- C) mod 2^M2EXP.
-
- The low bits of X in this algorithm are not very random. The least
- significant bit will have a period no more than 2, and the second
- bit no more than 4, etc. For this reason only the high half of
- each X is actually used.
-
- When a random number of more than M2EXP/2 bits is to be generated,
- multiple iterations of the recurrence are used and the results
- concatenated.
-
- -- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
- mp_bitcnt_t SIZE)
- Initialize STATE for a linear congruential algorithm as per
- `gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
- chosen so that SIZE bits (or more) of each X will be used, ie.
- M2EXP/2 >= SIZE.
-
- If successful the return value is non-zero. If SIZE is bigger
- than the table data provides then the return value is zero. The
- maximum SIZE currently supported is 128.
-
- -- Function: void gmp_randinit_set (gmp_randstate_t ROP,
- gmp_randstate_t OP)
- Initialize ROP with a copy of the algorithm and state from OP.
-
- -- Function: void gmp_randinit (gmp_randstate_t STATE,
- gmp_randalg_t ALG, ...)
- *This function is obsolete.*
-
- Initialize STATE with an algorithm selected by ALG. The only
- choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size'
- described above. A third parameter of type `unsigned long' is
- required, this is the SIZE for that function.
- `GMP_RAND_ALG_DEFAULT' or 0 are the same as `GMP_RAND_ALG_LC'.
-
- `gmp_randinit' sets bits in the global variable `gmp_errno' to
- indicate an error. `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is
- unsupported, or `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter
- is too big. It may be noted this error reporting is not thread
- safe (a good reason to use `gmp_randinit_lc_2exp_size' instead).
-
- -- Function: void gmp_randclear (gmp_randstate_t STATE)
- Free all memory occupied by STATE.
-
-\1f
-File: gmp.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
-
-9.2 Random State Seeding
-========================
-
- -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
- -- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
- unsigned long int SEED)
- Set an initial seed value into STATE.
-
- The size of a seed determines how many different sequences of
- random numbers that it's possible to generate. The "quality" of
- the seed is the randomness of a given seed compared to the
- previous seed used, and this affects the randomness of separate
- number sequences. The method for choosing a seed is critical if
- the generated numbers are to be used for important applications,
- such as generating cryptographic keys.
-
- Traditionally the system time has been used to seed, but care
- needs to be taken with this. If an application seeds often and
- the resolution of the system clock is low, then the same sequence
- of numbers might be repeated. Also, the system time is quite easy
- to guess, so if unpredictability is required then it should
- definitely not be the only source for the seed value. On some
- systems there's a special device `/dev/random' which provides
- random data better suited for use as a seed.
-
-\1f
-File: gmp.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
-
-9.3 Random State Miscellaneous
-==============================
-
- -- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
- unsigned long N)
- Return a uniformly distributed random number of N bits, ie. in the
- range 0 to 2^N-1 inclusive. N must be less than or equal to the
- number of bits in an `unsigned long'.
-
- -- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE,
- unsigned long N)
- Return a uniformly distributed random number in the range 0 to
- N-1, inclusive.
-
-\1f
-File: gmp.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
-
-10 Formatted Output
-*******************
-
-* Menu:
-
-* Formatted Output Strings::
-* Formatted Output Functions::
-* C++ Formatted Output::
-
-\1f
-File: gmp.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output
-
-10.1 Format Strings
-===================
-
-`gmp_printf' and friends accept format strings similar to the standard C
-`printf' (*note Formatted Output: (libc)Formatted Output.). A format
-specification is of the form
-
- % [flags] [width] [.[precision]] [type] conv
-
- GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
-respectively, `M' for `mp_limb_t', and `N' for an `mp_limb_t' array.
-`Z', `Q', `M' and `N' behave like integers. `Q' will print a `/' and a
-denominator, if needed. `F' behaves like a float. For example,
-
- mpz_t z;
- gmp_printf ("%s is an mpz %Zd\n", "here", z);
-
- mpq_t q;
- gmp_printf ("a hex rational: %#40Qx\n", q);
-
- mpf_t f;
- int n;
- gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
-
- mp_limb_t l;
- gmp_printf ("limb %Mu\n", l);
-
- const mp_limb_t *ptr;
- mp_size_t size;
- gmp_printf ("limb array %Nx\n", ptr, size);
-
- For `N' the limbs are expected least significant first, as per the
-`mpn' functions (*note Low-level Functions::). A negative size can be
-given to print the value as a negative.
-
- All the standard C `printf' types behave the same as the C library
-`printf', and can be freely intermixed with the GMP extensions. In the
-current implementation the standard parts of the format string are
-simply handed to `printf' and only the GMP extensions handled directly.
-
- The flags accepted are as follows. GLIBC style ' is only for the
-standard C types (not the GMP types), and only if the C library
-supports it.
-
- 0 pad with zeros (rather than spaces)
- # show the base with `0x', `0X' or `0'
- + always show a sign
- (space) show a space or a `-' sign
- ' group digits, GLIBC style (not GMP types)
-
- The optional width and precision can be given as a number within the
-format string, or as a `*' to take an extra parameter of type `int', the
-same as the standard `printf'.
-
- The standard types accepted are as follows. `h' and `l' are
-portable, the rest will depend on the compiler (or include files) for
-the type and the C library for the output.
-
- h short
- hh char
- j intmax_t or uintmax_t
- l long or wchar_t
- ll long long
- L long double
- q quad_t or u_quad_t
- t ptrdiff_t
- z size_t
-
-The GMP types are
-
- F mpf_t, float conversions
- Q mpq_t, integer conversions
- M mp_limb_t, integer conversions
- N mp_limb_t array, integer conversions
- Z mpz_t, integer conversions
-
- The conversions accepted are as follows. `a' and `A' are always
-supported for `mpf_t' but depend on the C library for standard C float
-types. `m' and `p' depend on the C library.
-
- a A hex floats, C99 style
- c character
- d decimal integer
- e E scientific format float
- f fixed point float
- i same as d
- g G fixed or scientific float
- m `strerror' string, GLIBC style
- n store characters written so far
- o octal integer
- p pointer
- s string
- u unsigned integer
- x X hex integer
-
- `o', `x' and `X' are unsigned for the standard C types, but for
-types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z',
-`Q' and `N'.
-
- `M' is a proxy for the C library `l' or `L', according to the size
-of `mp_limb_t'. Unsigned conversions will be usual, but a signed
-conversion can be used and will interpret the value as a twos complement
-negative.
-
- `n' can be used with any type, even the GMP types.
-
- Other types or conversions that might be accepted by the C library
-`printf' cannot be used through `gmp_printf', this includes for
-instance extensions registered with GLIBC `register_printf_function'.
-Also currently there's no support for POSIX `$' style numbered arguments
-(perhaps this will be added in the future).
-
- The precision field has it's usual meaning for integer `Z' and float
-`F' types, but is currently undefined for `Q' and should not be used
-with that.
-
- `mpf_t' conversions only ever generate as many digits as can be
-accurately represented by the operand, the same as `mpf_get_str' does.
-Zeros will be used if necessary to pad to the requested precision. This
-happens even for an `f' conversion of an `mpf_t' which is an integer,
-for instance 2^1024 in an `mpf_t' of 128 bits precision will only
-produce about 40 digits, then pad with zeros to the decimal point. An
-empty precision field like `%.Fe' or `%.Ff' can be used to specifically
-request just the significant digits.
-
- The decimal point character (or string) is taken from the current
-locale settings on systems which provide `localeconv' (*note Locales
-and Internationalization: (libc)Locales.). The C library will normally
-do the same for standard float output.
-
- The format string is only interpreted as plain `char's, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-\1f
-File: gmp.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output
-
-10.2 Functions
-==============
-
-Each of the following functions is similar to the corresponding C
-library function. The basic `printf' forms take a variable argument
-list. The `vprintf' forms take an argument pointer, see *Note Variadic
-Functions: (libc)Variadic Functions, or `man 3 va_start'.
-
- It should be emphasised that if a format string is invalid, or the
-arguments don't match what the format specifies, then the behaviour of
-any of these functions will be unpredictable. GCC format string
-checking is not available, since it doesn't recognise the GMP
-extensions.
-
- The file based functions `gmp_printf' and `gmp_fprintf' will return
--1 to indicate a write error. Output is not "atomic", so partial
-output may be produced if a write error occurs. All the functions can
-return -1 if the C library `printf' variant in use returns -1, but this
-shouldn't normally occur.
-
- -- Function: int gmp_printf (const char *FMT, ...)
- -- Function: int gmp_vprintf (const char *FMT, va_list AP)
- Print to the standard output `stdout'. Return the number of
- characters written, or -1 if an error occurred.
-
- -- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
- -- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
- Print to the stream FP. Return the number of characters written,
- or -1 if an error occurred.
-
- -- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
- -- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
- Form a null-terminated string in BUF. Return the number of
- characters written, excluding the terminating null.
-
- No overlap is permitted between the space at BUF and the string
- FMT.
-
- These functions are not recommended, since there's no protection
- against exceeding the space available at BUF.
-
- -- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char
- *FMT, ...)
- -- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
- *FMT, va_list AP)
- Form a null-terminated string in BUF. No more than SIZE bytes
- will be written. To get the full output, SIZE must be enough for
- the string and null-terminator.
-
- The return value is the total number of characters which ought to
- have been produced, excluding the terminating null. If RETVAL >=
- SIZE then the actual output has been truncated to the first SIZE-1
- characters, and a null appended.
-
- No overlap is permitted between the region {BUF,SIZE} and the FMT
- string.
-
- Notice the return value is in ISO C99 `snprintf' style. This is
- so even if the C library `vsnprintf' is the older GLIBC 2.0.x
- style.
-
- -- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
- -- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP)
- Form a null-terminated string in a block of memory obtained from
- the current memory allocation function (*note Custom
- Allocation::). The block will be the size of the string and
- null-terminator. The address of the block in stored to *PP. The
- return value is the number of characters produced, excluding the
- null-terminator.
-
- Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
- if there's no more memory available, it lets the current allocation
- function handle that.
-
- -- Function: int gmp_obstack_printf (struct obstack *OB, const char
- *FMT, ...)
- -- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
- *FMT, va_list AP)
- Append to the current object in OB. The return value is the
- number of characters written. A null-terminator is not written.
-
- FMT cannot be within the current object in OB, since that object
- might move as it grows.
-
- These functions are available only when the C library provides the
- obstack feature, which probably means only on GNU systems, see
- *Note Obstacks: (libc)Obstacks.
-
-\1f
-File: gmp.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
-
-10.3 C++ Formatted Output
-=========================
-
-The following functions are provided in `libgmpxx' (*note Headers and
-Libraries::), which is built if C++ support is enabled (*note Build
-Options::). Prototypes are available from `<gmp.h>'.
-
- -- Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- In hex or octal, OP is printed as a signed number, the same as for
- decimal. This is unlike the standard `operator<<' routines on
- `int' etc, which instead give twos complement.
-
- -- Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- Output will be a fraction like `5/9', or if the denominator is 1
- then just a plain integer like `123'.
-
- In hex or octal, OP is printed as a signed value, the same as for
- decimal. If `ios::showbase' is set then a base indicator is shown
- on both the numerator and denominator (if the denominator is
- required).
-
- -- Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- The decimal point follows the standard library float `operator<<',
- which on recent systems means the `std::locale' imbued on STREAM.
-
- Hex and octal are supported, unlike the standard `operator<<' on
- `double'. The mantissa will be in hex or octal, the exponent will
- be in decimal. For hex the exponent delimiter is an `@'. This is
- as per `mpf_out_str'.
-
- `ios::showbase' is supported, and will put a base on the mantissa,
- for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
- This last form is slightly strange, but at least differentiates
- itself from decimal.
-
- These operators mean that GMP types can be printed in the usual C++
-way, for example,
-
- mpz_t z;
- int n;
- ...
- cout << "iteration " << n << " value " << z << "\n";
-
- But note that `ostream' output (and `istream' input, *note C++
-Formatted Input::) is the only overloading available for the GMP types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *Note C++ Class Interface::.
-
-\1f
-File: gmp.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
-
-11 Formatted Input
-******************
-
-* Menu:
-
-* Formatted Input Strings::
-* Formatted Input Functions::
-* C++ Formatted Input::
-
-\1f
-File: gmp.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input
-
-11.1 Formatted Input Strings
-============================
-
-`gmp_scanf' and friends accept format strings similar to the standard C
-`scanf' (*note Formatted Input: (libc)Formatted Input.). A format
-specification is of the form
-
- % [flags] [width] [type] conv
-
- GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
-respectively. `Z' and `Q' behave like integers. `Q' will read a `/'
-and a denominator, if present. `F' behaves like a float.
-
- GMP variables don't require an `&' when passed to `gmp_scanf', since
-they're already "call-by-reference". For example,
-
- /* to read say "a(5) = 1234" */
- int n;
- mpz_t z;
- gmp_scanf ("a(%d) = %Zd\n", &n, z);
-
- mpq_t q1, q2;
- gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
-
- /* to read say "topleft (1.55,-2.66)" */
- mpf_t x, y;
- char buf[32];
- gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
-
- All the standard C `scanf' types behave the same as in the C library
-`scanf', and can be freely intermixed with the GMP extensions. In the
-current implementation the standard parts of the format string are
-simply handed to `scanf' and only the GMP extensions handled directly.
-
- The flags accepted are as follows. `a' and `'' will depend on
-support from the C library, and `'' cannot be used with GMP types.
-
- * read but don't store
- a allocate a buffer (string conversions)
- ' grouped digits, GLIBC style (not GMP
- types)
-
- The standard types accepted are as follows. `h' and `l' are
-portable, the rest will depend on the compiler (or include files) for
-the type and the C library for the input.
-
- h short
- hh char
- j intmax_t or uintmax_t
- l long int, double or wchar_t
- ll long long
- L long double
- q quad_t or u_quad_t
- t ptrdiff_t
- z size_t
-
-The GMP types are
-
- F mpf_t, float conversions
- Q mpq_t, integer conversions
- Z mpz_t, integer conversions
-
- The conversions accepted are as follows. `p' and `[' will depend on
-support from the C library, the rest are standard.
-
- c character or characters
- d decimal integer
- e E f g G float
- i integer with base indicator
- n characters read so far
- o octal integer
- p pointer
- s string of non-whitespace characters
- u decimal integer
- x X hex integer
- [ string of characters in a set
-
- `e', `E', `f', `g' and `G' are identical, they all read either fixed
-point or scientific format, and either upper or lower case `e' for the
-exponent in scientific format.
-
- C99 style hex float format (`printf %a', *note Formatted Output
-Strings::) is always accepted for `mpf_t', but for the standard float
-types it will depend on the C library.
-
- `x' and `X' are identical, both accept both upper and lower case
-hexadecimal.
-
- `o', `u', `x' and `X' all read positive or negative values. For the
-standard C types these are described as "unsigned" conversions, but
-that merely affects certain overflow handling, negatives are still
-allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of
-Integers.). For GMP types there are no overflows, so `d' and `u' are
-identical.
-
- `Q' type reads the numerator and (optional) denominator as given.
-If the value might not be in canonical form then `mpq_canonicalize'
-must be called before using it in any calculations (*note Rational
-Number Functions::).
-
- `Qi' will read a base specification separately for the numerator and
-denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11'
-would be 16/17.
-
- `n' can be used with any of the types above, even the GMP types.
-`*' to suppress assignment is allowed, though in that case it would do
-nothing at all.
-
- Other conversions or types that might be accepted by the C library
-`scanf' cannot be used through `gmp_scanf'.
-
- Whitespace is read and discarded before a field, except for `c' and
-`[' conversions.
-
- For float conversions, the decimal point character (or string)
-expected is taken from the current locale settings on systems which
-provide `localeconv' (*note Locales and Internationalization:
-(libc)Locales.). The C library will normally do the same for standard
-float input.
-
- The format string is only interpreted as plain `char's, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
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-File: gmp.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input
-
-11.2 Formatted Input Functions
-==============================
-
-Each of the following functions is similar to the corresponding C
-library function. The plain `scanf' forms take a variable argument
-list. The `vscanf' forms take an argument pointer, see *Note Variadic
-Functions: (libc)Variadic Functions, or `man 3 va_start'.
-
- It should be emphasised that if a format string is invalid, or the
-arguments don't match what the format specifies, then the behaviour of
-any of these functions will be unpredictable. GCC format string
-checking is not available, since it doesn't recognise the GMP
-extensions.
-
- No overlap is permitted between the FMT string and any of the results
-produced.
-
- -- Function: int gmp_scanf (const char *FMT, ...)
- -- Function: int gmp_vscanf (const char *FMT, va_list AP)
- Read from the standard input `stdin'.
-
- -- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
- -- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
- Read from the stream FP.
-
- -- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
- -- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
- AP)
- Read from a null-terminated string S.
-
- The return value from each of these functions is the same as the
-standard C99 `scanf', namely the number of fields successfully parsed
-and stored. `%n' fields and fields read but suppressed by `*' don't
-count towards the return value.
-
- If end of input (or a file error) is reached before a character for
-a field or a literal, and if no previous non-suppressed fields have
-matched, then the return value is `EOF' instead of 0. A whitespace
-character in the format string is only an optional match and doesn't
-induce an `EOF' in this fashion. Leading whitespace read and discarded
-for a field don't count as characters for that field.
-
- For the GMP types, input parsing follows C99 rules, namely one
-character of lookahead is used and characters are read while they
-continue to meet the format requirements. If this doesn't provide a
-complete number then the function terminates, with that field not
-stored nor counted towards the return value. For instance with `mpf_t'
-an input `1.23e-XYZ' would be read up to the `X' and that character
-pushed back since it's not a digit. The string `1.23e-' would then be
-considered invalid since an `e' must be followed by at least one digit.
-
- For the standard C types, in the current implementation GMP calls
-the C library `scanf' functions, which might have looser rules about
-what constitutes a valid input.
-
- Note that `gmp_sscanf' is the same as `gmp_fscanf' and only does one
-character of lookahead when parsing. Although clearly it could look at
-its entire input, it is deliberately made identical to `gmp_fscanf',
-the same way C99 `sscanf' is the same as `fscanf'.
-
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-File: gmp.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
-
-11.3 C++ Formatted Input
-========================
-
-The following functions are provided in `libgmpxx' (*note Headers and
-Libraries::), which is built only if C++ support is enabled (*note
-Build Options::). Prototypes are available from `<gmp.h>'.
-
- -- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
-
- -- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
- An integer like `123' will be read, or a fraction like `5/9'. No
- whitespace is allowed around the `/'. If the fraction is not in
- canonical form then `mpq_canonicalize' must be called (*note
- Rational Number Functions::) before operating on it.
-
- As per integer input, an `0' or `0x' base indicator is read when
- none of `ios::dec', `ios::oct' or `ios::hex' are set. This is
- done separately for numerator and denominator, so that for instance
- `0x10/11' is 16/11 and `0x10/0x11' is 16/17.
-
- -- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
-
- Hex or octal floats are not supported, but might be in the future,
- or perhaps it's best to accept only what the standard float
- `operator>>' does.
-
- Note that digit grouping specified by the `istream' locale is
-currently not accepted. Perhaps this will change in the future.
-
-
- These operators mean that GMP types can be read in the usual C++
-way, for example,
-
- mpz_t z;
- ...
- cin >> z;
-
- But note that `istream' input (and `ostream' output, *note C++
-Formatted Output::) is the only overloading available for the GMP types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *Note C++ Class Interface::.
-
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-File: gmp.info, Node: C++ Class Interface, Next: BSD Compatible Functions, Prev: Formatted Input, Up: Top
-
-12 C++ Class Interface
-**********************
-
-This chapter describes the C++ class based interface to GMP.
-
- All GMP C language types and functions can be used in C++ programs,
-since `gmp.h' has `extern "C"' qualifiers, but the class interface
-offers overloaded functions and operators which may be more convenient.
-
- Due to the implementation of this interface, a reasonably recent C++
-compiler is required, one supporting namespaces, partial specialization
-of templates and member templates. For GCC this means version 2.91 or
-later.
-
- *Everything described in this chapter is to be considered preliminary
-and might be subject to incompatible changes if some unforeseen
-difficulty reveals itself.*
-
-* Menu:
-
-* C++ Interface General::
-* C++ Interface Integers::
-* C++ Interface Rationals::
-* C++ Interface Floats::
-* C++ Interface Random Numbers::
-* C++ Interface Limitations::
-
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-File: gmp.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface
-
-12.1 C++ Interface General
-==========================
-
-All the C++ classes and functions are available with
-
- #include <gmpxx.h>
-
- Programs should be linked with the `libgmpxx' and `libgmp'
-libraries. For example,
-
- g++ mycxxprog.cc -lgmpxx -lgmp
-
-The classes defined are
-
- -- Class: mpz_class
- -- Class: mpq_class
- -- Class: mpf_class
-
- The standard operators and various standard functions are overloaded
-to allow arithmetic with these classes. For example,
-
- int
- main (void)
- {
- mpz_class a, b, c;
-
- a = 1234;
- b = "-5678";
- c = a+b;
- cout << "sum is " << c << "\n";
- cout << "absolute value is " << abs(c) << "\n";
-
- return 0;
- }
-
- An important feature of the implementation is that an expression like
-`a=b+c' results in a single call to the corresponding `mpz_add',
-without using a temporary for the `b+c' part. Expressions which by
-their nature imply intermediate values, like `a=b*c+d*e', still use
-temporaries though.
-
- The classes can be freely intermixed in expressions, as can the
-classes and the standard types `long', `unsigned long' and `double'.
-Smaller types like `int' or `float' can also be intermixed, since C++
-will promote them.
-
- Note that `bool' is not accepted directly, but must be explicitly
-cast to an `int' first. This is because C++ will automatically convert
-any pointer to a `bool', so if GMP accepted `bool' it would make all
-sorts of invalid class and pointer combinations compile but almost
-certainly not do anything sensible.
-
- Conversions back from the classes to standard C++ types aren't done
-automatically, instead member functions like `get_si' are provided (see
-the following sections for details).
-
- Also there are no automatic conversions from the classes to the
-corresponding GMP C types, instead a reference to the underlying C
-object can be obtained with the following functions,
-
- -- Function: mpz_t mpz_class::get_mpz_t ()
- -- Function: mpq_t mpq_class::get_mpq_t ()
- -- Function: mpf_t mpf_class::get_mpf_t ()
-
- These can be used to call a C function which doesn't have a C++ class
-interface. For example to set `a' to the GCD of `b' and `c',
-
- mpz_class a, b, c;
- ...
- mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
-
- In the other direction, a class can be initialized from the
-corresponding GMP C type, or assigned to if an explicit constructor is
-used. In both cases this makes a copy of the value, it doesn't create
-any sort of association. For example,
-
- mpz_t z;
- // ... init and calculate z ...
- mpz_class x(z);
- mpz_class y;
- y = mpz_class (z);
-
- There are no namespace setups in `gmpxx.h', all types and functions
-are simply put into the global namespace. This is what `gmp.h' has
-done in the past, and continues to do for compatibility. The extras
-provided by `gmpxx.h' follow GMP naming conventions and are unlikely to
-clash with anything.
-
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-File: gmp.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface
-
-12.2 C++ Interface Integers
-===========================
-
- -- Function: void mpz_class::mpz_class (type N)
- Construct an `mpz_class'. All the standard C++ types may be used,
- except `long long' and `long double', and all the GMP C++ classes
- can be used. Any necessary conversion follows the corresponding C
- function, for example `double' follows `mpz_set_d' (*note
- Assigning Integers::).
-
- -- Function: void mpz_class::mpz_class (mpz_t Z)
- Construct an `mpz_class' from an `mpz_t'. The value in Z is
- copied into the new `mpz_class', there won't be any permanent
- association between it and Z.
-
- -- Function: void mpz_class::mpz_class (const char *S)
- -- Function: void mpz_class::mpz_class (const char *S, int BASE = 0)
- -- Function: void mpz_class::mpz_class (const string& S)
- -- Function: void mpz_class::mpz_class (const string& S, int BASE = 0)
- Construct an `mpz_class' converted from a string using
- `mpz_set_str' (*note Assigning Integers::).
-
- If the string is not a valid integer, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: mpz_class operator/ (mpz_class A, mpz_class D)
- -- Function: mpz_class operator% (mpz_class A, mpz_class D)
- Divisions involving `mpz_class' round towards zero, as per the
- `mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::).
- This is the same as the C99 `/' and `%' operators.
-
- The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called
- directly if desired. For example,
-
- mpz_class q, a, d;
- ...
- mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
-
- -- Function: mpz_class abs (mpz_class OP1)
- -- Function: int cmp (mpz_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpz_class OP2)
- -- Function: bool mpz_class::fits_sint_p (void)
- -- Function: bool mpz_class::fits_slong_p (void)
- -- Function: bool mpz_class::fits_sshort_p (void)
- -- Function: bool mpz_class::fits_uint_p (void)
- -- Function: bool mpz_class::fits_ulong_p (void)
- -- Function: bool mpz_class::fits_ushort_p (void)
- -- Function: double mpz_class::get_d (void)
- -- Function: long mpz_class::get_si (void)
- -- Function: string mpz_class::get_str (int BASE = 10)
- -- Function: unsigned long mpz_class::get_ui (void)
- -- Function: int mpz_class::set_str (const char *STR, int BASE)
- -- Function: int mpz_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpz_class OP)
- -- Function: mpz_class sqrt (mpz_class OP)
- These functions provide a C++ class interface to the corresponding
- GMP C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
-
-
- Overloaded operators for combinations of `mpz_class' and `double'
-are provided for completeness, but it should be noted that if the given
-`double' is not an integer then the way any rounding is done is
-currently unspecified. The rounding might take place at the start, in
-the middle, or at the end of the operation, and it might change in the
-future.
-
- Conversions between `mpz_class' and `double', however, are defined
-to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
-And comparisons are always made exactly, as per `mpz_cmp_d'.
-
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-File: gmp.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface
-
-12.3 C++ Interface Rationals
-============================
-
-In all the following constructors, if a fraction is given then it
-should be in canonical form, or if not then `mpq_class::canonicalize'
-called.
-
- -- Function: void mpq_class::mpq_class (type OP)
- -- Function: void mpq_class::mpq_class (integer NUM, integer DEN)
- Construct an `mpq_class'. The initial value can be a single value
- of any type, or a pair of integers (`mpz_class' or standard C++
- integer types) representing a fraction, except that `long long'
- and `long double' are not supported. For example,
-
- mpq_class q (99);
- mpq_class q (1.75);
- mpq_class q (1, 3);
-
- -- Function: void mpq_class::mpq_class (mpq_t Q)
- Construct an `mpq_class' from an `mpq_t'. The value in Q is
- copied into the new `mpq_class', there won't be any permanent
- association between it and Q.
-
- -- Function: void mpq_class::mpq_class (const char *S)
- -- Function: void mpq_class::mpq_class (const char *S, int BASE = 0)
- -- Function: void mpq_class::mpq_class (const string& S)
- -- Function: void mpq_class::mpq_class (const string& S, int BASE = 0)
- Construct an `mpq_class' converted from a string using
- `mpq_set_str' (*note Initializing Rationals::).
-
- If the string is not a valid rational, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: void mpq_class::canonicalize ()
- Put an `mpq_class' into canonical form, as per *Note Rational
- Number Functions::. All arithmetic operators require their
- operands in canonical form, and will return results in canonical
- form.
-
- -- Function: mpq_class abs (mpq_class OP)
- -- Function: int cmp (mpq_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpq_class OP2)
- -- Function: double mpq_class::get_d (void)
- -- Function: string mpq_class::get_str (int BASE = 10)
- -- Function: int mpq_class::set_str (const char *STR, int BASE)
- -- Function: int mpq_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpq_class OP)
- These functions provide a C++ class interface to the corresponding
- GMP C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
-
- -- Function: mpz_class& mpq_class::get_num ()
- -- Function: mpz_class& mpq_class::get_den ()
- Get a reference to an `mpz_class' which is the numerator or
- denominator of an `mpq_class'. This can be used both for read and
- write access. If the object returned is modified, it modifies the
- original `mpq_class'.
-
- If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
-
- -- Function: mpz_t mpq_class::get_num_mpz_t ()
- -- Function: mpz_t mpq_class::get_den_mpz_t ()
- Get a reference to the underlying `mpz_t' numerator or denominator
- of an `mpq_class'. This can be passed to C functions expecting an
- `mpz_t'. Any modifications made to the `mpz_t' will modify the
- original `mpq_class'.
-
- If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
-
- -- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
- Read ROP from STREAM, using its `ios' formatting settings, the
- same as `mpq_t operator>>' (*note C++ Formatted Input::).
-
- If the ROP read might not be in canonical form then
- `mpq_class::canonicalize' must be called.
-
-\1f
-File: gmp.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
-
-12.4 C++ Interface Floats
-=========================
-
-When an expression requires the use of temporary intermediate
-`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
-same precision as the destination `f'. Explicit constructors can be
-used if this doesn't suit.
-
- -- Function: mpf_class::mpf_class (type OP)
- -- Function: mpf_class::mpf_class (type OP, unsigned long PREC)
- Construct an `mpf_class'. Any standard C++ type can be used,
- except `long long' and `long double', and any of the GMP C++
- classes can be used.
-
- If PREC is given, the initial precision is that value, in bits. If
- PREC is not given, then the initial precision is determined by the
- type of OP given. An `mpz_class', `mpq_class', or C++ builtin
- type will give the default `mpf' precision (*note Initializing
- Floats::). An `mpf_class' or expression will give the precision
- of that value. The precision of a binary expression is the higher
- of the two operands.
-
- mpf_class f(1.5); // default precision
- mpf_class f(1.5, 500); // 500 bits (at least)
- mpf_class f(x); // precision of x
- mpf_class f(abs(x)); // precision of x
- mpf_class f(-g, 1000); // 1000 bits (at least)
- mpf_class f(x+y); // greater of precisions of x and y
-
- -- Function: void mpf_class::mpf_class (const char *S)
- -- Function: void mpf_class::mpf_class (const char *S, unsigned long
- PREC, int BASE = 0)
- -- Function: void mpf_class::mpf_class (const string& S)
- -- Function: void mpf_class::mpf_class (const string& S, unsigned long
- PREC, int BASE = 0)
- Construct an `mpf_class' converted from a string using
- `mpf_set_str' (*note Assigning Floats::). If PREC is given, the
- initial precision is that value, in bits. If not, the default
- `mpf' precision (*note Initializing Floats::) is used.
-
- If the string is not a valid float, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: mpf_class& mpf_class::operator= (type OP)
- Convert and store the given OP value to an `mpf_class' object. The
- same types are accepted as for the constructors above.
-
- Note that `operator=' only stores a new value, it doesn't copy or
- change the precision of the destination, instead the value is
- truncated if necessary. This is the same as `mpf_set' etc. Note
- in particular this means for `mpf_class' a copy constructor is not
- the same as a default constructor plus assignment.
-
- mpf_class x (y); // x created with precision of y
-
- mpf_class x; // x created with default precision
- x = y; // value truncated to that precision
-
- Applications using templated code may need to be careful about the
- assumptions the code makes in this area, when working with
- `mpf_class' values of various different or non-default precisions.
- For instance implementations of the standard `complex' template
- have been seen in both styles above, though of course `complex' is
- normally only actually specified for use with the builtin float
- types.
-
- -- Function: mpf_class abs (mpf_class OP)
- -- Function: mpf_class ceil (mpf_class OP)
- -- Function: int cmp (mpf_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpf_class OP2)
- -- Function: bool mpf_class::fits_sint_p (void)
- -- Function: bool mpf_class::fits_slong_p (void)
- -- Function: bool mpf_class::fits_sshort_p (void)
- -- Function: bool mpf_class::fits_uint_p (void)
- -- Function: bool mpf_class::fits_ulong_p (void)
- -- Function: bool mpf_class::fits_ushort_p (void)
- -- Function: mpf_class floor (mpf_class OP)
- -- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
- -- Function: double mpf_class::get_d (void)
- -- Function: long mpf_class::get_si (void)
- -- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
- size_t DIGITS = 0)
- -- Function: unsigned long mpf_class::get_ui (void)
- -- Function: int mpf_class::set_str (const char *STR, int BASE)
- -- Function: int mpf_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpf_class OP)
- -- Function: mpf_class sqrt (mpf_class OP)
- -- Function: mpf_class trunc (mpf_class OP)
- These functions provide a C++ class interface to the corresponding
- GMP C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
-
- The accuracy provided by `hypot' is not currently guaranteed.
-
- -- Function: mp_bitcnt_t mpf_class::get_prec ()
- -- Function: void mpf_class::set_prec (mp_bitcnt_t PREC)
- -- Function: void mpf_class::set_prec_raw (mp_bitcnt_t PREC)
- Get or set the current precision of an `mpf_class'.
-
- The restrictions described for `mpf_set_prec_raw' (*note
- Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note
- in particular that the `mpf_class' must be restored to it's
- allocated precision before being destroyed. This must be done by
- application code, there's no automatic mechanism for it.
-
-\1f
-File: gmp.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
-
-12.5 C++ Interface Random Numbers
-=================================
-
- -- Class: gmp_randclass
- The C++ class interface to the GMP random number functions uses
- `gmp_randclass' to hold an algorithm selection and current state,
- as per `gmp_randstate_t'.
-
- -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
- (gmp_randstate_t, ...), ...)
- Construct a `gmp_randclass', using a call to the given RANDINIT
- function (*note Random State Initialization::). The arguments
- expected are the same as RANDINIT, but with `mpz_class' instead of
- `mpz_t'. For example,
-
- gmp_randclass r1 (gmp_randinit_default);
- gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
- gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
- gmp_randclass r4 (gmp_randinit_mt);
-
- `gmp_randinit_lc_2exp_size' will fail if the size requested is too
- big, an `std::length_error' exception is thrown in that case.
-
- -- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
- Construct a `gmp_randclass' using the same parameters as
- `gmp_randinit' (*note Random State Initialization::). This
- function is obsolete and the above RANDINIT style should be
- preferred.
-
- -- Function: void gmp_randclass::seed (unsigned long int S)
- -- Function: void gmp_randclass::seed (mpz_class S)
- Seed a random number generator. See *note Random Number
- Functions::, for how to choose a good seed.
-
- -- Function: mpz_class gmp_randclass::get_z_bits (unsigned long BITS)
- -- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
- Generate a random integer with a specified number of bits.
-
- -- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
- Generate a random integer in the range 0 to N-1 inclusive.
-
- -- Function: mpf_class gmp_randclass::get_f ()
- -- Function: mpf_class gmp_randclass::get_f (unsigned long PREC)
- Generate a random float F in the range 0 <= F < 1. F will be to
- PREC bits precision, or if PREC is not given then to the precision
- of the destination. For example,
-
- gmp_randclass r;
- ...
- mpf_class f (0, 512); // 512 bits precision
- f = r.get_f(); // random number, 512 bits
-
-\1f
-File: gmp.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface
-
-12.6 C++ Interface Limitations
-==============================
-
-`mpq_class' and Templated Reading
- A generic piece of template code probably won't know that
- `mpq_class' requires a `canonicalize' call if inputs read with
- `operator>>' might be non-canonical. This can lead to incorrect
- results.
-
- `operator>>' behaves as it does for reasons of efficiency. A
- canonicalize can be quite time consuming on large operands, and is
- best avoided if it's not necessary.
-
- But this potential difficulty reduces the usefulness of
- `mpq_class'. Perhaps a mechanism to tell `operator>>' what to do
- will be adopted in the future, maybe a preprocessor define, a
- global flag, or an `ios' flag pressed into service. Or maybe, at
- the risk of inconsistency, the `mpq_class' `operator>>' could
- canonicalize and leave `mpq_t' `operator>>' not doing so, for use
- on those occasions when that's acceptable. Send feedback or
- alternate ideas to <gmp-bugs@gmplib.org>.
-
-Subclassing
- Subclassing the GMP C++ classes works, but is not currently
- recommended.
-
- Expressions involving subclasses resolve correctly (or seem to),
- but in normal C++ fashion the subclass doesn't inherit
- constructors and assignments. There's many of those in the GMP
- classes, and a good way to reestablish them in a subclass is not
- yet provided.
-
-Templated Expressions
- A subtle difficulty exists when using expressions together with
- application-defined template functions. Consider the following,
- with `T' intended to be some numeric type,
-
- template <class T>
- T fun (const T &, const T &);
-
- When used with, say, plain `mpz_class' variables, it works fine:
- `T' is resolved as `mpz_class'.
-
- mpz_class f(1), g(2);
- fun (f, g); // Good
-
- But when one of the arguments is an expression, it doesn't work.
-
- mpz_class f(1), g(2), h(3);
- fun (f, g+h); // Bad
-
- This is because `g+h' ends up being a certain expression template
- type internal to `gmpxx.h', which the C++ template resolution
- rules are unable to automatically convert to `mpz_class'. The
- workaround is simply to add an explicit cast.
-
- mpz_class f(1), g(2), h(3);
- fun (f, mpz_class(g+h)); // Good
-
- Similarly, within `fun' it may be necessary to cast an expression
- to type `T' when calling a templated `fun2'.
-
- template <class T>
- void fun (T f, T g)
- {
- fun2 (f, f+g); // Bad
- }
-
- template <class T>
- void fun (T f, T g)
- {
- fun2 (f, T(f+g)); // Good
- }
-
-\1f
-File: gmp.info, Node: BSD Compatible Functions, Next: Custom Allocation, Prev: C++ Class Interface, Up: Top
-
-13 Berkeley MP Compatible Functions
-***********************************
-
-These functions are intended to be fully compatible with the Berkeley MP
-library which is available on many BSD derived U*ix systems. The
-`--enable-mpbsd' option must be used when building GNU MP to make these
-available (*note Installing GMP::).
-
- The original Berkeley MP library has a usage restriction: you cannot
-use the same variable as both source and destination in a single
-function call. The compatible functions in GNU MP do not share this
-restriction--inputs and outputs may overlap.
-
- It is not recommended that new programs are written using these
-functions. Apart from the incomplete set of functions, the interface
-for initializing `MINT' objects is more error prone, and the `pow'
-function collides with `pow' in `libm.a'.
-
- Include the header `mp.h' to get the definition of the necessary
-types and functions. If you are on a BSD derived system, make sure to
-include GNU `mp.h' if you are going to link the GNU `libmp.a' to your
-program. This means that you probably need to give the `-I<dir>'
-option to the compiler, where `<dir>' is the directory where you have
-GNU `mp.h'.
-
- -- Function: MINT * itom (signed short int INITIAL_VALUE)
- Allocate an integer consisting of a `MINT' object and dynamic limb
- space. Initialize the integer to INITIAL_VALUE. Return a pointer
- to the `MINT' object.
-
- -- Function: MINT * xtom (char *INITIAL_VALUE)
- Allocate an integer consisting of a `MINT' object and dynamic limb
- space. Initialize the integer from INITIAL_VALUE, a hexadecimal,
- null-terminated C string. Return a pointer to the `MINT' object.
-
- -- Function: void move (MINT *SRC, MINT *DEST)
- Set DEST to SRC by copying. Both variables must be previously
- initialized.
-
- -- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Add SRC_1 and SRC_2 and put the sum in DESTINATION.
-
- -- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.
-
- -- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Multiply SRC_1 and SRC_2 and put the product in DESTINATION.
-
- -- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,
- MINT *REMAINDER)
- -- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT
- *QUOTIENT, signed short int *REMAINDER)
- Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod
- DIVISOR. The quotient is rounded towards zero; the remainder has
- the same sign as the dividend unless it is zero.
-
- Some implementations of these functions work differently--or not
- at all--for negative arguments.
-
- -- Function: void msqrt (MINT *OP, MINT *ROOT, MINT *REMAINDER)
- Set ROOT to the truncated integer part of the square root of OP,
- like `mpz_sqrt'. Set REMAINDER to OP-ROOT*ROOT, i.e. zero if OP
- is a perfect square.
-
- If ROOT and REMAINDER are the same variable, the results are
- undefined.
-
- -- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
- Set DEST to (BASE raised to EXP) modulo MOD.
-
- Note that the name `pow' clashes with `pow' from the standard C
- math library (*note Exponentiation and Logarithms: (libc)Exponents
- and Logarithms.). An application will only be able to use one or
- the other.
-
- -- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
- Set DEST to BASE raised to EXP.
-
- -- Function: void gcd (MINT *OP1, MINT *OP2, MINT *RES)
- Set RES to the greatest common divisor of OP1 and OP2.
-
- -- Function: int mcmp (MINT *OP1, MINT *OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- -- Function: void min (MINT *DEST)
- Input a decimal string from `stdin', and put the read integer in
- DEST. SPC and TAB are allowed in the number string, and are
- ignored.
-
- -- Function: void mout (MINT *SRC)
- Output SRC to `stdout', as a decimal string. Also output a
- newline.
-
- -- Function: char * mtox (MINT *OP)
- Convert OP to a hexadecimal string, and return a pointer to the
- string. The returned string is allocated using the default memory
- allocation function, `malloc' by default. It will be
- `strlen(str)+1' bytes, that being exactly enough for the string
- and null-terminator.
-
- -- Function: void mfree (MINT *OP)
- De-allocate, the space used by OP. *This function should only be
- passed a value returned by `itom' or `xtom'.*
-
-\1f
-File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: BSD Compatible Functions, Up: Top
-
-14 Custom Allocation
-********************
-
-By default GMP uses `malloc', `realloc' and `free' for memory
-allocation, and if they fail GMP prints a message to the standard error
-output and terminates the program.
-
- Alternate functions can be specified, to allocate memory in a
-different way or to have a different error action on running out of
-memory.
-
- This feature is available in the Berkeley compatibility library
-(*note BSD Compatible Functions::) as well as the main GMP library.
-
- -- Function: void mp_set_memory_functions (
- void *(*ALLOC_FUNC_PTR) (size_t),
- void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
- void (*FREE_FUNC_PTR) (void *, size_t))
- Replace the current allocation functions from the arguments. If
- an argument is `NULL', the corresponding default function is used.
-
- These functions will be used for all memory allocation done by
- GMP, apart from temporary space from `alloca' if that function is
- available and GMP is configured to use it (*note Build Options::).
-
- *Be sure to call `mp_set_memory_functions' only when there are no
- active GMP objects allocated using the previous memory functions!
- Usually that means calling it before any other GMP function.*
-
- The functions supplied should fit the following declarations:
-
- -- Function: void * allocate_function (size_t ALLOC_SIZE)
- Return a pointer to newly allocated space with at least ALLOC_SIZE
- bytes.
-
- -- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
- size_t NEW_SIZE)
- Resize a previously allocated block PTR of OLD_SIZE bytes to be
- NEW_SIZE bytes.
-
- The block may be moved if necessary or if desired, and in that
- case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
- the new location. The return value is a pointer to the resized
- block, that being the new location if moved or just PTR if not.
-
- PTR is never `NULL', it's always a previously allocated block.
- NEW_SIZE may be bigger or smaller than OLD_SIZE.
-
- -- Function: void free_function (void *PTR, size_t SIZE)
- De-allocate the space pointed to by PTR.
-
- PTR is never `NULL', it's always a previously allocated block of
- SIZE bytes.
-
- A "byte" here means the unit used by the `sizeof' operator.
-
- The OLD_SIZE parameters to REALLOCATE_FUNCTION and FREE_FUNCTION are
-passed for convenience, but of course can be ignored if not needed.
-The default functions using `malloc' and friends for instance don't use
-them.
-
- No error return is allowed from any of these functions, if they
-return then they must have performed the specified operation. In
-particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
-return `NULL'.
-
- Getting a different fatal error action is a good use for custom
-allocation functions, for example giving a graphical dialog rather than
-the default print to `stderr'. How much is possible when genuinely out
-of memory is another question though.
-
- There's currently no defined way for the allocation functions to
-recover from an error such as out of memory, they must terminate
-program execution. A `longjmp' or throwing a C++ exception will have
-undefined results. This may change in the future.
-
- GMP may use allocated blocks to hold pointers to other allocated
-blocks. This will limit the assumptions a conservative garbage
-collection scheme can make.
-
- Since the default GMP allocation uses `malloc' and friends, those
-functions will be linked in even if the first thing a program does is an
-`mp_set_memory_functions'. It's necessary to change the GMP sources if
-this is a problem.
-
-
- -- Function: void mp_get_memory_functions (
- void *(**ALLOC_FUNC_PTR) (size_t),
- void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t),
- void (**FREE_FUNC_PTR) (void *, size_t))
- Get the current allocation functions, storing function pointers to
- the locations given by the arguments. If an argument is `NULL',
- that function pointer is not stored.
-
- For example, to get just the current free function,
-
- void (*freefunc) (void *, size_t);
-
- mp_get_memory_functions (NULL, NULL, &freefunc);
-
-\1f
-File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
-
-15 Language Bindings
-********************
-
-The following packages and projects offer access to GMP from languages
-other than C, though perhaps with varying levels of functionality and
-efficiency.
-
-
-C++
- * GMP C++ class interface, *note C++ Class Interface::
- Straightforward interface, expression templates to eliminate
- temporaries.
-
- * ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
- Linear algebra and polynomials using templates.
-
- * Arithmos `http://www.win.ua.ac.be/~cant/arithmos/'
- Rationals with infinities and square roots.
-
- * CLN `http://www.ginac.de/CLN/'
- High level classes for arithmetic.
-
- * LiDIA `http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/'
- A C++ library for computational number theory.
-
- * Linbox `http://www.linalg.org/'
- Sparse vectors and matrices.
-
- * NTL `http://www.shoup.net/ntl/'
- A C++ number theory library.
-
-Fortran
- * Omni F77 `http://phase.hpcc.jp/Omni/home.html'
- Arbitrary precision floats.
-
-Haskell
- * Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
-
-Java
- * Kaffe `http://www.kaffe.org/'
-
- * Kissme `http://kissme.sourceforge.net/'
-
-Lisp
- * GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html'
-
- * Librep `http://librep.sourceforge.net/'
-
- * XEmacs (21.5.18 beta and up) `http://www.xemacs.org'
- Optional big integers, rationals and floats using GMP.
-
-M4
- * GNU m4 betas `http://www.seindal.dk/rene/gnu/'
- Optionally provides an arbitrary precision `mpeval'.
-
-ML
- * MLton compiler `http://mlton.org/'
-
-Objective Caml
- * MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en'
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
- Optionally using GMP.
-
-Oz
- * Mozart `http://www.mozart-oz.org/'
-
-Pascal
- * GNU Pascal Compiler `http://www.gnu-pascal.de/'
- GMP unit.
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
- For Free Pascal, optionally using GMP.
-
-Perl
- * GMP module, see `demos/perl' in the GMP sources (*note
- Demonstration Programs::).
-
- * Math::GMP `http://www.cpan.org/'
- Compatible with Math::BigInt, but not as many functions as
- the GMP module above.
-
- * Math::BigInt::GMP `http://www.cpan.org/'
- Plug Math::GMP into normal Math::BigInt operations.
-
-Pike
- * mpz module in the standard distribution,
- `http://pike.ida.liu.se/'
-
-Prolog
- * SWI Prolog `http://www.swi-prolog.org/'
- Arbitrary precision floats.
-
-Python
- * mpz module in the standard distribution,
- `http://www.python.org/'
-
- * GMPY `http://gmpy.sourceforge.net/'
-
-Scheme
- * GNU Guile (upcoming 1.8)
- `http://www.gnu.org/software/guile/guile.html'
-
- * RScheme `http://www.rscheme.org/'
-
- * STklos `http://www.stklos.org/'
-
-Smalltalk
- * GNU Smalltalk
- `http://www.smalltalk.org/versions/GNUSmalltalk.html'
-
-Other
- * Axiom `http://savannah.nongnu.org/projects/axiom'
- Computer algebra using GCL.
-
- * DrGenius `http://drgenius.seul.org/'
- Geometry system and mathematical programming language.
-
- * GiNaC `http://www.ginac.de/'
- C++ computer algebra using CLN.
-
- * GOO `http://www.googoogaga.org/'
- Dynamic object oriented language.
-
- * Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
- Macsyma computer algebra using GCL.
-
- * Q `http://q-lang.sourceforge.net/'
- Equational programming system.
-
- * Regina `http://regina.sourceforge.net/'
- Topological calculator.
-
- * Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
- Yet another computer algebra system.
-
-
-\1f
-File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
-
-16 Algorithms
-*************
-
-This chapter is an introduction to some of the algorithms used for
-various GMP operations. The code is likely to be hard to understand
-without knowing something about the algorithms.
-
- Some GMP internals are mentioned, but applications that expect to be
-compatible with future GMP releases should take care to use only the
-documented functions.
-
-* Menu:
-
-* Multiplication Algorithms::
-* Division Algorithms::
-* Greatest Common Divisor Algorithms::
-* Powering Algorithms::
-* Root Extraction Algorithms::
-* Radix Conversion Algorithms::
-* Other Algorithms::
-* Assembly Coding::
-
-\1f
-File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
-
-16.1 Multiplication
-===================
-
-NxN limb multiplications and squares are done using one of five
-algorithms, as the size N increases.
-
- Algorithm Threshold
- Basecase (none)
- Karatsuba `MUL_TOOM22_THRESHOLD'
- Toom-3 `MUL_TOOM33_THRESHOLD'
- Toom-4 `MUL_TOOM44_THRESHOLD'
- FFT `MUL_FFT_THRESHOLD'
-
- Similarly for squaring, with the `SQR' thresholds.
-
- NxM multiplications of operands with different sizes above
-`MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired
-algorithms or directly with FFT, depending on operand size (*note
-Unbalanced Multiplication::).
-
-* Menu:
-
-* Basecase Multiplication::
-* Karatsuba Multiplication::
-* Toom 3-Way Multiplication::
-* Toom 4-Way Multiplication::
-* FFT Multiplication::
-* Other Multiplication::
-* Unbalanced Multiplication::
-
-\1f
-File: gmp.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
-
-16.1.1 Basecase Multiplication
-------------------------------
-
-Basecase NxM multiplication is a straightforward rectangular set of
-cross-products, the same as long multiplication done by hand and for
-that reason sometimes known as the schoolbook or grammar school method.
-This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M
-(*note References::), and the `mpn/generic/mul_basecase.c' code.
-
- Assembly implementations of `mpn_mul_basecase' are essentially the
-same as the generic C code, but have all the usual assembly tricks and
-obscurities introduced for speed.
-
- A square can be done in roughly half the time of a multiply, by
-using the fact that the cross products above and below the diagonal are
-the same. A triangle of products below the diagonal is formed, doubled
-(left shift by one bit), and then the products on the diagonal added.
-This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembly
-implementations take essentially the same approach.
-
- u0 u1 u2 u3 u4
- +---+---+---+---+---+
- u0 | d | | | | |
- +---+---+---+---+---+
- u1 | | d | | | |
- +---+---+---+---+---+
- u2 | | | d | | |
- +---+---+---+---+---+
- u3 | | | | d | |
- +---+---+---+---+---+
- u4 | | | | | d |
- +---+---+---+---+---+
-
- In practice squaring isn't a full 2x faster than multiplying, it's
-usually around 1.5x. Less than 1.5x probably indicates
-`mpn_sqr_basecase' wants improving on that CPU.
-
- On some CPUs `mpn_mul_basecase' can be faster than the generic C
-`mpn_sqr_basecase' on some small sizes. `SQR_BASECASE_THRESHOLD' is
-the size at which to use `mpn_sqr_basecase', this will be zero if that
-routine should be used always.
-
-\1f
-File: gmp.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
-
-16.1.2 Karatsuba Multiplication
--------------------------------
-
-The Karatsuba multiplication algorithm is described in Knuth section
-4.3.3 part A, and various other textbooks. A brief description is
-given here.
-
- The inputs x and y are treated as each split into two parts of equal
-length (or the most significant part one limb shorter if N is odd).
-
- high low
- +----------+----------+
- | x1 | x0 |
- +----------+----------+
-
- +----------+----------+
- | y1 | y0 |
- +----------+----------+
-
- Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
-(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
-y=y1*b+y0, and the following holds,
-
- x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
-
- This formula means doing only three multiplies of (N/2)x(N/2) limbs,
-whereas a basecase multiply of NxN limbs is equivalent to four
-multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
-positions where the three products must be added.
-
- high low
- +--------+--------+ +--------+--------+
- | x1*y1 | | x0*y0 |
- +--------+--------+ +--------+--------+
- +--------+--------+
- add | x1*y1 |
- +--------+--------+
- +--------+--------+
- add | x0*y0 |
- +--------+--------+
- +--------+--------+
- sub | (x1-x0)*(y1-y0) |
- +--------+--------+
-
- The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
-and the sign used to choose to add or subtract. Notice the sum
-high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
-additions, rather than 6*k, but in GMP extra function call overheads
-outweigh the saving.
-
- Squaring is similar to multiplying, but with x=y the formula reduces
-to an equivalent with three squares,
-
- x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
-
- The final result is accumulated from those three squares the same
-way as for the three multiplies above. The middle term (x1-x0)^2 is now
-always positive.
-
- A similar formula for both multiplying and squaring can be
-constructed with a middle term (x1+x0)*(y1+y0). But those sums can
-exceed k limbs, leading to more carry handling and additions than the
-form above.
-
- Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
-the exponent being log(3)/log(2), representing 3 multiplies each 1/2
-the size of the inputs. This is a big improvement over the basecase
-multiply at O(N^2) and the advantage soon overcomes the extra additions
-Karatsuba performs. `MUL_TOOM22_THRESHOLD' can be as little as 10
-limbs. The `SQR' threshold is usually about twice the `MUL'.
-
- The basecase algorithm will take a time of the form M(N) = a*N^2 +
-b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which
-expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4
-for a means per-crossproduct speedups in the basecase code will
-increase the threshold since they benefit M(N) more than K(N). And
-conversely the 3/2 for b means linear style speedups of b will increase
-the threshold since they benefit K(N) more than M(N). The latter can
-be seen for instance when adding an optimized `mpn_sqr_diagonal' to
-`mpn_sqr_basecase'. Of course all speedups reduce total time, and in
-that sense the algorithm thresholds are merely of academic interest.
-
-\1f
-File: gmp.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
-
-16.1.3 Toom 3-Way Multiplication
---------------------------------
-
-The Karatsuba formula is the simplest case of a general approach to
-splitting inputs that leads to both Toom and FFT algorithms. A
-description of Toom can be found in Knuth section 4.3.3, with an
-example 3-way calculation after Theorem A. The 3-way form used in GMP
-is described here.
-
- The operands are each considered split into 3 pieces of equal length
-(or the most significant part 1 or 2 limbs shorter than the other two).
-
- high low
- +----------+----------+----------+
- | x2 | x1 | x0 |
- +----------+----------+----------+
-
- +----------+----------+----------+
- | y2 | y1 | y0 |
- +----------+----------+----------+
-
-These parts are treated as the coefficients of two polynomials
-
- X(t) = x2*t^2 + x1*t + x0
- Y(t) = y2*t^2 + y1*t + y0
-
- Let b equal the power of 2 which is the size of the x0, x1, y0 and
-y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb).
-With this x=X(b) and y=Y(b).
-
- Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
-
- W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
-
- The w[i] are going to be determined, and when they are they'll give
-the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
-coefficients will be roughly b^2 each, and the final W(b) will be an
-addition like,
-
- high low
- +-------+-------+
- | w4 |
- +-------+-------+
- +--------+-------+
- | w3 |
- +--------+-------+
- +--------+-------+
- | w2 |
- +--------+-------+
- +--------+-------+
- | w1 |
- +--------+-------+
- +-------+-------+
- | w0 |
- +-------+-------+
-
- The w[i] coefficients could be formed by a simple set of cross
-products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
-this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
-equivalent merely to a basecase multiply. Instead the following
-approach is used.
-
- X(t) and Y(t) are evaluated and multiplied at 5 points, giving
-values of W(t) at those points. In GMP the following points are used,
-
- Point Value
- t=0 x0 * y0, which gives w0 immediately
- t=1 (x2+x1+x0) * (y2+y1+y0)
- t=-1 (x2-x1+x0) * (y2-y1+y0)
- t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
- t=inf x2 * y2, which gives w4 immediately
-
- At t=-1 the values can be negative and that's handled using the
-absolute values and tracking the sign separately. At t=inf the value
-is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
-it's much easier to think of as simply x2*y2 giving w4 immediately
-(much like x0*y0 at t=0 gives w0 immediately).
-
- Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
-linear combination of the w[i] coefficients, and the value of those
-combinations has just been calculated.
-
- W(0) = w0
- W(1) = w4 + w3 + w2 + w1 + w0
- W(-1) = w4 - w3 + w2 - w1 + w0
- W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
- W(inf) = w4
-
- This is a set of five equations in five unknowns, and some
-elementary linear algebra quickly isolates each w[i]. This involves
-adding or subtracting one W(t) value from another, and a couple of
-divisions by powers of 2 and one division by 3, the latter using the
-special `mpn_divexact_by3' (*note Exact Division::).
-
- The conversion of W(t) values to the coefficients is interpolation.
-A polynomial of degree 4 like W(t) is uniquely determined by values
-known at 5 different points. The points are arbitrary and can be
-chosen to make the linear equations come out with a convenient set of
-steps for quickly isolating the w[i].
-
- Squaring follows the same procedure as multiplication, but there's
-only one X(t) and it's evaluated at the 5 points, and those values
-squared to give values of W(t). The interpolation is then identical,
-and in fact the same `toom3_interpolate' subroutine is used for both
-squaring and multiplying.
-
- Toom-3 is asymptotically O(N^1.465), the exponent being
-log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
-size each. This is an improvement over Karatsuba at O(N^1.585), though
-Toom does more work in the evaluation and interpolation and so it only
-realizes its advantage above a certain size.
-
- Near the crossover between Toom-3 and Karatsuba there's generally a
-range of sizes where the difference between the two is small.
-`MUL_TOOM33_THRESHOLD' is a somewhat arbitrary point in that range and
-successive runs of the tune program can give different values due to
-small variations in measuring. A graph of time versus size for the two
-shows the effect, see `tune/README'.
-
- At the fairly small sizes where the Toom-3 thresholds occur it's
-worth remembering that the asymptotic behaviour for Karatsuba and
-Toom-3 can't be expected to make accurate predictions, due of course to
-the big influence of all sorts of overheads, and the fact that only a
-few recursions of each are being performed. Even at large sizes
-there's a good chance machine dependent effects like cache architecture
-will mean actual performance deviates from what might be predicted.
-
- The formula given for the Karatsuba algorithm (*note Karatsuba
-Multiplication::) has an equivalent for Toom-3 involving only five
-multiplies, but this would be complicated and unenlightening.
-
- An alternate view of Toom-3 can be found in Zuras (*note
-References::), using a vector to represent the x and y splits and a
-matrix multiplication for the evaluation and interpolation stages. The
-matrix inverses are not meant to be actually used, and they have
-elements with values much greater than in fact arise in the
-interpolation steps. The diagram shown for the 3-way is attractive,
-but again doesn't have to be implemented that way and for example with
-a bit of rearrangement just one division by 6 can be done.
-
-\1f
-File: gmp.info, Node: Toom 4-Way Multiplication, Next: FFT Multiplication, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
-
-16.1.4 Toom 4-Way Multiplication
---------------------------------
-
-Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
-respectively. Toom-4 analogously splits the operands into 4
-coefficients. Using the notation from the section on Toom-3
-multiplication, we form two polynomials:
-
- X(t) = x3*t^3 + x2*t^2 + x1*t + x0
- Y(t) = y3*t^3 + y2*t^2 + y1*t + y0
-
- X(t) and Y(t) are evaluated and multiplied at 7 points, giving
-values of W(t) at those points. In GMP the following points are used,
-
- Point Value
- t=0 x0 * y0, which gives w0 immediately
- t=1/2 (x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)
- t=-1/2 (-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)
- t=1 (x3+x2+x1+x0) * (y3+y2+y1+y0)
- t=-1 (-x3+x2-x1+x0) * (-y3+y2-y1+y0)
- t=2 (8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)
- t=inf x3 * y3, which gives w6 immediately
-
- The number of additions and subtractions for Toom-4 is much larger
-than for Toom-3. But several subexpressions occur multiple times, for
-example x2+x0, occurs for both t=1 and t=-1.
-
- Toom-4 is asymptotically O(N^1.404), the exponent being
-log(7)/log(4), representing 7 recursive multiplies of 1/4 the original
-size each.
-
-\1f
-File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
-
-16.1.5 FFT Multiplication
--------------------------
-
-At large to very large sizes a Fermat style FFT multiplication is used,
-following Scho"nhage and Strassen (*note References::). Descriptions
-of FFTs in various forms can be found in many textbooks, for instance
-Knuth section 4.3.3 part C or Lipson chapter IX. A brief description
-of the form used in GMP is given here.
-
- The multiplication done is x*y mod 2^N+1, for a given N. A full
-product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x
-and y with high zero limbs. The modular product is the native form for
-the algorithm, so padding to get a full product is unavoidable.
-
- The algorithm follows a split, evaluate, pointwise multiply,
-interpolate and combine similar to that described above for Karatsuba
-and Toom-3. A k parameter controls the split, with an FFT-k splitting
-into 2^k pieces of M=N/2^k bits each. N must be a multiple of
-(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
-bit shifts in the split and combine stages.
-
- The evaluations, pointwise multiplications, and interpolation, are
-all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
-2^k and of `mp_bits_per_limb'. The results of interpolation will be
-the following negacyclic convolution of the input pieces, and the
-choice of N' ensures these sums aren't truncated.
-
- ---
- \ b
- w[n] = / (-1) * x[i] * y[j]
- ---
- i+j==b*2^k+n
- b=0,1
-
- The points used for the evaluation are g^i for i=0 to 2^k-1 where
-g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
-necessary cancellations at the interpolation stage, and it's also a
-power of 2 so the fast Fourier transforms used for the evaluation and
-interpolation do only shifts, adds and negations.
-
- The pointwise multiplications are done modulo 2^N'+1 and either
-recurse into a further FFT or use a plain multiplication (Toom-3,
-Karatsuba or basecase), whichever is optimal at the size N'. The
-interpolation is an inverse fast Fourier transform. The resulting set
-of sums of x[i]*y[j] are added at appropriate offsets to give the final
-result.
-
- Squaring is the same, but x is the only input so it's one transform
-at the evaluate stage and the pointwise multiplies are squares. The
-interpolation is the same.
-
- For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm,
-the exponent representing 2^k recursed modular multiplies each
-1/2^(k-1) the size of the original. Each successive k is an asymptotic
-improvement, but overheads mean each is only faster at bigger and
-bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the
-thresholds where each k is used. Each new k effectively swaps some
-multiplying for some shifts, adds and overheads.
-
- A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
-plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
-A k=4 FFT at O(N^1.333) can be expected to be the first faster than
-Toom-3 at O(N^1.465). In practice this is what's found, with
-`MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300
-and 1000 limbs, depending on the CPU. So far it's been found that only
-very large FFTs recurse into pointwise multiplies above these sizes.
-
- When an FFT is to give a full product, the change of N to 2N doesn't
-alter the theoretical complexity for a given k, but for the purposes of
-considering where an FFT might be first used it can be assumed that the
-FFT is recursing into a normal multiply and that on that basis it's
-doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
-making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
-first FFT faster than Toom-3. In practice `MUL_FFT_THRESHOLD' and
-`SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere
-between 3000 and 10000 limbs.
-
- The way N is split into 2^k pieces and then 2M+k+3 is rounded up to
-a multiple of 2^k and `mp_bits_per_limb' means that when
-2^k>=mp_bits_per_limb the effective N is a multiple of 2^(2k-1) bits.
-The +k+3 means some values of N just under such a multiple will be
-rounded to the next. The complexity calculations above assume that a
-favourable size is used, meaning one which isn't padded through
-rounding, and it's also assumed that the extra +k+3 bits are negligible
-at typical FFT sizes.
-
- The practical effect of the 2^(2k-1) constraint is to introduce a
-step-effect into measured speeds. For example k=8 will round N up to a
-multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
-groups of sizes for which `mpn_mul_n' runs at the same speed. Or for
-k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice
-it's been found each k is used at quite small multiples of its size
-constraint and so the step effect is quite noticeable in a time versus
-size graph.
-
- The threshold determinations currently measure at the mid-points of
-size steps, but this is sub-optimal since at the start of a new step it
-can happen that it's better to go back to the previous k for a while.
-Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE'
-will be needed.
-
-\1f
-File: gmp.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
-
-16.1.6 Other Multiplication
----------------------------
-
-The Toom algorithms described above (*note Toom 3-Way Multiplication::,
-*note Toom 4-Way Multiplication::) generalizes to split into an
-arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C.
-This is not currently used. The notes here are merely for interest.
-
- In general a split into r+1 pieces is made, and evaluations and
-pointwise multiplications done at 2*r+1 points. A 4-way split does 7
-pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
-algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
-multiplications count towards big-O complexity, but the time spent in
-the evaluate and interpolate stages grows with r and has a significant
-practical impact, with the asymptotic advantage of each r realized only
-at bigger and bigger sizes. The overheads grow as O(N*r), whereas in
-an r=2^k FFT they grow only as O(N*log(r)).
-
- Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
-uses -r,...,0,...,r and the latter saves some small multiplies in the
-evaluate stage (or rather trades them for additions), and has a further
-saving of nearly half the interpolate steps. The idea is to separate
-odd and even final coefficients and then perform algorithm C steps C7
-and C8 on them separately. The divisors at step C7 become j^2 and the
-multipliers at C8 become 2*t*j-j^2.
-
- Splitting odd and even parts through positive and negative points
-can be thought of as using -1 as a square root of unity. If a 4th root
-of unity was available then a further split and speedup would be
-possible, but no such root exists for plain integers. Going to complex
-integers with i=sqrt(-1) doesn't help, essentially because in Cartesian
-form it takes three real multiplies to do a complex multiply. The
-existence of 2^k'th roots of unity in a suitable ring or field lets the
-fast Fourier transform keep splitting and get to O(N*log(r)).
-
- Floating point FFTs use complex numbers approximating Nth roots of
-unity. Some processors have special support for such FFTs. But these
-are not used in GMP since it's very difficult to guarantee an exact
-result (to some number of bits). An occasional difference of 1 in the
-last bit might not matter to a typical signal processing algorithm, but
-is of course of vital importance to GMP.
-
-\1f
-File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
-
-16.1.7 Unbalanced Multiplication
---------------------------------
-
-Multiplication of operands with different sizes, both below
-`MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication
-(*note Basecase Multiplication::).
-
- For really large operands, we invoke FFT directly.
-
- For operands between these sizes, we use Toom inspired algorithms
-suggested by Alberto Zanoni and Marco Bodrato. The idea is to split
-the operands into polynomials of different degree. GMP currently
-splits the smaller operand onto 2 coefficients, i.e., a polynomial of
-degree 1, but the larger operand can be split into 2, 3, or 4
-coefficients, i.e., a polynomial of degree 1 to 3.
-
-\1f
-File: gmp.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
-
-16.2 Division Algorithms
-========================
-
-* Menu:
-
-* Single Limb Division::
-* Basecase Division::
-* Divide and Conquer Division::
-* Block-Wise Barrett Division::
-* Exact Division::
-* Exact Remainder::
-* Small Quotient Division::
-
-\1f
-File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
-
-16.2.1 Single Limb Division
----------------------------
-
-Nx1 division is implemented using repeated 2x1 divisions from high to
-low, either with a hardware divide instruction or a multiplication by
-inverse, whichever is best on a given CPU.
-
- The multiply by inverse follows "Improved division by invariant
-integers" by Mo"ller and Granlund (*note References::) and is
-implemented as `udiv_qrnnd_preinv' in `gmp-impl.h'. The idea is to
-have a fixed-point approximation to 1/d (see `invert_limb') and then
-multiply by the high limb (plus one bit) of the dividend to get a
-quotient q. With d normalized (high bit set), q is no more than 1 too
-small. Subtracting q*d from the dividend gives a remainder, and
-reveals whether q or q-1 is correct.
-
- The result is a division done with two multiplications and four or
-five arithmetic operations. On CPUs with low latency multipliers this
-can be much faster than a hardware divide, though the cost of
-calculating the inverse at the start may mean it's only better on
-inputs bigger than say 4 or 5 limbs.
-
- When a divisor must be normalized, either for the generic C
-`__udiv_qrnnd_c' or the multiply by inverse, the division performed is
-actually a*2^k by d*2^k where a is the dividend and k is the power
-necessary to have the high bit of d*2^k set. The bit shifts for the
-dividend are usually accomplished "on the fly" meaning by extracting
-the appropriate bits at each step. Done this way the quotient limbs
-come out aligned ready to store. When only the remainder is wanted, an
-alternative is to take the dividend limbs unshifted and calculate r = a
-mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
-help on CPUs with poor bit shifts or few registers.
-
- The multiply by inverse can be done two limbs at a time. The
-calculation is basically the same, but the inverse is two limbs and the
-divisor treated as if padded with a low zero limb. This means more
-work, since the inverse will need a 2x2 multiply, but the four 1x1s to
-do that are independent and can therefore be done partly or wholly in
-parallel. Likewise for a 2x1 calculating q*d. The net effect is to
-process two limbs with roughly the same two multiplies worth of latency
-that one limb at a time gives. This extends to 3 or 4 limbs at a time,
-though the extra work to apply the inverse will almost certainly soon
-reach the limits of multiplier throughput.
-
- A similar approach in reverse can be taken to process just half a
-limb at a time if the divisor is only a half limb. In this case the
-1x1 multiply for the inverse effectively becomes two (1/2)x1 for each
-limb, which can be a saving on CPUs with a fast half limb multiply, or
-in fact if the only multiply is a half limb, and especially if it's not
-pipelined.
-
-\1f
-File: gmp.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
-
-16.2.2 Basecase Division
-------------------------
-
-Basecase NxM division is like long division done by hand, but in base
-2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and
-`mpn/generic/sb_divrem_mn.c'.
-
- Briefly stated, while the dividend remains larger than the divisor,
-a high quotient limb is formed and the Nx1 product q*d subtracted at
-the top end of the dividend. With a normalized divisor (most
-significant bit set), each quotient limb can be formed with a 2x1
-division and a 1x1 multiplication plus some subtractions. The 2x1
-division is by the high limb of the divisor and is done either with a
-hardware divide or a multiply by inverse (the same as in *Note Single
-Limb Division::) whichever is faster. Such a quotient is sometimes one
-too big, requiring an addback of the divisor, but that happens rarely.
-
- With Q=N-M being the number of quotient limbs, this is an O(Q*M)
-algorithm and will run at a speed similar to a basecase QxM
-multiplication, differing in fact only in the extra multiply and divide
-for each of the Q quotient limbs.
-
-\1f
-File: gmp.info, Node: Divide and Conquer Division, Next: Block-Wise Barrett Division, Prev: Basecase Division, Up: Division Algorithms
-
-16.2.3 Divide and Conquer Division
-----------------------------------
-
-For divisors larger than `DC_DIV_QR_THRESHOLD', division is done by
-dividing. Or to be precise by a recursive divide and conquer algorithm
-based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
-(*note References::).
-
- The algorithm consists essentially of recognising that a 2NxN
-division can be done with the basecase division algorithm (*note
-Basecase Division::), but using N/2 limbs as a base, not just a single
-limb. This way the multiplications that arise are (N/2)x(N/2) and can
-take advantage of Karatsuba and higher multiplication algorithms (*note
-Multiplication Algorithms::). The two "digits" of the quotient are
-formed by recursive Nx(N/2) divisions.
-
- If the (N/2)x(N/2) multiplies are done with a basecase multiplication
-then the work is about the same as a basecase division, but with more
-function call overheads and with some subtractions separated from the
-multiplies. These overheads mean that it's only when N/2 is above
-`MUL_TOOM22_THRESHOLD' that divide and conquer is of use.
-
- `DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be
-somewhere above twice `MUL_TOOM22_THRESHOLD', but how much above
-depends on the CPU. An optimized `mpn_mul_basecase' can lower
-`DC_DIV_QR_THRESHOLD' a little by offering a ready-made advantage over
-repeated `mpn_submul_1' calls.
-
- Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
-the time for an NxN multiplication done with FFTs. The actual time is
-a sum over multiplications of the recursed sizes, as can be seen near
-the end of section 2.2 of Burnikel and Ziegler. For example, within
-the Toom-3 range, divide and conquer is 2.63*M(N). With higher
-algorithms the M(N) term improves and the multiplier tends to log(N).
-In practice, at moderate to large sizes, a 2NxN division is about 2 to
-4 times slower than an NxN multiplication.
-
-\1f
-File: gmp.info, Node: Block-Wise Barrett Division, Next: Exact Division, Prev: Divide and Conquer Division, Up: Division Algorithms
-
-16.2.4 Block-Wise Barrett Division
-----------------------------------
-
-For the largest divisions, a block-wise Barrett division algorithm is
-used. Here, the divisor is inverted to a precision determined by the
-relative size of the dividend and divisor. Blocks of quotient limbs
-are then generated by multiplying blocks from the dividend by the
-inverse.
-
- Our block-wise algorithm computes a smaller inverse than in the
-plain Barrett algorithm. For a 2n/n division, the inverse will be just
-ceil(n/2) limbs.
-
-\1f
-File: gmp.info, Node: Exact Division, Next: Exact Remainder, Prev: Block-Wise Barrett Division, Up: Division Algorithms
-
-16.2.5 Exact Division
----------------------
-
-A so-called exact division is when the dividend is known to be an exact
-multiple of the divisor. Jebelean's exact division algorithm uses this
-knowledge to make some significant optimizations (*note References::).
-
- The idea can be illustrated in decimal for example with 368154
-divided by 543. Because the low digit of the dividend is 4, the low
-digit of the quotient must be 8. This is arrived at from 4*7 mod 10,
-using the fact 7 is the modular inverse of 3 (the low digit of the
-divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from
-the dividend leaving 363810. Notice the low digit has become zero.
-
- The procedure is repeated at the second digit, with the next
-quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving
-325800. And finally at the third digit with quotient digit 6 (8*7 mod
-10), subtracting 6*543=3258 leaving 0. So the quotient is 678.
-
- Notice however that the multiplies and subtractions don't need to
-extend past the low three digits of the dividend, since that's enough
-to determine the three quotient digits. For the last quotient digit no
-subtraction is needed at all. On a 2NxN division like this one, only
-about half the work of a normal basecase division is necessary.
-
- For an NxM exact division producing Q=N-M quotient limbs, the saving
-over a normal basecase division is in two parts. Firstly, each of the
-Q quotient limbs needs only one multiply, not a 2x1 divide and
-multiply. Secondly, the crossproducts are reduced when Q>M to
-Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are
-complementary. If Q is big then many divisions are saved, or if Q is
-small then the crossproducts reduce to a small number.
-
- The modular inverse used is calculated efficiently by `binvert_limb'
-in `gmp-impl.h'. This does four multiplies for a 32-bit limb, or six
-for a 64-bit limb. `tune/modlinv.c' has some alternate implementations
-that might suit processors better at bit twiddling than multiplying.
-
- The sub-quadratic exact division described by Jebelean in "Exact
-Division with Karatsuba Complexity" is not currently implemented. It
-uses a rearrangement similar to the divide and conquer for normal
-division (*note Divide and Conquer Division::), but operating from low
-to high. A further possibility not currently implemented is
-"Bidirectional Exact Integer Division" by Krandick and Jebelean which
-forms quotient limbs from both the high and low ends of the dividend,
-and can halve once more the number of crossproducts needed in a 2NxN
-division.
-
- A special case exact division by 3 exists in `mpn_divexact_by3',
-supporting Toom-3 multiplication and `mpq' canonicalizations. It forms
-quotient digits with a multiply by the modular inverse of 3 (which is
-`0xAA..AAB') and uses two comparisons to determine a borrow for the next
-limb. The multiplications don't need to be on the dependent chain, as
-long as the effect of the borrows is applied, which can help chips with
-pipelined multipliers.
-
-\1f
-File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
-
-16.2.6 Exact Remainder
-----------------------
-
-If the exact division algorithm is done with a full subtraction at each
-stage and the dividend isn't a multiple of the divisor, then low zero
-limbs are produced but with a remainder in the high limbs. For
-dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
-remainder r is of the form
-
- a = q*d + r*b^n
-
- n represents the number of zero limbs produced by the subtractions,
-that being the number of limbs produced for q. r will be in the range
-0<=r<d and can be viewed as a remainder, but one shifted up by a factor
-of b^n.
-
- Carrying out full subtractions at each stage means the same number
-of cross products must be done as a normal division, but there's still
-some single limb divisions saved. When d is a single limb some
-simplifications arise, providing good speedups on a number of
-processors.
-
- `mpn_divexact_by3', `mpn_modexact_1_odd' and the `mpn_redc_X'
-functions differ subtly in how they return r, leading to some negations
-in the above formula, but all are essentially the same.
-
- Clearly r is zero when a is a multiple of d, and this leads to
-divisibility or congruence tests which are potentially more efficient
-than a normal division.
-
- The factor of b^n on r can be ignored in a GCD when d is odd, hence
-the use of `mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui'
-etc (*note Greatest Common Divisor Algorithms::).
-
- Montgomery's REDC method for modular multiplications uses operands
-of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
-uses the factor of b^n in the exact remainder to reach a product in the
-same form (x*y)*b^-n (*note Modular Powering Algorithm::).
-
- Notice that r generally gives no useful information about the
-ordinary remainder a mod d since b^n mod d could be anything. If
-however b^n == 1 mod d, then r is the negative of the ordinary
-remainder. This occurs whenever d is a factor of b^n-1, as for example
-with 3 in `mpn_divexact_by3'. For a 32 or 64 bit limb other such
-factors include 5, 17 and 257, but no particular use has been found for
-this.
-
-\1f
-File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
-
-16.2.7 Small Quotient Division
-------------------------------
-
-An NxM division where the number of quotient limbs Q=N-M is small can
-be optimized somewhat.
-
- An ordinary basecase division normalizes the divisor by shifting it
-to make the high bit set, shifting the dividend accordingly, and
-shifting the remainder back down at the end of the calculation. This
-is wasteful if only a few quotient limbs are to be formed. Instead a
-division of just the top 2*Q limbs of the dividend by the top Q limbs
-of the divisor can be used to form a trial quotient. This requires
-only those limbs normalized, not the whole of the divisor and dividend.
-
- A multiply and subtract then applies the trial quotient to the M-Q
-unused limbs of the divisor and N-Q dividend limbs (which includes Q
-limbs remaining from the trial quotient division). The starting trial
-quotient can be 1 or 2 too big, but all cases of 2 too big and most
-cases of 1 too big are detected by first comparing the most significant
-limbs that will arise from the subtraction. An addback is done if the
-quotient still turns out to be 1 too big.
-
- This whole procedure is essentially the same as one step of the
-basecase algorithm done in a Q limb base, though with the trial
-quotient test done only with the high limbs, not an entire Q limb
-"digit" product. The correctness of this weaker test can be
-established by following the argument of Knuth section 4.3.1 exercise
-20 but with the v2*q>b*r+u2 condition appropriately relaxed.
-
-\1f
-File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
-
-16.3 Greatest Common Divisor
-============================
-
-* Menu:
-
-* Binary GCD::
-* Lehmer's Algorithm::
-* Subquadratic GCD::
-* Extended GCD::
-* Jacobi Symbol::
-
-\1f
-File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
-
-16.3.1 Binary GCD
------------------
-
-At small sizes GMP uses an O(N^2) binary style GCD. This is described
-in many textbooks, for example Knuth section 4.5.2 algorithm B. It
-simply consists of successively reducing odd operands a and b using
-
- a,b = abs(a-b),min(a,b)
- strip factors of 2 from a
-
- The Euclidean GCD algorithm, as per Knuth algorithms E and A,
-repeatedly computes the quotient q = floor(a/b) and replaces a,b by v,
-u - q v. The binary algorithm has so far been found to be faster than
-the Euclidean algorithm everywhere. One reason the binary method does
-well is that the implied quotient at each step is usually small, so
-often only one or two subtractions are needed to get the same effect as
-a division. Quotients 1, 2 and 3 for example occur 67.7% of the time,
-see Knuth section 4.5.3 Theorem E.
-
- When the implied quotient is large, meaning b is much smaller than
-a, then a division is worthwhile. This is the basis for the initial a
-mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
-and 1x1 cases). But after that initial reduction, big quotients occur
-too rarely to make it worth checking for them.
-
-
- The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
-described above. For two N-bit operands, the algorithm takes about
-0.68 iterations per bit. For optimum performance some attention needs
-to be paid to the way the factors of 2 are stripped from a.
-
- Firstly it may be noted that in twos complement the number of low
-zero bits on a-b is the same as b-a, so counting or testing can begin on
-a-b without waiting for abs(a-b) to be determined.
-
- A loop stripping low zero bits tends not to branch predict well,
-since the condition is data dependent. But on average there's only a
-few low zeros, so an option is to strip one or two bits arithmetically
-then loop for more (as done for AMD K6). Or use a lookup table to get
-a count for several bits then loop for more (as done for AMD K7). An
-alternative approach is to keep just one of a or b odd and iterate
-
- a,b = abs(a-b), min(a,b)
- a = a/2 if even
- b = b/2 if even
-
- This requires about 1.25 iterations per bit, but stripping of a
-single bit at each step avoids any branching. Repeating the bit strip
-reduces to about 0.9 iterations per bit, which may be a worthwhile
-tradeoff.
-
- Generally with the above approaches a speed of perhaps 6 cycles per
-bit can be achieved, which is still not terribly fast with for instance
-a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
-means it's not usually advantageous to combine a set of divisibility
-tests into a GCD.
-
- Currently, the binary algorithm is used for GCD only when N < 3.
-
-\1f
-File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.2 Lehmer's algorithm
--------------------------
-
-Lehmer's improvement of the Euclidean algorithms is based on the
-observation that the initial part of the quotient sequence depends only
-on the most significant parts of the inputs. The variant of Lehmer's
-algorithm used in GMP splits off the most significant two limbs, as
-suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by
-Jebelean (*note References::). The quotients of two double-limb inputs
-are collected as a 2 by 2 matrix with single-limb elements. This is
-done by the function `mpn_hgcd2'. The resulting matrix is applied to
-the inputs using `mpn_mul_1' and `mpn_submul_1'. Each iteration usually
-reduces the inputs by almost one limb. In the rare case of a large
-quotient, no progress can be made by examining just the most
-significant two limbs, and the quotient is computing using plain
-division.
-
- The resulting algorithm is asymptotically O(N^2), just as the
-Euclidean algorithm and the binary algorithm. The quadratic part of the
-work are the calls to `mpn_mul_1' and `mpn_submul_1'. For small sizes,
-the linear work is also significant. There are roughly N calls to the
-`mpn_hgcd2' function. This function uses a couple of important
-optimizations:
-
- * It uses the same relaxed notion of correctness as `mpn_hgcd' (see
- next section). This means that when called with the most
- significant two limbs of two large numbers, the returned matrix
- does not always correspond exactly to the initial quotient
- sequence for the two large numbers; the final quotient may
- sometimes be one off.
-
- * It takes advantage of the fact the quotients are usually small.
- The division operator is not used, since the corresponding
- assembler instruction is very slow on most architectures. (This
- code could probably be improved further, it uses many branches
- that are unfriendly to prediction).
-
- * It switches from double-limb calculations to single-limb
- calculations half-way through, when the input numbers have been
- reduced in size from two limbs to one and a half.
-
-
-\1f
-File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms
-
-16.3.3 Subquadratic GCD
------------------------
-
-For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD
-(Half GCD) function, as a generalization to Lehmer's algorithm.
-
- Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
-Then HGCD(a,b) returns a transformation matrix T with non-negative
-elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers
-c,d must be larger than S limbs, while their difference abs(c-d) must
-fit in S limbs. The matrix elements will also be of size roughly N/2.
-
- The HGCD base case uses Lehmer's algorithm, but with the above stop
-condition that returns reduced numbers and the corresponding
-transformation matrix half-way through. For inputs larger than
-`HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
-conquer algorithm in "On Scho"nhage's algorithm and subquadratic
-integer GCD computation" by Mo"ller (*note References::). The recursive
-algorithm consists of these main steps.
-
- * Call HGCD recursively, on the most significant N/2 limbs. Apply the
- resulting matrix T_1 to the full numbers, reducing them to a size
- just above 3N/2.
-
- * Perform a small number of division or subtraction steps to reduce
- the numbers to size below 3N/2. This is essential mainly for the
- unlikely case of large quotients.
-
- * Call HGCD recursively, on the most significant N/2 limbs of the
- reduced numbers. Apply the resulting matrix T_2 to the full
- numbers, reducing them to a size just above N/2.
-
- * Compute T = T_1 T_2.
-
- * Perform a small number of division and subtraction steps to
- satisfy the requirements, and return.
-
- GCD is then implemented as a loop around HGCD, similarly to Lehmer's
-algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
-`mpn_hgcd2', and applies the resulting matrix to the full numbers, the
-subquadratic GCD chops off the most significant third of the limbs (the
-proportion is a tuning parameter, and 1/3 seems to be more efficient
-than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix.
-Once the input numbers are reduced to size below `GCD_DC_THRESHOLD',
-Lehmer's algorithm is used for the rest of the work.
-
- The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
-where M(N) is the time for multiplying two N-limb numbers.
-
-\1f
-File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.4 Extended GCD
--------------------
-
-The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
-cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
-for plain GCD are extended to handle this case. The binary algorithm is
-used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes
-up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is
-implemented as a loop around HGCD, but with more book-keeping to keep
-track of the cofactors. This gives the same asymptotic running time as
-for GCD and HGCD, O(M(N)*log(N))
-
- One difference to plain GCD is that while the inputs a and b are
-reduced as the algorithm proceeds, the cofactors x and y grow in size.
-This makes the tuning of the chopping-point more difficult. The current
-code chops off the most significant half of the inputs for the call to
-HGCD in the first iteration, and the most significant two thirds for
-the remaining calls. This strategy could surely be improved. Also the
-stop condition for the loop, where Lehmer's algorithm is invoked once
-the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be
-improved by taking into account the current size of the cofactors.
-
-\1f
-File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.5 Jacobi Symbol
---------------------
-
-`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
-simple binary algorithm similar to that described for the GCDs (*note
-Binary GCD::). They're not very fast when both inputs are large.
-Lehmer's multi-step improvement or a binary based multi-step algorithm
-is likely to be better.
-
- When one operand fits a single limb, and that includes
-`mpz_kronecker_ui' and friends, an initial reduction is done with
-either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
-algorithm on a single limb. The binary algorithm is well suited to a
-single limb, and the whole calculation in this case is quite efficient.
-
- In all the routines sign changes for the result are accumulated
-using some bit twiddling, avoiding table lookups or conditional jumps.
-
+++ /dev/null
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-END-INFO-DIR-ENTRY
-
-\1f
-File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
-
-16.4 Powering Algorithms
-========================
-
-* Menu:
-
-* Normal Powering Algorithm::
-* Modular Powering Algorithm::
-
-\1f
-File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
-
-16.4.1 Normal Powering
-----------------------
-
-Normal `mpz' or `mpf' powering uses a simple binary algorithm,
-successively squaring and then multiplying by the base when a 1 bit is
-seen in the exponent, as per Knuth section 4.6.3. The "left to right"
-variant described there is used rather than algorithm A, since it's
-just as easy and can be done with somewhat less temporary memory.
-
-\1f
-File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
-
-16.4.2 Modular Powering
------------------------
-
-Modular powering is implemented using a 2^k-ary sliding window
-algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
-(*note References::). k is chosen according to the size of the
-exponent. Larger exponents use larger values of k, the choice being
-made to minimize the average number of multiplications that must
-supplement the squaring.
-
- The modular multiplies and squares use either a simple division or
-the REDC method by Montgomery (*note References::). REDC is a little
-faster, essentially saving N single limb divisions in a fashion similar
-to an exact remainder (*note Exact Remainder::).
-
-\1f
-File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
-
-16.5 Root Extraction Algorithms
-===============================
-
-* Menu:
-
-* Square Root Algorithm::
-* Nth Root Algorithm::
-* Perfect Square Algorithm::
-* Perfect Power Algorithm::
-
-\1f
-File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
-
-16.5.1 Square Root
-------------------
-
-Square roots are taken using the "Karatsuba Square Root" algorithm by
-Paul Zimmermann (*note References::).
-
- An input n is split into four parts of k bits each, so with b=2^k we
-have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
-that either the high or second highest bit is set. In GMP, k is kept
-on a limb boundary and the input is left shifted (by an even number of
-bits) to normalize.
-
- The square root of the high two parts is taken, by recursive
-application of the algorithm (bottoming out in a one-limb Newton's
-method),
-
- s1,r1 = sqrtrem (a3*b + a2)
-
- This is an approximation to the desired root and is extended by a
-division to give s,r,
-
- q,u = divrem (r1*b + a1, 2*s1)
- s = s1*b + q
- r = u*b + a0 - q^2
-
- The normalization requirement on a3 means at this point s is either
-correct or 1 too big. r is negative in the latter case, so
-
- if r < 0 then
- r = r + 2*s - 1
- s = s - 1
-
- The algorithm is expressed in a divide and conquer form, but as
-noted in the paper it can also be viewed as a discrete variant of
-Newton's method, or as a variation on the schoolboy method (no longer
-taught) for square roots two digits at a time.
-
- If the remainder r is not required then usually only a few high limbs
-of r and u need to be calculated to determine whether an adjustment to
-s is required. This optimization is not currently implemented.
-
- In the Karatsuba multiplication range this algorithm is
-O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
-limbs. In the FFT multiplication range this grows to a bound of
-O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
-Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
-
- The algorithm does all its calculations in integers and the resulting
-`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
-precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
-
-\1f
-File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
-
-16.5.2 Nth Root
----------------
-
-Integer Nth roots are taken using Newton's method with the following
-iteration, where A is the input and n is the root to be taken.
-
- 1 A
- a[i+1] = - * ( --------- + (n-1)*a[i] )
- n a[i]^(n-1)
-
- The initial approximation a[1] is generated bitwise by successively
-powering a trial root with or without new 1 bits, aiming to be just
-above the true root. The iteration converges quadratically when
-started from a good approximation. When n is large more initial bits
-are needed to get good convergence. The current implementation is not
-particularly well optimized.
-
-\1f
-File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
-
-16.5.3 Perfect Square
----------------------
-
-A significant fraction of non-squares can be quickly identified by
-checking whether the input is a quadratic residue modulo small integers.
-
- `mpz_perfect_square_p' first tests the input mod 256, which means
-just examining the low byte. Only 44 different values occur for
-squares mod 256, so 82.8% of inputs can be immediately identified as
-non-squares.
-
- On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
-for a total 99.25% of inputs identified as non-squares. On a 64-bit
-system 97 is tested too, for a total 99.62%.
-
- These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
-for 64-bits), and such a remainder can be quickly taken just using
-additions (see `mpn_mod_34lsub1').
-
- When nails are in use moduli are instead selected by the `gen-psqr.c'
-program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
-could be done with nails using some extra bit shifts, but this is not
-currently implemented.
-
- In any case each modulus is applied to the `mpn_mod_34lsub1' or
-`mpn_mod_1' remainder and a table lookup identifies non-squares. By
-using a "modexact" style calculation, and suitably permuted tables,
-just one multiply each is required, see the code for details. Moduli
-are also combined to save operations, so long as the lookup tables
-don't become too big. `gen-psqr.c' does all the pre-calculations.
-
- A square root must still be taken for any value that passes these
-tests, to verify it's really a square and not one of the small fraction
-of non-squares that get through (ie. a pseudo-square to all the tested
-bases).
-
- Clearly more residue tests could be done, `mpz_perfect_square_p' only
-uses a compact and efficient set. Big inputs would probably benefit
-from more residue testing, small inputs might be better off with less.
-The assumed distribution of squares versus non-squares in the input
-would affect such considerations.
-
-\1f
-File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
-
-16.5.4 Perfect Power
---------------------
-
-Detecting perfect powers is required by some factorization algorithms.
-Currently `mpz_perfect_power_p' is implemented using repeated Nth root
-extractions, though naturally only prime roots need to be considered.
-(*Note Nth Root Algorithm::.)
-
- If a prime divisor p with multiplicity e can be found, then only
-roots which are divisors of e need to be considered, much reducing the
-work necessary. To this end divisibility by a set of small primes is
-checked.
-
-\1f
-File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
-
-16.6 Radix Conversion
-=====================
-
-Radix conversions are less important than other algorithms. A program
-dominated by conversions should probably use a different data
-representation.
-
-* Menu:
-
-* Binary to Radix::
-* Radix to Binary::
-
-\1f
-File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
-
-16.6.1 Binary to Radix
-----------------------
-
-Conversions from binary to a power-of-2 radix use a simple and fast
-O(N) bit extraction algorithm.
-
- Conversions from binary to other radices use one of two algorithms.
-Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
-Repeated divisions by b^n are made, where b is the radix and n is the
-biggest power that fits in a limb. But instead of simply using the
-remainder r from such divisions, an extra divide step is done to give a
-fractional limb representing r/b^n. The digits of r can then be
-extracted using multiplications by b rather than divisions. Special
-case code is provided for decimal, allowing multiplications by 10 to
-optimize to shifts and adds.
-
- Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
-used. For an input t, powers b^(n*2^i) of the radix are calculated,
-until a power between t and sqrt(t) is reached. t is then divided by
-that largest power, giving a quotient which is the digits above that
-power, and a remainder which is those below. These two parts are in
-turn divided by the second highest power, and so on recursively. When
-a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
-limbs, the basecase algorithm described above is used.
-
- The advantage of this algorithm is that big divisions can make use
-of the sub-quadratic divide and conquer division (*note Divide and
-Conquer Division::), and big divisions tend to have less overheads than
-lots of separate single limb divisions anyway. But in any case the
-cost of calculating the powers b^(n*2^i) must first be overcome.
-
- `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
-the same basic thing, the point where it becomes worth doing a big
-division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
-includes the cost of calculating the radix power required, whereas
-`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
-case when recursing.
-
- Since the base case produces digits from least to most significant
-but they want to be stored from most to least, it's necessary to
-calculate in advance how many digits there will be, or at least be sure
-not to underestimate that. For GMP the number of input bits is
-multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
-The result is either correct or one too big.
-
- Examining some of the high bits of the input could increase the
-chance of getting the exact number of digits, but an exact result every
-time would not be practical, since in general the difference between
-numbers 100... and 99... is only in the last few bits and the work to
-identify 99... might well be almost as much as a full conversion.
-
- `mpf_get_str' doesn't currently use the algorithm described here, it
-multiplies or divides by a power of b to move the radix point to the
-just above the highest non-zero digit (or at worst one above that
-location), then multiplies by b^n to bring out digits. This is O(N^2)
-and is certainly not optimal.
-
- The r/b^n scheme described above for using multiplications to bring
-out digits might be useful for more than a single limb. Some brief
-experiments with it on the base case when recursing didn't give a
-noticeable improvement, but perhaps that was only due to the
-implementation. Something similar would work for the sub-quadratic
-divisions too, though there would be the cost of calculating a bigger
-radix power.
-
- Another possible improvement for the sub-quadratic part would be to
-arrange for radix powers that balanced the sizes of quotient and
-remainder produced, ie. the highest power would be an b^(n*k)
-approximately equal to sqrt(t), not restricted to a 2^i factor. That
-ought to smooth out a graph of times against sizes, but may or may not
-be a net speedup.
-
-\1f
-File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
-
-16.6.2 Radix to Binary
-----------------------
-
-*This section needs to be rewritten, it currently describes the
-algorithms used before GMP 4.3.*
-
- Conversions from a power-of-2 radix into binary use a simple and fast
-O(N) bitwise concatenation algorithm.
-
- Conversions from other radices use one of two algorithms. Sizes
-below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups
-of n digits are converted to limbs, where n is the biggest power of the
-base b which will fit in a limb, then those groups are accumulated into
-the result by multiplying by b^n and adding. This saves
-multi-precision operations, as per Knuth section 4.4 part E (*note
-References::). Some special case code is provided for decimal, giving
-the compiler a chance to optimize multiplications by 10.
-
- Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
-used. First groups of n digits are converted into limbs. Then adjacent
-limbs are combined into limb pairs with x*b^n+y, where x and y are the
-limbs. Adjacent limb pairs are combined into quads similarly with
-x*b^(2n)+y. This continues until a single block remains, that being
-the result.
-
- The advantage of this method is that the multiplications for each x
-are big blocks, allowing Karatsuba and higher algorithms to be used.
-But the cost of calculating the powers b^(n*2^i) must be overcome.
-`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
-digits, and on some processors much bigger still.
-
- `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
-tuned for decimal), though it might be better based on a limb count, so
-as to be independent of the base. But that sort of count isn't used by
-the base case and so would need some sort of initial calculation or
-estimate.
-
- The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger
-than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that
-`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5,
-or more).
-
-\1f
-File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
-
-16.7 Other Algorithms
-=====================
-
-* Menu:
-
-* Prime Testing Algorithm::
-* Factorial Algorithm::
-* Binomial Coefficients Algorithm::
-* Fibonacci Numbers Algorithm::
-* Lucas Numbers Algorithm::
-* Random Number Algorithms::
-
-\1f
-File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
-
-16.7.1 Prime Testing
---------------------
-
-The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
-Functions::) first does some trial division by small factors and then
-uses the Miller-Rabin probabilistic primality testing algorithm, as
-described in Knuth section 4.5.4 algorithm P (*note References::).
-
- For an odd input n, and with n = q*2^k+1 where q is odd, this
-algorithm selects a random base x and tests whether x^q mod n is 1 or
--1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably
-prime, if not then n is definitely composite.
-
- Any prime n will pass the test, but some composites do too. Such
-composites are known as strong pseudoprimes to base x. No n is a
-strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
-22), hence with x chosen at random there's no more than a 1/4 chance a
-"probable prime" will in fact be composite.
-
- In fact strong pseudoprimes are quite rare, making the test much more
-powerful than this analysis would suggest, but 1/4 is all that's proven
-for an arbitrary n.
-
-\1f
-File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
-
-16.7.2 Factorial
-----------------
-
-Factorials are calculated by a combination of removal of twos,
-powering, and binary splitting. The procedure can be best illustrated
-with an example,
-
- 23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23
-
-has factors of two removed,
-
- 23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23
-
-and the resulting terms collected up according to their multiplicity,
-
- 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)
-
- Each sequence such as 13.15.17.19.21.23 is evaluated by splitting
-into every second term, as for instance (13.17.21).(15.19.23), and the
-same recursively on each half. This is implemented iteratively using
-some bit twiddling.
-
- Such splitting is more efficient than repeated Nx1 multiplies since
-it forms big multiplies, allowing Karatsuba and higher algorithms to be
-used. And even below the Karatsuba threshold a big block of work can
-be more efficient for the basecase algorithm.
-
- Splitting into subsequences of every second term keeps the resulting
-products more nearly equal in size than would the simpler approach of
-say taking the first half and second half of the sequence. Nearly
-equal products are more efficient for the current multiply
-implementation.
-
-\1f
-File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
-
-16.7.3 Binomial Coefficients
-----------------------------
-
-Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
-using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
-product simply from i=2 to i=k.
-
- k (n-k+i)
- C(n,k) = (n-k+1) * prod -------
- i=2 i
-
- It's easy to show that each denominator i will divide the product so
-far, so the exact division algorithm is used (*note Exact Division::).
-
- The numerators n-k+i and denominators i are first accumulated into
-as many fit a limb, to save multi-precision operations, though for
-`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
-and n-k+i in general won't fit in a limb at all.
-
-\1f
-File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
-
-16.7.4 Fibonacci Numbers
-------------------------
-
-The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
-calculating isolated F[n] or F[n],F[n-1] values efficiently.
-
- For small n, a table of single limb values in `__gmp_fib_table' is
-used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
-to F[93]. For convenience the table starts at F[-1].
-
- Beyond the table, values are generated with a binary powering
-algorithm, calculating a pair F[n] and F[n-1] working from high to low
-across the bits of n. The formulas used are
-
- F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
- F[2k-1] = F[k]^2 + F[k-1]^2
-
- F[2k] = F[2k+1] - F[2k-1]
-
- At each step, k is the high b bits of n. If the next bit of n is 0
-then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
-and the process repeated until all bits of n are incorporated. Notice
-these formulas require just two squares per bit of n.
-
- It'd be possible to handle the first few n above the single limb
-table with simple additions, using the defining Fibonacci recurrence
-F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
-be faster for only about 10 or 20 values of n, and including a block of
-code for just those doesn't seem worthwhile. If they really mattered
-it'd be better to extend the data table.
-
- Using a table avoids lots of calculations on small numbers, and
-makes small n go fast. A bigger table would make more small n go fast,
-it's just a question of balancing size against desired speed. For GMP
-the code is kept compact, with the emphasis primarily on a good
-powering algorithm.
-
- `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
-interested in F[n]. In this case the last step of the algorithm can
-become one multiply instead of two squares. One of the following two
-formulas is used, according as n is odd or even.
-
- F[2k] = F[k]*(F[k]+2F[k-1])
-
- F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
-
- F[2k+1] here is the same as above, just rearranged to be a multiply.
-For interest, the 2*(-1)^k term both here and above can be applied
-just to the low limb of the calculation, without a carry or borrow into
-further limbs, which saves some code size. See comments with
-`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
-
-\1f
-File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
-
-16.7.5 Lucas Numbers
---------------------
-
-`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
-Fibonacci numbers with the following simple formulas.
-
- L[k] = F[k] + 2*F[k-1]
- L[k-1] = 2*F[k] - F[k-1]
-
- `mpz_lucnum_ui' is only interested in L[n], and some work can be
-saved. Trailing zero bits on n can be handled with a single square
-each.
-
- L[2k] = L[k]^2 - 2*(-1)^k
-
- And the lowest 1 bit can be handled with one multiply of a pair of
-Fibonacci numbers, similar to what `mpz_fib_ui' does.
-
- L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
-
-\1f
-File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
-
-16.7.6 Random Numbers
----------------------
-
-For the `urandomb' functions, random numbers are generated simply by
-concatenating bits produced by the generator. As long as the generator
-has good randomness properties this will produce well-distributed N bit
-numbers.
-
- For the `urandomm' functions, random numbers in a range 0<=R<N are
-generated by taking values R of ceil(log2(N)) bits each until one
-satisfies R<N. This will normally require only one or two attempts,
-but the attempts are limited in case the generator is somehow
-degenerate and produces only 1 bits or similar.
-
- The Mersenne Twister generator is by Matsumoto and Nishimura (*note
-References::). It has a non-repeating period of 2^19937-1, which is a
-Mersenne prime, hence the name of the generator. The state is 624
-words of 32-bits each, which is iterated with one XOR and shift for each
-32-bit word generated, making the algorithm very fast. Randomness
-properties are also very good and this is the default algorithm used by
-GMP.
-
- Linear congruential generators are described in many text books, for
-instance Knuth volume 2 (*note References::). With a modulus M and
-parameters A and C, a integer state S is iterated by the formula S <-
-A*S+C mod M. At each step the new state is a linear function of the
-previous, mod M, hence the name of the generator.
-
- In GMP only moduli of the form 2^N are supported, and the current
-implementation is not as well optimized as it could be. Overheads are
-significant when N is small, and when N is large clearly the multiply
-at each step will become slow. This is not a big concern, since the
-Mersenne Twister generator is better in every respect and is therefore
-recommended for all normal applications.
-
- For both generators the current state can be deduced by observing
-enough output and applying some linear algebra (over GF(2) in the case
-of the Mersenne Twister). This generally means raw output is
-unsuitable for cryptographic applications without further hashing or
-the like.
-
-\1f
-File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms
-
-16.8 Assembly Coding
-====================
-
-The assembly subroutines in GMP are the most significant source of
-speed at small to moderate sizes. At larger sizes algorithm selection
-becomes more important, but of course speedups in low level routines
-will still speed up everything proportionally.
-
- Carry handling and widening multiplies that are important for GMP
-can't be easily expressed in C. GCC `asm' blocks help a lot and are
-provided in `longlong.h', but hand coding low level routines invariably
-offers a speedup over generic C by a factor of anything from 2 to 10.
-
-* Menu:
-
-* Assembly Code Organisation::
-* Assembly Basics::
-* Assembly Carry Propagation::
-* Assembly Cache Handling::
-* Assembly Functional Units::
-* Assembly Floating Point::
-* Assembly SIMD Instructions::
-* Assembly Software Pipelining::
-* Assembly Loop Unrolling::
-* Assembly Writing Guide::
-
-\1f
-File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding
-
-16.8.1 Code Organisation
-------------------------
-
-The various `mpn' subdirectories contain machine-dependent code, written
-in C or assembly. The `mpn/generic' subdirectory contains default code,
-used when there's no machine-specific version of a particular file.
-
- Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
-64-bit variants in a family cannot share code and have separate
-directories. Within a family further subdirectories may exist for CPU
-variants.
-
- In each directory a `nails' subdirectory may exist, holding code with
-nails support for that CPU variant. A `NAILS_SUPPORT' directive in each
-file indicates the nails values the code handles. Nails code only
-exists where it's faster, or promises to be faster, than plain code.
-There's no effort put into nails if they're not going to enhance a
-given CPU.
-
-\1f
-File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding
-
-16.8.2 Assembly Basics
-----------------------
-
-`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
-overall GMP performance. All multiplications and divisions come down to
-repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
-`mpn_rshift' are next most important.
-
- On some CPUs assembly versions of the internal functions
-`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
-mainly through avoiding function call overheads. They can also
-potentially make better use of a wide superscalar processor, as can
-bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.
-
- The restrictions on overlaps between sources and destinations (*note
-Low-level Functions::) are designed to facilitate a variety of
-implementations. For example, knowing `mpn_add_n' won't have partly
-overlapping sources and destination means reading can be done far ahead
-of writing on superscalar processors, and loops can be vectorized on a
-vector processor, depending on the carry handling.
-
-\1f
-File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding
-
-16.8.3 Carry Propagation
-------------------------
-
-The problem that presents most challenges in GMP is propagating carries
-from one limb to the next. In functions like `mpn_addmul_1' and
-`mpn_add_n', carries are the only dependencies between limb operations.
-
- On processors with carry flags, a straightforward CISC style `adc' is
-generally best. AMD K6 `mpn_addmul_1' however is an example of an
-unusual set of circumstances where a branch works out better.
-
- On RISC processors generally an add and compare for overflow is
-used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
-carry propagation schemes require 4 instructions, meaning at least 4
-cycles per limb, but other schemes may use just 1 or 2. On wide
-superscalar processors performance may be completely determined by the
-number of dependent instructions between carry-in and carry-out for
-each limb.
-
- On vector processors good use can be made of the fact that a carry
-bit only very rarely propagates more than one limb. When adding a
-single bit to a limb, there's only a carry out if that limb was
-`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
-`mpn/cray/add_n.c' is an example of this, it adds all limbs in
-parallel, adds one set of carry bits in parallel and then only rarely
-needs to fall through to a loop propagating further carries.
-
- On the x86s, GCC (as of version 2.95.2) doesn't generate
-particularly good code for the RISC style idioms that are necessary to
-handle carry bits in C. Often conditional jumps are generated where
-`adc' or `sbb' forms would be better. And so unfortunately almost any
-loop involving carry bits needs to be coded in assembly for best
-results.
-
-\1f
-File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding
-
-16.8.4 Cache Handling
----------------------
-
-GMP aims to perform well both on operands that fit entirely in L1 cache
-and those which don't.
-
- Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
-large operands, so L2 and main memory performance is important for them.
-`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
-basecases, so L1 performance matters most for them, unless assembly
-versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
-case the remaining uses are mostly for larger operands.
-
- For L2 or main memory operands, memory access times will almost
-certainly be more than the calculation time. The aim therefore is to
-maximize memory throughput, by starting a load of the next cache line
-while processing the contents of the previous one. Clearly this is
-only possible if the chip has a lock-up free cache or some sort of
-prefetch instruction. Most current chips have both these features.
-
- Prefetching sources combines well with loop unrolling, since a
-prefetch can be initiated once per unrolled loop (or more than once if
-the loop covers more than one cache line).
-
- On CPUs without write-allocate caches, prefetching destinations will
-ensure individual stores don't go further down the cache hierarchy,
-limiting bandwidth. Of course for calculations which are slow anyway,
-like `mpn_divrem_1', write-throughs might be fine.
-
- The distance ahead to prefetch will be determined by memory latency
-versus throughput. The aim of course is to have data arriving
-continuously, at peak throughput. Some CPUs have limits on the number
-of fetches or prefetches in progress.
-
- If a special prefetch instruction doesn't exist then a plain load
-can be used, but in that case care must be taken not to attempt to read
-past the end of an operand, since that might produce a segmentation
-violation.
-
- Some CPUs or systems have hardware that detects sequential memory
-accesses and initiates suitable cache movements automatically, making
-life easy.
-
-\1f
-File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding
-
-16.8.5 Functional Units
------------------------
-
-When choosing an approach for an assembly loop, consideration is given
-to what operations can execute simultaneously and what throughput can
-thereby be achieved. In some cases an algorithm can be tweaked to
-accommodate available resources.
-
- Loop control will generally require a counter and pointer updates,
-costing as much as 5 instructions, plus any delays a branch introduces.
-CPU addressing modes might reduce pointer updates, perhaps by allowing
-just one updating pointer and others expressed as offsets from it, or
-on CISC chips with all addressing done with the loop counter as a
-scaled index.
-
- The final loop control cost can be amortised by processing several
-limbs in each iteration (*note Assembly Loop Unrolling::). This at
-least ensures loop control isn't a big fraction the work done.
-
- Memory throughput is always a limit. If perhaps only one load or
-one store can be done per cycle then 3 cycles/limb will the top speed
-for "binary" operations like `mpn_add_n', and any code achieving that
-is optimal.
-
- Integer resources can be freed up by having the loop counter in a
-float register, or by pressing the float units into use for some
-multiplying, perhaps doing every second limb on the float side (*note
-Assembly Floating Point::).
-
- Float resources can be freed up by doing carry propagation on the
-integer side, or even by doing integer to float conversions in integers
-using bit twiddling.
-
-\1f
-File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding
-
-16.8.6 Floating Point
----------------------
-
-Floating point arithmetic is used in GMP for multiplications on CPUs
-with poor integer multipliers. It's mostly useful for `mpn_mul_1',
-`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
-`mpn_mul_basecase' on both 32-bit and 64-bit machines.
-
- With IEEE 53-bit double precision floats, integer multiplications
-producing up to 53 bits will give exact results. Breaking a 64x64
-multiplication into eight 16x32->48 bit pieces is convenient. With
-some care though six 21x32->53 bit products can be used, if one of the
-lower two 21-bit pieces also uses the sign bit.
-
- For the `mpn_mul_1' family of functions on a 64-bit machine, the
-invariant single limb is split at the start, into 3 or 4 pieces.
-Inside the loop, the bignum operand is split into 32-bit pieces. Fast
-conversion of these unsigned 32-bit pieces to floating point is highly
-machine-dependent. In some cases, reading the data into the integer
-unit, zero-extending to 64-bits, then transferring to the floating
-point unit back via memory is the only option.
-
- Converting partial products back to 64-bit limbs is usually best
-done as a signed conversion. Since all values are smaller than 2^53,
-signed and unsigned are the same, but most processors lack unsigned
-conversions.
-
-
-
- Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
-`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
-into four 16-bit parts. The multi-limb operand U is split in the loop
-into two 32-bit parts.
-
- +---+---+---+---+
- |v48|v32|v16|v00| V operand
- +---+---+---+---+
-
- +-------+---+---+
- x | u32 | u00 | U operand (one limb)
- +---------------+
-
- ---------------------------------
-
- +-----------+
- | u00 x v00 | p00 48-bit products
- +-----------+
- +-----------+
- | u00 x v16 | p16
- +-----------+
- +-----------+
- | u00 x v32 | p32
- +-----------+
- +-----------+
- | u00 x v48 | p48
- +-----------+
- +-----------+
- | u32 x v00 | r32
- +-----------+
- +-----------+
- | u32 x v16 | r48
- +-----------+
- +-----------+
- | u32 x v32 | r64
- +-----------+
- +-----------+
- | u32 x v48 | r80
- +-----------+
-
- p32 and r32 can be summed using floating-point addition, and
-likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from
-the previous iteration.
-
- For each loop then, four 49-bit quantities are transferred to the
-integer unit, aligned as follows,
-
- |-----64bits----|-----64bits----|
- +------------+
- | p00 + r64' | i00
- +------------+
- +------------+
- | p16 + r80' | i16
- +------------+
- +------------+
- | p32 + r32 | i32
- +------------+
- +------------+
- | p48 + r48 | i48
- +------------+
-
- The challenge then is to sum these efficiently and add in a carry
-limb, generating a low 64-bit result limb and a high 33-bit carry limb
-(i48 extends 33 bits into the high half).
-
-\1f
-File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding
-
-16.8.7 SIMD Instructions
-------------------------
-
-The single-instruction multiple-data support in current microprocessors
-is aimed at signal processing algorithms where each data point can be
-treated more or less independently. There's generally not much support
-for propagating the sort of carries that arise in GMP.
-
- SIMD multiplications of say four 16x16 bit multiplies only do as much
-work as one 32x32 from GMP's point of view, and need some shifts and
-adds besides. But of course if say the SIMD form is fully pipelined
-and uses less instruction decoding then it may still be worthwhile.
-
- On the x86 chips, MMX has so far found a use in `mpn_rshift' and
-`mpn_lshift', and is used in a special case for 16-bit multipliers in
-the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1',
-`mpn_addmul_1', and `mpn_submul_1'.
-
-\1f
-File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding
-
-16.8.8 Software Pipelining
---------------------------
-
-Software pipelining consists of scheduling instructions around the
-branch point in a loop. For example a loop might issue a load not for
-use in the present iteration but the next, thereby allowing extra
-cycles for the data to arrive from memory.
-
- Naturally this is wanted only when doing things like loads or
-multiplies that take several cycles to complete, and only where a CPU
-has multiple functional units so that other work can be done in the
-meantime.
-
- A pipeline with several stages will have a data value in progress at
-each stage and each loop iteration moves them along one stage. This is
-like juggling.
-
- If the latency of some instruction is greater than the loop time
-then it will be necessary to unroll, so one register has a result ready
-to use while another (or multiple others) are still in progress.
-(*note Assembly Loop Unrolling::).
-
-\1f
-File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding
-
-16.8.9 Loop Unrolling
----------------------
-
-Loop unrolling consists of replicating code so that several limbs are
-processed in each loop. At a minimum this reduces loop overheads by a
-corresponding factor, but it can also allow better register usage, for
-example alternately using one register combination and then another.
-Judicious use of `m4' macros can help avoid lots of duplication in the
-source code.
-
- Any amount of unrolling can be handled with a loop counter that's
-decremented by N each time, stopping when the remaining count is less
-than the further N the loop will process. Or by subtracting N at the
-start, the termination condition becomes when the counter C is less
-than 0 (and the count of remaining limbs is C+N).
-
- Alternately for a power of 2 unroll the loop count and remainder can
-be established with a shift and mask. This is convenient if also
-making a computed jump into the middle of a large loop.
-
- The limbs not a multiple of the unrolling can be handled in various
-ways, for example
-
- * A simple loop at the end (or the start) to process the excess.
- Care will be wanted that it isn't too much slower than the
- unrolled part.
-
- * A set of binary tests, for example after an 8-limb unrolling, test
- for 4 more limbs to process, then a further 2 more or not, and
- finally 1 more or not. This will probably take more code space
- than a simple loop.
-
- * A `switch' statement, providing separate code for each possible
- excess, for example an 8-limb unrolling would have separate code
- for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
- take a lot of code, but may be the best way to optimize all cases
- in combination with a deep pipelined loop.
-
- * A computed jump into the middle of the loop, thus making the first
- iteration handle the excess. This should make times smoothly
- increase with size, which is attractive, but setups for the jump
- and adjustments for pointers can be tricky and could become quite
- difficult in combination with deep pipelining.
-
-\1f
-File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding
-
-16.8.10 Writing Guide
----------------------
-
-This is a guide to writing software pipelined loops for processing limb
-vectors in assembly.
-
- First determine the algorithm and which instructions are needed.
-Code it without unrolling or scheduling, to make sure it works. On a
-3-operand CPU try to write each new value to a new register, this will
-greatly simplify later steps.
-
- Then note for each instruction the functional unit and/or issue port
-requirements. If an instruction can use either of two units, like U0
-or U1 then make a category "U0/U1". Count the total using each unit
-(or combined unit), and count all instructions.
-
- Figure out from those counts the best possible loop time. The goal
-will be to find a perfect schedule where instruction latencies are
-completely hidden. The total instruction count might be the limiting
-factor, or perhaps a particular functional unit. It might be possible
-to tweak the instructions to help the limiting factor.
-
- Suppose the loop time is N, then make N issue buckets, with the
-final loop branch at the end of the last. Now fill the buckets with
-dummy instructions using the functional units desired. Run this to
-make sure the intended speed is reached.
-
- Now replace the dummy instructions with the real instructions from
-the slow but correct loop you started with. The first will typically
-be a load instruction. Then the instruction using that value is placed
-in a bucket an appropriate distance down. Run the loop again, to check
-it still runs at target speed.
-
- Keep placing instructions, frequently measuring the loop. After a
-few you will need to wrap around from the last bucket back to the top
-of the loop. If you used the new-register for new-value strategy above
-then there will be no register conflicts. If not then take care not to
-clobber something already in use. Changing registers at this time is
-very error prone.
-
- The loop will overlap two or more of the original loop iterations,
-and the computation of one vector element result will be started in one
-iteration of the new loop, and completed one or several iterations
-later.
-
- The final step is to create feed-in and wind-down code for the loop.
-A good way to do this is to make a copy (or copies) of the loop at the
-start and delete those instructions which don't have valid antecedents,
-and at the end replicate and delete those whose results are unwanted
-(including any further loads).
-
- The loop will have a minimum number of limbs loaded and processed,
-so the feed-in code must test if the request size is smaller and skip
-either to a suitable part of the wind-down or to special code for small
-sizes.
-
-\1f
-File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top
-
-17 Internals
-************
-
-*This chapter is provided only for informational purposes and the
-various internals described here may change in future GMP releases.
-Applications expecting to be compatible with future releases should use
-only the documented interfaces described in previous chapters.*
-
-* Menu:
-
-* Integer Internals::
-* Rational Internals::
-* Float Internals::
-* Raw Output Internals::
-* C++ Interface Internals::
-
-\1f
-File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
-
-17.1 Integer Internals
-======================
-
-`mpz_t' variables represent integers using sign and magnitude, in space
-dynamically allocated and reallocated. The fields are as follows.
-
-`_mp_size'
- The number of limbs, or the negative of that when representing a
- negative integer. Zero is represented by `_mp_size' set to zero,
- in which case the `_mp_d' data is unused.
-
-`_mp_d'
- A pointer to an array of limbs which is the magnitude. These are
- stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
- is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
- most significant. Whenever `_mp_size' is non-zero, the most
- significant limb is non-zero.
-
- Currently there's always at least one limb allocated, so for
- instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
- can fetch `_mp_d[0]' unconditionally (though its value is then
- only wanted if `_mp_size' is non-zero).
-
-`_mp_alloc'
- `_mp_alloc' is the number of limbs currently allocated at `_mp_d',
- and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine
- is about to (or might be about to) increase `_mp_size', it checks
- `_mp_alloc' to see whether there's enough space, and reallocates
- if not. `MPZ_REALLOC' is generally used for this.
-
- The various bitwise logical functions like `mpz_and' behave as if
-negative values were twos complement. But sign and magnitude is always
-used internally, and necessary adjustments are made during the
-calculations. Sometimes this isn't pretty, but sign and magnitude are
-best for other routines.
-
- Some internal temporary variables are setup with `MPZ_TMP_INIT' and
-these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
-memory allocation functions. Care is taken to ensure that these are
-big enough that no reallocation is necessary (since it would have
-unpredictable consequences).
-
- `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is
-usually a `long'. This is done to make the fields just 32 bits on some
-64 bits systems, thereby saving a few bytes of data space but still
-providing plenty of range.
-
-\1f
-File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals
-
-17.2 Rational Internals
-=======================
-
-`mpq_t' variables represent rationals using an `mpz_t' numerator and
-denominator (*note Integer Internals::).
-
- The canonical form adopted is denominator positive (and non-zero),
-no common factors between numerator and denominator, and zero uniquely
-represented as 0/1.
-
- It's believed that casting out common factors at each stage of a
-calculation is best in general. A GCD is an O(N^2) operation so it's
-better to do a few small ones immediately than to delay and have to do
-a big one later. Knowing the numerator and denominator have no common
-factors can be used for example in `mpq_mul' to make only two cross
-GCDs necessary, not four.
-
- This general approach to common factors is badly sub-optimal in the
-presence of simple factorizations or little prospect for cancellation,
-but GMP has no way to know when this will occur. As per *Note
-Efficiency::, that's left to applications. The `mpq_t' framework might
-still suit, with `mpq_numref' and `mpq_denref' for direct access to the
-numerator and denominator, or of course `mpz_t' variables can be used
-directly.
-
-\1f
-File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals
-
-17.3 Float Internals
-====================
-
-Efficient calculation is the primary aim of GMP floats and the use of
-whole limbs and simple rounding facilitates this.
-
- `mpf_t' floats have a variable precision mantissa and a single
-machine word signed exponent. The mantissa is represented using sign
-and magnitude.
-
- most least
- significant significant
- limb limb
-
- _mp_d
- |---- _mp_exp ---> |
- _____ _____ _____ _____ _____
- |_____|_____|_____|_____|_____|
- . <------------ radix point
-
- <-------- _mp_size --------->
-
-The fields are as follows.
-
-`_mp_size'
- The number of limbs currently in use, or the negative of that when
- representing a negative value. Zero is represented by `_mp_size'
- and `_mp_exp' both set to zero, and in that case the `_mp_d' data
- is unused. (In the future `_mp_exp' might be undefined when
- representing zero.)
-
-`_mp_prec'
- The precision of the mantissa, in limbs. In any calculation the
- aim is to produce `_mp_prec' limbs of result (the most significant
- being non-zero).
-
-`_mp_d'
- A pointer to the array of limbs which is the absolute value of the
- mantissa. These are stored "little endian" as per the `mpn'
- functions, so `_mp_d[0]' is the least significant limb and
- `_mp_d[ABS(_mp_size)-1]' the most significant.
-
- The most significant limb is always non-zero, but there are no
- other restrictions on its value, in particular the highest 1 bit
- can be anywhere within the limb.
-
- `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
- for convenience (see below). There are no reallocations during a
- calculation, only in a change of precision with `mpf_set_prec'.
-
-`_mp_exp'
- The exponent, in limbs, determining the location of the implied
- radix point. Zero means the radix point is just above the most
- significant limb. Positive values mean a radix point offset
- towards the lower limbs and hence a value >= 1, as for example in
- the diagram above. Negative exponents mean a radix point further
- above the highest limb.
-
- Naturally the exponent can be any value, it doesn't have to fall
- within the limbs as the diagram shows, it can be a long way above
- or a long way below. Limbs other than those included in the
- `{_mp_d,_mp_size}' data are treated as zero.
-
- The `_mp_size' and `_mp_prec' fields are `int', although the
-`mp_size_t' type is usually a `long'. The `_mp_exp' field is usually
-`long'. This is done to make some fields just 32 bits on some 64 bits
-systems, thereby saving a few bytes of data space but still providing
-plenty of precision and a very large range.
-
-
-The following various points should be noted.
-
-Low Zeros
- The least significant limbs `_mp_d[0]' etc can be zero, though
- such low zeros can always be ignored. Routines likely to produce
- low zeros check and avoid them to save time in subsequent
- calculations, but for most routines they're quite unlikely and
- aren't checked.
-
-Mantissa Size Range
- The `_mp_size' count of limbs in use can be less than `_mp_prec' if
- the value can be represented in less. This means low precision
- values or small integers stored in a high precision `mpf_t' can
- still be operated on efficiently.
-
- `_mp_size' can also be greater than `_mp_prec'. Firstly a value is
- allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
- and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
- `_mp_size' unchanged and so the size can be arbitrarily bigger than
- `_mp_prec'.
-
-Rounding
- All rounding is done on limb boundaries. Calculating `_mp_prec'
- limbs with the high non-zero will ensure the application requested
- minimum precision is obtained.
-
- The use of simple "trunc" rounding towards zero is efficient,
- since there's no need to examine extra limbs and increment or
- decrement.
-
-Bit Shifts
- Since the exponent is in limbs, there are no bit shifts in basic
- operations like `mpf_add' and `mpf_mul'. When differing exponents
- are encountered all that's needed is to adjust pointers to line up
- the relevant limbs.
-
- Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
- shifts, but the choice is between an exponent in limbs which
- requires shifts there, or one in bits which requires them almost
- everywhere else.
-
-Use of `_mp_prec+1' Limbs
- The extra limb on `_mp_d' (`_mp_prec+1' rather than just
- `_mp_prec') helps when an `mpf' routine might get a carry from its
- operation. `mpf_add' for instance will do an `mpn_add' of
- `_mp_prec' limbs. If there's no carry then that's the result, but
- if there is a carry then it's stored in the extra limb of space and
- `_mp_size' becomes `_mp_prec+1'.
-
- Whenever `_mp_prec+1' limbs are held in a variable, the low limb
- is not needed for the intended precision, only the `_mp_prec' high
- limbs. But zeroing it out or moving the rest down is unnecessary.
- Subsequent routines reading the value will simply take the high
- limbs they need, and this will be `_mp_prec' if their target has
- that same precision. This is no more than a pointer adjustment,
- and must be checked anyway since the destination precision can be
- different from the sources.
-
- Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
- if available. This ensures that a variable which has `_mp_size'
- equal to `_mp_prec+1' will get its full exact value copied.
- Strictly speaking this is unnecessary since only `_mp_prec' limbs
- are needed for the application's requested precision, but it's
- considered that an `mpf_set' from one variable into another of the
- same precision ought to produce an exact copy.
-
-Application Precisions
- `__GMPF_BITS_TO_PREC' converts an application requested precision
- to an `_mp_prec'. The value in bits is rounded up to a whole limb
- then an extra limb is added since the most significant limb of
- `_mp_d' is only non-zero and therefore might contain only one bit.
-
- `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
- extra limb from `_mp_prec' before converting to bits. The net
- effect of reading back with `mpf_get_prec' is simply the precision
- rounded up to a multiple of `mp_bits_per_limb'.
-
- Note that the extra limb added here for the high only being
- non-zero is in addition to the extra limb allocated to `_mp_d'.
- For example with a 32-bit limb, an application request for 250
- bits will be rounded up to 8 limbs, then an extra added for the
- high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then
- gets 10 limbs allocated. Reading back with `mpf_get_prec' will
- take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
- bits.
-
- Strictly speaking, the fact the high limb has at least one bit
- means that a float with, say, 3 limbs of 32-bits each will be
- holding at least 65 bits, but for the purposes of `mpf_t' it's
- considered simply to be 64 bits, a nice multiple of the limb size.
-
-\1f
-File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals
-
-17.4 Raw Output Internals
-=========================
-
-`mpz_out_raw' uses the following format.
-
- +------+------------------------+
- | size | data bytes |
- +------+------------------------+
-
- The size is 4 bytes written most significant byte first, being the
-number of subsequent data bytes, or the twos complement negative of
-that when a negative integer is represented. The data bytes are the
-absolute value of the integer, written most significant byte first.
-
- The most significant data byte is always non-zero, so the output is
-the same on all systems, irrespective of limb size.
-
- In GMP 1, leading zero bytes were written to pad the data bytes to a
-multiple of the limb size. `mpz_inp_raw' will still accept this, for
-compatibility.
-
- The use of "big endian" for both the size and data fields is
-deliberate, it makes the data easy to read in a hex dump of a file.
-Unfortunately it also means that the limb data must be reversed when
-reading or writing, so neither a big endian nor little endian system
-can just read and write `_mp_d'.
-
-\1f
-File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals
-
-17.5 C++ Interface Internals
-============================
-
-A system of expression templates is used to ensure something like
-`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the
-scheme also ensures the precision of the final destination is used for
-any temporaries within a statement like `f=w*x+y*z'. These are
-important features which a naive implementation cannot provide.
-
- A simplified description of the scheme follows. The true scheme is
-complicated by the fact that expressions have different return types.
-For detailed information, refer to the source code.
-
- To perform an operation, say, addition, we first define a "function
-object" evaluating it,
-
- struct __gmp_binary_plus
- {
- static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
- };
-
-And an "additive expression" object,
-
- __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
- operator+(const mpf_class &f, const mpf_class &g)
- {
- return __gmp_expr
- <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
- }
-
- The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
-to encapsulate any possible kind of expression into a single template
-type. In fact even `mpf_class' etc are `typedef' specializations of
-`__gmp_expr'.
-
- Next we define assignment of `__gmp_expr' to `mpf_class'.
-
- template <class T>
- mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
- {
- expr.eval(this->get_mpf_t(), this->precision());
- return *this;
- }
-
- template <class Op>
- void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
- (mpf_t f, mp_bitcnt_t precision)
- {
- Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
- }
-
- where `expr.val1' and `expr.val2' are references to the expression's
-operands (here `expr' is the `__gmp_binary_expr' stored within the
-`__gmp_expr').
-
- This way, the expression is actually evaluated only at the time of
-assignment, when the required precision (that of `f') is known.
-Furthermore the target `mpf_t' is now available, thus we can call
-`mpf_add' directly with `f' as the output argument.
-
- Compound expressions are handled by defining operators taking
-subexpressions as their arguments, like this:
-
- template <class T, class U>
- __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
- {
- return __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- (expr1, expr2);
- }
-
- And the corresponding specializations of `__gmp_expr::eval':
-
- template <class T, class U, class Op>
- void __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
- (mpf_t f, mp_bitcnt_t precision)
- {
- // declare two temporaries
- mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
- Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
- }
-
- The expression is thus recursively evaluated to any level of
-complexity and all subexpressions are evaluated to the precision of `f'.
-
-\1f
-File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top
-
-Appendix A Contributors
-***********************
-
-Torbjo"rn Granlund wrote the original GMP library and is still the main
-developer. Code not explicitly attributed to others, was contributed by
-Torbjo"rn. Several other individuals and organizations have contributed
-GMP. Here is a list in chronological order on first contribution:
-
- Gunnar Sjo"din and Hans Riesel helped with mathematical problems in
-early versions of the library.
-
- Richard Stallman helped with the interface design and revised the
-first version of this manual.
-
- Brian Beuning and Doug Lea helped with testing of early versions of
-the library and made creative suggestions.
-
- John Amanatides of York University in Canada contributed the function
-`mpz_probab_prime_p'.
-
- Paul Zimmermann wrote the REDC-based mpz_powm code, the
-Scho"nhage-Strassen FFT multiply code, and the Karatsuba square root
-code. He also improved the Toom3 code for GMP 4.2. Paul sparked the
-development of GMP 2, with his comparisons between bignum packages.
-The ECMNET project Paul is organizing was a driving force behind many
-of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth
-root code (with Torbjo"rn).
-
- Ken Weber (Kent State University, Universidade Federal do Rio Grande
-do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact',
-`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil)
-grant 301314194-2.
-
- Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
-configure. He has also made valuable suggestions and tested numerous
-intermediary releases.
-
- Joachim Hollman was involved in the design of the `mpf' interface,
-and in the `mpz' design revisions for version 2.
-
- Bennet Yee contributed the initial versions of `mpz_jacobi' and
-`mpz_legendre'.
-
- Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
-`mpn/m68k/rshift.S' (now in `.asm' form).
-
- Robert Harley of Inria, France and David Seal of ARM, England,
-suggested clever improvements for population count. Robert also wrote
-highly optimized Karatsuba and 3-way Toom multiplication functions for
-GMP 3, and contributed the ARM assembly code.
-
- Torsten Ekedahl of the Mathematical department of Stockholm
-University provided significant inspiration during several phases of
-the GMP development. His mathematical expertise helped improve several
-algorithms.
-
- Linus Nordberg wrote the new configure system based on autoconf and
-implemented the new random functions.
-
- Kevin Ryde worked on a large number of things: optimized x86 code,
-m4 asm macros, parameter tuning, speed measuring, the configure system,
-function inlining, divisibility tests, bit scanning, Jacobi symbols,
-Fibonacci and Lucas number functions, printf and scanf functions, perl
-interface, demo expression parser, the algorithms chapter in the
-manual, `gmpasm-mode.el', and various miscellaneous improvements
-elsewhere.
-
- Kent Boortz made the Mac OS 9 port.
-
- Steve Root helped write the optimized alpha 21264 assembly code.
-
- Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
-`istream' input routines.
-
- Jason Moxham rewrote `mpz_fac_ui'.
-
- Pedro Gimeno implemented the Mersenne Twister and made other random
-number improvements.
-
- Niels Mo"ller wrote the sub-quadratic GCD and extended GCD code, the
-quadratic Hensel division code, and (with Torbjo"rn) the new divide and
-conquer division code for GMP 4.3. Niels also helped implement the new
-Toom multiply code for GMP 4.3 and implemented helper functions to
-simplify Toom evaluations for GMP 5.0. He wrote the original version
-of mpn_mulmod_bnm1.
-
- Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply
-strategy, and found the optimal strategies for evaluation and
-interpolation in Toom multiplication.
-
- Marco Bodrato helped implement the new Toom multiply code for GMP
-4.3 and implemented most of the new Toom multiply and squaring code for
-5.0. He is the main author of the current mpn_mulmod_bnm1 and
-mpn_mullo_n. Marco also wrote the functions mpn_invert and
-mpn_invertappr.
-
- David Harvey suggested the internal function `mpn_bdiv_dbm1',
-implementing division relevant to Toom multiplication. He also worked
-on fast assembly sequences, in particular on a fast AMD64
-`mpn_mul_basecase'.
-
- Martin Boij wrote `mpn_perfect_power_p'.
-
- (This list is chronological, not ordered after significance. If you
-have contributed to GMP but are not listed above, please tell
-<gmp-devel@gmplib.org> about the omission!)
-
- The development of floating point functions of GNU MP 2, were
-supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
-project POSSO (POlynomial System SOlving).
-
- The development of GMP 2, 3, and 4 was supported in part by the IDA
-Center for Computing Sciences.
-
- Thanks go to Hans Thorsen for donating an SGI system for the GMP
-test system environment.
-
-\1f
-File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
-
-Appendix B References
-*********************
-
-B.1 Books
-=========
-
- * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
- in Analytic Number Theory and Computational Complexity", Wiley,
- 1998.
-
- * Richard Crandall and Carl Pomerance, "Prime Numbers: A
- Computational Perspective", 2nd edition, Springer-Verlag, 2005.
- `http://math.dartmouth.edu/~carlp/'
-
- * Henri Cohen, "A Course in Computational Algebraic Number Theory",
- Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
- `http://www.math.u-bordeaux.fr/~cohen/'
-
- * Donald E. Knuth, "The Art of Computer Programming", volume 2,
- "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
- `http://www-cs-faculty.stanford.edu/~knuth/taocp.html'
-
- * John D. Lipson, "Elements of Algebra and Algebraic Computing", The
- Benjamin Cummings Publishing Company Inc, 1981.
-
- * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
- "Handbook of Applied Cryptography",
- `http://www.cacr.math.uwaterloo.ca/hac/'
-
- * Richard M. Stallman and the GCC Developer Community, "Using the
- GNU Compiler Collection", Free Software Foundation, 2008,
- available online `http://gcc.gnu.org/onlinedocs/', and in the GCC
- package `ftp://ftp.gnu.org/gnu/gcc/'
-
-B.2 Papers
-==========
-
- * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
- Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
- 225-252. Also available online as INRIA Research Report 4475,
- June 2001, `http://www.inria.fr/rrrt/rr-4475.html'
-
- * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
- Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
- `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'
-
- * Torbjo"rn Granlund and Peter L. Montgomery, "Division by Invariant
- Integers using Multiplication", in Proceedings of the SIGPLAN
- PLDI'94 Conference, June 1994. Also available
- `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz).
-
- * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
- invariant integers", to appear.
-
- * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large
- and small", to appear.
-
- * Tudor Jebelean, "An algorithm for exact division", Journal of
- Symbolic Computation, volume 15, 1993, pp. 169-180. Research
- report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
-
- * Tudor Jebelean, "Exact Division with Karatsuba Complexity -
- Extended Abstract", RISC-Linz technical report 96-31,
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'
-
- * Tudor Jebelean, "Practical Integer Division with Karatsuba
- Complexity", ISSAC 97, pp. 339-341. Technical report available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'
-
- * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
- ISSAC 93, pp. 111-116. Technical report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'
-
- * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
- Finding the GCD of Long Integers", Journal of Symbolic
- Computation, volume 19, 1995, pp. 145-157. Technical report
- version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'
-
- * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
- Division", Journal of Symbolic Computation, volume 21, 1996, pp.
- 441-455. Early technical report version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'
-
- * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
- 623-dimensionally equidistributed uniform pseudorandom number
- generator", ACM Transactions on Modelling and Computer Simulation,
- volume 8, January 1998, pp. 3-30. Available online
- `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz'
- (or .pdf)
-
- * R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
- Proceedings of the 13th Annual IEEE Symposium on Switching and
- Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast
- Modular Transforms", Journal of Computer and System Sciences,
- volume 8, number 3, June 1974, pp. 366-386.
-
- * Niels Mo"ller, "On Scho"nhage's algorithm and subquadratic integer
- GCD computation", in Mathematics of Computation, volume 77,
- January 2008, pp. 589-607.
-
- * Peter L. Montgomery, "Modular Multiplication Without Trial
- Division", in Mathematics of Computation, volume 44, number 170,
- April 1985.
-
- * Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation
- grosser Zahlen", Computing 7, 1971, pp. 281-292.
-
- * Kenneth Weber, "The accelerated integer GCD algorithm", ACM
- Transactions on Mathematical Software, volume 21, number 1, March
- 1995, pp. 111-122.
-
- * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
- 3805, November 1999, `http://www.inria.fr/rrrt/rr-3805.html'
-
- * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
- Implementations",
- `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
-
- * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
- IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
- Reprinted as "More on Multiplying and Squaring Large Integers",
- IEEE Transactions on Computers, volume 43, number 8, August 1994,
- pp. 899-908.
-
-\1f
-File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
-
-Appendix C GNU Free Documentation License
-*****************************************
-
- Version 1.3, 3 November 2008
-
- Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
- `http://fsf.org/'
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-
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- It complements the GNU General Public License, which is a copyleft
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- public wiki that anybody can edit is an example of such a server.
- A "Massive Multiauthor Collaboration" (or "MMC") contained in the
- site means any set of copyrightable works thus published on the MMC
- site.
-
- "CC-BY-SA" means the Creative Commons Attribution-Share Alike 3.0
- license published by Creative Commons Corporation, a not-for-profit
- corporation with a principal place of business in San Francisco,
- California, as well as future copyleft versions of that license
- published by that same organization.
-
- "Incorporate" means to publish or republish a Document, in whole or
- in part, as part of another Document.
-
- An MMC is "eligible for relicensing" if it is licensed under this
- License, and if all works that were first published under this
- License somewhere other than this MMC, and subsequently
- incorporated in whole or in part into the MMC, (1) had no cover
- texts or invariant sections, and (2) were thus incorporated prior
- to November 1, 2008.
-
- The operator of an MMC Site may republish an MMC contained in the
- site under CC-BY-SA on the same site at any time before August 1,
- 2009, provided the MMC is eligible for relicensing.
-
-
-ADDENDUM: How to use this License for your documents
-====================================================
-
-To use this License in a document you have written, include a copy of
-the License in the document and put the following copyright and license
-notices just after the title page:
-
- Copyright (C) YEAR YOUR NAME.
- Permission is granted to copy, distribute and/or modify this document
- under the terms of the GNU Free Documentation License, Version 1.3
- or any later version published by the Free Software Foundation;
- with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
- Texts. A copy of the license is included in the section entitled ``GNU
- Free Documentation License''.
-
- If you have Invariant Sections, Front-Cover Texts and Back-Cover
-Texts, replace the "with...Texts." line with this:
-
- with the Invariant Sections being LIST THEIR TITLES, with
- the Front-Cover Texts being LIST, and with the Back-Cover Texts
- being LIST.
-
- If you have Invariant Sections without Cover Texts, or some other
-combination of the three, merge those two alternatives to suit the
-situation.
-
- If your document contains nontrivial examples of program code, we
-recommend releasing these examples in parallel under your choice of
-free software license, such as the GNU General Public License, to
-permit their use in free software.
-
-\1f
-File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top
-
-Concept Index
-*************
-
-\0\b[index\0\b]
-* Menu:
-
-* #include: Headers and Libraries.
- (line 6)
-* --build: Build Options. (line 52)
-* --disable-fft: Build Options. (line 317)
-* --disable-shared: Build Options. (line 45)
-* --disable-static: Build Options. (line 45)
-* --enable-alloca: Build Options. (line 278)
-* --enable-assert: Build Options. (line 327)
-* --enable-cxx: Build Options. (line 230)
-* --enable-fat: Build Options. (line 164)
-* --enable-mpbsd: Build Options. (line 322)
-* --enable-profiling <1>: Profiling. (line 6)
-* --enable-profiling: Build Options. (line 331)
-* --exec-prefix: Build Options. (line 32)
-* --host: Build Options. (line 66)
-* --prefix: Build Options. (line 32)
-* -finstrument-functions: Profiling. (line 66)
-* 2exp functions: Efficiency. (line 43)
-* 68000: Notes for Particular Systems.
- (line 80)
-* 80x86: Notes for Particular Systems.
- (line 126)
-* ABI <1>: Build Options. (line 171)
-* ABI: ABI and ISA. (line 6)
-* About this manual: Introduction to GMP. (line 58)
-* AC_CHECK_LIB: Autoconf. (line 11)
-* AIX <1>: ABI and ISA. (line 184)
-* AIX <2>: Notes for Particular Systems.
- (line 7)
-* AIX: ABI and ISA. (line 169)
-* Algorithms: Algorithms. (line 6)
-* alloca: Build Options. (line 278)
-* Allocation of memory: Custom Allocation. (line 6)
-* AMD64: ABI and ISA. (line 44)
-* Anonymous FTP of latest version: Introduction to GMP. (line 38)
-* Application Binary Interface: ABI and ISA. (line 6)
-* Arithmetic functions <1>: Float Arithmetic. (line 6)
-* Arithmetic functions <2>: Integer Arithmetic. (line 6)
-* Arithmetic functions: Rational Arithmetic. (line 6)
-* ARM: Notes for Particular Systems.
- (line 20)
-* Assembly cache handling: Assembly Cache Handling.
- (line 6)
-* Assembly carry propagation: Assembly Carry Propagation.
- (line 6)
-* Assembly code organisation: Assembly Code Organisation.
- (line 6)
-* Assembly coding: Assembly Coding. (line 6)
-* Assembly floating Point: Assembly Floating Point.
- (line 6)
-* Assembly loop unrolling: Assembly Loop Unrolling.
- (line 6)
-* Assembly SIMD: Assembly SIMD Instructions.
- (line 6)
-* Assembly software pipelining: Assembly Software Pipelining.
- (line 6)
-* Assembly writing guide: Assembly Writing Guide.
- (line 6)
-* Assertion checking <1>: Debugging. (line 79)
-* Assertion checking: Build Options. (line 327)
-* Assignment functions <1>: Assigning Floats. (line 6)
-* Assignment functions <2>: Initializing Rationals.
- (line 6)
-* Assignment functions <3>: Simultaneous Integer Init & Assign.
- (line 6)
-* Assignment functions <4>: Simultaneous Float Init & Assign.
- (line 6)
-* Assignment functions: Assigning Integers. (line 6)
-* Autoconf: Autoconf. (line 6)
-* Basics: GMP Basics. (line 6)
-* Berkeley MP compatible functions <1>: Build Options. (line 322)
-* Berkeley MP compatible functions: BSD Compatible Functions.
- (line 6)
-* Binomial coefficient algorithm: Binomial Coefficients Algorithm.
- (line 6)
-* Binomial coefficient functions: Number Theoretic Functions.
- (line 100)
-* Binutils strip: Known Build Problems.
- (line 28)
-* Bit manipulation functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Bit scanning functions: Integer Logic and Bit Fiddling.
- (line 38)
-* Bit shift left: Integer Arithmetic. (line 35)
-* Bit shift right: Integer Division. (line 53)
-* Bits per limb: Useful Macros and Constants.
- (line 7)
-* BSD MP compatible functions <1>: Build Options. (line 322)
-* BSD MP compatible functions: BSD Compatible Functions.
- (line 6)
-* Bug reporting: Reporting Bugs. (line 6)
-* Build directory: Build Options. (line 19)
-* Build notes for binary packaging: Notes for Package Builds.
- (line 6)
-* Build notes for particular systems: Notes for Particular Systems.
- (line 6)
-* Build options: Build Options. (line 6)
-* Build problems known: Known Build Problems.
- (line 6)
-* Build system: Build Options. (line 52)
-* Building GMP: Installing GMP. (line 6)
-* Bus error: Debugging. (line 7)
-* C compiler: Build Options. (line 182)
-* C++ compiler: Build Options. (line 254)
-* C++ interface: C++ Class Interface. (line 6)
-* C++ interface internals: C++ Interface Internals.
- (line 6)
-* C++ istream input: C++ Formatted Input. (line 6)
-* C++ ostream output: C++ Formatted Output.
- (line 6)
-* C++ support: Build Options. (line 230)
-* CC: Build Options. (line 182)
-* CC_FOR_BUILD: Build Options. (line 217)
-* CFLAGS: Build Options. (line 182)
-* Checker: Debugging. (line 115)
-* checkergcc: Debugging. (line 122)
-* Code organisation: Assembly Code Organisation.
- (line 6)
-* Compaq C++: Notes for Particular Systems.
- (line 25)
-* Comparison functions <1>: Integer Comparisons. (line 6)
-* Comparison functions <2>: Comparing Rationals. (line 6)
-* Comparison functions: Float Comparison. (line 6)
-* Compatibility with older versions: Compatibility with older versions.
- (line 6)
-* Conditions for copying GNU MP: Copying. (line 6)
-* Configuring GMP: Installing GMP. (line 6)
-* Congruence algorithm: Exact Remainder. (line 29)
-* Congruence functions: Integer Division. (line 124)
-* Constants: Useful Macros and Constants.
- (line 6)
-* Contributors: Contributors. (line 6)
-* Conventions for parameters: Parameter Conventions.
- (line 6)
-* Conventions for variables: Variable Conventions.
- (line 6)
-* Conversion functions <1>: Converting Integers. (line 6)
-* Conversion functions <2>: Converting Floats. (line 6)
-* Conversion functions: Rational Conversions.
- (line 6)
-* Copying conditions: Copying. (line 6)
-* CPPFLAGS: Build Options. (line 208)
-* CPU types <1>: Introduction to GMP. (line 24)
-* CPU types: Build Options. (line 108)
-* Cross compiling: Build Options. (line 66)
-* Custom allocation: Custom Allocation. (line 6)
-* CXX: Build Options. (line 254)
-* CXXFLAGS: Build Options. (line 254)
-* Cygwin: Notes for Particular Systems.
- (line 43)
-* Darwin: Known Build Problems.
- (line 51)
-* Debugging: Debugging. (line 6)
-* Demonstration programs: Demonstration Programs.
- (line 6)
-* Digits in an integer: Miscellaneous Integer Functions.
- (line 23)
-* Divisibility algorithm: Exact Remainder. (line 29)
-* Divisibility functions: Integer Division. (line 124)
-* Divisibility testing: Efficiency. (line 91)
-* Division algorithms: Division Algorithms. (line 6)
-* Division functions <1>: Rational Arithmetic. (line 22)
-* Division functions <2>: Integer Division. (line 6)
-* Division functions: Float Arithmetic. (line 33)
-* DJGPP <1>: Notes for Particular Systems.
- (line 43)
-* DJGPP: Known Build Problems.
- (line 18)
-* DLLs: Notes for Particular Systems.
- (line 56)
-* DocBook: Build Options. (line 354)
-* Documentation formats: Build Options. (line 347)
-* Documentation license: GNU Free Documentation License.
- (line 6)
-* DVI: Build Options. (line 350)
-* Efficiency: Efficiency. (line 6)
-* Emacs: Emacs. (line 6)
-* Exact division functions: Integer Division. (line 102)
-* Exact remainder: Exact Remainder. (line 6)
-* Example programs: Demonstration Programs.
- (line 6)
-* Exec prefix: Build Options. (line 32)
-* Execution profiling <1>: Profiling. (line 6)
-* Execution profiling: Build Options. (line 331)
-* Exponentiation functions <1>: Integer Exponentiation.
- (line 6)
-* Exponentiation functions: Float Arithmetic. (line 41)
-* Export: Integer Import and Export.
- (line 45)
-* Expression parsing demo: Demonstration Programs.
- (line 18)
-* Extended GCD: Number Theoretic Functions.
- (line 45)
-* Factor removal functions: Number Theoretic Functions.
- (line 90)
-* Factorial algorithm: Factorial Algorithm. (line 6)
-* Factorial functions: Number Theoretic Functions.
- (line 95)
-* Factorization demo: Demonstration Programs.
- (line 25)
-* Fast Fourier Transform: FFT Multiplication. (line 6)
-* Fat binary: Build Options. (line 164)
-* FFT multiplication <1>: FFT Multiplication. (line 6)
-* FFT multiplication: Build Options. (line 317)
-* Fibonacci number algorithm: Fibonacci Numbers Algorithm.
- (line 6)
-* Fibonacci sequence functions: Number Theoretic Functions.
- (line 108)
-* Float arithmetic functions: Float Arithmetic. (line 6)
-* Float assignment functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Float assignment functions: Assigning Floats. (line 6)
-* Float comparison functions: Float Comparison. (line 6)
-* Float conversion functions: Converting Floats. (line 6)
-* Float functions: Floating-point Functions.
- (line 6)
-* Float initialization functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Float initialization functions: Initializing Floats. (line 6)
-* Float input and output functions: I/O of Floats. (line 6)
-* Float internals: Float Internals. (line 6)
-* Float miscellaneous functions: Miscellaneous Float Functions.
- (line 6)
-* Float random number functions: Miscellaneous Float Functions.
- (line 27)
-* Float rounding functions: Miscellaneous Float Functions.
- (line 9)
-* Float sign tests: Float Comparison. (line 33)
-* Floating point mode: Notes for Particular Systems.
- (line 34)
-* Floating-point functions: Floating-point Functions.
- (line 6)
-* Floating-point number: Nomenclature and Types.
- (line 21)
-* fnccheck: Profiling. (line 77)
-* Formatted input: Formatted Input. (line 6)
-* Formatted output: Formatted Output. (line 6)
-* Free Documentation License: GNU Free Documentation License.
- (line 6)
-* frexp <1>: Converting Floats. (line 23)
-* frexp: Converting Integers. (line 42)
-* FTP of latest version: Introduction to GMP. (line 38)
-* Function classes: Function Classes. (line 6)
-* FunctionCheck: Profiling. (line 77)
-* GCC Checker: Debugging. (line 115)
-* GCD algorithms: Greatest Common Divisor Algorithms.
- (line 6)
-* GCD extended: Number Theoretic Functions.
- (line 45)
-* GCD functions: Number Theoretic Functions.
- (line 30)
-* GDB: Debugging. (line 58)
-* Generic C: Build Options. (line 153)
-* GMP Perl module: Demonstration Programs.
- (line 35)
-* GMP version number: Useful Macros and Constants.
- (line 12)
-* gmp.h: Headers and Libraries.
- (line 6)
-* gmpxx.h: C++ Interface General.
- (line 8)
-* GNU Debugger: Debugging. (line 58)
-* GNU Free Documentation License: GNU Free Documentation License.
- (line 6)
-* GNU strip: Known Build Problems.
- (line 28)
-* gprof: Profiling. (line 41)
-* Greatest common divisor algorithms: Greatest Common Divisor Algorithms.
- (line 6)
-* Greatest common divisor functions: Number Theoretic Functions.
- (line 30)
-* Hardware floating point mode: Notes for Particular Systems.
- (line 34)
-* Headers: Headers and Libraries.
- (line 6)
-* Heap problems: Debugging. (line 24)
-* Home page: Introduction to GMP. (line 34)
-* Host system: Build Options. (line 66)
-* HP-UX: ABI and ISA. (line 107)
-* HPPA: ABI and ISA. (line 68)
-* I/O functions <1>: I/O of Integers. (line 6)
-* I/O functions <2>: I/O of Rationals. (line 6)
-* I/O functions: I/O of Floats. (line 6)
-* i386: Notes for Particular Systems.
- (line 126)
-* IA-64: ABI and ISA. (line 107)
-* Import: Integer Import and Export.
- (line 11)
-* In-place operations: Efficiency. (line 57)
-* Include files: Headers and Libraries.
- (line 6)
-* info-lookup-symbol: Emacs. (line 6)
-* Initialization functions <1>: Initializing Integers.
- (line 6)
-* Initialization functions <2>: Initializing Rationals.
- (line 6)
-* Initialization functions <3>: Random State Initialization.
- (line 6)
-* Initialization functions <4>: Simultaneous Float Init & Assign.
- (line 6)
-* Initialization functions <5>: Simultaneous Integer Init & Assign.
- (line 6)
-* Initialization functions: Initializing Floats. (line 6)
-* Initializing and clearing: Efficiency. (line 21)
-* Input functions <1>: I/O of Integers. (line 6)
-* Input functions <2>: I/O of Rationals. (line 6)
-* Input functions <3>: I/O of Floats. (line 6)
-* Input functions: Formatted Input Functions.
- (line 6)
-* Install prefix: Build Options. (line 32)
-* Installing GMP: Installing GMP. (line 6)
-* Instruction Set Architecture: ABI and ISA. (line 6)
-* instrument-functions: Profiling. (line 66)
-* Integer: Nomenclature and Types.
- (line 6)
-* Integer arithmetic functions: Integer Arithmetic. (line 6)
-* Integer assignment functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
-* Integer assignment functions: Assigning Integers. (line 6)
-* Integer bit manipulation functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Integer comparison functions: Integer Comparisons. (line 6)
-* Integer conversion functions: Converting Integers. (line 6)
-* Integer division functions: Integer Division. (line 6)
-* Integer exponentiation functions: Integer Exponentiation.
- (line 6)
-* Integer export: Integer Import and Export.
- (line 45)
-* Integer functions: Integer Functions. (line 6)
-* Integer import: Integer Import and Export.
- (line 11)
-* Integer initialization functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
-* Integer initialization functions: Initializing Integers.
- (line 6)
-* Integer input and output functions: I/O of Integers. (line 6)
-* Integer internals: Integer Internals. (line 6)
-* Integer logical functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Integer miscellaneous functions: Miscellaneous Integer Functions.
- (line 6)
-* Integer random number functions: Integer Random Numbers.
- (line 6)
-* Integer root functions: Integer Roots. (line 6)
-* Integer sign tests: Integer Comparisons. (line 28)
-* Integer special functions: Integer Special Functions.
- (line 6)
-* Interix: Notes for Particular Systems.
- (line 51)
-* Internals: Internals. (line 6)
-* Introduction: Introduction to GMP. (line 6)
-* Inverse modulo functions: Number Theoretic Functions.
- (line 60)
-* IRIX <1>: Known Build Problems.
- (line 38)
-* IRIX: ABI and ISA. (line 132)
-* ISA: ABI and ISA. (line 6)
-* istream input: C++ Formatted Input. (line 6)
-* Jacobi symbol algorithm: Jacobi Symbol. (line 6)
-* Jacobi symbol functions: Number Theoretic Functions.
- (line 66)
-* Karatsuba multiplication: Karatsuba Multiplication.
- (line 6)
-* Karatsuba square root algorithm: Square Root Algorithm.
- (line 6)
-* Kronecker symbol functions: Number Theoretic Functions.
- (line 78)
-* Language bindings: Language Bindings. (line 6)
-* Latest version of GMP: Introduction to GMP. (line 38)
-* LCM functions: Number Theoretic Functions.
- (line 55)
-* Least common multiple functions: Number Theoretic Functions.
- (line 55)
-* Legendre symbol functions: Number Theoretic Functions.
- (line 69)
-* libgmp: Headers and Libraries.
- (line 22)
-* libgmpxx: Headers and Libraries.
- (line 27)
-* Libraries: Headers and Libraries.
- (line 22)
-* Libtool: Headers and Libraries.
- (line 33)
-* Libtool versioning: Notes for Package Builds.
- (line 9)
-* License conditions: Copying. (line 6)
-* Limb: Nomenclature and Types.
- (line 31)
-* Limb size: Useful Macros and Constants.
- (line 7)
-* Linear congruential algorithm: Random Number Algorithms.
- (line 25)
-* Linear congruential random numbers: Random State Initialization.
- (line 32)
-* Linking: Headers and Libraries.
- (line 22)
-* Logical functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Low-level functions: Low-level Functions. (line 6)
-* Lucas number algorithm: Lucas Numbers Algorithm.
- (line 6)
-* Lucas number functions: Number Theoretic Functions.
- (line 119)
-* MacOS X: Known Build Problems.
- (line 51)
-* Mailing lists: Introduction to GMP. (line 45)
-* Malloc debugger: Debugging. (line 30)
-* Malloc problems: Debugging. (line 24)
-* Memory allocation: Custom Allocation. (line 6)
-* Memory management: Memory Management. (line 6)
-* Mersenne twister algorithm: Random Number Algorithms.
- (line 17)
-* Mersenne twister random numbers: Random State Initialization.
- (line 13)
-* MINGW: Notes for Particular Systems.
- (line 43)
-* MIPS: ABI and ISA. (line 132)
-* Miscellaneous float functions: Miscellaneous Float Functions.
- (line 6)
-* Miscellaneous integer functions: Miscellaneous Integer Functions.
- (line 6)
-* MMX: Notes for Particular Systems.
- (line 132)
-* Modular inverse functions: Number Theoretic Functions.
- (line 60)
-* Most significant bit: Miscellaneous Integer Functions.
- (line 34)
-* mp.h: BSD Compatible Functions.
- (line 21)
-* MPN_PATH: Build Options. (line 335)
-* MS Windows: Notes for Particular Systems.
- (line 56)
-* MS-DOS: Notes for Particular Systems.
- (line 43)
-* Multi-threading: Reentrancy. (line 6)
-* Multiplication algorithms: Multiplication Algorithms.
- (line 6)
-* Nails: Low-level Functions. (line 478)
-* Native compilation: Build Options. (line 52)
-* NeXT: Known Build Problems.
- (line 57)
-* Next prime function: Number Theoretic Functions.
- (line 23)
-* Nomenclature: Nomenclature and Types.
- (line 6)
-* Non-Unix systems: Build Options. (line 11)
-* Nth root algorithm: Nth Root Algorithm. (line 6)
-* Number sequences: Efficiency. (line 147)
-* Number theoretic functions: Number Theoretic Functions.
- (line 6)
-* Numerator and denominator: Applying Integer Functions.
- (line 6)
-* obstack output: Formatted Output Functions.
- (line 81)
-* OpenBSD: Notes for Particular Systems.
- (line 86)
-* Optimizing performance: Performance optimization.
- (line 6)
-* ostream output: C++ Formatted Output.
- (line 6)
-* Other languages: Language Bindings. (line 6)
-* Output functions <1>: I/O of Floats. (line 6)
-* Output functions <2>: I/O of Rationals. (line 6)
-* Output functions <3>: Formatted Output Functions.
- (line 6)
-* Output functions: I/O of Integers. (line 6)
-* Packaged builds: Notes for Package Builds.
- (line 6)
-* Parameter conventions: Parameter Conventions.
- (line 6)
-* Parsing expressions demo: Demonstration Programs.
- (line 21)
-* Particular systems: Notes for Particular Systems.
- (line 6)
-* Past GMP versions: Compatibility with older versions.
- (line 6)
-* PDF: Build Options. (line 350)
-* Perfect power algorithm: Perfect Power Algorithm.
- (line 6)
-* Perfect power functions: Integer Roots. (line 27)
-* Perfect square algorithm: Perfect Square Algorithm.
- (line 6)
-* Perfect square functions: Integer Roots. (line 36)
-* perl: Demonstration Programs.
- (line 35)
-* Perl module: Demonstration Programs.
- (line 35)
-* Postscript: Build Options. (line 350)
-* Power/PowerPC <1>: Known Build Problems.
- (line 63)
-* Power/PowerPC: Notes for Particular Systems.
- (line 92)
-* Powering algorithms: Powering Algorithms. (line 6)
-* Powering functions <1>: Float Arithmetic. (line 41)
-* Powering functions: Integer Exponentiation.
- (line 6)
-* PowerPC: ABI and ISA. (line 167)
-* Precision of floats: Floating-point Functions.
- (line 6)
-* Precision of hardware floating point: Notes for Particular Systems.
- (line 34)
-* Prefix: Build Options. (line 32)
-* Prime testing algorithms: Prime Testing Algorithm.
- (line 6)
-* Prime testing functions: Number Theoretic Functions.
- (line 7)
-* printf formatted output: Formatted Output. (line 6)
-* Probable prime testing functions: Number Theoretic Functions.
- (line 7)
-* prof: Profiling. (line 24)
-* Profiling: Profiling. (line 6)
-* Radix conversion algorithms: Radix Conversion Algorithms.
- (line 6)
-* Random number algorithms: Random Number Algorithms.
- (line 6)
-* Random number functions <1>: Integer Random Numbers.
- (line 6)
-* Random number functions <2>: Miscellaneous Float Functions.
- (line 27)
-* Random number functions: Random Number Functions.
- (line 6)
-* Random number seeding: Random State Seeding.
- (line 6)
-* Random number state: Random State Initialization.
- (line 6)
-* Random state: Nomenclature and Types.
- (line 46)
-* Rational arithmetic: Efficiency. (line 113)
-* Rational arithmetic functions: Rational Arithmetic. (line 6)
-* Rational assignment functions: Initializing Rationals.
- (line 6)
-* Rational comparison functions: Comparing Rationals. (line 6)
-* Rational conversion functions: Rational Conversions.
- (line 6)
-* Rational initialization functions: Initializing Rationals.
- (line 6)
-* Rational input and output functions: I/O of Rationals. (line 6)
-* Rational internals: Rational Internals. (line 6)
-* Rational number: Nomenclature and Types.
- (line 16)
-* Rational number functions: Rational Number Functions.
- (line 6)
-* Rational numerator and denominator: Applying Integer Functions.
- (line 6)
-* Rational sign tests: Comparing Rationals. (line 27)
-* Raw output internals: Raw Output Internals.
- (line 6)
-* Reallocations: Efficiency. (line 30)
-* Reentrancy: Reentrancy. (line 6)
-* References: References. (line 6)
-* Remove factor functions: Number Theoretic Functions.
- (line 90)
-* Reporting bugs: Reporting Bugs. (line 6)
-* Root extraction algorithm: Nth Root Algorithm. (line 6)
-* Root extraction algorithms: Root Extraction Algorithms.
- (line 6)
-* Root extraction functions <1>: Float Arithmetic. (line 37)
-* Root extraction functions: Integer Roots. (line 6)
-* Root testing functions: Integer Roots. (line 36)
-* Rounding functions: Miscellaneous Float Functions.
- (line 9)
-* Sample programs: Demonstration Programs.
- (line 6)
-* Scan bit functions: Integer Logic and Bit Fiddling.
- (line 38)
-* scanf formatted input: Formatted Input. (line 6)
-* SCO: Known Build Problems.
- (line 38)
-* Seeding random numbers: Random State Seeding.
- (line 6)
-* Segmentation violation: Debugging. (line 7)
-* Sequent Symmetry: Known Build Problems.
- (line 68)
-* Services for Unix: Notes for Particular Systems.
- (line 51)
-* Shared library versioning: Notes for Package Builds.
- (line 9)
-* Sign tests <1>: Float Comparison. (line 33)
-* Sign tests <2>: Integer Comparisons. (line 28)
-* Sign tests: Comparing Rationals. (line 27)
-* Size in digits: Miscellaneous Integer Functions.
- (line 23)
-* Small operands: Efficiency. (line 7)
-* Solaris <1>: ABI and ISA. (line 201)
-* Solaris: Known Build Problems.
- (line 78)
-* Sparc: Notes for Particular Systems.
- (line 108)
-* Sparc V9: ABI and ISA. (line 201)
-* Special integer functions: Integer Special Functions.
- (line 6)
-* Square root algorithm: Square Root Algorithm.
- (line 6)
-* SSE2: Notes for Particular Systems.
- (line 132)
-* Stack backtrace: Debugging. (line 50)
-* Stack overflow <1>: Debugging. (line 7)
-* Stack overflow: Build Options. (line 278)
-* Static linking: Efficiency. (line 14)
-* stdarg.h: Headers and Libraries.
- (line 17)
-* stdio.h: Headers and Libraries.
- (line 11)
-* Stripped libraries: Known Build Problems.
- (line 28)
-* Sun: ABI and ISA. (line 201)
-* SunOS: Notes for Particular Systems.
- (line 120)
-* Systems: Notes for Particular Systems.
- (line 6)
-* Temporary memory: Build Options. (line 278)
-* Texinfo: Build Options. (line 347)
-* Text input/output: Efficiency. (line 153)
-* Thread safety: Reentrancy. (line 6)
-* Toom multiplication <1>: Other Multiplication.
- (line 6)
-* Toom multiplication <2>: Toom 4-Way Multiplication.
- (line 6)
-* Toom multiplication: Toom 3-Way Multiplication.
- (line 6)
-* Types: Nomenclature and Types.
- (line 6)
-* ui and si functions: Efficiency. (line 50)
-* Unbalanced multiplication: Unbalanced Multiplication.
- (line 6)
-* Upward compatibility: Compatibility with older versions.
- (line 6)
-* Useful macros and constants: Useful Macros and Constants.
- (line 6)
-* User-defined precision: Floating-point Functions.
- (line 6)
-* Valgrind: Debugging. (line 130)
-* Variable conventions: Variable Conventions.
- (line 6)
-* Version number: Useful Macros and Constants.
- (line 12)
-* Web page: Introduction to GMP. (line 34)
-* Windows: Notes for Particular Systems.
- (line 56)
-* x86: Notes for Particular Systems.
- (line 126)
-* x87: Notes for Particular Systems.
- (line 34)
-* XML: Build Options. (line 354)
-
-\1f
-File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top
-
-Function and Type Index
-***********************
-
-\0\b[index\0\b]
-* Menu:
-
-* __GMP_CC: Useful Macros and Constants.
- (line 23)
-* __GMP_CFLAGS: Useful Macros and Constants.
- (line 24)
-* __GNU_MP_VERSION: Useful Macros and Constants.
- (line 10)
-* __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
- (line 11)
-* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
- (line 12)
-* _mpz_realloc: Integer Special Functions.
- (line 51)
-* abs <1>: C++ Interface Rationals.
- (line 43)
-* abs <2>: C++ Interface Integers.
- (line 42)
-* abs: C++ Interface Floats.
- (line 70)
-* ceil: C++ Interface Floats.
- (line 71)
-* cmp <1>: C++ Interface Floats.
- (line 72)
-* cmp <2>: C++ Interface Rationals.
- (line 44)
-* cmp <3>: C++ Interface Integers.
- (line 44)
-* cmp: C++ Interface Rationals.
- (line 45)
-* floor: C++ Interface Floats.
- (line 80)
-* gcd: BSD Compatible Functions.
- (line 82)
-* gmp_asprintf: Formatted Output Functions.
- (line 65)
-* gmp_errno: Random State Initialization.
- (line 55)
-* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization.
- (line 55)
-* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization.
- (line 55)
-* gmp_fprintf: Formatted Output Functions.
- (line 29)
-* gmp_fscanf: Formatted Input Functions.
- (line 25)
-* GMP_LIMB_BITS: Low-level Functions. (line 508)
-* GMP_NAIL_BITS: Low-level Functions. (line 506)
-* GMP_NAIL_MASK: Low-level Functions. (line 516)
-* GMP_NUMB_BITS: Low-level Functions. (line 507)
-* GMP_NUMB_MASK: Low-level Functions. (line 517)
-* GMP_NUMB_MAX: Low-level Functions. (line 525)
-* gmp_obstack_printf: Formatted Output Functions.
- (line 79)
-* gmp_obstack_vprintf: Formatted Output Functions.
- (line 81)
-* gmp_printf: Formatted Output Functions.
- (line 24)
-* GMP_RAND_ALG_DEFAULT: Random State Initialization.
- (line 49)
-* GMP_RAND_ALG_LC: Random State Initialization.
- (line 49)
-* gmp_randclass: C++ Interface Random Numbers.
- (line 7)
-* gmp_randclass::get_f: C++ Interface Random Numbers.
- (line 45)
-* gmp_randclass::get_z_bits: C++ Interface Random Numbers.
- (line 39)
-* gmp_randclass::get_z_range: C++ Interface Random Numbers.
- (line 42)
-* gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
- (line 13)
-* gmp_randclass::seed: C++ Interface Random Numbers.
- (line 33)
-* gmp_randclear: Random State Initialization.
- (line 62)
-* gmp_randinit: Random State Initialization.
- (line 47)
-* gmp_randinit_default: Random State Initialization.
- (line 7)
-* gmp_randinit_lc_2exp: Random State Initialization.
- (line 18)
-* gmp_randinit_lc_2exp_size: Random State Initialization.
- (line 32)
-* gmp_randinit_mt: Random State Initialization.
- (line 13)
-* gmp_randinit_set: Random State Initialization.
- (line 43)
-* gmp_randseed: Random State Seeding.
- (line 7)
-* gmp_randseed_ui: Random State Seeding.
- (line 9)
-* gmp_randstate_t: Nomenclature and Types.
- (line 46)
-* gmp_scanf: Formatted Input Functions.
- (line 21)
-* gmp_snprintf: Formatted Output Functions.
- (line 46)
-* gmp_sprintf: Formatted Output Functions.
- (line 34)
-* gmp_sscanf: Formatted Input Functions.
- (line 29)
-* gmp_urandomb_ui: Random State Miscellaneous.
- (line 8)
-* gmp_urandomm_ui: Random State Miscellaneous.
- (line 14)
-* gmp_vasprintf: Formatted Output Functions.
- (line 66)
-* gmp_version: Useful Macros and Constants.
- (line 18)
-* gmp_vfprintf: Formatted Output Functions.
- (line 30)
-* gmp_vfscanf: Formatted Input Functions.
- (line 26)
-* gmp_vprintf: Formatted Output Functions.
- (line 25)
-* gmp_vscanf: Formatted Input Functions.
- (line 22)
-* gmp_vsnprintf: Formatted Output Functions.
- (line 48)
-* gmp_vsprintf: Formatted Output Functions.
- (line 35)
-* gmp_vsscanf: Formatted Input Functions.
- (line 31)
-* hypot: C++ Interface Floats.
- (line 81)
-* itom: BSD Compatible Functions.
- (line 29)
-* madd: BSD Compatible Functions.
- (line 43)
-* mcmp: BSD Compatible Functions.
- (line 85)
-* mdiv: BSD Compatible Functions.
- (line 53)
-* mfree: BSD Compatible Functions.
- (line 105)
-* min: BSD Compatible Functions.
- (line 89)
-* MINT: BSD Compatible Functions.
- (line 21)
-* mout: BSD Compatible Functions.
- (line 94)
-* move: BSD Compatible Functions.
- (line 39)
-* mp_bitcnt_t: Nomenclature and Types.
- (line 42)
-* mp_bits_per_limb: Useful Macros and Constants.
- (line 7)
-* mp_exp_t: Nomenclature and Types.
- (line 27)
-* mp_get_memory_functions: Custom Allocation. (line 93)
-* mp_limb_t: Nomenclature and Types.
- (line 31)
-* mp_set_memory_functions: Custom Allocation. (line 21)
-* mp_size_t: Nomenclature and Types.
- (line 37)
-* mpf_abs: Float Arithmetic. (line 47)
-* mpf_add: Float Arithmetic. (line 7)
-* mpf_add_ui: Float Arithmetic. (line 9)
-* mpf_ceil: Miscellaneous Float Functions.
- (line 7)
-* mpf_class: C++ Interface General.
- (line 20)
-* mpf_class::fits_sint_p: C++ Interface Floats.
- (line 74)
-* mpf_class::fits_slong_p: C++ Interface Floats.
- (line 75)
-* mpf_class::fits_sshort_p: C++ Interface Floats.
- (line 76)
-* mpf_class::fits_uint_p: C++ Interface Floats.
- (line 77)
-* mpf_class::fits_ulong_p: C++ Interface Floats.
- (line 78)
-* mpf_class::fits_ushort_p: C++ Interface Floats.
- (line 79)
-* mpf_class::get_d: C++ Interface Floats.
- (line 82)
-* mpf_class::get_mpf_t: C++ Interface General.
- (line 66)
-* mpf_class::get_prec: C++ Interface Floats.
- (line 100)
-* mpf_class::get_si: C++ Interface Floats.
- (line 83)
-* mpf_class::get_str: C++ Interface Floats.
- (line 85)
-* mpf_class::get_ui: C++ Interface Floats.
- (line 86)
-* mpf_class::mpf_class: C++ Interface Floats.
- (line 38)
-* mpf_class::operator=: C++ Interface Floats.
- (line 47)
-* mpf_class::set_prec: C++ Interface Floats.
- (line 101)
-* mpf_class::set_prec_raw: C++ Interface Floats.
- (line 102)
-* mpf_class::set_str: C++ Interface Floats.
- (line 88)
-* mpf_clear: Initializing Floats. (line 37)
-* mpf_clears: Initializing Floats. (line 41)
-* mpf_cmp: Float Comparison. (line 7)
-* mpf_cmp_d: Float Comparison. (line 8)
-* mpf_cmp_si: Float Comparison. (line 10)
-* mpf_cmp_ui: Float Comparison. (line 9)
-* mpf_div: Float Arithmetic. (line 29)
-* mpf_div_2exp: Float Arithmetic. (line 53)
-* mpf_div_ui: Float Arithmetic. (line 33)
-* mpf_eq: Float Comparison. (line 17)
-* mpf_fits_sint_p: Miscellaneous Float Functions.
- (line 20)
-* mpf_fits_slong_p: Miscellaneous Float Functions.
- (line 18)
-* mpf_fits_sshort_p: Miscellaneous Float Functions.
- (line 22)
-* mpf_fits_uint_p: Miscellaneous Float Functions.
- (line 19)
-* mpf_fits_ulong_p: Miscellaneous Float Functions.
- (line 17)
-* mpf_fits_ushort_p: Miscellaneous Float Functions.
- (line 21)
-* mpf_floor: Miscellaneous Float Functions.
- (line 8)
-* mpf_get_d: Converting Floats. (line 7)
-* mpf_get_d_2exp: Converting Floats. (line 16)
-* mpf_get_default_prec: Initializing Floats. (line 12)
-* mpf_get_prec: Initializing Floats. (line 62)
-* mpf_get_si: Converting Floats. (line 27)
-* mpf_get_str: Converting Floats. (line 37)
-* mpf_get_ui: Converting Floats. (line 28)
-* mpf_init: Initializing Floats. (line 19)
-* mpf_init2: Initializing Floats. (line 26)
-* mpf_init_set: Simultaneous Float Init & Assign.
- (line 16)
-* mpf_init_set_d: Simultaneous Float Init & Assign.
- (line 19)
-* mpf_init_set_si: Simultaneous Float Init & Assign.
- (line 18)
-* mpf_init_set_str: Simultaneous Float Init & Assign.
- (line 25)
-* mpf_init_set_ui: Simultaneous Float Init & Assign.
- (line 17)
-* mpf_inits: Initializing Floats. (line 31)
-* mpf_inp_str: I/O of Floats. (line 37)
-* mpf_integer_p: Miscellaneous Float Functions.
- (line 14)
-* mpf_mul: Float Arithmetic. (line 19)
-* mpf_mul_2exp: Float Arithmetic. (line 50)
-* mpf_mul_ui: Float Arithmetic. (line 21)
-* mpf_neg: Float Arithmetic. (line 44)
-* mpf_out_str: I/O of Floats. (line 17)
-* mpf_pow_ui: Float Arithmetic. (line 41)
-* mpf_random2: Miscellaneous Float Functions.
- (line 36)
-* mpf_reldiff: Float Comparison. (line 29)
-* mpf_set: Assigning Floats. (line 10)
-* mpf_set_d: Assigning Floats. (line 13)
-* mpf_set_default_prec: Initializing Floats. (line 7)
-* mpf_set_prec: Initializing Floats. (line 65)
-* mpf_set_prec_raw: Initializing Floats. (line 72)
-* mpf_set_q: Assigning Floats. (line 15)
-* mpf_set_si: Assigning Floats. (line 12)
-* mpf_set_str: Assigning Floats. (line 18)
-* mpf_set_ui: Assigning Floats. (line 11)
-* mpf_set_z: Assigning Floats. (line 14)
-* mpf_sgn: Float Comparison. (line 33)
-* mpf_sqrt: Float Arithmetic. (line 36)
-* mpf_sqrt_ui: Float Arithmetic. (line 37)
-* mpf_sub: Float Arithmetic. (line 12)
-* mpf_sub_ui: Float Arithmetic. (line 16)
-* mpf_swap: Assigning Floats. (line 52)
-* mpf_t: Nomenclature and Types.
- (line 21)
-* mpf_trunc: Miscellaneous Float Functions.
- (line 9)
-* mpf_ui_div: Float Arithmetic. (line 31)
-* mpf_ui_sub: Float Arithmetic. (line 14)
-* mpf_urandomb: Miscellaneous Float Functions.
- (line 27)
-* mpn_add: Low-level Functions. (line 69)
-* mpn_add_1: Low-level Functions. (line 64)
-* mpn_add_n: Low-level Functions. (line 54)
-* mpn_addmul_1: Low-level Functions. (line 148)
-* mpn_and_n: Low-level Functions. (line 420)
-* mpn_andn_n: Low-level Functions. (line 435)
-* mpn_cmp: Low-level Functions. (line 284)
-* mpn_com: Low-level Functions. (line 460)
-* mpn_copyd: Low-level Functions. (line 469)
-* mpn_copyi: Low-level Functions. (line 465)
-* mpn_divexact_by3: Low-level Functions. (line 229)
-* mpn_divexact_by3c: Low-level Functions. (line 231)
-* mpn_divmod: Low-level Functions. (line 224)
-* mpn_divmod_1: Low-level Functions. (line 208)
-* mpn_divrem: Low-level Functions. (line 182)
-* mpn_divrem_1: Low-level Functions. (line 206)
-* mpn_gcd: Low-level Functions. (line 289)
-* mpn_gcd_1: Low-level Functions. (line 299)
-* mpn_gcdext: Low-level Functions. (line 305)
-* mpn_get_str: Low-level Functions. (line 346)
-* mpn_hamdist: Low-level Functions. (line 410)
-* mpn_ior_n: Low-level Functions. (line 425)
-* mpn_iorn_n: Low-level Functions. (line 440)
-* mpn_lshift: Low-level Functions. (line 260)
-* mpn_mod_1: Low-level Functions. (line 255)
-* mpn_mul: Low-level Functions. (line 114)
-* mpn_mul_1: Low-level Functions. (line 133)
-* mpn_mul_n: Low-level Functions. (line 103)
-* mpn_nand_n: Low-level Functions. (line 445)
-* mpn_neg: Low-level Functions. (line 98)
-* mpn_nior_n: Low-level Functions. (line 450)
-* mpn_perfect_square_p: Low-level Functions. (line 416)
-* mpn_popcount: Low-level Functions. (line 406)
-* mpn_random: Low-level Functions. (line 395)
-* mpn_random2: Low-level Functions. (line 396)
-* mpn_rshift: Low-level Functions. (line 272)
-* mpn_scan0: Low-level Functions. (line 380)
-* mpn_scan1: Low-level Functions. (line 388)
-* mpn_set_str: Low-level Functions. (line 361)
-* mpn_sqr: Low-level Functions. (line 125)
-* mpn_sqrtrem: Low-level Functions. (line 328)
-* mpn_sub: Low-level Functions. (line 90)
-* mpn_sub_1: Low-level Functions. (line 85)
-* mpn_sub_n: Low-level Functions. (line 76)
-* mpn_submul_1: Low-level Functions. (line 159)
-* mpn_tdiv_qr: Low-level Functions. (line 171)
-* mpn_xnor_n: Low-level Functions. (line 455)
-* mpn_xor_n: Low-level Functions. (line 430)
-* mpn_zero: Low-level Functions. (line 472)
-* mpq_abs: Rational Arithmetic. (line 31)
-* mpq_add: Rational Arithmetic. (line 7)
-* mpq_canonicalize: Rational Number Functions.
- (line 22)
-* mpq_class: C++ Interface General.
- (line 19)
-* mpq_class::canonicalize: C++ Interface Rationals.
- (line 37)
-* mpq_class::get_d: C++ Interface Rationals.
- (line 46)
-* mpq_class::get_den: C++ Interface Rationals.
- (line 58)
-* mpq_class::get_den_mpz_t: C++ Interface Rationals.
- (line 68)
-* mpq_class::get_mpq_t: C++ Interface General.
- (line 65)
-* mpq_class::get_num: C++ Interface Rationals.
- (line 57)
-* mpq_class::get_num_mpz_t: C++ Interface Rationals.
- (line 67)
-* mpq_class::get_str: C++ Interface Rationals.
- (line 47)
-* mpq_class::mpq_class: C++ Interface Rationals.
- (line 22)
-* mpq_class::set_str: C++ Interface Rationals.
- (line 49)
-* mpq_clear: Initializing Rationals.
- (line 16)
-* mpq_clears: Initializing Rationals.
- (line 20)
-* mpq_cmp: Comparing Rationals. (line 7)
-* mpq_cmp_si: Comparing Rationals. (line 17)
-* mpq_cmp_ui: Comparing Rationals. (line 15)
-* mpq_denref: Applying Integer Functions.
- (line 18)
-* mpq_div: Rational Arithmetic. (line 22)
-* mpq_div_2exp: Rational Arithmetic. (line 25)
-* mpq_equal: Comparing Rationals. (line 33)
-* mpq_get_d: Rational Conversions.
- (line 7)
-* mpq_get_den: Applying Integer Functions.
- (line 24)
-* mpq_get_num: Applying Integer Functions.
- (line 23)
-* mpq_get_str: Rational Conversions.
- (line 22)
-* mpq_init: Initializing Rationals.
- (line 7)
-* mpq_inits: Initializing Rationals.
- (line 12)
-* mpq_inp_str: I/O of Rationals. (line 23)
-* mpq_inv: Rational Arithmetic. (line 34)
-* mpq_mul: Rational Arithmetic. (line 15)
-* mpq_mul_2exp: Rational Arithmetic. (line 18)
-* mpq_neg: Rational Arithmetic. (line 28)
-* mpq_numref: Applying Integer Functions.
- (line 17)
-* mpq_out_str: I/O of Rationals. (line 15)
-* mpq_set: Initializing Rationals.
- (line 24)
-* mpq_set_d: Rational Conversions.
- (line 17)
-* mpq_set_den: Applying Integer Functions.
- (line 26)
-* mpq_set_f: Rational Conversions.
- (line 18)
-* mpq_set_num: Applying Integer Functions.
- (line 25)
-* mpq_set_si: Initializing Rationals.
- (line 31)
-* mpq_set_str: Initializing Rationals.
- (line 36)
-* mpq_set_ui: Initializing Rationals.
- (line 29)
-* mpq_set_z: Initializing Rationals.
- (line 25)
-* mpq_sgn: Comparing Rationals. (line 27)
-* mpq_sub: Rational Arithmetic. (line 11)
-* mpq_swap: Initializing Rationals.
- (line 56)
-* mpq_t: Nomenclature and Types.
- (line 16)
-* mpz_abs: Integer Arithmetic. (line 42)
-* mpz_add: Integer Arithmetic. (line 7)
-* mpz_add_ui: Integer Arithmetic. (line 9)
-* mpz_addmul: Integer Arithmetic. (line 25)
-* mpz_addmul_ui: Integer Arithmetic. (line 27)
-* mpz_and: Integer Logic and Bit Fiddling.
- (line 11)
-* mpz_array_init: Integer Special Functions.
- (line 11)
-* mpz_bin_ui: Number Theoretic Functions.
- (line 98)
-* mpz_bin_uiui: Number Theoretic Functions.
- (line 100)
-* mpz_cdiv_q: Integer Division. (line 13)
-* mpz_cdiv_q_2exp: Integer Division. (line 24)
-* mpz_cdiv_q_ui: Integer Division. (line 17)
-* mpz_cdiv_qr: Integer Division. (line 15)
-* mpz_cdiv_qr_ui: Integer Division. (line 21)
-* mpz_cdiv_r: Integer Division. (line 14)
-* mpz_cdiv_r_2exp: Integer Division. (line 25)
-* mpz_cdiv_r_ui: Integer Division. (line 19)
-* mpz_cdiv_ui: Integer Division. (line 23)
-* mpz_class: C++ Interface General.
- (line 18)
-* mpz_class::fits_sint_p: C++ Interface Integers.
- (line 45)
-* mpz_class::fits_slong_p: C++ Interface Integers.
- (line 46)
-* mpz_class::fits_sshort_p: C++ Interface Integers.
- (line 47)
-* mpz_class::fits_uint_p: C++ Interface Integers.
- (line 48)
-* mpz_class::fits_ulong_p: C++ Interface Integers.
- (line 49)
-* mpz_class::fits_ushort_p: C++ Interface Integers.
- (line 50)
-* mpz_class::get_d: C++ Interface Integers.
- (line 51)
-* mpz_class::get_mpz_t: C++ Interface General.
- (line 64)
-* mpz_class::get_si: C++ Interface Integers.
- (line 52)
-* mpz_class::get_str: C++ Interface Integers.
- (line 53)
-* mpz_class::get_ui: C++ Interface Integers.
- (line 54)
-* mpz_class::mpz_class: C++ Interface Integers.
- (line 7)
-* mpz_class::set_str: C++ Interface Integers.
- (line 56)
-* mpz_clear: Initializing Integers.
- (line 44)
-* mpz_clears: Initializing Integers.
- (line 48)
-* mpz_clrbit: Integer Logic and Bit Fiddling.
- (line 54)
-* mpz_cmp: Integer Comparisons. (line 7)
-* mpz_cmp_d: Integer Comparisons. (line 8)
-* mpz_cmp_si: Integer Comparisons. (line 9)
-* mpz_cmp_ui: Integer Comparisons. (line 10)
-* mpz_cmpabs: Integer Comparisons. (line 18)
-* mpz_cmpabs_d: Integer Comparisons. (line 19)
-* mpz_cmpabs_ui: Integer Comparisons. (line 20)
-* mpz_com: Integer Logic and Bit Fiddling.
- (line 20)
-* mpz_combit: Integer Logic and Bit Fiddling.
- (line 57)
-* mpz_congruent_2exp_p: Integer Division. (line 124)
-* mpz_congruent_p: Integer Division. (line 121)
-* mpz_congruent_ui_p: Integer Division. (line 123)
-* mpz_divexact: Integer Division. (line 101)
-* mpz_divexact_ui: Integer Division. (line 102)
-* mpz_divisible_2exp_p: Integer Division. (line 112)
-* mpz_divisible_p: Integer Division. (line 110)
-* mpz_divisible_ui_p: Integer Division. (line 111)
-* mpz_even_p: Miscellaneous Integer Functions.
- (line 18)
-* mpz_export: Integer Import and Export.
- (line 45)
-* mpz_fac_ui: Number Theoretic Functions.
- (line 95)
-* mpz_fdiv_q: Integer Division. (line 27)
-* mpz_fdiv_q_2exp: Integer Division. (line 38)
-* mpz_fdiv_q_ui: Integer Division. (line 31)
-* mpz_fdiv_qr: Integer Division. (line 29)
-* mpz_fdiv_qr_ui: Integer Division. (line 35)
-* mpz_fdiv_r: Integer Division. (line 28)
-* mpz_fdiv_r_2exp: Integer Division. (line 39)
-* mpz_fdiv_r_ui: Integer Division. (line 33)
-* mpz_fdiv_ui: Integer Division. (line 37)
-* mpz_fib2_ui: Number Theoretic Functions.
- (line 108)
-* mpz_fib_ui: Number Theoretic Functions.
- (line 106)
-* mpz_fits_sint_p: Miscellaneous Integer Functions.
- (line 10)
-* mpz_fits_slong_p: Miscellaneous Integer Functions.
- (line 8)
-* mpz_fits_sshort_p: Miscellaneous Integer Functions.
- (line 12)
-* mpz_fits_uint_p: Miscellaneous Integer Functions.
- (line 9)
-* mpz_fits_ulong_p: Miscellaneous Integer Functions.
- (line 7)
-* mpz_fits_ushort_p: Miscellaneous Integer Functions.
- (line 11)
-* mpz_gcd: Number Theoretic Functions.
- (line 30)
-* mpz_gcd_ui: Number Theoretic Functions.
- (line 35)
-* mpz_gcdext: Number Theoretic Functions.
- (line 45)
-* mpz_get_d: Converting Integers. (line 27)
-* mpz_get_d_2exp: Converting Integers. (line 35)
-* mpz_get_si: Converting Integers. (line 18)
-* mpz_get_str: Converting Integers. (line 46)
-* mpz_get_ui: Converting Integers. (line 11)
-* mpz_getlimbn: Integer Special Functions.
- (line 60)
-* mpz_hamdist: Integer Logic and Bit Fiddling.
- (line 29)
-* mpz_import: Integer Import and Export.
- (line 11)
-* mpz_init: Initializing Integers.
- (line 26)
-* mpz_init2: Initializing Integers.
- (line 33)
-* mpz_init_set: Simultaneous Integer Init & Assign.
- (line 27)
-* mpz_init_set_d: Simultaneous Integer Init & Assign.
- (line 30)
-* mpz_init_set_si: Simultaneous Integer Init & Assign.
- (line 29)
-* mpz_init_set_str: Simultaneous Integer Init & Assign.
- (line 34)
-* mpz_init_set_ui: Simultaneous Integer Init & Assign.
- (line 28)
-* mpz_inits: Initializing Integers.
- (line 29)
-* mpz_inp_raw: I/O of Integers. (line 59)
-* mpz_inp_str: I/O of Integers. (line 28)
-* mpz_invert: Number Theoretic Functions.
- (line 60)
-* mpz_ior: Integer Logic and Bit Fiddling.
- (line 14)
-* mpz_jacobi: Number Theoretic Functions.
- (line 66)
-* mpz_kronecker: Number Theoretic Functions.
- (line 74)
-* mpz_kronecker_si: Number Theoretic Functions.
- (line 75)
-* mpz_kronecker_ui: Number Theoretic Functions.
- (line 76)
-* mpz_lcm: Number Theoretic Functions.
- (line 54)
-* mpz_lcm_ui: Number Theoretic Functions.
- (line 55)
-* mpz_legendre: Number Theoretic Functions.
- (line 69)
-* mpz_lucnum2_ui: Number Theoretic Functions.
- (line 119)
-* mpz_lucnum_ui: Number Theoretic Functions.
- (line 117)
-* mpz_mod: Integer Division. (line 91)
-* mpz_mod_ui: Integer Division. (line 93)
-* mpz_mul: Integer Arithmetic. (line 19)
-* mpz_mul_2exp: Integer Arithmetic. (line 35)
-* mpz_mul_si: Integer Arithmetic. (line 20)
-* mpz_mul_ui: Integer Arithmetic. (line 22)
-* mpz_neg: Integer Arithmetic. (line 39)
-* mpz_nextprime: Number Theoretic Functions.
- (line 23)
-* mpz_odd_p: Miscellaneous Integer Functions.
- (line 17)
-* mpz_out_raw: I/O of Integers. (line 43)
-* mpz_out_str: I/O of Integers. (line 16)
-* mpz_perfect_power_p: Integer Roots. (line 27)
-* mpz_perfect_square_p: Integer Roots. (line 36)
-* mpz_popcount: Integer Logic and Bit Fiddling.
- (line 23)
-* mpz_pow_ui: Integer Exponentiation.
- (line 31)
-* mpz_powm: Integer Exponentiation.
- (line 8)
-* mpz_powm_sec: Integer Exponentiation.
- (line 18)
-* mpz_powm_ui: Integer Exponentiation.
- (line 10)
-* mpz_probab_prime_p: Number Theoretic Functions.
- (line 7)
-* mpz_random: Integer Random Numbers.
- (line 42)
-* mpz_random2: Integer Random Numbers.
- (line 51)
-* mpz_realloc2: Initializing Integers.
- (line 52)
-* mpz_remove: Number Theoretic Functions.
- (line 90)
-* mpz_root: Integer Roots. (line 7)
-* mpz_rootrem: Integer Roots. (line 13)
-* mpz_rrandomb: Integer Random Numbers.
- (line 31)
-* mpz_scan0: Integer Logic and Bit Fiddling.
- (line 37)
-* mpz_scan1: Integer Logic and Bit Fiddling.
- (line 38)
-* mpz_set: Assigning Integers. (line 10)
-* mpz_set_d: Assigning Integers. (line 13)
-* mpz_set_f: Assigning Integers. (line 15)
-* mpz_set_q: Assigning Integers. (line 14)
-* mpz_set_si: Assigning Integers. (line 12)
-* mpz_set_str: Assigning Integers. (line 21)
-* mpz_set_ui: Assigning Integers. (line 11)
-* mpz_setbit: Integer Logic and Bit Fiddling.
- (line 51)
-* mpz_sgn: Integer Comparisons. (line 28)
-* mpz_si_kronecker: Number Theoretic Functions.
- (line 77)
-* mpz_size: Integer Special Functions.
- (line 68)
-* mpz_sizeinbase: Miscellaneous Integer Functions.
- (line 23)
-* mpz_sqrt: Integer Roots. (line 17)
-* mpz_sqrtrem: Integer Roots. (line 20)
-* mpz_sub: Integer Arithmetic. (line 12)
-* mpz_sub_ui: Integer Arithmetic. (line 14)
-* mpz_submul: Integer Arithmetic. (line 30)
-* mpz_submul_ui: Integer Arithmetic. (line 32)
-* mpz_swap: Assigning Integers. (line 37)
-* mpz_t: Nomenclature and Types.
- (line 6)
-* mpz_tdiv_q: Integer Division. (line 41)
-* mpz_tdiv_q_2exp: Integer Division. (line 52)
-* mpz_tdiv_q_ui: Integer Division. (line 45)
-* mpz_tdiv_qr: Integer Division. (line 43)
-* mpz_tdiv_qr_ui: Integer Division. (line 49)
-* mpz_tdiv_r: Integer Division. (line 42)
-* mpz_tdiv_r_2exp: Integer Division. (line 53)
-* mpz_tdiv_r_ui: Integer Division. (line 47)
-* mpz_tdiv_ui: Integer Division. (line 51)
-* mpz_tstbit: Integer Logic and Bit Fiddling.
- (line 60)
-* mpz_ui_kronecker: Number Theoretic Functions.
- (line 78)
-* mpz_ui_pow_ui: Integer Exponentiation.
- (line 33)
-* mpz_ui_sub: Integer Arithmetic. (line 16)
-* mpz_urandomb: Integer Random Numbers.
- (line 14)
-* mpz_urandomm: Integer Random Numbers.
- (line 23)
-* mpz_xor: Integer Logic and Bit Fiddling.
- (line 17)
-* msqrt: BSD Compatible Functions.
- (line 63)
-* msub: BSD Compatible Functions.
- (line 46)
-* mtox: BSD Compatible Functions.
- (line 98)
-* mult: BSD Compatible Functions.
- (line 49)
-* operator%: C++ Interface Integers.
- (line 30)
-* operator/: C++ Interface Integers.
- (line 29)
-* operator<<: C++ Formatted Output.
- (line 20)
-* operator>> <1>: C++ Formatted Input. (line 11)
-* operator>>: C++ Interface Rationals.
- (line 77)
-* pow: BSD Compatible Functions.
- (line 71)
-* rpow: BSD Compatible Functions.
- (line 79)
-* sdiv: BSD Compatible Functions.
- (line 55)
-* sgn <1>: C++ Interface Rationals.
- (line 50)
-* sgn <2>: C++ Interface Integers.
- (line 57)
-* sgn: C++ Interface Floats.
- (line 89)
-* sqrt <1>: C++ Interface Integers.
- (line 58)
-* sqrt: C++ Interface Floats.
- (line 90)
-* trunc: C++ Interface Floats.
- (line 91)
-* xtom: BSD Compatible Functions.
- (line 34)
-
-
+++ /dev/null
-/*
- * FILE: d0.h
- * AUTHOR: Rudolf Polzer - divVerent@xonotic.org
- *
- * Copyright (c) 2010, Rudolf Polzer
- * All rights reserved.
- *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions
- * are met:
- * 1. Redistributions of source code must retain the above copyright
- * notice, this list of conditions and the following disclaimer.
- * 2. Redistributions in binary form must reproduce the above copyright
- * notice, this list of conditions and the following disclaimer in the
- * documentation and/or other materials provided with the distribution.
- * 3. Neither the name of the copyright holder nor the names of contributors
- * may be used to endorse or promote products derived from this software
- * without specific prior written permission.
- *
- * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTOR(S) ``AS IS'' AND
- * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
- * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
- * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTOR(S) BE LIABLE
- * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
- * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
- * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
- * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
- * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
- * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
- * SUCH DAMAGE.
- *
- * $Format:commit %H$
- * $Id: 6c55afeb50f24bd316079ae46582e65f8020b19b $
- */
-
-#ifndef __D0_H__
-#define __D0_H__
-
-#include <unistd.h> // size_t
-
-#define D0_EXPORT __attribute__((__visibility__("default")))
-#define D0_USED __attribute__((used))
-#define D0_WARN_UNUSED_RESULT __attribute__((warn_unused_result))
-#define D0_BOOL int
-
-typedef void *(d0_malloc_t)(size_t len);
-typedef void (d0_free_t)(void *p);
-typedef void *(d0_createmutex_t)(void);
-typedef void (d0_destroymutex_t)(void *);
-typedef int (d0_lockmutex_t)(void *); // zero on success
-typedef int (d0_unlockmutex_t)(void *); // zero on success
-
-extern d0_malloc_t *d0_malloc;
-extern d0_free_t *d0_free;
-extern d0_createmutex_t *d0_createmutex;
-extern d0_destroymutex_t *d0_destroymutex;
-extern d0_lockmutex_t *d0_lockmutex;
-extern d0_unlockmutex_t *d0_unlockmutex;
-
-void d0_setmallocfuncs(d0_malloc_t *m, d0_free_t *f);
-void d0_setmutexfuncs(d0_createmutex_t *c, d0_destroymutex_t *d, d0_lockmutex_t *l, d0_unlockmutex_t *u);
-void d0_initfuncs(void); // initializes them, this needs to be only called internally once
-
-extern const char *d0_bsd_license_notice;
-
-#endif
+++ /dev/null
-/*
- * FILE: d0_blind_id.h
- * AUTHOR: Rudolf Polzer - divVerent@xonotic.org
- *
- * Copyright (c) 2010, Rudolf Polzer
- * All rights reserved.
- *
- * Redistribution and use in source and binary forms, with or without
- * modification, are permitted provided that the following conditions
- * are met:
- * 1. Redistributions of source code must retain the above copyright
- * notice, this list of conditions and the following disclaimer.
- * 2. Redistributions in binary form must reproduce the above copyright
- * notice, this list of conditions and the following disclaimer in the
- * documentation and/or other materials provided with the distribution.
- * 3. Neither the name of the copyright holder nor the names of contributors
- * may be used to endorse or promote products derived from this software
- * without specific prior written permission.
- *
- * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTOR(S) ``AS IS'' AND
- * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
- * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
- * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTOR(S) BE LIABLE
- * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
- * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
- * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
- * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
- * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
- * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
- * SUCH DAMAGE.
- *
- * $Format:commit %H$
- * $Id: bf838f43093aceadcd2d20071684f1e7148a4332 $
- */
-
-#ifndef __D0_BLIND_ID_H__
-#define __D0_BLIND_ID_H__
-
-#include "d0.h"
-
-typedef struct d0_blind_id_s d0_blind_id_t;
-typedef D0_BOOL (*d0_fastreject_function) (const d0_blind_id_t *ctx, void *pass);
-
-D0_EXPORT D0_WARN_UNUSED_RESULT d0_blind_id_t *d0_blind_id_new(void);
-D0_EXPORT void d0_blind_id_free(d0_blind_id_t *a);
-D0_EXPORT void d0_blind_id_clear(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_copy(d0_blind_id_t *ctx, const d0_blind_id_t *src);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_key(d0_blind_id_t *ctx, int k);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_key_fastreject(d0_blind_id_t *ctx, int k, d0_fastreject_function reject, void *pass);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_key(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_public_key(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_key(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_public_key(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_fingerprint64_public_key(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_id_modulus(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_id_modulus(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_id_modulus(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_id_start(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_generate_private_id_request(d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_answer_private_id_request(const d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_finish_private_id_request(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_id_request_camouflage(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_id_request_camouflage(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_private_id(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_read_public_id(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_private_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_write_public_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_start(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL send_modulus, const char *message, size_t msglen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_challenge(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL recv_modulus, const char *inbuf, size_t inbuflen, char *outbuf, size_t *outbuflen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_response(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_verify(d0_blind_id_t *ctx, const char *inbuf, size_t inbuflen, char *msg, size_t *msglen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_authenticate_with_private_id_generate_missing_signature(d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_sign(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL send_modulus, const char *message, size_t msglen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_sign_detached(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL send_modulus, const char *message, size_t msglen, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_verify(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL recv_modulus, const char *inbuf, size_t inbuflen, char *msg, size_t *msglen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sign_with_private_id_verify_detached(d0_blind_id_t *ctx, D0_BOOL is_first, D0_BOOL recv_modulus, const char *inbuf, size_t inbuflen, const char *msg, size_t msglen, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_fingerprint64_public_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_verify_public_id(const d0_blind_id_t *ctx, D0_BOOL *status);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_verify_private_id(const d0_blind_id_t *ctx);
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_sessionkey_public_id(const d0_blind_id_t *ctx, char *outbuf, size_t *outbuflen); // can only be done after successful key exchange, this performs a modpow; key length is limited by SHA_DIGESTSIZE for now; also ONLY valid after successful d0_blind_id_authenticate_with_private_id_verify/d0_blind_id_fingerprint64_public_id
-
-D0_EXPORT D0_WARN_UNUSED_RESULT D0_BOOL d0_blind_id_INITIALIZE(void);
-D0_EXPORT void d0_blind_id_SHUTDOWN(void);
-
-D0_EXPORT void d0_blind_id_util_sha256(char *out, const char *in, size_t n);
-
-// for exporting
-D0_EXPORT void d0_blind_id_setmallocfuncs(d0_malloc_t *m, d0_free_t *f);
-D0_EXPORT void d0_blind_id_setmutexfuncs(d0_createmutex_t *c, d0_destroymutex_t *d, d0_lockmutex_t *l, d0_unlockmutex_t *u);
-
-#endif
+++ /dev/null
-// from http://www.efgh.com/software/rijndael.htm (public domain)
-
-#ifndef H__RIJNDAEL
-#define H__RIJNDAEL
-
-#include "d0.h"
-
-D0_EXPORT int d0_rijndael_setup_encrypt(unsigned long *rk, const unsigned char *key,
- int keybits);
-D0_EXPORT int d0_rijndael_setup_decrypt(unsigned long *rk, const unsigned char *key,
- int keybits);
-D0_EXPORT void d0_rijndael_encrypt(const unsigned long *rk, int nrounds,
- const unsigned char plaintext[16], unsigned char ciphertext[16]);
-D0_EXPORT void d0_rijndael_decrypt(const unsigned long *rk, int nrounds,
- const unsigned char ciphertext[16], unsigned char plaintext[16]);
-
-#define D0_RIJNDAEL_KEYLENGTH(keybits) ((keybits)/8)
-#define D0_RIJNDAEL_RKLENGTH(keybits) ((keybits)/8+28)
-#define D0_RIJNDAEL_NROUNDS(keybits) ((keybits)/32+6)
-
-#endif
+++ /dev/null
-# libd0_blind_id.la - a libtool library file
-# Generated by ltmain.sh (GNU libtool) 2.2.6b Debian-2.2.6b-2ubuntu1
-#
-# Please DO NOT delete this file!
-# It is necessary for linking the library.
-
-# The name that we can dlopen(3).
-dlname=''
-
-# Names of this library.
-library_names=''
-
-# The name of the static archive.
-old_library='libd0_blind_id.a'
-
-# Linker flags that can not go in dependency_libs.
-inherited_linker_flags=''
-
-# Libraries that this one depends upon.
-dependency_libs=' -L/tmp/d0_blind_id.deps/lib/ /tmp/g/lib/libgmp.la'
-
-# Names of additional weak libraries provided by this library
-weak_library_names=''
-
-# Version information for libd0_blind_id.
-current=0
-age=0
-revision=0
-
-# Is this an already installed library?
-installed=yes
-
-# Should we warn about portability when linking against -modules?
-shouldnotlink=no
-
-# Files to dlopen/dlpreopen
-dlopen=''
-dlpreopen=''
-
-# Directory that this library needs to be installed in:
-libdir='/usr/local/lib'
+++ /dev/null
-libd0_blind_id.so.0.0.0
\ No newline at end of file
+++ /dev/null
-libd0_blind_id.so.0.0.0
\ No newline at end of file
+++ /dev/null
-# libd0_rijndael.la - a libtool library file
-# Generated by ltmain.sh (GNU libtool) 2.2.6b Debian-2.2.6b-2ubuntu1
-#
-# Please DO NOT delete this file!
-# It is necessary for linking the library.
-
-# The name that we can dlopen(3).
-dlname=''
-
-# Names of this library.
-library_names=''
-
-# The name of the static archive.
-old_library='libd0_rijndael.a'
-
-# Linker flags that can not go in dependency_libs.
-inherited_linker_flags=''
-
-# Libraries that this one depends upon.
-dependency_libs=' -L/tmp/d0_blind_id.deps/lib/ /tmp/g/lib/libgmp.la'
-
-# Names of additional weak libraries provided by this library
-weak_library_names=''
-
-# Version information for libd0_rijndael.
-current=0
-age=0
-revision=0
-
-# Is this an already installed library?
-installed=yes
-
-# Should we warn about portability when linking against -modules?
-shouldnotlink=no
-
-# Files to dlopen/dlpreopen
-dlopen=''
-dlpreopen=''
-
-# Directory that this library needs to be installed in:
-libdir='/usr/local/lib'
+++ /dev/null
-libd0_rijndael.so.0.0.0
\ No newline at end of file
+++ /dev/null
-libd0_rijndael.so.0.0.0
\ No newline at end of file
+++ /dev/null
-prefix=/usr/local
-exec_prefix=${prefix}
-libdir=${exec_prefix}/lib
-includedir=${prefix}/include
-
-Name: Blind-ID
-Description: Library for user identification using RSA blind signatures
-Requires:
-Version: 0.5
-Libs: -L${libdir} -ld0_blind_id
-Cflags: -I${includedir}/d0_blind_id
+++ /dev/null
-prefix=/usr/local
-exec_prefix=${prefix}
-libdir=${exec_prefix}/lib
-includedir=${prefix}/include
-
-Name: Rijndael
-Description: Library for Rijndael encryption
-Requires:
-Version: 0.5
-Libs: -L${libdir} -ld0_rijndael
-Cflags: -I${includedir}/d0_blind_id
+++ /dev/null
-/* Definitions for GNU multiple precision functions. -*- mode: c -*-
-
-Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1999, 2000, 2001, 2002, 2003,
-2004, 2005, 2006, 2007, 2008, 2009 Free Software Foundation, Inc.
-
-This file is part of the GNU MP Library.
-
-The GNU MP Library is free software; you can redistribute it and/or modify
-it under the terms of the GNU Lesser General Public License as published by
-the Free Software Foundation; either version 3 of the License, or (at your
-option) any later version.
-
-The GNU MP Library is distributed in the hope that it will be useful, but
-WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
-or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public
-License for more details.
-
-You should have received a copy of the GNU Lesser General Public License
-along with the GNU MP Library. If not, see http://www.gnu.org/licenses/. */
-
-#ifndef __GMP_H__
-
-#if defined (__cplusplus)
-#include <iosfwd> /* for std::istream, std::ostream, std::string */
-#include <cstdio>
-#endif
-
-
-/* Instantiated by configure. */
-#if ! defined (__GMP_WITHIN_CONFIGURE)
-#define __GMP_HAVE_HOST_CPU_FAMILY_power 0
-#define __GMP_HAVE_HOST_CPU_FAMILY_powerpc 0
-#define GMP_LIMB_BITS 64
-#define GMP_NAIL_BITS 0
-#endif
-#define GMP_NUMB_BITS (GMP_LIMB_BITS - GMP_NAIL_BITS)
-#define GMP_NUMB_MASK ((~ __GMP_CAST (mp_limb_t, 0)) >> GMP_NAIL_BITS)
-#define GMP_NUMB_MAX GMP_NUMB_MASK
-#define GMP_NAIL_MASK (~ GMP_NUMB_MASK)
-
-
-/* The following (everything under ifndef __GNU_MP__) must be identical in
- gmp.h and mp.h to allow both to be included in an application or during
- the library build. */
-#ifndef __GNU_MP__
-#define __GNU_MP__ 5
-
-#define __need_size_t /* tell gcc stddef.h we only want size_t */
-#if defined (__cplusplus)
-#include <cstddef> /* for size_t */
-#else
-#include <stddef.h> /* for size_t */
-#endif
-#undef __need_size_t
-
-/* Instantiated by configure. */
-#if ! defined (__GMP_WITHIN_CONFIGURE)
-/* #undef _LONG_LONG_LIMB */
-#define __GMP_LIBGMP_DLL 0
-#endif
-
-
-/* __STDC__ - some ANSI compilers define this only to 0, hence the use of
- "defined" and not "__STDC__-0". In particular Sun workshop C 5.0
- sets __STDC__ to 0, but requires "##" for token pasting.
-
- _AIX - gnu ansidecl.h asserts that all known AIX compilers are ANSI but
- don't always define __STDC__.
-
- __DECC - current versions of DEC C (5.9 for instance) for alpha are ANSI,
- but don't define __STDC__ in their default mode. Don't know if old
- versions might have been K&R, but let's not worry about that unless
- someone is still using one.
-
- _mips - gnu ansidecl.h says the RISC/OS MIPS compiler is ANSI in SVR4
- mode, but doesn't define __STDC__.
-
- _MSC_VER - Microsoft C is ANSI, but __STDC__ is undefined unless the /Za
- option is given (in which case it's 1).
-
- _WIN32 - tested for by gnu ansidecl.h, no doubt on the assumption that
- all w32 compilers are ansi.
-
- Note: This same set of tests is used by gen-psqr.c and
- demos/expr/expr-impl.h, so if anything needs adding, then be sure to
- update those too. */
-
-#if defined (__STDC__) \
- || defined (__cplusplus) \
- || defined (_AIX) \
- || defined (__DECC) \
- || (defined (__mips) && defined (_SYSTYPE_SVR4)) \
- || defined (_MSC_VER) \
- || defined (_WIN32)
-#define __GMP_HAVE_CONST 1
-#define __GMP_HAVE_PROTOTYPES 1
-#define __GMP_HAVE_TOKEN_PASTE 1
-#else
-#define __GMP_HAVE_CONST 0
-#define __GMP_HAVE_PROTOTYPES 0
-#define __GMP_HAVE_TOKEN_PASTE 0
-#endif
-
-
-#if __GMP_HAVE_CONST
-#define __gmp_const const
-#define __gmp_signed signed
-#else
-#define __gmp_const
-#define __gmp_signed
-#endif
-
-
-/* __GMP_DECLSPEC supports Windows DLL versions of libgmp, and is empty in
- all other circumstances.
-
- When compiling objects for libgmp, __GMP_DECLSPEC is an export directive,
- or when compiling for an application it's an import directive. The two
- cases are differentiated by __GMP_WITHIN_GMP defined by the GMP Makefiles
- (and not defined from an application).
-
- __GMP_DECLSPEC_XX is similarly used for libgmpxx. __GMP_WITHIN_GMPXX
- indicates when building libgmpxx, and in that case libgmpxx functions are
- exports, but libgmp functions which might get called are imports.
-
- libmp.la uses __GMP_DECLSPEC, just as if it were libgmp.la. libgmp and
- libmp don't call each other, so there's no conflict or confusion.
-
- Libtool DLL_EXPORT define is not used.
-
- There's no attempt to support GMP built both static and DLL. Doing so
- would mean applications would have to tell us which of the two is going
- to be used when linking, and that seems very tedious and error prone if
- using GMP by hand, and equally tedious from a package since autoconf and
- automake don't give much help.
-
- __GMP_DECLSPEC is required on all documented global functions and
- variables, the various internals in gmp-impl.h etc can be left unadorned.
- But internals used by the test programs or speed measuring programs
- should have __GMP_DECLSPEC, and certainly constants or variables must
- have it or the wrong address will be resolved.
-
- In gcc __declspec can go at either the start or end of a prototype.
-
- In Microsoft C __declspec must go at the start, or after the type like
- void __declspec(...) *foo()". There's no __dllexport or anything to
- guard against someone foolish #defining dllexport. _export used to be
- available, but no longer.
-
- In Borland C _export still exists, but needs to go after the type, like
- "void _export foo();". Would have to change the __GMP_DECLSPEC syntax to
- make use of that. Probably more trouble than it's worth. */
-
-#if defined (__GNUC__)
-#define __GMP_DECLSPEC_EXPORT __declspec(__dllexport__)
-#define __GMP_DECLSPEC_IMPORT __declspec(__dllimport__)
-#endif
-#if defined (_MSC_VER) || defined (__BORLANDC__)
-#define __GMP_DECLSPEC_EXPORT __declspec(dllexport)
-#define __GMP_DECLSPEC_IMPORT __declspec(dllimport)
-#endif
-#ifdef __WATCOMC__
-#define __GMP_DECLSPEC_EXPORT __export
-#define __GMP_DECLSPEC_IMPORT __import
-#endif
-#ifdef __IBMC__
-#define __GMP_DECLSPEC_EXPORT _Export
-#define __GMP_DECLSPEC_IMPORT _Import
-#endif
-
-#if __GMP_LIBGMP_DLL
-#if __GMP_WITHIN_GMP
-/* compiling to go into a DLL libgmp */
-#define __GMP_DECLSPEC __GMP_DECLSPEC_EXPORT
-#else
-/* compiling to go into an application which will link to a DLL libgmp */
-#define __GMP_DECLSPEC __GMP_DECLSPEC_IMPORT
-#endif
-#else
-/* all other cases */
-#define __GMP_DECLSPEC
-#endif
-
-
-#ifdef __GMP_SHORT_LIMB
-typedef unsigned int mp_limb_t;
-typedef int mp_limb_signed_t;
-#else
-#ifdef _LONG_LONG_LIMB
-typedef unsigned long long int mp_limb_t;
-typedef long long int mp_limb_signed_t;
-#else
-typedef unsigned long int mp_limb_t;
-typedef long int mp_limb_signed_t;
-#endif
-#endif
-typedef unsigned long int mp_bitcnt_t;
-
-/* For reference, note that the name __mpz_struct gets into C++ mangled
- function names, which means although the "__" suggests an internal, we
- must leave this name for binary compatibility. */
-typedef struct
-{
- int _mp_alloc; /* Number of *limbs* allocated and pointed
- to by the _mp_d field. */
- int _mp_size; /* abs(_mp_size) is the number of limbs the
- last field points to. If _mp_size is
- negative this is a negative number. */
- mp_limb_t *_mp_d; /* Pointer to the limbs. */
-} __mpz_struct;
-
-#endif /* __GNU_MP__ */
-
-
-typedef __mpz_struct MP_INT; /* gmp 1 source compatibility */
-typedef __mpz_struct mpz_t[1];
-
-typedef mp_limb_t * mp_ptr;
-typedef __gmp_const mp_limb_t * mp_srcptr;
-#if defined (_CRAY) && ! defined (_CRAYMPP)
-/* plain `int' is much faster (48 bits) */
-#define __GMP_MP_SIZE_T_INT 1
-typedef int mp_size_t;
-typedef int mp_exp_t;
-#else
-#define __GMP_MP_SIZE_T_INT 0
-typedef long int mp_size_t;
-typedef long int mp_exp_t;
-#endif
-
-typedef struct
-{
- __mpz_struct _mp_num;
- __mpz_struct _mp_den;
-} __mpq_struct;
-
-typedef __mpq_struct MP_RAT; /* gmp 1 source compatibility */
-typedef __mpq_struct mpq_t[1];
-
-typedef struct
-{
- int _mp_prec; /* Max precision, in number of `mp_limb_t's.
- Set by mpf_init and modified by
- mpf_set_prec. The area pointed to by the
- _mp_d field contains `prec' + 1 limbs. */
- int _mp_size; /* abs(_mp_size) is the number of limbs the
- last field points to. If _mp_size is
- negative this is a negative number. */
- mp_exp_t _mp_exp; /* Exponent, in the base of `mp_limb_t'. */
- mp_limb_t *_mp_d; /* Pointer to the limbs. */
-} __mpf_struct;
-
-/* typedef __mpf_struct MP_FLOAT; */
-typedef __mpf_struct mpf_t[1];
-
-/* Available random number generation algorithms. */
-typedef enum
-{
- GMP_RAND_ALG_DEFAULT = 0,
- GMP_RAND_ALG_LC = GMP_RAND_ALG_DEFAULT /* Linear congruential. */
-} gmp_randalg_t;
-
-/* Random state struct. */
-typedef struct
-{
- mpz_t _mp_seed; /* _mp_d member points to state of the generator. */
- gmp_randalg_t _mp_alg; /* Currently unused. */
- union {
- void *_mp_lc; /* Pointer to function pointers structure. */
- } _mp_algdata;
-} __gmp_randstate_struct;
-typedef __gmp_randstate_struct gmp_randstate_t[1];
-
-/* Types for function declarations in gmp files. */
-/* ??? Should not pollute user name space with these ??? */
-typedef __gmp_const __mpz_struct *mpz_srcptr;
-typedef __mpz_struct *mpz_ptr;
-typedef __gmp_const __mpf_struct *mpf_srcptr;
-typedef __mpf_struct *mpf_ptr;
-typedef __gmp_const __mpq_struct *mpq_srcptr;
-typedef __mpq_struct *mpq_ptr;
-
-
-/* This is not wanted in mp.h, so put it outside the __GNU_MP__ common
- section. */
-#if __GMP_LIBGMP_DLL
-#if __GMP_WITHIN_GMPXX
-/* compiling to go into a DLL libgmpxx */
-#define __GMP_DECLSPEC_XX __GMP_DECLSPEC_EXPORT
-#else
-/* compiling to go into a application which will link to a DLL libgmpxx */
-#define __GMP_DECLSPEC_XX __GMP_DECLSPEC_IMPORT
-#endif
-#else
-/* all other cases */
-#define __GMP_DECLSPEC_XX
-#endif
-
-
-#if __GMP_HAVE_PROTOTYPES
-#define __GMP_PROTO(x) x
-#else
-#define __GMP_PROTO(x) ()
-#endif
-
-#ifndef __MPN
-#if __GMP_HAVE_TOKEN_PASTE
-#define __MPN(x) __gmpn_##x
-#else
-#define __MPN(x) __gmpn_/**/x
-#endif
-#endif
-
-/* For reference, "defined(EOF)" cannot be used here. In g++ 2.95.4,
- <iostream> defines EOF but not FILE. */
-#if defined (FILE) \
- || defined (H_STDIO) \
- || defined (_H_STDIO) /* AIX */ \
- || defined (_STDIO_H) /* glibc, Sun, SCO */ \
- || defined (_STDIO_H_) /* BSD, OSF */ \
- || defined (__STDIO_H) /* Borland */ \
- || defined (__STDIO_H__) /* IRIX */ \
- || defined (_STDIO_INCLUDED) /* HPUX */ \
- || defined (__dj_include_stdio_h_) /* DJGPP */ \
- || defined (_FILE_DEFINED) /* Microsoft */ \
- || defined (__STDIO__) /* Apple MPW MrC */ \
- || defined (_MSL_STDIO_H) /* Metrowerks */ \
- || defined (_STDIO_H_INCLUDED) /* QNX4 */ \
- || defined (_ISO_STDIO_ISO_H) /* Sun C++ */
-#define _GMP_H_HAVE_FILE 1
-#endif
-
-/* In ISO C, if a prototype involving "struct obstack *" is given without
- that structure defined, then the struct is scoped down to just the
- prototype, causing a conflict if it's subsequently defined for real. So
- only give prototypes if we've got obstack.h. */
-#if defined (_OBSTACK_H) /* glibc <obstack.h> */
-#define _GMP_H_HAVE_OBSTACK 1
-#endif
-
-/* The prototypes for gmp_vprintf etc are provided only if va_list is
- available, via an application having included <stdarg.h> or <varargs.h>.
- Usually va_list is a typedef so can't be tested directly, but C99
- specifies that va_start is a macro (and it was normally a macro on past
- systems too), so look for that.
-
- <stdio.h> will define some sort of va_list for vprintf and vfprintf, but
- let's not bother trying to use that since it's not standard and since
- application uses for gmp_vprintf etc will almost certainly require the
- whole <stdarg.h> or <varargs.h> anyway. */
-
-#ifdef va_start
-#define _GMP_H_HAVE_VA_LIST 1
-#endif
-
-/* Test for gcc >= maj.min, as per __GNUC_PREREQ in glibc */
-#if defined (__GNUC__) && defined (__GNUC_MINOR__)
-#define __GMP_GNUC_PREREQ(maj, min) \
- ((__GNUC__ << 16) + __GNUC_MINOR__ >= ((maj) << 16) + (min))
-#else
-#define __GMP_GNUC_PREREQ(maj, min) 0
-#endif
-
-/* "pure" is in gcc 2.96 and up, see "(gcc)Function Attributes". Basically
- it means a function does nothing but examine its arguments and memory
- (global or via arguments) to generate a return value, but changes nothing
- and has no side-effects. __GMP_NO_ATTRIBUTE_CONST_PURE lets
- tune/common.c etc turn this off when trying to write timing loops. */
-#if __GMP_GNUC_PREREQ (2,96) && ! defined (__GMP_NO_ATTRIBUTE_CONST_PURE)
-#define __GMP_ATTRIBUTE_PURE __attribute__ ((__pure__))
-#else
-#define __GMP_ATTRIBUTE_PURE
-#endif
-
-
-/* __GMP_CAST allows us to use static_cast in C++, so our macros are clean
- to "g++ -Wold-style-cast".
-
- Casts in "extern inline" code within an extern "C" block don't induce
- these warnings, so __GMP_CAST only needs to be used on documented
- macros. */
-
-#ifdef __cplusplus
-#define __GMP_CAST(type, expr) (static_cast<type> (expr))
-#else
-#define __GMP_CAST(type, expr) ((type) (expr))
-#endif
-
-
-/* An empty "throw ()" means the function doesn't throw any C++ exceptions,
- this can save some stack frame info in applications.
-
- Currently it's given only on functions which never divide-by-zero etc,
- don't allocate memory, and are expected to never need to allocate memory.
- This leaves open the possibility of a C++ throw from a future GMP
- exceptions scheme.
-
- mpz_set_ui etc are omitted to leave open the lazy allocation scheme
- described in doc/tasks.html. mpz_get_d etc are omitted to leave open
- exceptions for float overflows.
-
- Note that __GMP_NOTHROW must be given on any inlines the same as on their
- prototypes (for g++ at least, where they're used together). Note also
- that g++ 3.0 demands that __GMP_NOTHROW is before other attributes like
- __GMP_ATTRIBUTE_PURE. */
-
-#if defined (__cplusplus)
-#define __GMP_NOTHROW throw ()
-#else
-#define __GMP_NOTHROW
-#endif
-
-
-/* PORTME: What other compilers have a useful "extern inline"? "static
- inline" would be an acceptable substitute if the compiler (or linker)
- discards unused statics. */
-
- /* gcc has __inline__ in all modes, including strict ansi. Give a prototype
- for an inline too, so as to correctly specify "dllimport" on windows, in
- case the function is called rather than inlined.
- GCC 4.3 and above with -std=c99 or -std=gnu99 implements ISO C99
- inline semantics, unless -fgnu89-inline is used. */
-#ifdef __GNUC__
-#if (defined __GNUC_STDC_INLINE__) || (__GNUC__ == 4 && __GNUC_MINOR__ == 2)
-#define __GMP_EXTERN_INLINE extern __inline__ __attribute__ ((__gnu_inline__))
-#else
-#define __GMP_EXTERN_INLINE extern __inline__
-#endif
-#define __GMP_INLINE_PROTOTYPES 1
-#endif
-
-/* DEC C (eg. version 5.9) supports "static __inline foo()", even in -std1
- strict ANSI mode. Inlining is done even when not optimizing (ie. -O0
- mode, which is the default), but an unnecessary local copy of foo is
- emitted unless -O is used. "extern __inline" is accepted, but the
- "extern" appears to be ignored, ie. it becomes a plain global function
- but which is inlined within its file. Don't know if all old versions of
- DEC C supported __inline, but as a start let's do the right thing for
- current versions. */
-#ifdef __DECC
-#define __GMP_EXTERN_INLINE static __inline
-#endif
-
-/* SCO OpenUNIX 8 cc supports "static inline foo()" but not in -Xc strict
- ANSI mode (__STDC__ is 1 in that mode). Inlining only actually takes
- place under -O. Without -O "foo" seems to be emitted whether it's used
- or not, which is wasteful. "extern inline foo()" isn't useful, the
- "extern" is apparently ignored, so foo is inlined if possible but also
- emitted as a global, which causes multiple definition errors when
- building a shared libgmp. */
-#ifdef __SCO_VERSION__
-#if __SCO_VERSION__ > 400000000 && __STDC__ != 1 \
- && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE static inline
-#endif
-#endif
-
-/* Microsoft's C compiler accepts __inline */
-#ifdef _MSC_VER
-#define __GMP_EXTERN_INLINE __inline
-#endif
-
-/* Recent enough Sun C compilers want "inline" */
-#if defined (__SUNPRO_C) && __SUNPRO_C >= 0x560 \
- && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE inline
-#endif
-
-/* Somewhat older Sun C compilers want "static inline" */
-#if defined (__SUNPRO_C) && __SUNPRO_C >= 0x540 \
- && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE static inline
-#endif
-
-
-/* C++ always has "inline" and since it's a normal feature the linker should
- discard duplicate non-inlined copies, or if it doesn't then that's a
- problem for everyone, not just GMP. */
-#if defined (__cplusplus) && ! defined (__GMP_EXTERN_INLINE)
-#define __GMP_EXTERN_INLINE inline
-#endif
-
-/* Don't do any inlining within a configure run, since if the compiler ends
- up emitting copies of the code into the object file it can end up
- demanding the various support routines (like mpn_popcount) for linking,
- making the "alloca" test and perhaps others fail. And on hppa ia64 a
- pre-release gcc 3.2 was seen not respecting the "extern" in "extern
- __inline__", triggering this problem too. */
-#if defined (__GMP_WITHIN_CONFIGURE) && ! __GMP_WITHIN_CONFIGURE_INLINE
-#undef __GMP_EXTERN_INLINE
-#endif
-
-/* By default, don't give a prototype when there's going to be an inline
- version. Note in particular that Cray C++ objects to the combination of
- prototype and inline. */
-#ifdef __GMP_EXTERN_INLINE
-#ifndef __GMP_INLINE_PROTOTYPES
-#define __GMP_INLINE_PROTOTYPES 0
-#endif
-#else
-#define __GMP_INLINE_PROTOTYPES 1
-#endif
-
-
-#define __GMP_ABS(x) ((x) >= 0 ? (x) : -(x))
-#define __GMP_MAX(h,i) ((h) > (i) ? (h) : (i))
-
-/* __GMP_USHRT_MAX is not "~ (unsigned short) 0" because short is promoted
- to int by "~". */
-#define __GMP_UINT_MAX (~ (unsigned) 0)
-#define __GMP_ULONG_MAX (~ (unsigned long) 0)
-#define __GMP_USHRT_MAX ((unsigned short) ~0)
-
-
-/* __builtin_expect is in gcc 3.0, and not in 2.95. */
-#if __GMP_GNUC_PREREQ (3,0)
-#define __GMP_LIKELY(cond) __builtin_expect ((cond) != 0, 1)
-#define __GMP_UNLIKELY(cond) __builtin_expect ((cond) != 0, 0)
-#else
-#define __GMP_LIKELY(cond) (cond)
-#define __GMP_UNLIKELY(cond) (cond)
-#endif
-
-#ifdef _CRAY
-#define __GMP_CRAY_Pragma(str) _Pragma (str)
-#else
-#define __GMP_CRAY_Pragma(str)
-#endif
-
-
-/* Allow direct user access to numerator and denominator of a mpq_t object. */
-#define mpq_numref(Q) (&((Q)->_mp_num))
-#define mpq_denref(Q) (&((Q)->_mp_den))
-
-
-#if defined (__cplusplus)
-extern "C" {
-using std::FILE;
-#endif
-
-#define mp_set_memory_functions __gmp_set_memory_functions
-__GMP_DECLSPEC void mp_set_memory_functions __GMP_PROTO ((void *(*) (size_t),
- void *(*) (void *, size_t, size_t),
- void (*) (void *, size_t))) __GMP_NOTHROW;
-
-#define mp_get_memory_functions __gmp_get_memory_functions
-__GMP_DECLSPEC void mp_get_memory_functions __GMP_PROTO ((void *(**) (size_t),
- void *(**) (void *, size_t, size_t),
- void (**) (void *, size_t))) __GMP_NOTHROW;
-
-#define mp_bits_per_limb __gmp_bits_per_limb
-__GMP_DECLSPEC extern __gmp_const int mp_bits_per_limb;
-
-#define gmp_errno __gmp_errno
-__GMP_DECLSPEC extern int gmp_errno;
-
-#define gmp_version __gmp_version
-__GMP_DECLSPEC extern __gmp_const char * __gmp_const gmp_version;
-
-
-/**************** Random number routines. ****************/
-
-/* obsolete */
-#define gmp_randinit __gmp_randinit
-__GMP_DECLSPEC void gmp_randinit __GMP_PROTO ((gmp_randstate_t, gmp_randalg_t, ...));
-
-#define gmp_randinit_default __gmp_randinit_default
-__GMP_DECLSPEC void gmp_randinit_default __GMP_PROTO ((gmp_randstate_t));
-
-#define gmp_randinit_lc_2exp __gmp_randinit_lc_2exp
-__GMP_DECLSPEC void gmp_randinit_lc_2exp __GMP_PROTO ((gmp_randstate_t,
- mpz_srcptr, unsigned long int,
- mp_bitcnt_t));
-
-#define gmp_randinit_lc_2exp_size __gmp_randinit_lc_2exp_size
-__GMP_DECLSPEC int gmp_randinit_lc_2exp_size __GMP_PROTO ((gmp_randstate_t, mp_bitcnt_t));
-
-#define gmp_randinit_mt __gmp_randinit_mt
-__GMP_DECLSPEC void gmp_randinit_mt __GMP_PROTO ((gmp_randstate_t));
-
-#define gmp_randinit_set __gmp_randinit_set
-__GMP_DECLSPEC void gmp_randinit_set __GMP_PROTO ((gmp_randstate_t, __gmp_const __gmp_randstate_struct *));
-
-#define gmp_randseed __gmp_randseed
-__GMP_DECLSPEC void gmp_randseed __GMP_PROTO ((gmp_randstate_t, mpz_srcptr));
-
-#define gmp_randseed_ui __gmp_randseed_ui
-__GMP_DECLSPEC void gmp_randseed_ui __GMP_PROTO ((gmp_randstate_t, unsigned long int));
-
-#define gmp_randclear __gmp_randclear
-__GMP_DECLSPEC void gmp_randclear __GMP_PROTO ((gmp_randstate_t));
-
-#define gmp_urandomb_ui __gmp_urandomb_ui
-__GMP_DECLSPEC unsigned long gmp_urandomb_ui __GMP_PROTO ((gmp_randstate_t, unsigned long));
-
-#define gmp_urandomm_ui __gmp_urandomm_ui
-__GMP_DECLSPEC unsigned long gmp_urandomm_ui __GMP_PROTO ((gmp_randstate_t, unsigned long));
-
-
-/**************** Formatted output routines. ****************/
-
-#define gmp_asprintf __gmp_asprintf
-__GMP_DECLSPEC int gmp_asprintf __GMP_PROTO ((char **, __gmp_const char *, ...));
-
-#define gmp_fprintf __gmp_fprintf
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC int gmp_fprintf __GMP_PROTO ((FILE *, __gmp_const char *, ...));
-#endif
-
-#define gmp_obstack_printf __gmp_obstack_printf
-#if defined (_GMP_H_HAVE_OBSTACK)
-__GMP_DECLSPEC int gmp_obstack_printf __GMP_PROTO ((struct obstack *, __gmp_const char *, ...));
-#endif
-
-#define gmp_obstack_vprintf __gmp_obstack_vprintf
-#if defined (_GMP_H_HAVE_OBSTACK) && defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_obstack_vprintf __GMP_PROTO ((struct obstack *, __gmp_const char *, va_list));
-#endif
-
-#define gmp_printf __gmp_printf
-__GMP_DECLSPEC int gmp_printf __GMP_PROTO ((__gmp_const char *, ...));
-
-#define gmp_snprintf __gmp_snprintf
-__GMP_DECLSPEC int gmp_snprintf __GMP_PROTO ((char *, size_t, __gmp_const char *, ...));
-
-#define gmp_sprintf __gmp_sprintf
-__GMP_DECLSPEC int gmp_sprintf __GMP_PROTO ((char *, __gmp_const char *, ...));
-
-#define gmp_vasprintf __gmp_vasprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vasprintf __GMP_PROTO ((char **, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vfprintf __gmp_vfprintf
-#if defined (_GMP_H_HAVE_FILE) && defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vfprintf __GMP_PROTO ((FILE *, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vprintf __gmp_vprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vprintf __GMP_PROTO ((__gmp_const char *, va_list));
-#endif
-
-#define gmp_vsnprintf __gmp_vsnprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vsnprintf __GMP_PROTO ((char *, size_t, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vsprintf __gmp_vsprintf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vsprintf __GMP_PROTO ((char *, __gmp_const char *, va_list));
-#endif
-
-
-/**************** Formatted input routines. ****************/
-
-#define gmp_fscanf __gmp_fscanf
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC int gmp_fscanf __GMP_PROTO ((FILE *, __gmp_const char *, ...));
-#endif
-
-#define gmp_scanf __gmp_scanf
-__GMP_DECLSPEC int gmp_scanf __GMP_PROTO ((__gmp_const char *, ...));
-
-#define gmp_sscanf __gmp_sscanf
-__GMP_DECLSPEC int gmp_sscanf __GMP_PROTO ((__gmp_const char *, __gmp_const char *, ...));
-
-#define gmp_vfscanf __gmp_vfscanf
-#if defined (_GMP_H_HAVE_FILE) && defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vfscanf __GMP_PROTO ((FILE *, __gmp_const char *, va_list));
-#endif
-
-#define gmp_vscanf __gmp_vscanf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vscanf __GMP_PROTO ((__gmp_const char *, va_list));
-#endif
-
-#define gmp_vsscanf __gmp_vsscanf
-#if defined (_GMP_H_HAVE_VA_LIST)
-__GMP_DECLSPEC int gmp_vsscanf __GMP_PROTO ((__gmp_const char *, __gmp_const char *, va_list));
-#endif
-
-
-/**************** Integer (i.e. Z) routines. ****************/
-
-#define _mpz_realloc __gmpz_realloc
-#define mpz_realloc __gmpz_realloc
-__GMP_DECLSPEC void *_mpz_realloc __GMP_PROTO ((mpz_ptr, mp_size_t));
-
-#define mpz_abs __gmpz_abs
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_abs)
-__GMP_DECLSPEC void mpz_abs __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-#endif
-
-#define mpz_add __gmpz_add
-__GMP_DECLSPEC void mpz_add __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_add_ui __gmpz_add_ui
-__GMP_DECLSPEC void mpz_add_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_addmul __gmpz_addmul
-__GMP_DECLSPEC void mpz_addmul __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_addmul_ui __gmpz_addmul_ui
-__GMP_DECLSPEC void mpz_addmul_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_and __gmpz_and
-__GMP_DECLSPEC void mpz_and __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_array_init __gmpz_array_init
-__GMP_DECLSPEC void mpz_array_init __GMP_PROTO ((mpz_ptr, mp_size_t, mp_size_t));
-
-#define mpz_bin_ui __gmpz_bin_ui
-__GMP_DECLSPEC void mpz_bin_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_bin_uiui __gmpz_bin_uiui
-__GMP_DECLSPEC void mpz_bin_uiui __GMP_PROTO ((mpz_ptr, unsigned long int, unsigned long int));
-
-#define mpz_cdiv_q __gmpz_cdiv_q
-__GMP_DECLSPEC void mpz_cdiv_q __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_cdiv_q_2exp __gmpz_cdiv_q_2exp
-__GMP_DECLSPEC void mpz_cdiv_q_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long));
-
-#define mpz_cdiv_q_ui __gmpz_cdiv_q_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_q_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_cdiv_qr __gmpz_cdiv_qr
-__GMP_DECLSPEC void mpz_cdiv_qr __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_cdiv_qr_ui __gmpz_cdiv_qr_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_qr_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_cdiv_r __gmpz_cdiv_r
-__GMP_DECLSPEC void mpz_cdiv_r __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_cdiv_r_2exp __gmpz_cdiv_r_2exp
-__GMP_DECLSPEC void mpz_cdiv_r_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_cdiv_r_ui __gmpz_cdiv_r_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_r_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_cdiv_ui __gmpz_cdiv_ui
-__GMP_DECLSPEC unsigned long int mpz_cdiv_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_clear __gmpz_clear
-__GMP_DECLSPEC void mpz_clear __GMP_PROTO ((mpz_ptr));
-
-#define mpz_clears __gmpz_clears
-__GMP_DECLSPEC void mpz_clears __GMP_PROTO ((mpz_ptr, ...));
-
-#define mpz_clrbit __gmpz_clrbit
-__GMP_DECLSPEC void mpz_clrbit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_cmp __gmpz_cmp
-__GMP_DECLSPEC int mpz_cmp __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmp_d __gmpz_cmp_d
-__GMP_DECLSPEC int mpz_cmp_d __GMP_PROTO ((mpz_srcptr, double)) __GMP_ATTRIBUTE_PURE;
-
-#define _mpz_cmp_si __gmpz_cmp_si
-__GMP_DECLSPEC int _mpz_cmp_si __GMP_PROTO ((mpz_srcptr, signed long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define _mpz_cmp_ui __gmpz_cmp_ui
-__GMP_DECLSPEC int _mpz_cmp_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmpabs __gmpz_cmpabs
-__GMP_DECLSPEC int mpz_cmpabs __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmpabs_d __gmpz_cmpabs_d
-__GMP_DECLSPEC int mpz_cmpabs_d __GMP_PROTO ((mpz_srcptr, double)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_cmpabs_ui __gmpz_cmpabs_ui
-__GMP_DECLSPEC int mpz_cmpabs_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_com __gmpz_com
-__GMP_DECLSPEC void mpz_com __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_combit __gmpz_combit
-__GMP_DECLSPEC void mpz_combit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_congruent_p __gmpz_congruent_p
-__GMP_DECLSPEC int mpz_congruent_p __GMP_PROTO ((mpz_srcptr, mpz_srcptr, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_congruent_2exp_p __gmpz_congruent_2exp_p
-__GMP_DECLSPEC int mpz_congruent_2exp_p __GMP_PROTO ((mpz_srcptr, mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_congruent_ui_p __gmpz_congruent_ui_p
-__GMP_DECLSPEC int mpz_congruent_ui_p __GMP_PROTO ((mpz_srcptr, unsigned long, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_divexact __gmpz_divexact
-__GMP_DECLSPEC void mpz_divexact __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_divexact_ui __gmpz_divexact_ui
-__GMP_DECLSPEC void mpz_divexact_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long));
-
-#define mpz_divisible_p __gmpz_divisible_p
-__GMP_DECLSPEC int mpz_divisible_p __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_divisible_ui_p __gmpz_divisible_ui_p
-__GMP_DECLSPEC int mpz_divisible_ui_p __GMP_PROTO ((mpz_srcptr, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_divisible_2exp_p __gmpz_divisible_2exp_p
-__GMP_DECLSPEC int mpz_divisible_2exp_p __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_dump __gmpz_dump
-__GMP_DECLSPEC void mpz_dump __GMP_PROTO ((mpz_srcptr));
-
-#define mpz_export __gmpz_export
-__GMP_DECLSPEC void *mpz_export __GMP_PROTO ((void *, size_t *, int, size_t, int, size_t, mpz_srcptr));
-
-#define mpz_fac_ui __gmpz_fac_ui
-__GMP_DECLSPEC void mpz_fac_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_fdiv_q __gmpz_fdiv_q
-__GMP_DECLSPEC void mpz_fdiv_q __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_fdiv_q_2exp __gmpz_fdiv_q_2exp
-__GMP_DECLSPEC void mpz_fdiv_q_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_fdiv_q_ui __gmpz_fdiv_q_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_q_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_fdiv_qr __gmpz_fdiv_qr
-__GMP_DECLSPEC void mpz_fdiv_qr __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_fdiv_qr_ui __gmpz_fdiv_qr_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_qr_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_fdiv_r __gmpz_fdiv_r
-__GMP_DECLSPEC void mpz_fdiv_r __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_fdiv_r_2exp __gmpz_fdiv_r_2exp
-__GMP_DECLSPEC void mpz_fdiv_r_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_fdiv_r_ui __gmpz_fdiv_r_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_r_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_fdiv_ui __gmpz_fdiv_ui
-__GMP_DECLSPEC unsigned long int mpz_fdiv_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fib_ui __gmpz_fib_ui
-__GMP_DECLSPEC void mpz_fib_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_fib2_ui __gmpz_fib2_ui
-__GMP_DECLSPEC void mpz_fib2_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, unsigned long int));
-
-#define mpz_fits_sint_p __gmpz_fits_sint_p
-__GMP_DECLSPEC int mpz_fits_sint_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fits_slong_p __gmpz_fits_slong_p
-__GMP_DECLSPEC int mpz_fits_slong_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fits_sshort_p __gmpz_fits_sshort_p
-__GMP_DECLSPEC int mpz_fits_sshort_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_fits_uint_p __gmpz_fits_uint_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_fits_uint_p)
-__GMP_DECLSPEC int mpz_fits_uint_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_fits_ulong_p __gmpz_fits_ulong_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_fits_ulong_p)
-__GMP_DECLSPEC int mpz_fits_ulong_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_fits_ushort_p __gmpz_fits_ushort_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_fits_ushort_p)
-__GMP_DECLSPEC int mpz_fits_ushort_p __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_gcd __gmpz_gcd
-__GMP_DECLSPEC void mpz_gcd __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_gcd_ui __gmpz_gcd_ui
-__GMP_DECLSPEC unsigned long int mpz_gcd_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_gcdext __gmpz_gcdext
-__GMP_DECLSPEC void mpz_gcdext __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_get_d __gmpz_get_d
-__GMP_DECLSPEC double mpz_get_d __GMP_PROTO ((mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_get_d_2exp __gmpz_get_d_2exp
-__GMP_DECLSPEC double mpz_get_d_2exp __GMP_PROTO ((signed long int *, mpz_srcptr));
-
-#define mpz_get_si __gmpz_get_si
-__GMP_DECLSPEC /* signed */ long int mpz_get_si __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_get_str __gmpz_get_str
-__GMP_DECLSPEC char *mpz_get_str __GMP_PROTO ((char *, int, mpz_srcptr));
-
-#define mpz_get_ui __gmpz_get_ui
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_get_ui)
-__GMP_DECLSPEC unsigned long int mpz_get_ui __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_getlimbn __gmpz_getlimbn
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_getlimbn)
-__GMP_DECLSPEC mp_limb_t mpz_getlimbn __GMP_PROTO ((mpz_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_hamdist __gmpz_hamdist
-__GMP_DECLSPEC mp_bitcnt_t mpz_hamdist __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_import __gmpz_import
-__GMP_DECLSPEC void mpz_import __GMP_PROTO ((mpz_ptr, size_t, int, size_t, int, size_t, __gmp_const void *));
-
-#define mpz_init __gmpz_init
-__GMP_DECLSPEC void mpz_init __GMP_PROTO ((mpz_ptr));
-
-#define mpz_init2 __gmpz_init2
-__GMP_DECLSPEC void mpz_init2 __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_inits __gmpz_inits
-__GMP_DECLSPEC void mpz_inits __GMP_PROTO ((mpz_ptr, ...));
-
-#define mpz_init_set __gmpz_init_set
-__GMP_DECLSPEC void mpz_init_set __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_init_set_d __gmpz_init_set_d
-__GMP_DECLSPEC void mpz_init_set_d __GMP_PROTO ((mpz_ptr, double));
-
-#define mpz_init_set_si __gmpz_init_set_si
-__GMP_DECLSPEC void mpz_init_set_si __GMP_PROTO ((mpz_ptr, signed long int));
-
-#define mpz_init_set_str __gmpz_init_set_str
-__GMP_DECLSPEC int mpz_init_set_str __GMP_PROTO ((mpz_ptr, __gmp_const char *, int));
-
-#define mpz_init_set_ui __gmpz_init_set_ui
-__GMP_DECLSPEC void mpz_init_set_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_inp_raw __gmpz_inp_raw
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_inp_raw __GMP_PROTO ((mpz_ptr, FILE *));
-#endif
-
-#define mpz_inp_str __gmpz_inp_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_inp_str __GMP_PROTO ((mpz_ptr, FILE *, int));
-#endif
-
-#define mpz_invert __gmpz_invert
-__GMP_DECLSPEC int mpz_invert __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_ior __gmpz_ior
-__GMP_DECLSPEC void mpz_ior __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_jacobi __gmpz_jacobi
-__GMP_DECLSPEC int mpz_jacobi __GMP_PROTO ((mpz_srcptr, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_kronecker mpz_jacobi /* alias */
-
-#define mpz_kronecker_si __gmpz_kronecker_si
-__GMP_DECLSPEC int mpz_kronecker_si __GMP_PROTO ((mpz_srcptr, long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_kronecker_ui __gmpz_kronecker_ui
-__GMP_DECLSPEC int mpz_kronecker_ui __GMP_PROTO ((mpz_srcptr, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_si_kronecker __gmpz_si_kronecker
-__GMP_DECLSPEC int mpz_si_kronecker __GMP_PROTO ((long, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_ui_kronecker __gmpz_ui_kronecker
-__GMP_DECLSPEC int mpz_ui_kronecker __GMP_PROTO ((unsigned long, mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_lcm __gmpz_lcm
-__GMP_DECLSPEC void mpz_lcm __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_lcm_ui __gmpz_lcm_ui
-__GMP_DECLSPEC void mpz_lcm_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long));
-
-#define mpz_legendre mpz_jacobi /* alias */
-
-#define mpz_lucnum_ui __gmpz_lucnum_ui
-__GMP_DECLSPEC void mpz_lucnum_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_lucnum2_ui __gmpz_lucnum2_ui
-__GMP_DECLSPEC void mpz_lucnum2_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, unsigned long int));
-
-#define mpz_millerrabin __gmpz_millerrabin
-__GMP_DECLSPEC int mpz_millerrabin __GMP_PROTO ((mpz_srcptr, int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_mod __gmpz_mod
-__GMP_DECLSPEC void mpz_mod __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_mod_ui mpz_fdiv_r_ui /* same as fdiv_r because divisor unsigned */
-
-#define mpz_mul __gmpz_mul
-__GMP_DECLSPEC void mpz_mul __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_mul_2exp __gmpz_mul_2exp
-__GMP_DECLSPEC void mpz_mul_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_mul_si __gmpz_mul_si
-__GMP_DECLSPEC void mpz_mul_si __GMP_PROTO ((mpz_ptr, mpz_srcptr, long int));
-
-#define mpz_mul_ui __gmpz_mul_ui
-__GMP_DECLSPEC void mpz_mul_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_neg __gmpz_neg
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_neg)
-__GMP_DECLSPEC void mpz_neg __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-#endif
-
-#define mpz_nextprime __gmpz_nextprime
-__GMP_DECLSPEC void mpz_nextprime __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_out_raw __gmpz_out_raw
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_out_raw __GMP_PROTO ((FILE *, mpz_srcptr));
-#endif
-
-#define mpz_out_str __gmpz_out_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpz_out_str __GMP_PROTO ((FILE *, int, mpz_srcptr));
-#endif
-
-#define mpz_perfect_power_p __gmpz_perfect_power_p
-__GMP_DECLSPEC int mpz_perfect_power_p __GMP_PROTO ((mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_perfect_square_p __gmpz_perfect_square_p
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_perfect_square_p)
-__GMP_DECLSPEC int mpz_perfect_square_p __GMP_PROTO ((mpz_srcptr)) __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_popcount __gmpz_popcount
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_popcount)
-__GMP_DECLSPEC mp_bitcnt_t mpz_popcount __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_pow_ui __gmpz_pow_ui
-__GMP_DECLSPEC void mpz_pow_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_powm __gmpz_powm
-__GMP_DECLSPEC void mpz_powm __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_powm_sec __gmpz_powm_sec
-__GMP_DECLSPEC void mpz_powm_sec __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_powm_ui __gmpz_powm_ui
-__GMP_DECLSPEC void mpz_powm_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int, mpz_srcptr));
-
-#define mpz_probab_prime_p __gmpz_probab_prime_p
-__GMP_DECLSPEC int mpz_probab_prime_p __GMP_PROTO ((mpz_srcptr, int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_random __gmpz_random
-__GMP_DECLSPEC void mpz_random __GMP_PROTO ((mpz_ptr, mp_size_t));
-
-#define mpz_random2 __gmpz_random2
-__GMP_DECLSPEC void mpz_random2 __GMP_PROTO ((mpz_ptr, mp_size_t));
-
-#define mpz_realloc2 __gmpz_realloc2
-__GMP_DECLSPEC void mpz_realloc2 __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_remove __gmpz_remove
-__GMP_DECLSPEC unsigned long int mpz_remove __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_root __gmpz_root
-__GMP_DECLSPEC int mpz_root __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_rootrem __gmpz_rootrem
-__GMP_DECLSPEC void mpz_rootrem __GMP_PROTO ((mpz_ptr,mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_rrandomb __gmpz_rrandomb
-__GMP_DECLSPEC void mpz_rrandomb __GMP_PROTO ((mpz_ptr, gmp_randstate_t, mp_bitcnt_t));
-
-#define mpz_scan0 __gmpz_scan0
-__GMP_DECLSPEC mp_bitcnt_t mpz_scan0 __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_scan1 __gmpz_scan1
-__GMP_DECLSPEC mp_bitcnt_t mpz_scan1 __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_set __gmpz_set
-__GMP_DECLSPEC void mpz_set __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_set_d __gmpz_set_d
-__GMP_DECLSPEC void mpz_set_d __GMP_PROTO ((mpz_ptr, double));
-
-#define mpz_set_f __gmpz_set_f
-__GMP_DECLSPEC void mpz_set_f __GMP_PROTO ((mpz_ptr, mpf_srcptr));
-
-#define mpz_set_q __gmpz_set_q
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_set_q)
-__GMP_DECLSPEC void mpz_set_q __GMP_PROTO ((mpz_ptr, mpq_srcptr));
-#endif
-
-#define mpz_set_si __gmpz_set_si
-__GMP_DECLSPEC void mpz_set_si __GMP_PROTO ((mpz_ptr, signed long int));
-
-#define mpz_set_str __gmpz_set_str
-__GMP_DECLSPEC int mpz_set_str __GMP_PROTO ((mpz_ptr, __gmp_const char *, int));
-
-#define mpz_set_ui __gmpz_set_ui
-__GMP_DECLSPEC void mpz_set_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
-
-#define mpz_setbit __gmpz_setbit
-__GMP_DECLSPEC void mpz_setbit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
-
-#define mpz_size __gmpz_size
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpz_size)
-__GMP_DECLSPEC size_t mpz_size __GMP_PROTO ((mpz_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpz_sizeinbase __gmpz_sizeinbase
-__GMP_DECLSPEC size_t mpz_sizeinbase __GMP_PROTO ((mpz_srcptr, int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_sqrt __gmpz_sqrt
-__GMP_DECLSPEC void mpz_sqrt __GMP_PROTO ((mpz_ptr, mpz_srcptr));
-
-#define mpz_sqrtrem __gmpz_sqrtrem
-__GMP_DECLSPEC void mpz_sqrtrem __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr));
-
-#define mpz_sub __gmpz_sub
-__GMP_DECLSPEC void mpz_sub __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_sub_ui __gmpz_sub_ui
-__GMP_DECLSPEC void mpz_sub_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_ui_sub __gmpz_ui_sub
-__GMP_DECLSPEC void mpz_ui_sub __GMP_PROTO ((mpz_ptr, unsigned long int, mpz_srcptr));
-
-#define mpz_submul __gmpz_submul
-__GMP_DECLSPEC void mpz_submul __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_submul_ui __gmpz_submul_ui
-__GMP_DECLSPEC void mpz_submul_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_swap __gmpz_swap
-__GMP_DECLSPEC void mpz_swap __GMP_PROTO ((mpz_ptr, mpz_ptr)) __GMP_NOTHROW;
-
-#define mpz_tdiv_ui __gmpz_tdiv_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_ui __GMP_PROTO ((mpz_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpz_tdiv_q __gmpz_tdiv_q
-__GMP_DECLSPEC void mpz_tdiv_q __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_tdiv_q_2exp __gmpz_tdiv_q_2exp
-__GMP_DECLSPEC void mpz_tdiv_q_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_tdiv_q_ui __gmpz_tdiv_q_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_q_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_tdiv_qr __gmpz_tdiv_qr
-__GMP_DECLSPEC void mpz_tdiv_qr __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_tdiv_qr_ui __gmpz_tdiv_qr_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_qr_ui __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_tdiv_r __gmpz_tdiv_r
-__GMP_DECLSPEC void mpz_tdiv_r __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-#define mpz_tdiv_r_2exp __gmpz_tdiv_r_2exp
-__GMP_DECLSPEC void mpz_tdiv_r_2exp __GMP_PROTO ((mpz_ptr, mpz_srcptr, mp_bitcnt_t));
-
-#define mpz_tdiv_r_ui __gmpz_tdiv_r_ui
-__GMP_DECLSPEC unsigned long int mpz_tdiv_r_ui __GMP_PROTO ((mpz_ptr, mpz_srcptr, unsigned long int));
-
-#define mpz_tstbit __gmpz_tstbit
-__GMP_DECLSPEC int mpz_tstbit __GMP_PROTO ((mpz_srcptr, mp_bitcnt_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpz_ui_pow_ui __gmpz_ui_pow_ui
-__GMP_DECLSPEC void mpz_ui_pow_ui __GMP_PROTO ((mpz_ptr, unsigned long int, unsigned long int));
-
-#define mpz_urandomb __gmpz_urandomb
-__GMP_DECLSPEC void mpz_urandomb __GMP_PROTO ((mpz_ptr, gmp_randstate_t, mp_bitcnt_t));
-
-#define mpz_urandomm __gmpz_urandomm
-__GMP_DECLSPEC void mpz_urandomm __GMP_PROTO ((mpz_ptr, gmp_randstate_t, mpz_srcptr));
-
-#define mpz_xor __gmpz_xor
-#define mpz_eor __gmpz_xor
-__GMP_DECLSPEC void mpz_xor __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
-
-
-/**************** Rational (i.e. Q) routines. ****************/
-
-#define mpq_abs __gmpq_abs
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpq_abs)
-__GMP_DECLSPEC void mpq_abs __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-#endif
-
-#define mpq_add __gmpq_add
-__GMP_DECLSPEC void mpq_add __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_canonicalize __gmpq_canonicalize
-__GMP_DECLSPEC void mpq_canonicalize __GMP_PROTO ((mpq_ptr));
-
-#define mpq_clear __gmpq_clear
-__GMP_DECLSPEC void mpq_clear __GMP_PROTO ((mpq_ptr));
-
-#define mpq_clears __gmpq_clears
-__GMP_DECLSPEC void mpq_clears __GMP_PROTO ((mpq_ptr, ...));
-
-#define mpq_cmp __gmpq_cmp
-__GMP_DECLSPEC int mpq_cmp __GMP_PROTO ((mpq_srcptr, mpq_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define _mpq_cmp_si __gmpq_cmp_si
-__GMP_DECLSPEC int _mpq_cmp_si __GMP_PROTO ((mpq_srcptr, long, unsigned long)) __GMP_ATTRIBUTE_PURE;
-
-#define _mpq_cmp_ui __gmpq_cmp_ui
-__GMP_DECLSPEC int _mpq_cmp_ui __GMP_PROTO ((mpq_srcptr, unsigned long int, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpq_div __gmpq_div
-__GMP_DECLSPEC void mpq_div __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_div_2exp __gmpq_div_2exp
-__GMP_DECLSPEC void mpq_div_2exp __GMP_PROTO ((mpq_ptr, mpq_srcptr, mp_bitcnt_t));
-
-#define mpq_equal __gmpq_equal
-__GMP_DECLSPEC int mpq_equal __GMP_PROTO ((mpq_srcptr, mpq_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpq_get_num __gmpq_get_num
-__GMP_DECLSPEC void mpq_get_num __GMP_PROTO ((mpz_ptr, mpq_srcptr));
-
-#define mpq_get_den __gmpq_get_den
-__GMP_DECLSPEC void mpq_get_den __GMP_PROTO ((mpz_ptr, mpq_srcptr));
-
-#define mpq_get_d __gmpq_get_d
-__GMP_DECLSPEC double mpq_get_d __GMP_PROTO ((mpq_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpq_get_str __gmpq_get_str
-__GMP_DECLSPEC char *mpq_get_str __GMP_PROTO ((char *, int, mpq_srcptr));
-
-#define mpq_init __gmpq_init
-__GMP_DECLSPEC void mpq_init __GMP_PROTO ((mpq_ptr));
-
-#define mpq_inits __gmpq_inits
-__GMP_DECLSPEC void mpq_inits __GMP_PROTO ((mpq_ptr, ...));
-
-#define mpq_inp_str __gmpq_inp_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpq_inp_str __GMP_PROTO ((mpq_ptr, FILE *, int));
-#endif
-
-#define mpq_inv __gmpq_inv
-__GMP_DECLSPEC void mpq_inv __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-
-#define mpq_mul __gmpq_mul
-__GMP_DECLSPEC void mpq_mul __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_mul_2exp __gmpq_mul_2exp
-__GMP_DECLSPEC void mpq_mul_2exp __GMP_PROTO ((mpq_ptr, mpq_srcptr, mp_bitcnt_t));
-
-#define mpq_neg __gmpq_neg
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpq_neg)
-__GMP_DECLSPEC void mpq_neg __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-#endif
-
-#define mpq_out_str __gmpq_out_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpq_out_str __GMP_PROTO ((FILE *, int, mpq_srcptr));
-#endif
-
-#define mpq_set __gmpq_set
-__GMP_DECLSPEC void mpq_set __GMP_PROTO ((mpq_ptr, mpq_srcptr));
-
-#define mpq_set_d __gmpq_set_d
-__GMP_DECLSPEC void mpq_set_d __GMP_PROTO ((mpq_ptr, double));
-
-#define mpq_set_den __gmpq_set_den
-__GMP_DECLSPEC void mpq_set_den __GMP_PROTO ((mpq_ptr, mpz_srcptr));
-
-#define mpq_set_f __gmpq_set_f
-__GMP_DECLSPEC void mpq_set_f __GMP_PROTO ((mpq_ptr, mpf_srcptr));
-
-#define mpq_set_num __gmpq_set_num
-__GMP_DECLSPEC void mpq_set_num __GMP_PROTO ((mpq_ptr, mpz_srcptr));
-
-#define mpq_set_si __gmpq_set_si
-__GMP_DECLSPEC void mpq_set_si __GMP_PROTO ((mpq_ptr, signed long int, unsigned long int));
-
-#define mpq_set_str __gmpq_set_str
-__GMP_DECLSPEC int mpq_set_str __GMP_PROTO ((mpq_ptr, __gmp_const char *, int));
-
-#define mpq_set_ui __gmpq_set_ui
-__GMP_DECLSPEC void mpq_set_ui __GMP_PROTO ((mpq_ptr, unsigned long int, unsigned long int));
-
-#define mpq_set_z __gmpq_set_z
-__GMP_DECLSPEC void mpq_set_z __GMP_PROTO ((mpq_ptr, mpz_srcptr));
-
-#define mpq_sub __gmpq_sub
-__GMP_DECLSPEC void mpq_sub __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
-
-#define mpq_swap __gmpq_swap
-__GMP_DECLSPEC void mpq_swap __GMP_PROTO ((mpq_ptr, mpq_ptr)) __GMP_NOTHROW;
-
-
-/**************** Float (i.e. F) routines. ****************/
-
-#define mpf_abs __gmpf_abs
-__GMP_DECLSPEC void mpf_abs __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_add __gmpf_add
-__GMP_DECLSPEC void mpf_add __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_add_ui __gmpf_add_ui
-__GMP_DECLSPEC void mpf_add_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-#define mpf_ceil __gmpf_ceil
-__GMP_DECLSPEC void mpf_ceil __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_clear __gmpf_clear
-__GMP_DECLSPEC void mpf_clear __GMP_PROTO ((mpf_ptr));
-
-#define mpf_clears __gmpf_clears
-__GMP_DECLSPEC void mpf_clears __GMP_PROTO ((mpf_ptr, ...));
-
-#define mpf_cmp __gmpf_cmp
-__GMP_DECLSPEC int mpf_cmp __GMP_PROTO ((mpf_srcptr, mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_cmp_d __gmpf_cmp_d
-__GMP_DECLSPEC int mpf_cmp_d __GMP_PROTO ((mpf_srcptr, double)) __GMP_ATTRIBUTE_PURE;
-
-#define mpf_cmp_si __gmpf_cmp_si
-__GMP_DECLSPEC int mpf_cmp_si __GMP_PROTO ((mpf_srcptr, signed long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_cmp_ui __gmpf_cmp_ui
-__GMP_DECLSPEC int mpf_cmp_ui __GMP_PROTO ((mpf_srcptr, unsigned long int)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_div __gmpf_div
-__GMP_DECLSPEC void mpf_div __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_div_2exp __gmpf_div_2exp
-__GMP_DECLSPEC void mpf_div_2exp __GMP_PROTO ((mpf_ptr, mpf_srcptr, mp_bitcnt_t));
-
-#define mpf_div_ui __gmpf_div_ui
-__GMP_DECLSPEC void mpf_div_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_dump __gmpf_dump
-__GMP_DECLSPEC void mpf_dump __GMP_PROTO ((mpf_srcptr));
-
-#define mpf_eq __gmpf_eq
-__GMP_DECLSPEC int mpf_eq __GMP_PROTO ((mpf_srcptr, mpf_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_sint_p __gmpf_fits_sint_p
-__GMP_DECLSPEC int mpf_fits_sint_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_slong_p __gmpf_fits_slong_p
-__GMP_DECLSPEC int mpf_fits_slong_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_sshort_p __gmpf_fits_sshort_p
-__GMP_DECLSPEC int mpf_fits_sshort_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_uint_p __gmpf_fits_uint_p
-__GMP_DECLSPEC int mpf_fits_uint_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_ulong_p __gmpf_fits_ulong_p
-__GMP_DECLSPEC int mpf_fits_ulong_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_fits_ushort_p __gmpf_fits_ushort_p
-__GMP_DECLSPEC int mpf_fits_ushort_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_floor __gmpf_floor
-__GMP_DECLSPEC void mpf_floor __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_get_d __gmpf_get_d
-__GMP_DECLSPEC double mpf_get_d __GMP_PROTO ((mpf_srcptr)) __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_d_2exp __gmpf_get_d_2exp
-__GMP_DECLSPEC double mpf_get_d_2exp __GMP_PROTO ((signed long int *, mpf_srcptr));
-
-#define mpf_get_default_prec __gmpf_get_default_prec
-__GMP_DECLSPEC mp_bitcnt_t mpf_get_default_prec __GMP_PROTO ((void)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_prec __gmpf_get_prec
-__GMP_DECLSPEC mp_bitcnt_t mpf_get_prec __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_si __gmpf_get_si
-__GMP_DECLSPEC long mpf_get_si __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_get_str __gmpf_get_str
-__GMP_DECLSPEC char *mpf_get_str __GMP_PROTO ((char *, mp_exp_t *, int, size_t, mpf_srcptr));
-
-#define mpf_get_ui __gmpf_get_ui
-__GMP_DECLSPEC unsigned long mpf_get_ui __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_init __gmpf_init
-__GMP_DECLSPEC void mpf_init __GMP_PROTO ((mpf_ptr));
-
-#define mpf_init2 __gmpf_init2
-__GMP_DECLSPEC void mpf_init2 __GMP_PROTO ((mpf_ptr, mp_bitcnt_t));
-
-#define mpf_inits __gmpf_inits
-__GMP_DECLSPEC void mpf_inits __GMP_PROTO ((mpf_ptr, ...));
-
-#define mpf_init_set __gmpf_init_set
-__GMP_DECLSPEC void mpf_init_set __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_init_set_d __gmpf_init_set_d
-__GMP_DECLSPEC void mpf_init_set_d __GMP_PROTO ((mpf_ptr, double));
-
-#define mpf_init_set_si __gmpf_init_set_si
-__GMP_DECLSPEC void mpf_init_set_si __GMP_PROTO ((mpf_ptr, signed long int));
-
-#define mpf_init_set_str __gmpf_init_set_str
-__GMP_DECLSPEC int mpf_init_set_str __GMP_PROTO ((mpf_ptr, __gmp_const char *, int));
-
-#define mpf_init_set_ui __gmpf_init_set_ui
-__GMP_DECLSPEC void mpf_init_set_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
-
-#define mpf_inp_str __gmpf_inp_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpf_inp_str __GMP_PROTO ((mpf_ptr, FILE *, int));
-#endif
-
-#define mpf_integer_p __gmpf_integer_p
-__GMP_DECLSPEC int mpf_integer_p __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_mul __gmpf_mul
-__GMP_DECLSPEC void mpf_mul __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_mul_2exp __gmpf_mul_2exp
-__GMP_DECLSPEC void mpf_mul_2exp __GMP_PROTO ((mpf_ptr, mpf_srcptr, mp_bitcnt_t));
-
-#define mpf_mul_ui __gmpf_mul_ui
-__GMP_DECLSPEC void mpf_mul_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_neg __gmpf_neg
-__GMP_DECLSPEC void mpf_neg __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_out_str __gmpf_out_str
-#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC size_t mpf_out_str __GMP_PROTO ((FILE *, int, size_t, mpf_srcptr));
-#endif
-
-#define mpf_pow_ui __gmpf_pow_ui
-__GMP_DECLSPEC void mpf_pow_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_random2 __gmpf_random2
-__GMP_DECLSPEC void mpf_random2 __GMP_PROTO ((mpf_ptr, mp_size_t, mp_exp_t));
-
-#define mpf_reldiff __gmpf_reldiff
-__GMP_DECLSPEC void mpf_reldiff __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_set __gmpf_set
-__GMP_DECLSPEC void mpf_set __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_set_d __gmpf_set_d
-__GMP_DECLSPEC void mpf_set_d __GMP_PROTO ((mpf_ptr, double));
-
-#define mpf_set_default_prec __gmpf_set_default_prec
-__GMP_DECLSPEC void mpf_set_default_prec __GMP_PROTO ((mp_bitcnt_t)) __GMP_NOTHROW;
-
-#define mpf_set_prec __gmpf_set_prec
-__GMP_DECLSPEC void mpf_set_prec __GMP_PROTO ((mpf_ptr, mp_bitcnt_t));
-
-#define mpf_set_prec_raw __gmpf_set_prec_raw
-__GMP_DECLSPEC void mpf_set_prec_raw __GMP_PROTO ((mpf_ptr, mp_bitcnt_t)) __GMP_NOTHROW;
-
-#define mpf_set_q __gmpf_set_q
-__GMP_DECLSPEC void mpf_set_q __GMP_PROTO ((mpf_ptr, mpq_srcptr));
-
-#define mpf_set_si __gmpf_set_si
-__GMP_DECLSPEC void mpf_set_si __GMP_PROTO ((mpf_ptr, signed long int));
-
-#define mpf_set_str __gmpf_set_str
-__GMP_DECLSPEC int mpf_set_str __GMP_PROTO ((mpf_ptr, __gmp_const char *, int));
-
-#define mpf_set_ui __gmpf_set_ui
-__GMP_DECLSPEC void mpf_set_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
-
-#define mpf_set_z __gmpf_set_z
-__GMP_DECLSPEC void mpf_set_z __GMP_PROTO ((mpf_ptr, mpz_srcptr));
-
-#define mpf_size __gmpf_size
-__GMP_DECLSPEC size_t mpf_size __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpf_sqrt __gmpf_sqrt
-__GMP_DECLSPEC void mpf_sqrt __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_sqrt_ui __gmpf_sqrt_ui
-__GMP_DECLSPEC void mpf_sqrt_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
-
-#define mpf_sub __gmpf_sub
-__GMP_DECLSPEC void mpf_sub __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
-
-#define mpf_sub_ui __gmpf_sub_ui
-__GMP_DECLSPEC void mpf_sub_ui __GMP_PROTO ((mpf_ptr, mpf_srcptr, unsigned long int));
-
-#define mpf_swap __gmpf_swap
-__GMP_DECLSPEC void mpf_swap __GMP_PROTO ((mpf_ptr, mpf_ptr)) __GMP_NOTHROW;
-
-#define mpf_trunc __gmpf_trunc
-__GMP_DECLSPEC void mpf_trunc __GMP_PROTO ((mpf_ptr, mpf_srcptr));
-
-#define mpf_ui_div __gmpf_ui_div
-__GMP_DECLSPEC void mpf_ui_div __GMP_PROTO ((mpf_ptr, unsigned long int, mpf_srcptr));
-
-#define mpf_ui_sub __gmpf_ui_sub
-__GMP_DECLSPEC void mpf_ui_sub __GMP_PROTO ((mpf_ptr, unsigned long int, mpf_srcptr));
-
-#define mpf_urandomb __gmpf_urandomb
-__GMP_DECLSPEC void mpf_urandomb __GMP_PROTO ((mpf_t, gmp_randstate_t, mp_bitcnt_t));
-
-
-/************ Low level positive-integer (i.e. N) routines. ************/
-
-/* This is ugly, but we need to make user calls reach the prefixed function. */
-
-#define mpn_add __MPN(add)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_add)
-__GMP_DECLSPEC mp_limb_t mpn_add __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_srcptr,mp_size_t));
-#endif
-
-#define mpn_add_1 __MPN(add_1)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_add_1)
-__GMP_DECLSPEC mp_limb_t mpn_add_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t)) __GMP_NOTHROW;
-#endif
-
-#define mpn_add_n __MPN(add_n)
-__GMP_DECLSPEC mp_limb_t mpn_add_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_addmul_1 __MPN(addmul_1)
-__GMP_DECLSPEC mp_limb_t mpn_addmul_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_cmp __MPN(cmp)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_cmp)
-__GMP_DECLSPEC int mpn_cmp __GMP_PROTO ((mp_srcptr, mp_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-#endif
-
-#define mpn_divexact_by3(dst,src,size) \
- mpn_divexact_by3c (dst, src, size, __GMP_CAST (mp_limb_t, 0))
-
-#define mpn_divexact_by3c __MPN(divexact_by3c)
-__GMP_DECLSPEC mp_limb_t mpn_divexact_by3c __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_divmod_1(qp,np,nsize,dlimb) \
- mpn_divrem_1 (qp, __GMP_CAST (mp_size_t, 0), np, nsize, dlimb)
-
-#define mpn_divrem __MPN(divrem)
-__GMP_DECLSPEC mp_limb_t mpn_divrem __GMP_PROTO ((mp_ptr, mp_size_t, mp_ptr, mp_size_t, mp_srcptr, mp_size_t));
-
-#define mpn_divrem_1 __MPN(divrem_1)
-__GMP_DECLSPEC mp_limb_t mpn_divrem_1 __GMP_PROTO ((mp_ptr, mp_size_t, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_divrem_2 __MPN(divrem_2)
-__GMP_DECLSPEC mp_limb_t mpn_divrem_2 __GMP_PROTO ((mp_ptr, mp_size_t, mp_ptr, mp_size_t, mp_srcptr));
-
-#define mpn_gcd __MPN(gcd)
-__GMP_DECLSPEC mp_size_t mpn_gcd __GMP_PROTO ((mp_ptr, mp_ptr, mp_size_t, mp_ptr, mp_size_t));
-
-#define mpn_gcd_1 __MPN(gcd_1)
-__GMP_DECLSPEC mp_limb_t mpn_gcd_1 __GMP_PROTO ((mp_srcptr, mp_size_t, mp_limb_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_gcdext_1 __MPN(gcdext_1)
-__GMP_DECLSPEC mp_limb_t mpn_gcdext_1 __GMP_PROTO ((mp_limb_signed_t *, mp_limb_signed_t *, mp_limb_t, mp_limb_t));
-
-#define mpn_gcdext __MPN(gcdext)
-__GMP_DECLSPEC mp_size_t mpn_gcdext __GMP_PROTO ((mp_ptr, mp_ptr, mp_size_t *, mp_ptr, mp_size_t, mp_ptr, mp_size_t));
-
-#define mpn_get_str __MPN(get_str)
-__GMP_DECLSPEC size_t mpn_get_str __GMP_PROTO ((unsigned char *, int, mp_ptr, mp_size_t));
-
-#define mpn_hamdist __MPN(hamdist)
-__GMP_DECLSPEC mp_bitcnt_t mpn_hamdist __GMP_PROTO ((mp_srcptr, mp_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpn_lshift __MPN(lshift)
-__GMP_DECLSPEC mp_limb_t mpn_lshift __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, unsigned int));
-
-#define mpn_mod_1 __MPN(mod_1)
-__GMP_DECLSPEC mp_limb_t mpn_mod_1 __GMP_PROTO ((mp_srcptr, mp_size_t, mp_limb_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_mul __MPN(mul)
-__GMP_DECLSPEC mp_limb_t mpn_mul __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t));
-
-#define mpn_mul_1 __MPN(mul_1)
-__GMP_DECLSPEC mp_limb_t mpn_mul_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_mul_n __MPN(mul_n)
-__GMP_DECLSPEC void mpn_mul_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_sqr __MPN(sqr)
-__GMP_DECLSPEC void mpn_sqr __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-
-#define mpn_neg __MPN(neg)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_neg)
-__GMP_DECLSPEC mp_limb_t mpn_neg __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#endif
-
-#define mpn_com __MPN(com)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_com)
-__GMP_DECLSPEC void mpn_com __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#endif
-
-#define mpn_perfect_square_p __MPN(perfect_square_p)
-__GMP_DECLSPEC int mpn_perfect_square_p __GMP_PROTO ((mp_srcptr, mp_size_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_perfect_power_p __MPN(perfect_power_p)
-__GMP_DECLSPEC int mpn_perfect_power_p __GMP_PROTO ((mp_srcptr, mp_size_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_popcount __MPN(popcount)
-__GMP_DECLSPEC mp_bitcnt_t mpn_popcount __GMP_PROTO ((mp_srcptr, mp_size_t)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
-
-#define mpn_pow_1 __MPN(pow_1)
-__GMP_DECLSPEC mp_size_t mpn_pow_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t, mp_ptr));
-
-/* undocumented now, but retained here for upward compatibility */
-#define mpn_preinv_mod_1 __MPN(preinv_mod_1)
-__GMP_DECLSPEC mp_limb_t mpn_preinv_mod_1 __GMP_PROTO ((mp_srcptr, mp_size_t, mp_limb_t, mp_limb_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_random __MPN(random)
-__GMP_DECLSPEC void mpn_random __GMP_PROTO ((mp_ptr, mp_size_t));
-
-#define mpn_random2 __MPN(random2)
-__GMP_DECLSPEC void mpn_random2 __GMP_PROTO ((mp_ptr, mp_size_t));
-
-#define mpn_rshift __MPN(rshift)
-__GMP_DECLSPEC mp_limb_t mpn_rshift __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, unsigned int));
-
-#define mpn_scan0 __MPN(scan0)
-__GMP_DECLSPEC mp_bitcnt_t mpn_scan0 __GMP_PROTO ((mp_srcptr, mp_bitcnt_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_scan1 __MPN(scan1)
-__GMP_DECLSPEC mp_bitcnt_t mpn_scan1 __GMP_PROTO ((mp_srcptr, mp_bitcnt_t)) __GMP_ATTRIBUTE_PURE;
-
-#define mpn_set_str __MPN(set_str)
-__GMP_DECLSPEC mp_size_t mpn_set_str __GMP_PROTO ((mp_ptr, __gmp_const unsigned char *, size_t, int));
-
-#define mpn_sqrtrem __MPN(sqrtrem)
-__GMP_DECLSPEC mp_size_t mpn_sqrtrem __GMP_PROTO ((mp_ptr, mp_ptr, mp_srcptr, mp_size_t));
-
-#define mpn_sub __MPN(sub)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_sub)
-__GMP_DECLSPEC mp_limb_t mpn_sub __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_srcptr,mp_size_t));
-#endif
-
-#define mpn_sub_1 __MPN(sub_1)
-#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_sub_1)
-__GMP_DECLSPEC mp_limb_t mpn_sub_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t)) __GMP_NOTHROW;
-#endif
-
-#define mpn_sub_n __MPN(sub_n)
-__GMP_DECLSPEC mp_limb_t mpn_sub_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_submul_1 __MPN(submul_1)
-__GMP_DECLSPEC mp_limb_t mpn_submul_1 __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t, mp_limb_t));
-
-#define mpn_tdiv_qr __MPN(tdiv_qr)
-__GMP_DECLSPEC void mpn_tdiv_qr __GMP_PROTO ((mp_ptr, mp_ptr, mp_size_t, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t));
-
-#define mpn_and_n __MPN(and_n)
-__GMP_DECLSPEC void mpn_and_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_andn_n __MPN(andn_n)
-__GMP_DECLSPEC void mpn_andn_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_nand_n __MPN(nand_n)
-__GMP_DECLSPEC void mpn_nand_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_ior_n __MPN(ior_n)
-__GMP_DECLSPEC void mpn_ior_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_iorn_n __MPN(iorn_n)
-__GMP_DECLSPEC void mpn_iorn_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_nior_n __MPN(nior_n)
-__GMP_DECLSPEC void mpn_nior_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_xor_n __MPN(xor_n)
-__GMP_DECLSPEC void mpn_xor_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-#define mpn_xnor_n __MPN(xnor_n)
-__GMP_DECLSPEC void mpn_xnor_n __GMP_PROTO ((mp_ptr, mp_srcptr, mp_srcptr, mp_size_t));
-
-#define mpn_copyi __MPN(copyi)
-__GMP_DECLSPEC void mpn_copyi __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#define mpn_copyd __MPN(copyd)
-__GMP_DECLSPEC void mpn_copyd __GMP_PROTO ((mp_ptr, mp_srcptr, mp_size_t));
-#define mpn_zero __MPN(zero)
-__GMP_DECLSPEC void mpn_zero __GMP_PROTO ((mp_ptr, mp_size_t));
-
-/**************** mpz inlines ****************/
-
-/* The following are provided as inlines where possible, but always exist as
- library functions too, for binary compatibility.
-
- Within gmp itself this inlining generally isn't relied on, since it
- doesn't get done for all compilers, whereas if something is worth
- inlining then it's worth arranging always.
-
- There are two styles of inlining here. When the same bit of code is
- wanted for the inline as for the library version, then __GMP_FORCE_foo
- arranges for that code to be emitted and the __GMP_EXTERN_INLINE
- directive suppressed, eg. mpz_fits_uint_p. When a different bit of code
- is wanted for the inline than for the library version, then
- __GMP_FORCE_foo arranges the inline to be suppressed, eg. mpz_abs. */
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpz_abs)
-__GMP_EXTERN_INLINE void
-mpz_abs (mpz_ptr __gmp_w, mpz_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpz_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_size = __GMP_ABS (__gmp_w->_mp_size);
-}
-#endif
-
-#if GMP_NAIL_BITS == 0
-#define __GMPZ_FITS_UTYPE_P(z,maxval) \
- mp_size_t __gmp_n = z->_mp_size; \
- mp_ptr __gmp_p = z->_mp_d; \
- return (__gmp_n == 0 || (__gmp_n == 1 && __gmp_p[0] <= maxval));
-#else
-#define __GMPZ_FITS_UTYPE_P(z,maxval) \
- mp_size_t __gmp_n = z->_mp_size; \
- mp_ptr __gmp_p = z->_mp_d; \
- return (__gmp_n == 0 || (__gmp_n == 1 && __gmp_p[0] <= maxval) \
- || (__gmp_n == 2 && __gmp_p[1] <= ((mp_limb_t) maxval >> GMP_NUMB_BITS)));
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_fits_uint_p)
-#if ! defined (__GMP_FORCE_mpz_fits_uint_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_fits_uint_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_UINT_MAX);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_fits_ulong_p)
-#if ! defined (__GMP_FORCE_mpz_fits_ulong_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_fits_ulong_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_ULONG_MAX);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_fits_ushort_p)
-#if ! defined (__GMP_FORCE_mpz_fits_ushort_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_fits_ushort_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_USHRT_MAX);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_get_ui)
-#if ! defined (__GMP_FORCE_mpz_get_ui)
-__GMP_EXTERN_INLINE
-#endif
-unsigned long
-mpz_get_ui (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- mp_ptr __gmp_p = __gmp_z->_mp_d;
- mp_size_t __gmp_n = __gmp_z->_mp_size;
- mp_limb_t __gmp_l = __gmp_p[0];
- /* This is a "#if" rather than a plain "if" so as to avoid gcc warnings
- about "<< GMP_NUMB_BITS" exceeding the type size, and to avoid Borland
- C++ 6.0 warnings about condition always true for something like
- "__GMP_ULONG_MAX < GMP_NUMB_MASK". */
-#if GMP_NAIL_BITS == 0 || defined (_LONG_LONG_LIMB)
- /* limb==long and no nails, or limb==longlong, one limb is enough */
- return (__gmp_n != 0 ? __gmp_l : 0);
-#else
- /* limb==long and nails, need two limbs when available */
- __gmp_n = __GMP_ABS (__gmp_n);
- if (__gmp_n <= 1)
- return (__gmp_n != 0 ? __gmp_l : 0);
- else
- return __gmp_l + (__gmp_p[1] << GMP_NUMB_BITS);
-#endif
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_getlimbn)
-#if ! defined (__GMP_FORCE_mpz_getlimbn)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpz_getlimbn (mpz_srcptr __gmp_z, mp_size_t __gmp_n) __GMP_NOTHROW
-{
- mp_limb_t __gmp_result = 0;
- if (__GMP_LIKELY (__gmp_n >= 0 && __gmp_n < __GMP_ABS (__gmp_z->_mp_size)))
- __gmp_result = __gmp_z->_mp_d[__gmp_n];
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpz_neg)
-__GMP_EXTERN_INLINE void
-mpz_neg (mpz_ptr __gmp_w, mpz_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpz_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_size = - __gmp_w->_mp_size;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_perfect_square_p)
-#if ! defined (__GMP_FORCE_mpz_perfect_square_p)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpz_perfect_square_p (mpz_srcptr __gmp_a)
-{
- mp_size_t __gmp_asize;
- int __gmp_result;
-
- __gmp_asize = __gmp_a->_mp_size;
- __gmp_result = (__gmp_asize >= 0); /* zero is a square, negatives are not */
- if (__GMP_LIKELY (__gmp_asize > 0))
- __gmp_result = mpn_perfect_square_p (__gmp_a->_mp_d, __gmp_asize);
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_popcount)
-#if ! defined (__GMP_FORCE_mpz_popcount)
-__GMP_EXTERN_INLINE
-#endif
-mp_bitcnt_t
-mpz_popcount (mpz_srcptr __gmp_u) __GMP_NOTHROW
-{
- mp_size_t __gmp_usize;
- mp_bitcnt_t __gmp_result;
-
- __gmp_usize = __gmp_u->_mp_size;
- __gmp_result = (__gmp_usize < 0 ? __GMP_ULONG_MAX : 0);
- if (__GMP_LIKELY (__gmp_usize > 0))
- __gmp_result = mpn_popcount (__gmp_u->_mp_d, __gmp_usize);
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_set_q)
-#if ! defined (__GMP_FORCE_mpz_set_q)
-__GMP_EXTERN_INLINE
-#endif
-void
-mpz_set_q (mpz_ptr __gmp_w, mpq_srcptr __gmp_u)
-{
- mpz_tdiv_q (__gmp_w, mpq_numref (__gmp_u), mpq_denref (__gmp_u));
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpz_size)
-#if ! defined (__GMP_FORCE_mpz_size)
-__GMP_EXTERN_INLINE
-#endif
-size_t
-mpz_size (mpz_srcptr __gmp_z) __GMP_NOTHROW
-{
- return __GMP_ABS (__gmp_z->_mp_size);
-}
-#endif
-
-
-/**************** mpq inlines ****************/
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpq_abs)
-__GMP_EXTERN_INLINE void
-mpq_abs (mpq_ptr __gmp_w, mpq_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpq_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_num._mp_size = __GMP_ABS (__gmp_w->_mp_num._mp_size);
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) && ! defined (__GMP_FORCE_mpq_neg)
-__GMP_EXTERN_INLINE void
-mpq_neg (mpq_ptr __gmp_w, mpq_srcptr __gmp_u)
-{
- if (__gmp_w != __gmp_u)
- mpq_set (__gmp_w, __gmp_u);
- __gmp_w->_mp_num._mp_size = - __gmp_w->_mp_num._mp_size;
-}
-#endif
-
-
-/**************** mpn inlines ****************/
-
-/* The comments with __GMPN_ADD_1 below apply here too.
-
- The test for FUNCTION returning 0 should predict well. If it's assumed
- {yp,ysize} will usually have a random number of bits then the high limb
- won't be full and a carry out will occur a good deal less than 50% of the
- time.
-
- ysize==0 isn't a documented feature, but is used internally in a few
- places.
-
- Producing cout last stops it using up a register during the main part of
- the calculation, though gcc (as of 3.0) on an "if (mpn_add (...))"
- doesn't seem able to move the true and false legs of the conditional up
- to the two places cout is generated. */
-
-#define __GMPN_AORS(cout, wp, xp, xsize, yp, ysize, FUNCTION, TEST) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x; \
- \
- /* ASSERT ((ysize) >= 0); */ \
- /* ASSERT ((xsize) >= (ysize)); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE2_P (wp, xsize, xp, xsize)); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE2_P (wp, xsize, yp, ysize)); */ \
- \
- __gmp_i = (ysize); \
- if (__gmp_i != 0) \
- { \
- if (FUNCTION (wp, xp, yp, __gmp_i)) \
- { \
- do \
- { \
- if (__gmp_i >= (xsize)) \
- { \
- (cout) = 1; \
- goto __gmp_done; \
- } \
- __gmp_x = (xp)[__gmp_i]; \
- } \
- while (TEST); \
- } \
- } \
- if ((wp) != (xp)) \
- __GMPN_COPY_REST (wp, xp, xsize, __gmp_i); \
- (cout) = 0; \
- __gmp_done: \
- ; \
- } while (0)
-
-#define __GMPN_ADD(cout, wp, xp, xsize, yp, ysize) \
- __GMPN_AORS (cout, wp, xp, xsize, yp, ysize, mpn_add_n, \
- (((wp)[__gmp_i++] = (__gmp_x + 1) & GMP_NUMB_MASK) == 0))
-#define __GMPN_SUB(cout, wp, xp, xsize, yp, ysize) \
- __GMPN_AORS (cout, wp, xp, xsize, yp, ysize, mpn_sub_n, \
- (((wp)[__gmp_i++] = (__gmp_x - 1) & GMP_NUMB_MASK), __gmp_x == 0))
-
-
-/* The use of __gmp_i indexing is designed to ensure a compile time src==dst
- remains nice and clear to the compiler, so that __GMPN_COPY_REST can
- disappear, and the load/add/store gets a chance to become a
- read-modify-write on CISC CPUs.
-
- Alternatives:
-
- Using a pair of pointers instead of indexing would be possible, but gcc
- isn't able to recognise compile-time src==dst in that case, even when the
- pointers are incremented more or less together. Other compilers would
- very likely have similar difficulty.
-
- gcc could use "if (__builtin_constant_p(src==dst) && src==dst)" or
- similar to detect a compile-time src==dst. This works nicely on gcc
- 2.95.x, it's not good on gcc 3.0 where __builtin_constant_p(p==p) seems
- to be always false, for a pointer p. But the current code form seems
- good enough for src==dst anyway.
-
- gcc on x86 as usual doesn't give particularly good flags handling for the
- carry/borrow detection. It's tempting to want some multi instruction asm
- blocks to help it, and this was tried, but in truth there's only a few
- instructions to save and any gain is all too easily lost by register
- juggling setting up for the asm. */
-
-#if GMP_NAIL_BITS == 0
-#define __GMPN_AORS_1(cout, dst, src, n, v, OP, CB) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x, __gmp_r; \
- \
- /* ASSERT ((n) >= 1); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE_P (dst, src, n)); */ \
- \
- __gmp_x = (src)[0]; \
- __gmp_r = __gmp_x OP (v); \
- (dst)[0] = __gmp_r; \
- if (CB (__gmp_r, __gmp_x, (v))) \
- { \
- (cout) = 1; \
- for (__gmp_i = 1; __gmp_i < (n);) \
- { \
- __gmp_x = (src)[__gmp_i]; \
- __gmp_r = __gmp_x OP 1; \
- (dst)[__gmp_i] = __gmp_r; \
- ++__gmp_i; \
- if (!CB (__gmp_r, __gmp_x, 1)) \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, __gmp_i); \
- (cout) = 0; \
- break; \
- } \
- } \
- } \
- else \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, 1); \
- (cout) = 0; \
- } \
- } while (0)
-#endif
-
-#if GMP_NAIL_BITS >= 1
-#define __GMPN_AORS_1(cout, dst, src, n, v, OP, CB) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x, __gmp_r; \
- \
- /* ASSERT ((n) >= 1); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE_P (dst, src, n)); */ \
- \
- __gmp_x = (src)[0]; \
- __gmp_r = __gmp_x OP (v); \
- (dst)[0] = __gmp_r & GMP_NUMB_MASK; \
- if (__gmp_r >> GMP_NUMB_BITS != 0) \
- { \
- (cout) = 1; \
- for (__gmp_i = 1; __gmp_i < (n);) \
- { \
- __gmp_x = (src)[__gmp_i]; \
- __gmp_r = __gmp_x OP 1; \
- (dst)[__gmp_i] = __gmp_r & GMP_NUMB_MASK; \
- ++__gmp_i; \
- if (__gmp_r >> GMP_NUMB_BITS == 0) \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, __gmp_i); \
- (cout) = 0; \
- break; \
- } \
- } \
- } \
- else \
- { \
- if ((src) != (dst)) \
- __GMPN_COPY_REST (dst, src, n, 1); \
- (cout) = 0; \
- } \
- } while (0)
-#endif
-
-#define __GMPN_ADDCB(r,x,y) ((r) < (y))
-#define __GMPN_SUBCB(r,x,y) ((x) < (y))
-
-#define __GMPN_ADD_1(cout, dst, src, n, v) \
- __GMPN_AORS_1(cout, dst, src, n, v, +, __GMPN_ADDCB)
-#define __GMPN_SUB_1(cout, dst, src, n, v) \
- __GMPN_AORS_1(cout, dst, src, n, v, -, __GMPN_SUBCB)
-
-
-/* Compare {xp,size} and {yp,size}, setting "result" to positive, zero or
- negative. size==0 is allowed. On random data usually only one limb will
- need to be examined to get a result, so it's worth having it inline. */
-#define __GMPN_CMP(result, xp, yp, size) \
- do { \
- mp_size_t __gmp_i; \
- mp_limb_t __gmp_x, __gmp_y; \
- \
- /* ASSERT ((size) >= 0); */ \
- \
- (result) = 0; \
- __gmp_i = (size); \
- while (--__gmp_i >= 0) \
- { \
- __gmp_x = (xp)[__gmp_i]; \
- __gmp_y = (yp)[__gmp_i]; \
- if (__gmp_x != __gmp_y) \
- { \
- /* Cannot use __gmp_x - __gmp_y, may overflow an "int" */ \
- (result) = (__gmp_x > __gmp_y ? 1 : -1); \
- break; \
- } \
- } \
- } while (0)
-
-
-#if defined (__GMPN_COPY) && ! defined (__GMPN_COPY_REST)
-#define __GMPN_COPY_REST(dst, src, size, start) \
- do { \
- /* ASSERT ((start) >= 0); */ \
- /* ASSERT ((start) <= (size)); */ \
- __GMPN_COPY ((dst)+(start), (src)+(start), (size)-(start)); \
- } while (0)
-#endif
-
-/* Copy {src,size} to {dst,size}, starting at "start". This is designed to
- keep the indexing dst[j] and src[j] nice and simple for __GMPN_ADD_1,
- __GMPN_ADD, etc. */
-#if ! defined (__GMPN_COPY_REST)
-#define __GMPN_COPY_REST(dst, src, size, start) \
- do { \
- mp_size_t __gmp_j; \
- /* ASSERT ((size) >= 0); */ \
- /* ASSERT ((start) >= 0); */ \
- /* ASSERT ((start) <= (size)); */ \
- /* ASSERT (MPN_SAME_OR_SEPARATE_P (dst, src, size)); */ \
- __GMP_CRAY_Pragma ("_CRI ivdep"); \
- for (__gmp_j = (start); __gmp_j < (size); __gmp_j++) \
- (dst)[__gmp_j] = (src)[__gmp_j]; \
- } while (0)
-#endif
-
-/* Enhancement: Use some of the smarter code from gmp-impl.h. Maybe use
- mpn_copyi if there's a native version, and if we don't mind demanding
- binary compatibility for it (on targets which use it). */
-
-#if ! defined (__GMPN_COPY)
-#define __GMPN_COPY(dst, src, size) __GMPN_COPY_REST (dst, src, size, 0)
-#endif
-
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_add)
-#if ! defined (__GMP_FORCE_mpn_add)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_add (mp_ptr __gmp_wp, mp_srcptr __gmp_xp, mp_size_t __gmp_xsize, mp_srcptr __gmp_yp, mp_size_t __gmp_ysize)
-{
- mp_limb_t __gmp_c;
- __GMPN_ADD (__gmp_c, __gmp_wp, __gmp_xp, __gmp_xsize, __gmp_yp, __gmp_ysize);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_add_1)
-#if ! defined (__GMP_FORCE_mpn_add_1)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_add_1 (mp_ptr __gmp_dst, mp_srcptr __gmp_src, mp_size_t __gmp_size, mp_limb_t __gmp_n) __GMP_NOTHROW
-{
- mp_limb_t __gmp_c;
- __GMPN_ADD_1 (__gmp_c, __gmp_dst, __gmp_src, __gmp_size, __gmp_n);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_cmp)
-#if ! defined (__GMP_FORCE_mpn_cmp)
-__GMP_EXTERN_INLINE
-#endif
-int
-mpn_cmp (mp_srcptr __gmp_xp, mp_srcptr __gmp_yp, mp_size_t __gmp_size) __GMP_NOTHROW
-{
- int __gmp_result;
- __GMPN_CMP (__gmp_result, __gmp_xp, __gmp_yp, __gmp_size);
- return __gmp_result;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_sub)
-#if ! defined (__GMP_FORCE_mpn_sub)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_sub (mp_ptr __gmp_wp, mp_srcptr __gmp_xp, mp_size_t __gmp_xsize, mp_srcptr __gmp_yp, mp_size_t __gmp_ysize)
-{
- mp_limb_t __gmp_c;
- __GMPN_SUB (__gmp_c, __gmp_wp, __gmp_xp, __gmp_xsize, __gmp_yp, __gmp_ysize);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_sub_1)
-#if ! defined (__GMP_FORCE_mpn_sub_1)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_sub_1 (mp_ptr __gmp_dst, mp_srcptr __gmp_src, mp_size_t __gmp_size, mp_limb_t __gmp_n) __GMP_NOTHROW
-{
- mp_limb_t __gmp_c;
- __GMPN_SUB_1 (__gmp_c, __gmp_dst, __gmp_src, __gmp_size, __gmp_n);
- return __gmp_c;
-}
-#endif
-
-#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_neg)
-#if ! defined (__GMP_FORCE_mpn_neg)
-__GMP_EXTERN_INLINE
-#endif
-mp_limb_t
-mpn_neg (mp_ptr __gmp_rp, mp_srcptr __gmp_up, mp_size_t __gmp_n)
-{
- mp_limb_t __gmp_ul, __gmp_cy;
- __gmp_cy = 0;
- do {
- __gmp_ul = *__gmp_up++;
- *__gmp_rp++ = -__gmp_ul - __gmp_cy;
- __gmp_cy |= __gmp_ul != 0;
- } while (--__gmp_n != 0);
- return __gmp_cy;
-}
-#endif
-
-#if defined (__cplusplus)
-}
-#endif
-
-
-/* Allow faster testing for negative, zero, and positive. */
-#define mpz_sgn(Z) ((Z)->_mp_size < 0 ? -1 : (Z)->_mp_size > 0)
-#define mpf_sgn(F) ((F)->_mp_size < 0 ? -1 : (F)->_mp_size > 0)
-#define mpq_sgn(Q) ((Q)->_mp_num._mp_size < 0 ? -1 : (Q)->_mp_num._mp_size > 0)
-
-/* When using GCC, optimize certain common comparisons. */
-#if defined (__GNUC__) && __GNUC__ >= 2
-#define mpz_cmp_ui(Z,UI) \
- (__builtin_constant_p (UI) && (UI) == 0 \
- ? mpz_sgn (Z) : _mpz_cmp_ui (Z,UI))
-#define mpz_cmp_si(Z,SI) \
- (__builtin_constant_p (SI) && (SI) == 0 ? mpz_sgn (Z) \
- : __builtin_constant_p (SI) && (SI) > 0 \
- ? _mpz_cmp_ui (Z, __GMP_CAST (unsigned long int, SI)) \
- : _mpz_cmp_si (Z,SI))
-#define mpq_cmp_ui(Q,NUI,DUI) \
- (__builtin_constant_p (NUI) && (NUI) == 0 \
- ? mpq_sgn (Q) : _mpq_cmp_ui (Q,NUI,DUI))
-#define mpq_cmp_si(q,n,d) \
- (__builtin_constant_p ((n) >= 0) && (n) >= 0 \
- ? mpq_cmp_ui (q, __GMP_CAST (unsigned long, n), d) \
- : _mpq_cmp_si (q, n, d))
-#else
-#define mpz_cmp_ui(Z,UI) _mpz_cmp_ui (Z,UI)
-#define mpz_cmp_si(Z,UI) _mpz_cmp_si (Z,UI)
-#define mpq_cmp_ui(Q,NUI,DUI) _mpq_cmp_ui (Q,NUI,DUI)
-#define mpq_cmp_si(q,n,d) _mpq_cmp_si(q,n,d)
-#endif
-
-
-/* Using "&" rather than "&&" means these can come out branch-free. Every
- mpz_t has at least one limb allocated, so fetching the low limb is always
- allowed. */
-#define mpz_odd_p(z) (((z)->_mp_size != 0) & __GMP_CAST (int, (z)->_mp_d[0]))
-#define mpz_even_p(z) (! mpz_odd_p (z))
-
-
-/**************** C++ routines ****************/
-
-#ifdef __cplusplus
-__GMP_DECLSPEC_XX std::ostream& operator<< (std::ostream &, mpz_srcptr);
-__GMP_DECLSPEC_XX std::ostream& operator<< (std::ostream &, mpq_srcptr);
-__GMP_DECLSPEC_XX std::ostream& operator<< (std::ostream &, mpf_srcptr);
-__GMP_DECLSPEC_XX std::istream& operator>> (std::istream &, mpz_ptr);
-__GMP_DECLSPEC_XX std::istream& operator>> (std::istream &, mpq_ptr);
-__GMP_DECLSPEC_XX std::istream& operator>> (std::istream &, mpf_ptr);
-#endif
-
-
-/* Source-level compatibility with GMP 2 and earlier. */
-#define mpn_divmod(qp,np,nsize,dp,dsize) \
- mpn_divrem (qp, __GMP_CAST (mp_size_t, 0), np, nsize, dp, dsize)
-
-/* Source-level compatibility with GMP 1. */
-#define mpz_mdiv mpz_fdiv_q
-#define mpz_mdivmod mpz_fdiv_qr
-#define mpz_mmod mpz_fdiv_r
-#define mpz_mdiv_ui mpz_fdiv_q_ui
-#define mpz_mdivmod_ui(q,r,n,d) \
- (((r) == 0) ? mpz_fdiv_q_ui (q,n,d) : mpz_fdiv_qr_ui (q,r,n,d))
-#define mpz_mmod_ui(r,n,d) \
- (((r) == 0) ? mpz_fdiv_ui (n,d) : mpz_fdiv_r_ui (r,n,d))
-
-/* Useful synonyms, but not quite compatible with GMP 1. */
-#define mpz_div mpz_fdiv_q
-#define mpz_divmod mpz_fdiv_qr
-#define mpz_div_ui mpz_fdiv_q_ui
-#define mpz_divmod_ui mpz_fdiv_qr_ui
-#define mpz_div_2exp mpz_fdiv_q_2exp
-#define mpz_mod_2exp mpz_fdiv_r_2exp
-
-enum
-{
- GMP_ERROR_NONE = 0,
- GMP_ERROR_UNSUPPORTED_ARGUMENT = 1,
- GMP_ERROR_DIVISION_BY_ZERO = 2,
- GMP_ERROR_SQRT_OF_NEGATIVE = 4,
- GMP_ERROR_INVALID_ARGUMENT = 8
-};
-
-/* Define CC and CFLAGS which were used to build this version of GMP */
-#define __GMP_CC "gcc -std=gnu99"
-#define __GMP_CFLAGS "-O2 -pedantic -m64 -mtune=k8 -march=k8"
-
-/* Major version number is the value of __GNU_MP__ too, above and in mp.h. */
-#define __GNU_MP_VERSION 5
-#define __GNU_MP_VERSION_MINOR 0
-#define __GNU_MP_VERSION_PATCHLEVEL 1
-#define __GMP_MP_RELEASE (__GNU_MP_VERSION * 10000 + __GNU_MP_VERSION_MINOR * 100 + __GNU_MP_VERSION_PATCHLEVEL)
-
-#define __GMP_H__
-#endif /* __GMP_H__ */
+++ /dev/null
-# libgmp.la - a libtool library file
-# Generated by ltmain.sh (GNU libtool) 2.2.6b
-#
-# Please DO NOT delete this file!
-# It is necessary for linking the library.
-
-# The name that we can dlopen(3).
-dlname=''
-
-# Names of this library.
-library_names=''
-
-# The name of the static archive.
-old_library='libgmp.a'
-
-# Linker flags that can not go in dependency_libs.
-inherited_linker_flags=''
-
-# Libraries that this one depends upon.
-dependency_libs=''
-
-# Names of additional weak libraries provided by this library
-weak_library_names=''
-
-# Version information for libgmp.
-current=10
-age=0
-revision=1
-
-# Is this an already installed library?
-installed=yes
-
-# Should we warn about portability when linking against -modules?
-shouldnotlink=no
-
-# Files to dlopen/dlpreopen
-dlopen=''
-dlpreopen=''
-
-# Directory that this library needs to be installed in:
-libdir='/tmp/g/lib'
+++ /dev/null
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-END-INFO-DIR-ENTRY
-
-\1f
-Indirect:
-gmp.info-1: 981
-gmp.info-2: 300864
-\1f
-Tag Table:
-(Indirect)
-Node: Top\7f981
-Node: Copying\7f3211
-Node: Introduction to GMP\7f5062
-Node: Installing GMP\7f7773
-Node: Build Options\7f8505
-Node: ABI and ISA\7f24573
-Node: Notes for Package Builds\7f34251
-Node: Notes for Particular Systems\7f37338
-Node: Known Build Problems\7f43895
-Node: Performance optimization\7f47429
-Node: GMP Basics\7f48563
-Node: Headers and Libraries\7f49211
-Node: Nomenclature and Types\7f50635
-Node: Function Classes\7f52632
-Node: Variable Conventions\7f54325
-Node: Parameter Conventions\7f55934
-Node: Memory Management\7f57990
-Node: Reentrancy\7f59118
-Node: Useful Macros and Constants\7f60991
-Node: Compatibility with older versions\7f61989
-Node: Demonstration Programs\7f62950
-Node: Efficiency\7f64815
-Node: Debugging\7f72439
-Node: Profiling\7f78997
-Node: Autoconf\7f82988
-Node: Emacs\7f84767
-Node: Reporting Bugs\7f85373
-Node: Integer Functions\7f87916
-Node: Initializing Integers\7f88692
-Node: Assigning Integers\7f90839
-Node: Simultaneous Integer Init & Assign\7f92426
-Node: Converting Integers\7f94051
-Node: Integer Arithmetic\7f96973
-Node: Integer Division\7f98559
-Node: Integer Exponentiation\7f104869
-Node: Integer Roots\7f106309
-Node: Number Theoretic Functions\7f107983
-Node: Integer Comparisons\7f114126
-Node: Integer Logic and Bit Fiddling\7f115504
-Node: I/O of Integers\7f118051
-Node: Integer Random Numbers\7f120935
-Node: Integer Import and Export\7f123546
-Node: Miscellaneous Integer Functions\7f127556
-Node: Integer Special Functions\7f129416
-Node: Rational Number Functions\7f132503
-Node: Initializing Rationals\7f133696
-Node: Rational Conversions\7f136157
-Node: Rational Arithmetic\7f137888
-Node: Comparing Rationals\7f139192
-Node: Applying Integer Functions\7f140559
-Node: I/O of Rationals\7f142042
-Node: Floating-point Functions\7f143902
-Node: Initializing Floats\7f146787
-Node: Assigning Floats\7f150874
-Node: Simultaneous Float Init & Assign\7f153441
-Node: Converting Floats\7f154969
-Node: Float Arithmetic\7f158217
-Node: Float Comparison\7f160230
-Node: I/O of Floats\7f161811
-Node: Miscellaneous Float Functions\7f164409
-Node: Low-level Functions\7f166303
-Node: Random Number Functions\7f190437
-Node: Random State Initialization\7f191505
-Node: Random State Seeding\7f194363
-Node: Random State Miscellaneous\7f195752
-Node: Formatted Output\7f196393
-Node: Formatted Output Strings\7f196638
-Node: Formatted Output Functions\7f201852
-Node: C++ Formatted Output\7f205927
-Node: Formatted Input\7f208609
-Node: Formatted Input Strings\7f208845
-Node: Formatted Input Functions\7f213497
-Node: C++ Formatted Input\7f216466
-Node: C++ Class Interface\7f218369
-Node: C++ Interface General\7f219370
-Node: C++ Interface Integers\7f222440
-Node: C++ Interface Rationals\7f225871
-Node: C++ Interface Floats\7f229548
-Node: C++ Interface Random Numbers\7f234830
-Node: C++ Interface Limitations\7f237236
-Node: BSD Compatible Functions\7f240056
-Node: Custom Allocation\7f244767
-Node: Language Bindings\7f249085
-Node: Algorithms\7f253038
-Node: Multiplication Algorithms\7f253738
-Node: Basecase Multiplication\7f254710
-Node: Karatsuba Multiplication\7f256618
-Node: Toom 3-Way Multiplication\7f260243
-Node: Toom 4-Way Multiplication\7f266657
-Node: FFT Multiplication\7f268029
-Node: Other Multiplication\7f273365
-Node: Unbalanced Multiplication\7f275839
-Node: Division Algorithms\7f276627
-Node: Single Limb Division\7f277006
-Node: Basecase Division\7f279897
-Node: Divide and Conquer Division\7f281100
-Node: Block-Wise Barrett Division\7f283169
-Node: Exact Division\7f283821
-Node: Exact Remainder\7f286986
-Node: Small Quotient Division\7f289213
-Node: Greatest Common Divisor Algorithms\7f290811
-Node: Binary GCD\7f291108
-Node: Lehmer's Algorithm\7f293957
-Node: Subquadratic GCD\7f296177
-Node: Extended GCD\7f298636
-Node: Jacobi Symbol\7f299948
-Node: Powering Algorithms\7f300864
-Node: Normal Powering Algorithm\7f301127
-Node: Modular Powering Algorithm\7f301655
-Node: Root Extraction Algorithms\7f302435
-Node: Square Root Algorithm\7f302750
-Node: Nth Root Algorithm\7f304891
-Node: Perfect Square Algorithm\7f305676
-Node: Perfect Power Algorithm\7f307762
-Node: Radix Conversion Algorithms\7f308383
-Node: Binary to Radix\7f308759
-Node: Radix to Binary\7f312688
-Node: Other Algorithms\7f314776
-Node: Prime Testing Algorithm\7f315128
-Node: Factorial Algorithm\7f316312
-Node: Binomial Coefficients Algorithm\7f317715
-Node: Fibonacci Numbers Algorithm\7f318609
-Node: Lucas Numbers Algorithm\7f321083
-Node: Random Number Algorithms\7f321804
-Node: Assembly Coding\7f323925
-Node: Assembly Code Organisation\7f324885
-Node: Assembly Basics\7f325852
-Node: Assembly Carry Propagation\7f327002
-Node: Assembly Cache Handling\7f328833
-Node: Assembly Functional Units\7f330994
-Node: Assembly Floating Point\7f332607
-Node: Assembly SIMD Instructions\7f336385
-Node: Assembly Software Pipelining\7f337367
-Node: Assembly Loop Unrolling\7f338429
-Node: Assembly Writing Guide\7f340644
-Node: Internals\7f343409
-Node: Integer Internals\7f343921
-Node: Rational Internals\7f346177
-Node: Float Internals\7f347415
-Node: Raw Output Internals\7f354829
-Node: C++ Interface Internals\7f356023
-Node: Contributors\7f359309
-Node: References\7f364267
-Node: GNU Free Documentation License\7f369925
-Node: Concept Index\7f395094
-Node: Function Index\7f441276
-\1f
-End Tag Table
+++ /dev/null
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-END-INFO-DIR-ENTRY
-
-\1f
-File: gmp.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
-
-GNU MP
-******
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-
-* Menu:
-
-* Copying:: GMP Copying Conditions (LGPL).
-* Introduction to GMP:: Brief introduction to GNU MP.
-* Installing GMP:: How to configure and compile the GMP library.
-* GMP Basics:: What every GMP user should know.
-* Reporting Bugs:: How to usefully report bugs.
-* Integer Functions:: Functions for arithmetic on signed integers.
-* Rational Number Functions:: Functions for arithmetic on rational numbers.
-* Floating-point Functions:: Functions for arithmetic on floats.
-* Low-level Functions:: Fast functions for natural numbers.
-* Random Number Functions:: Functions for generating random numbers.
-* Formatted Output:: `printf' style output.
-* Formatted Input:: `scanf' style input.
-* C++ Class Interface:: Class wrappers around GMP types.
-* BSD Compatible Functions:: All functions found in BSD MP.
-* Custom Allocation:: How to customize the internal allocation.
-* Language Bindings:: Using GMP from other languages.
-* Algorithms:: What happens behind the scenes.
-* Internals:: How values are represented behind the scenes.
-
-* Contributors:: Who brings you this library?
-* References:: Some useful papers and books to read.
-* GNU Free Documentation License::
-* Concept Index::
-* Function Index::
-
-\1f
-File: gmp.info, Node: Copying, Next: Introduction to GMP, Prev: Top, Up: Top
-
-GNU MP Copying Conditions
-*************************
-
-This library is "free"; this means that everyone is free to use it and
-free to redistribute it on a free basis. The library is not in the
-public domain; it is copyrighted and there are restrictions on its
-distribution, but these restrictions are designed to permit everything
-that a good cooperating citizen would want to do. What is not allowed
-is to try to prevent others from further sharing any version of this
-library that they might get from you.
-
- Specifically, we want to make sure that you have the right to give
-away copies of the library, that you receive source code or else can
-get it if you want it, that you can change this library or use pieces
-of it in new free programs, and that you know you can do these things.
-
- To make sure that everyone has such rights, we have to forbid you to
-deprive anyone else of these rights. For example, if you distribute
-copies of the GNU MP library, you must give the recipients all the
-rights that you have. You must make sure that they, too, receive or
-can get the source code. And you must tell them their rights.
-
- Also, for our own protection, we must make certain that everyone
-finds out that there is no warranty for the GNU MP library. If it is
-modified by someone else and passed on, we want their recipients to
-know that what they have is not what we distributed, so that any
-problems introduced by others will not reflect on our reputation.
-
- The precise conditions of the license for the GNU MP library are
-found in the Lesser General Public License version 3 that accompanies
-the source code, see `COPYING.LIB'. Certain demonstration programs are
-provided under the terms of the plain General Public License version 3,
-see `COPYING'.
-
-\1f
-File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top
-
-1 Introduction to GNU MP
-************************
-
-GNU MP is a portable library written in C for arbitrary precision
-arithmetic on integers, rational numbers, and floating-point numbers.
-It aims to provide the fastest possible arithmetic for all applications
-that need higher precision than is directly supported by the basic C
-types.
-
- Many applications use just a few hundred bits of precision; but some
-applications may need thousands or even millions of bits. GMP is
-designed to give good performance for both, by choosing algorithms
-based on the sizes of the operands, and by carefully keeping the
-overhead at a minimum.
-
- The speed of GMP is achieved by using fullwords as the basic
-arithmetic type, by using sophisticated algorithms, by including
-carefully optimized assembly code for the most common inner loops for
-many different CPUs, and by a general emphasis on speed (as opposed to
-simplicity or elegance).
-
- There is assembly code for these CPUs: ARM, DEC Alpha 21064, 21164,
-and 21264, AMD 29000, AMD K6, K6-2, Athlon, and Athlon64, Hitachi
-SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium
-Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola
-MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64,
-National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC,
-generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some
-optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and
-Pyramid AP/XP.
-
-For up-to-date information on GMP, please see the GMP web pages at
-
- `http://gmplib.org/'
-
-The latest version of the library is available at
-
- `ftp://ftp.gnu.org/gnu/gmp/'
-
- Many sites around the world mirror `ftp.gnu.org', please use a mirror
-near you, see `http://www.gnu.org/order/ftp.html' for a full list.
-
- There are three public mailing lists of interest. One for release
-announcements, one for general questions and discussions about usage of
-the GMP library and one for bug reports. For more information, see
-
- `http://gmplib.org/mailman/listinfo/'.
-
- The proper place for bug reports is <gmp-bugs@gmplib.org>. See
-*Note Reporting Bugs:: for information about reporting bugs.
-
-
-1.1 How to use this Manual
-==========================
-
-Everyone should read *Note GMP Basics::. If you need to install the
-library yourself, then read *Note Installing GMP::. If you have a
-system with multiple ABIs, then read *Note ABI and ISA::, for the
-compiler options that must be used on applications.
-
- The rest of the manual can be used for later reference, although it
-is probably a good idea to glance through it.
-
-\1f
-File: gmp.info, Node: Installing GMP, Next: GMP Basics, Prev: Introduction to GMP, Up: Top
-
-2 Installing GMP
-****************
-
-GMP has an autoconf/automake/libtool based configuration system. On a
-Unix-like system a basic build can be done with
-
- ./configure
- make
-
-Some self-tests can be run with
-
- make check
-
-And you can install (under `/usr/local' by default) with
-
- make install
-
- If you experience problems, please report them to
-<gmp-bugs@gmplib.org>. See *Note Reporting Bugs::, for information on
-what to include in useful bug reports.
-
-* Menu:
-
-* Build Options::
-* ABI and ISA::
-* Notes for Package Builds::
-* Notes for Particular Systems::
-* Known Build Problems::
-* Performance optimization::
-
-\1f
-File: gmp.info, Node: Build Options, Next: ABI and ISA, Prev: Installing GMP, Up: Installing GMP
-
-2.1 Build Options
-=================
-
-All the usual autoconf configure options are available, run `./configure
---help' for a summary. The file `INSTALL.autoconf' has some generic
-installation information too.
-
-Tools
- `configure' requires various Unix-like tools. See *Note Notes for
- Particular Systems::, for some options on non-Unix systems.
-
- It might be possible to build without the help of `configure',
- certainly all the code is there, but unfortunately you'll be on
- your own.
-
-Build Directory
- To compile in a separate build directory, `cd' to that directory,
- and prefix the configure command with the path to the GMP source
- directory. For example
-
- cd /my/build/dir
- /my/sources/gmp-5.0.1/configure
-
- Not all `make' programs have the necessary features (`VPATH') to
- support this. In particular, SunOS and Slowaris `make' have bugs
- that make them unable to build in a separate directory. Use GNU
- `make' instead.
-
-`--prefix' and `--exec-prefix'
- The `--prefix' option can be used in the normal way to direct GMP
- to install under a particular tree. The default is `/usr/local'.
-
- `--exec-prefix' can be used to direct architecture-dependent files
- like `libgmp.a' to a different location. This can be used to share
- architecture-independent parts like the documentation, but
- separate the dependent parts. Note however that `gmp.h' and
- `mp.h' are architecture-dependent since they encode certain
- aspects of `libgmp', so it will be necessary to ensure both
- `$prefix/include' and `$exec_prefix/include' are available to the
- compiler.
-
-`--disable-shared', `--disable-static'
- By default both shared and static libraries are built (where
- possible), but one or other can be disabled. Shared libraries
- result in smaller executables and permit code sharing between
- separate running processes, but on some CPUs are slightly slower,
- having a small cost on each function call.
-
-Native Compilation, `--build=CPU-VENDOR-OS'
- For normal native compilation, the system can be specified with
- `--build'. By default `./configure' uses the output from running
- `./config.guess'. On some systems `./config.guess' can determine
- the exact CPU type, on others it will be necessary to give it
- explicitly. For example,
-
- ./configure --build=ultrasparc-sun-solaris2.7
-
- In all cases the `OS' part is important, since it controls how
- libtool generates shared libraries. Running `./config.guess' is
- the simplest way to see what it should be, if you don't know
- already.
-
-Cross Compilation, `--host=CPU-VENDOR-OS'
- When cross-compiling, the system used for compiling is given by
- `--build' and the system where the library will run is given by
- `--host'. For example when using a FreeBSD Athlon system to build
- GNU/Linux m68k binaries,
-
- ./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
-
- Compiler tools are sought first with the host system type as a
- prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
- plain `ranlib'. This makes it possible for a set of
- cross-compiling tools to co-exist with native tools. The prefix
- is the argument to `--host', and this can be an alias, such as
- `m68k-linux'. But note that tools don't have to be setup this
- way, it's enough to just have a `PATH' with a suitable
- cross-compiling `cc' etc.
-
- Compiling for a different CPU in the same family as the build
- system is a form of cross-compilation, though very possibly this
- would merely be special options on a native compiler. In any case
- `./configure' avoids depending on being able to run code on the
- build system, which is important when creating binaries for a
- newer CPU since they very possibly won't run on the build system.
-
- In all cases the compiler must be able to produce an executable
- (of whatever format) from a standard C `main'. Although only
- object files will go to make up `libgmp', `./configure' uses
- linking tests for various purposes, such as determining what
- functions are available on the host system.
-
- Currently a warning is given unless an explicit `--build' is used
- when cross-compiling, because it may not be possible to correctly
- guess the build system type if the `PATH' has only a
- cross-compiling `cc'.
-
- Note that the `--target' option is not appropriate for GMP. It's
- for use when building compiler tools, with `--host' being where
- they will run, and `--target' what they'll produce code for.
- Ordinary programs or libraries like GMP are only interested in the
- `--host' part, being where they'll run. (Some past versions of
- GMP used `--target' incorrectly.)
-
-CPU types
- In general, if you want a library that runs as fast as possible,
- you should configure GMP for the exact CPU type your system uses.
- However, this may mean the binaries won't run on older members of
- the family, and might run slower on other members, older or newer.
- The best idea is always to build GMP for the exact machine type
- you intend to run it on.
-
- The following CPUs have specific support. See `configure.in' for
- details of what code and compiler options they select.
-
- * Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
- alphaev6, alphaev67, alphaev68 alphaev7
-
- * Cray: c90, j90, t90, sv1
-
- * HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64
-
- * IA-64: ia64, itanium, itanium2
-
- * MIPS: mips, mips3, mips64
-
- * Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
- m68060, m68302, m68360, m88k, m88110
-
- * POWER: power, power1, power2, power2sc
-
- * PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
- powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
- powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
- powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
- powerpc821, powerpc823, powerpc860, powerpc970
-
- * SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
- ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
-
- * x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro,
- pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64,
- viac3, viac32
-
- * Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
- z8k
-
- CPUs not listed will use generic C code.
-
-Generic C Build
- If some of the assembly code causes problems, or if otherwise
- desired, the generic C code can be selected with CPU `none'. For
- example,
-
- ./configure --host=none-unknown-freebsd3.5
-
- Note that this will run quite slowly, but it should be portable
- and should at least make it possible to get something running if
- all else fails.
-
-Fat binary, `--enable-fat'
- Using `--enable-fat' selects a "fat binary" build on x86, where
- optimized low level subroutines are chosen at runtime according to
- the CPU detected. This means more code, but gives good
- performance on all x86 chips. (This option might become available
- for more architectures in the future.)
-
-`ABI'
- On some systems GMP supports multiple ABIs (application binary
- interfaces), meaning data type sizes and calling conventions. By
- default GMP chooses the best ABI available, but a particular ABI
- can be selected. For example
-
- ./configure --host=mips64-sgi-irix6 ABI=n32
-
- See *Note ABI and ISA::, for the available choices on relevant
- CPUs, and what applications need to do.
-
-`CC', `CFLAGS'
- By default the C compiler used is chosen from among some likely
- candidates, with `gcc' normally preferred if it's present. The
- usual `CC=whatever' can be passed to `./configure' to choose
- something different.
-
- For various systems, default compiler flags are set based on the
- CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to
- `./configure' to use something different or to set good flags for
- systems GMP doesn't otherwise know.
-
- The `CC' and `CFLAGS' used are printed during `./configure', and
- can be found in each generated `Makefile'. This is the easiest way
- to check the defaults when considering changing or adding
- something.
-
- Note that when `CC' and `CFLAGS' are specified on a system
- supporting multiple ABIs it's important to give an explicit
- `ABI=whatever', since GMP can't determine the ABI just from the
- flags and won't be able to select the correct assembly code.
-
- If just `CC' is selected then normal default `CFLAGS' for that
- compiler will be used (if GMP recognises it). For example
- `CC=gcc' can be used to force the use of GCC, with default flags
- (and default ABI).
-
-`CPPFLAGS'
- Any flags like `-D' defines or `-I' includes required by the
- preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
- Compiling is done with both `CPPFLAGS' and `CFLAGS', but
- preprocessing uses just `CPPFLAGS'. This distinction is because
- most preprocessors won't accept all the flags the compiler does.
- Preprocessing is done separately in some configure tests, and in
- the `ansi2knr' support for K&R compilers.
-
-`CC_FOR_BUILD'
- Some build-time programs are compiled and run to generate
- host-specific data tables. `CC_FOR_BUILD' is the compiler used
- for this. It doesn't need to be in any particular ABI or mode, it
- merely needs to generate executables that can run. The default is
- to try the selected `CC' and some likely candidates such as `cc'
- and `gcc', looking for something that works.
-
- No flags are used with `CC_FOR_BUILD' because a simple invocation
- like `cc foo.c' should be enough. If some particular options are
- required they can be included as for instance `CC_FOR_BUILD="cc
- -whatever"'.
-
-C++ Support, `--enable-cxx'
- C++ support in GMP can be enabled with `--enable-cxx', in which
- case a C++ compiler will be required. As a convenience
- `--enable-cxx=detect' can be used to enable C++ support only if a
- compiler can be found. The C++ support consists of a library
- `libgmpxx.la' and header file `gmpxx.h' (*note Headers and
- Libraries::).
-
- A separate `libgmpxx.la' has been adopted rather than having C++
- objects within `libgmp.la' in order to ensure dynamic linked C
- programs aren't bloated by a dependency on the C++ standard
- library, and to avoid any chance that the C++ compiler could be
- required when linking plain C programs.
-
- `libgmpxx.la' will use certain internals from `libgmp.la' and can
- only be expected to work with `libgmp.la' from the same GMP
- version. Future changes to the relevant internals will be
- accompanied by renaming, so a mismatch will cause unresolved
- symbols rather than perhaps mysterious misbehaviour.
-
- In general `libgmpxx.la' will be usable only with the C++ compiler
- that built it, since name mangling and runtime support are usually
- incompatible between different compilers.
-
-`CXX', `CXXFLAGS'
- When C++ support is enabled, the C++ compiler and its flags can be
- set with variables `CXX' and `CXXFLAGS' in the usual way. The
- default for `CXX' is the first compiler that works from a list of
- likely candidates, with `g++' normally preferred when available.
- The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
- `-g', then for `g++' either `-g -O2' or `-O2', or for other
- compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
- when using `gcc' and `g++' together, since the flags for `gcc' will
- usually suit `g++'.
-
- It's important that the C and C++ compilers match, meaning their
- startup and runtime support routines are compatible and that they
- generate code in the same ABI (if there's a choice of ABIs on the
- system). `./configure' isn't currently able to check these things
- very well itself, so for that reason `--disable-cxx' is the
- default, to avoid a build failure due to a compiler mismatch.
- Perhaps this will change in the future.
-
- Incidentally, it's normally not good enough to set `CXX' to the
- same as `CC'. Although `gcc' for instance recognises `foo.cc' as
- C++ code, only `g++' will invoke the linker the right way when
- building an executable or shared library from C++ object files.
-
-Temporary Memory, `--enable-alloca=<choice>'
- GMP allocates temporary workspace using one of the following three
- methods, which can be selected with for instance
- `--enable-alloca=malloc-reentrant'.
-
- * `alloca' - C library or compiler builtin.
-
- * `malloc-reentrant' - the heap, in a re-entrant fashion.
-
- * `malloc-notreentrant' - the heap, with global variables.
-
- For convenience, the following choices are also available.
- `--disable-alloca' is the same as `no'.
-
- * `yes' - a synonym for `alloca'.
-
- * `no' - a synonym for `malloc-reentrant'.
-
- * `reentrant' - `alloca' if available, otherwise
- `malloc-reentrant'. This is the default.
-
- * `notreentrant' - `alloca' if available, otherwise
- `malloc-notreentrant'.
-
- `alloca' is reentrant and fast, and is recommended. It actually
- allocates just small blocks on the stack; larger ones use
- malloc-reentrant.
-
- `malloc-reentrant' is, as the name suggests, reentrant and thread
- safe, but `malloc-notreentrant' is faster and should be used if
- reentrancy is not required.
-
- The two malloc methods in fact use the memory allocation functions
- selected by `mp_set_memory_functions', these being `malloc' and
- friends by default. *Note Custom Allocation::.
-
- An additional choice `--enable-alloca=debug' is available, to help
- when debugging memory related problems (*note Debugging::).
-
-FFT Multiplication, `--disable-fft'
- By default multiplications are done using Karatsuba, 3-way Toom,
- and Fermat FFT. The FFT is only used on large to very large
- operands and can be disabled to save code size if desired.
-
-Berkeley MP, `--enable-mpbsd'
- The Berkeley MP compatibility library (`libmp') and header file
- (`mp.h') are built and installed only if `--enable-mpbsd' is used.
- *Note BSD Compatible Functions::.
-
-Assertion Checking, `--enable-assert'
- This option enables some consistency checking within the library.
- This can be of use while debugging, *note Debugging::.
-
-Execution Profiling, `--enable-profiling=prof/gprof/instrument'
- Enable profiling support, in one of various styles, *note
- Profiling::.
-
-`MPN_PATH'
- Various assembly versions of each mpn subroutines are provided.
- For a given CPU, a search is made though a path to choose a
- version of each. For example `sparcv8' has
-
- MPN_PATH="sparc32/v8 sparc32 generic"
-
- which means look first for v8 code, then plain sparc32 (which is
- v7), and finally fall back on generic C. Knowledgeable users with
- special requirements can specify a different path. Normally this
- is completely unnecessary.
-
-Documentation
- The source for the document you're now reading is `doc/gmp.texi',
- in Texinfo format, see *Note Texinfo: (texinfo)Top.
-
- Info format `doc/gmp.info' is included in the distribution. The
- usual automake targets are available to make PostScript, DVI, PDF
- and HTML (these will require various TeX and Texinfo tools).
-
- DocBook and XML can be generated by the Texinfo `makeinfo' program
- too, see *Note Options for `makeinfo': (texinfo)makeinfo options.
-
- Some supplementary notes can also be found in the `doc'
- subdirectory.
-
-
-\1f
-File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP
-
-2.2 ABI and ISA
-===============
-
-ABI (Application Binary Interface) refers to the calling conventions
-between functions, meaning what registers are used and what sizes the
-various C data types are. ISA (Instruction Set Architecture) refers to
-the instructions and registers a CPU has available.
-
- Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
-defined, the latter for compatibility with older CPUs in the family.
-GMP supports some CPUs like this in both ABIs. In fact within GMP
-`ABI' means a combination of chip ABI, plus how GMP chooses to use it.
-For example in some 32-bit ABIs, GMP may support a limb as either a
-32-bit `long' or a 64-bit `long long'.
-
- By default GMP chooses the best ABI available for a given system,
-and this generally gives significantly greater speed. But an ABI can
-be chosen explicitly to make GMP compatible with other libraries, or
-particular application requirements. For example,
-
- ./configure ABI=32
-
- In all cases it's vital that all object code used in a given program
-is compiled for the same ABI.
-
- Usually a limb is implemented as a `long'. When a `long long' limb
-is used this is encoded in the generated `gmp.h'. This is convenient
-for applications, but it does mean that `gmp.h' will vary, and can't be
-just copied around. `gmp.h' remains compiler independent though, since
-all compilers for a particular ABI will be expected to use the same
-limb type.
-
- Currently no attempt is made to follow whatever conventions a system
-has for installing library or header files built for a particular ABI.
-This will probably only matter when installing multiple builds of GMP,
-and it might be as simple as configuring with a special `libdir', or it
-might require more than that. Note that builds for different ABIs need
-to done separately, with a fresh `./configure' and `make' each.
-
-
-AMD64 (`x86_64')
- On AMD64 systems supporting both 32-bit and 64-bit modes for
- applications, the following ABI choices are available.
-
- `ABI=64'
- The 64-bit ABI uses 64-bit limbs and pointers and makes full
- use of the chip architecture. This is the default.
- Applications will usually not need special compiler flags,
- but for reference the option is
-
- gcc -m64
-
- `ABI=32'
- The 32-bit ABI is the usual i386 conventions. This will be
- slower, and is not recommended except for inter-operating
- with other code not yet 64-bit capable. Applications must be
- compiled with
-
- gcc -m32
-
- (In GCC 2.95 and earlier there's no `-m32' option, it's the
- only mode.)
-
-
-HPPA 2.0 (`hppa2.0*', `hppa64')
-
- `ABI=2.0w'
- The 2.0w ABI uses 64-bit limbs and pointers and is available
- on HP-UX 11 or up. Applications must be compiled with
-
- gcc [built for 2.0w]
- cc +DD64
-
- `ABI=2.0n'
- The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal
- calling conventions, but with 64-bit instructions permitted
- within functions. GMP uses a 64-bit `long long' for a limb.
- This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or
- higher. Applications must be compiled with
-
- gcc [built for 2.0n]
- cc +DA2.0 +e
-
- Note that current versions of GCC (eg. 3.2) don't generate
- 64-bit instructions for `long long' operations and so may be
- slower than for 2.0w. (The GMP assembly code is the same
- though.)
-
- `ABI=1.0'
- HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit
- HPPA 1.0 ABI. No special compiler options are needed for
- applications.
-
- All three ABIs are available for CPU types `hppa2.0w', `hppa2.0'
- and `hppa64', but for CPU type `hppa2.0n' only 2.0n or 1.0 are
- considered.
-
- Note that GCC on HP-UX has no options to choose between 2.0n and
- 2.0w modes, unlike HP `cc'. Instead it must be built for one or
- the other ABI. GMP will detect how it was built, and skip to the
- corresponding `ABI'.
-
-
-IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
- HP-UX supports two ABIs for IA-64. GMP performance is the same in
- both.
-
- `ABI=32'
- In the 32-bit ABI, pointers, `int's and `long's are 32 bits
- and GMP uses a 64 bit `long long' for a limb. Applications
- can be compiled without any special flags since this ABI is
- the default in both HP C and GCC, but for reference the flags
- are
-
- gcc -milp32
- cc +DD32
-
- `ABI=64'
- In the 64-bit ABI, `long's and pointers are 64 bits and GMP
- uses a `long' for a limb. Applications must be compiled with
-
- gcc -mlp64
- cc +DD64
-
- On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the
- only choice.
-
-
-MIPS under IRIX 6 (`mips*-*-irix[6789]')
- IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs
- o32, n32, and 64. n32 or 64 are recommended, and GMP performance
- will be the same in each. The default is n32.
-
- `ABI=o32'
- The o32 ABI is 32-bit pointers and integers, and no 64-bit
- operations. GMP will be slower than in n32 or 64, this
- option only exists to support old compilers, eg. GCC 2.7.2.
- Applications can be compiled with no special flags on an old
- compiler, or on a newer compiler with
-
- gcc -mabi=32
- cc -32
-
- `ABI=n32'
- The n32 ABI is 32-bit pointers and integers, but with a
- 64-bit limb using a `long long'. Applications must be
- compiled with
-
- gcc -mabi=n32
- cc -n32
-
- `ABI=64'
- The 64-bit ABI is 64-bit pointers and integers. Applications
- must be compiled with
-
- gcc -mabi=64
- cc -64
-
- Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have
- the necessary support for n32 or 64 and so only gets a 32-bit limb
- and the MIPS 2 code.
-
-
-PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970', `power4', `power5')
-
- `ABI=aix64'
- The AIX 64 ABI uses 64-bit limbs and pointers and is the
- default on PowerPC 64 `*-*-aix*' systems. Applications must
- be compiled with
-
- gcc -maix64
- xlc -q64
-
- `ABI=mode64'
- The `mode64' ABI uses 64-bit limbs and pointers, and is the
- default on 64-bit GNU/Linux, BSD, and Mac OS X/Darwin
- systems. Applications must be compiled with
-
- gcc -m64
-
- `ABI=mode32'
- The `mode32' ABI uses a 64-bit `long long' limb but with the
- chip still in 32-bit mode and using 32-bit calling
- conventions. This is the default on for systems where the
- true 64-bit ABIs are unavailable. No special compiler
- options are needed for applications.
-
- `ABI=32'
- This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
- special compiler options are needed for applications.
-
- GMP speed is greatest in `aix64' and `mode32'. In `ABI=32' only
- the 32-bit ISA is used and this doesn't make full use of a 64-bit
- chip. On a suitable system we could perhaps use more of the ISA,
- but there are no plans to do so.
-
-
-Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
-
- `ABI=64'
- The 64-bit V9 ABI is available on the various BSD sparc64
- ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
- and up (when the kernel is in 64-bit mode). GCC 3.2 or
- higher, or Sun `cc' is required. On GNU/Linux, depending on
- the default `gcc' mode, applications must be compiled with
-
- gcc -m64
-
- On Solaris applications must be compiled with
-
- gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
- cc -xarch=v9
-
- On the BSD sparc64 systems no special options are required,
- since 64-bits is the only ABI available.
-
- `ABI=32'
- For the basic 32-bit ABI, GMP still uses as much of the V9
- ISA as it can. In the Sun documentation this combination is
- known as "v8plus". On GNU/Linux, depending on the default
- `gcc' mode, applications may need to be compiled with
-
- gcc -m32
-
- On Solaris, no special compiler options are required for
- applications, though using something like the following is
- recommended. (`gcc' 2.8 and earlier only support `-mv8'
- though.)
-
- gcc -mv8plus
- cc -xarch=v8plus
-
- GMP speed is greatest in `ABI=64', so it's the default where
- available. The speed is partly because there are extra registers
- available and partly because 64-bits is considered the more
- important case and has therefore had better code written for it.
-
- Don't be confused by the names of the `-m' and `-x' compiler
- options, they're called `arch' but effectively control both ABI
- and ISA.
-
- On Solaris 2.6 and earlier, only `ABI=32' is available since the
- kernel doesn't save all registers.
-
- On Solaris 2.7 with the kernel in 32-bit mode, a normal native
- build will reject `ABI=64' because the resulting executables won't
- run. `ABI=64' can still be built if desired by making it look
- like a cross-compile, for example
-
- ./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
-
-\1f
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-
-2.3 Notes for Package Builds
-============================
-
-GMP should present no great difficulties for packaging in a binary
-distribution.
-
- Libtool is used to build the library and `-version-info' is set
-appropriately, having started from `3:0:0' in GMP 3.0 (*note Library
-interface versions: (libtool)Versioning.).
-
- The GMP 4 series will be upwardly binary compatible in each release
-and will be upwardly binary compatible with all of the GMP 3 series.
-Additional function interfaces may be added in each release, so on
-systems where libtool versioning is not fully checked by the loader an
-auxiliary mechanism may be needed to express that a dynamic linked
-application depends on a new enough GMP.
-
- An auxiliary mechanism may also be needed to express that
-`libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires
-`libgmp.la' from the same GMP version, since this is not done by the
-libtool versioning, nor otherwise. A mismatch will result in
-unresolved symbols from the linker, or perhaps the loader.
-
- When building a package for a CPU family, care should be taken to use
-`--host' (or `--build') to choose the least common denominator among
-the CPUs which might use the package. For example this might mean plain
-`sparc' (meaning V7) for SPARCs.
-
- For x86s, `--enable-fat' sets things up for a fat binary build,
-making a runtime selection of optimized low level routines. This is a
-good choice for packaging to run on a range of x86 chips.
-
- Users who care about speed will want GMP built for their exact CPU
-type, to make best use of the available optimizations. Providing a way
-to suitably rebuild a package may be useful. This could be as simple
-as making it possible for a user to omit `--build' (and `--host') so
-`./config.guess' will detect the CPU. But a way to manually specify a
-`--build' will be wanted for systems where `./config.guess' is inexact.
-
- On systems with multiple ABIs, a packaged build will need to decide
-which among the choices is to be provided, see *Note ABI and ISA::. A
-given run of `./configure' etc will only build one ABI. If a second
-ABI is also required then a second run of `./configure' etc must be
-made, starting from a clean directory tree (`make distclean').
-
- As noted under "ABI and ISA", currently no attempt is made to follow
-system conventions for install locations that vary with ABI, such as
-`/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'.
-A package build can override `libdir' and other standard variables as
-necessary.
-
- Note that `gmp.h' is a generated file, and will be architecture and
-ABI dependent. When attempting to install two ABIs simultaneously it
-will be important that an application compile gets the correct `gmp.h'
-for its desired ABI. If compiler include paths don't vary with ABI
-options then it might be necessary to create a `/usr/include/gmp.h'
-which tests preprocessor symbols and chooses the correct actual `gmp.h'.
-
-\1f
-File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP
-
-2.4 Notes for Particular Systems
-================================
-
-AIX 3 and 4
- On systems `*-*-aix[34]*' shared libraries are disabled by
- default, since some versions of the native `ar' fail on the
- convenience libraries used. A shared build can be attempted with
-
- ./configure --enable-shared --disable-static
-
- Note that the `--disable-static' is necessary because in a shared
- build libtool makes `libgmp.a' a symlink to `libgmp.so',
- apparently for the benefit of old versions of `ld' which only
- recognise `.a', but unfortunately this is done even if a fully
- functional `ld' is available.
-
-ARM
- On systems `arm*-*-*', versions of GCC up to and including 2.95.3
- have a bug in unsigned division, giving wrong results for some
- operands. GMP `./configure' will demand GCC 2.95.4 or later.
-
-Compaq C++
- Compaq C++ on OSF 5.1 has two flavours of `iostream', a standard
- one and an old pre-standard one (see `man iostream_intro'). GMP
- can only use the standard one, which unfortunately is not the
- default but must be selected by defining `__USE_STD_IOSTREAM'.
- Configure with for instance
-
- ./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
-
-Floating Point Mode
- On some systems, the hardware floating point has a control mode
- which can set all operations to be done in a particular precision,
- for instance single, double or extended on x86 systems (x87
- floating point). The GMP functions involving a `double' cannot be
- expected to operate to their full precision when the hardware is
- in single precision mode. Of course this affects all code,
- including application code, not just GMP.
-
-MS-DOS and MS Windows
- On an MS-DOS system DJGPP can be used to build GMP, and on an MS
- Windows system Cygwin, DJGPP and MINGW can be used. All three are
- excellent ports of GCC and the various GNU tools.
-
- `http://www.cygwin.com/'
- `http://www.delorie.com/djgpp/'
- `http://www.mingw.org/'
-
- Microsoft also publishes an Interix "Services for Unix" which can
- be used to build GMP on Windows (with a normal `./configure'), but
- it's not free software.
-
-MS Windows DLLs
- On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
- GMP builds only a static library, but a DLL can be built instead
- using
-
- ./configure --disable-static --enable-shared
-
- Static and DLL libraries can't both be built, since certain export
- directives in `gmp.h' must be different.
-
- A MINGW DLL build of GMP can be used with Microsoft C. Libtool
- doesn't install a `.lib' format import library, but it can be
- created with MS `lib' as follows, and copied to the install
- directory. Similarly for `libmp' and `libgmpxx'.
-
- cd .libs
- lib /def:libgmp-3.dll.def /out:libgmp-3.lib
-
- MINGW uses the C runtime library `msvcrt.dll' for I/O, so
- applications wanting to use the GMP I/O routines must be compiled
- with `cl /MD' to do the same. If one of the other C runtime
- library choices provided by MS C is desired then the suggestion is
- to use the GMP string functions and confine I/O to the application.
-
-Motorola 68k CPU Types
- `m68k' is taken to mean 68000. `m68020' or higher will give a
- performance boost on applicable CPUs. `m68360' can be used for
- CPU32 series chips. `m68302' can be used for "Dragonball" series
- chips, though this is merely a synonym for `m68000'.
-
-OpenBSD 2.6
- `m4' in this release of OpenBSD has a bug in `eval' that makes it
- unsuitable for `.asm' file processing. `./configure' will detect
- the problem and either abort or choose another m4 in the `PATH'.
- The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
-
-Power CPU Types
- In GMP, CPU types `power*' and `powerpc*' will each use
- instructions not available on the other, so it's important to
- choose the right one for the CPU that will be used. Currently GMP
- has no assembly code support for using just the common instruction
- subset. To get executables that run on both, the current
- suggestion is to use the generic C code (CPU `none'), possibly
- with appropriate compiler options (like `-mcpu=common' for `gcc').
- CPU `rs6000' (which is not a CPU but a family of workstations) is
- accepted by `config.sub', but is currently equivalent to `none'.
-
-Sparc CPU Types
- `sparcv8' or `supersparc' on relevant systems will give a
- significant performance increase over the V7 code selected by plain
- `sparc'.
-
-Sparc App Regs
- The GMP assembly code for both 32-bit and 64-bit Sparc clobbers the
- "application registers" `g2', `g3' and `g4', the same way that the
- GCC default `-mapp-regs' does (*note SPARC Options: (gcc)SPARC
- Options.).
-
- This makes that code unsuitable for use with the special V9
- `-mcmodel=embmedany' (which uses `g4' as a data segment pointer),
- and for applications wanting to use those registers for special
- purposes. In these cases the only suggestion currently is to
- build GMP with CPU `none' to avoid the assembly code.
-
-SunOS 4
- `/usr/bin/m4' lacks various features needed to process `.asm'
- files, and instead `./configure' will automatically use
- `/usr/5bin/m4', which we believe is always available (if not then
- use GNU m4).
-
-x86 CPU Types
- `i586', `pentium' or `pentiummmx' code is good for its intended P5
- Pentium chips, but quite slow when run on Intel P6 class chips
- (PPro, P-II, P-III). `i386' is a better choice when making
- binaries that must run on both.
-
-x86 MMX and SSE2 Code
- If the CPU selected has MMX code but the assembler doesn't support
- it, a warning is given and non-MMX code is used instead. This
- will be an inferior build, since the MMX code that's present is
- there because it's faster than the corresponding plain integer
- code. The same applies to SSE2.
-
- Old versions of `gas' don't support MMX instructions, in particular
- version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent
- OpenBSD 3.1 doesn't.
-
- Solaris 2.6 and 2.7 `as' generate incorrect object code for
- register to register `movq' instructions, and so can't be used for
- MMX code. Install a recent `gas' if MMX code is wanted on these
- systems.
-
-\1f
-File: gmp.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing GMP
-
-2.5 Known Build Problems
-========================
-
-You might find more up-to-date information at `http://gmplib.org/'.
-
-Compiler link options
- The version of libtool currently in use rather aggressively strips
- compiler options when linking a shared library. This will
- hopefully be relaxed in the future, but for now if this is a
- problem the suggestion is to create a little script to hide them,
- and for instance configure with
-
- ./configure CC=gcc-with-my-options
-
-DJGPP (`*-*-msdosdjgpp*')
- The DJGPP port of `bash' 2.03 is unable to run the `configure'
- script, it exits silently, having died writing a preamble to
- `config.log'. Use `bash' 2.04 or higher.
-
- `make all' was found to run out of memory during the final
- `libgmp.la' link on one system tested, despite having 64Mb
- available. Running `make libgmp.la' directly helped, perhaps
- recursing into the various subdirectories uses up memory.
-
-GNU binutils `strip' prior to 2.12
- `strip' from GNU binutils 2.11 and earlier should not be used on
- the static libraries `libgmp.a' and `libmp.a' since it will
- discard all but the last of multiple archive members with the same
- name, like the three versions of `init.o' in `libgmp.a'. Binutils
- 2.12 or higher can be used successfully.
-
- The shared libraries `libgmp.so' and `libmp.so' are not affected by
- this and any version of `strip' can be used on them.
-
-`make' syntax error
- On certain versions of SCO OpenServer 5 and IRIX 6.5 the native
- `make' is unable to handle the long dependencies list for
- `libgmp.la'. The symptom is a "syntax error" on the following
- line of the top-level `Makefile'.
-
- libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
-
- Either use GNU Make, or as a workaround remove
- `$(libgmp_la_DEPENDENCIES)' from that line (which will make the
- initial build work, but if any recompiling is done `libgmp.la'
- might not be rebuilt).
-
-MacOS X (`*-*-darwin*')
- Libtool currently only knows how to create shared libraries on
- MacOS X using the native `cc' (which is a modified GCC), not a
- plain GCC. A static-only build should work though
- (`--disable-shared').
-
-NeXT prior to 3.3
- The system compiler on old versions of NeXT was a massacred and
- old GCC, even if it called itself `cc'. This compiler cannot be
- used to build GMP, you need to get a real GCC, and install that.
- (NeXT may have fixed this in release 3.3 of their system.)
-
-POWER and PowerPC
- Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP
- on POWER or PowerPC. If you want to use GCC for these machines,
- get GCC 2.7.2.1 (or later).
-
-Sequent Symmetry
- Use the GNU assembler instead of the system assembler, since the
- latter has serious bugs.
-
-Solaris 2.6
- The system `sed' prints an error "Output line too long" when
- libtool builds `libgmp.la'. This doesn't seem to cause any
- obvious ill effects, but GNU `sed' is recommended, to avoid any
- doubt.
-
-Sparc Solaris 2.7 with gcc 2.95.2 in `ABI=32'
- A shared library build of GMP seems to fail in this combination,
- it builds but then fails the tests, apparently due to some
- incorrect data relocations within `gmp_randinit_lc_2exp_size'.
- The exact cause is unknown, `--disable-shared' is recommended.
-
-\1f
-File: gmp.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing GMP
-
-2.6 Performance optimization
-============================
-
-For optimal performance, build GMP for the exact CPU type of the target
-computer, see *Note Build Options::.
-
- Unlike what is the case for most other programs, the compiler
-typically doesn't matter much, since GMP uses assembly language for the
-most critical operation.
-
- In particular for long-running GMP applications, and applications
-demanding extremely large numbers, building and running the `tuneup'
-program in the `tune' subdirectory, can be important. For example,
-
- cd tune
- make tuneup
- ./tuneup
-
- will generate better contents for the `gmp-mparam.h' parameter file.
-
- To use the results, put the output in the file file indicated in the
-`Parameters for ...' header. Then recompile from scratch.
-
- The `tuneup' program takes one useful parameter, `-f NNN', which
-instructs the program how long to check FFT multiply parameters. If
-you're going to use GMP for extremely large numbers, you may want to
-run `tuneup' with a large NNN value.
-
-\1f
-File: gmp.info, Node: GMP Basics, Next: Reporting Bugs, Prev: Installing GMP, Up: Top
-
-3 GMP Basics
-************
-
-*Using functions, macros, data types, etc. not documented in this
-manual is strongly discouraged. If you do so your application is
-guaranteed to be incompatible with future versions of GMP.*
-
-* Menu:
-
-* Headers and Libraries::
-* Nomenclature and Types::
-* Function Classes::
-* Variable Conventions::
-* Parameter Conventions::
-* Memory Management::
-* Reentrancy::
-* Useful Macros and Constants::
-* Compatibility with older versions::
-* Demonstration Programs::
-* Efficiency::
-* Debugging::
-* Profiling::
-* Autoconf::
-* Emacs::
-
-\1f
-File: gmp.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: GMP Basics, Up: GMP Basics
-
-3.1 Headers and Libraries
-=========================
-
-All declarations needed to use GMP are collected in the include file
-`gmp.h'. It is designed to work with both C and C++ compilers.
-
- #include <gmp.h>
-
- Note however that prototypes for GMP functions with `FILE *'
-parameters are only provided if `<stdio.h>' is included too.
-
- #include <stdio.h>
- #include <gmp.h>
-
- Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes
-with `va_list' parameters, such as `gmp_vprintf'. And `<obstack.h>'
-for prototypes with `struct obstack' parameters, such as
-`gmp_obstack_printf', when available.
-
- All programs using GMP must link against the `libgmp' library. On a
-typical Unix-like system this can be done with `-lgmp', for example
-
- gcc myprogram.c -lgmp
-
- GMP C++ functions are in a separate `libgmpxx' library. This is
-built and installed if C++ support has been enabled (*note Build
-Options::). For example,
-
- g++ mycxxprog.cc -lgmpxx -lgmp
-
- GMP is built using Libtool and an application can use that to link
-if desired, *note GNU Libtool: (libtool)Top.
-
- If GMP has been installed to a non-standard location then it may be
-necessary to use `-I' and `-L' compiler options to point to the right
-directories, and some sort of run-time path for a shared library.
-
-\1f
-File: gmp.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: GMP Basics
-
-3.2 Nomenclature and Types
-==========================
-
-In this manual, "integer" usually means a multiple precision integer, as
-defined by the GMP library. The C data type for such integers is
-`mpz_t'. Here are some examples of how to declare such integers:
-
- mpz_t sum;
-
- struct foo { mpz_t x, y; };
-
- mpz_t vec[20];
-
- "Rational number" means a multiple precision fraction. The C data
-type for these fractions is `mpq_t'. For example:
-
- mpq_t quotient;
-
- "Floating point number" or "Float" for short, is an arbitrary
-precision mantissa with a limited precision exponent. The C data type
-for such objects is `mpf_t'. For example:
-
- mpf_t fp;
-
- The floating point functions accept and return exponents in the C
-type `mp_exp_t'. Currently this is usually a `long', but on some
-systems it's an `int' for efficiency.
-
- A "limb" means the part of a multi-precision number that fits in a
-single machine word. (We chose this word because a limb of the human
-body is analogous to a digit, only larger, and containing several
-digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
-is `mp_limb_t'.
-
- Counts of limbs of a multi-precision number represented in the C type
-`mp_size_t'. Currently this is normally a `long', but on some systems
-it's an `int' for efficiency, and on some systems it will be `long
-long' in the future.
-
- Counts of bits of a multi-precision number are represented in the C
-type `mp_bitcnt_t'. Currently this is always an `unsigned long', but on
-some systems it will be an `unsigned long long' in the future .
-
- "Random state" means an algorithm selection and current state data.
-The C data type for such objects is `gmp_randstate_t'. For example:
-
- gmp_randstate_t rstate;
-
- Also, in general `mp_bitcnt_t' is used for bit counts and ranges, and
-`size_t' is used for byte or character counts.
-
-\1f
-File: gmp.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics
-
-3.3 Function Classes
-====================
-
-There are six classes of functions in the GMP library:
-
- 1. Functions for signed integer arithmetic, with names beginning with
- `mpz_'. The associated type is `mpz_t'. There are about 150
- functions in this class. (*note Integer Functions::)
-
- 2. Functions for rational number arithmetic, with names beginning with
- `mpq_'. The associated type is `mpq_t'. There are about 40
- functions in this class, but the integer functions can be used for
- arithmetic on the numerator and denominator separately. (*note
- Rational Number Functions::)
-
- 3. Functions for floating-point arithmetic, with names beginning with
- `mpf_'. The associated type is `mpf_t'. There are about 60
- functions is this class. (*note Floating-point Functions::)
-
- 4. Functions compatible with Berkeley MP, such as `itom', `madd', and
- `mult'. The associated type is `MINT'. (*note BSD Compatible
- Functions::)
-
- 5. Fast low-level functions that operate on natural numbers. These
- are used by the functions in the preceding groups, and you can
- also call them directly from very time-critical user programs.
- These functions' names begin with `mpn_'. The associated type is
- array of `mp_limb_t'. There are about 30 (hard-to-use) functions
- in this class. (*note Low-level Functions::)
-
- 6. Miscellaneous functions. Functions for setting up custom
- allocation and functions for generating random numbers. (*note
- Custom Allocation::, and *note Random Number Functions::)
-
-\1f
-File: gmp.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: GMP Basics
-
-3.4 Variable Conventions
-========================
-
-GMP functions generally have output arguments before input arguments.
-This notation is by analogy with the assignment operator. The BSD MP
-compatibility functions are exceptions, having the output arguments
-last.
-
- GMP lets you use the same variable for both input and output in one
-call. For example, the main function for integer multiplication,
-`mpz_mul', can be used to square `x' and put the result back in `x' with
-
- mpz_mul (x, x, x);
-
- Before you can assign to a GMP variable, you need to initialize it
-by calling one of the special initialization functions. When you're
-done with a variable, you need to clear it out, using one of the
-functions for that purpose. Which function to use depends on the type
-of variable. See the chapters on integer functions, rational number
-functions, and floating-point functions for details.
-
- A variable should only be initialized once, or at least cleared
-between each initialization. After a variable has been initialized, it
-may be assigned to any number of times.
-
- For efficiency reasons, avoid excessive initializing and clearing.
-In general, initialize near the start of a function and clear near the
-end. For example,
-
- void
- foo (void)
- {
- mpz_t n;
- int i;
- mpz_init (n);
- for (i = 1; i < 100; i++)
- {
- mpz_mul (n, ...);
- mpz_fdiv_q (n, ...);
- ...
- }
- mpz_clear (n);
- }
-
-\1f
-File: gmp.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics
-
-3.5 Parameter Conventions
-=========================
-
-When a GMP variable is used as a function parameter, it's effectively a
-call-by-reference, meaning if the function stores a value there it will
-change the original in the caller. Parameters which are input-only can
-be designated `const' to provoke a compiler error or warning on
-attempting to modify them.
-
- When a function is going to return a GMP result, it should designate
-a parameter that it sets, like the library functions do. More than one
-value can be returned by having more than one output parameter, again
-like the library functions. A `return' of an `mpz_t' etc doesn't
-return the object, only a pointer, and this is almost certainly not
-what's wanted.
-
- Here's an example accepting an `mpz_t' parameter, doing a
-calculation, and storing the result to the indicated parameter.
-
- void
- foo (mpz_t result, const mpz_t param, unsigned long n)
- {
- unsigned long i;
- mpz_mul_ui (result, param, n);
- for (i = 1; i < n; i++)
- mpz_add_ui (result, result, i*7);
- }
-
- int
- main (void)
- {
- mpz_t r, n;
- mpz_init (r);
- mpz_init_set_str (n, "123456", 0);
- foo (r, n, 20L);
- gmp_printf ("%Zd\n", r);
- return 0;
- }
-
- `foo' works even if the mainline passes the same variable for
-`param' and `result', just like the library functions. But sometimes
-it's tricky to make that work, and an application might not want to
-bother supporting that sort of thing.
-
- For interest, the GMP types `mpz_t' etc are implemented as
-one-element arrays of certain structures. This is why declaring a
-variable creates an object with the fields GMP needs, but then using it
-as a parameter passes a pointer to the object. Note that the actual
-fields in each `mpz_t' etc are for internal use only and should not be
-accessed directly by code that expects to be compatible with future GMP
-releases.
-
-\1f
-File: gmp.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: GMP Basics
-
-3.6 Memory Management
-=====================
-
-The GMP types like `mpz_t' are small, containing only a couple of sizes,
-and pointers to allocated data. Once a variable is initialized, GMP
-takes care of all space allocation. Additional space is allocated
-whenever a variable doesn't have enough.
-
- `mpz_t' and `mpq_t' variables never reduce their allocated space.
-Normally this is the best policy, since it avoids frequent reallocation.
-Applications that need to return memory to the heap at some particular
-point can use `mpz_realloc2', or clear variables no longer needed.
-
- `mpf_t' variables, in the current implementation, use a fixed amount
-of space, determined by the chosen precision and allocated at
-initialization, so their size doesn't change.
-
- All memory is allocated using `malloc' and friends by default, but
-this can be changed, see *Note Custom Allocation::. Temporary memory
-on the stack is also used (via `alloca'), but this can be changed at
-build-time if desired, see *Note Build Options::.
-
-\1f
-File: gmp.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: GMP Basics
-
-3.7 Reentrancy
-==============
-
-GMP is reentrant and thread-safe, with some exceptions:
-
- * If configured with `--enable-alloca=malloc-notreentrant' (or with
- `--enable-alloca=notreentrant' when `alloca' is not available),
- then naturally GMP is not reentrant.
-
- * `mpf_set_default_prec' and `mpf_init' use a global variable for the
- selected precision. `mpf_init2' can be used instead, and in the
- C++ interface an explicit precision to the `mpf_class' constructor.
-
- * `mpz_random' and the other old random number functions use a global
- random state and are hence not reentrant. The newer random number
- functions that accept a `gmp_randstate_t' parameter can be used
- instead.
-
- * `gmp_randinit' (obsolete) returns an error indication through a
- global variable, which is not thread safe. Applications are
- advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp'
- instead.
-
- * `mp_set_memory_functions' uses global variables to store the
- selected memory allocation functions.
-
- * If the memory allocation functions set by a call to
- `mp_set_memory_functions' (or `malloc' and friends by default) are
- not reentrant, then GMP will not be reentrant either.
-
- * If the standard I/O functions such as `fwrite' are not reentrant
- then the GMP I/O functions using them will not be reentrant either.
-
- * It's safe for two threads to read from the same GMP variable
- simultaneously, but it's not safe for one to read while the
- another might be writing, nor for two threads to write
- simultaneously. It's not safe for two threads to generate a
- random number from the same `gmp_randstate_t' simultaneously,
- since this involves an update of that variable.
-
-\1f
-File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: GMP Basics
-
-3.8 Useful Macros and Constants
-===============================
-
- -- Global Constant: const int mp_bits_per_limb
- The number of bits per limb.
-
- -- Macro: __GNU_MP_VERSION
- -- Macro: __GNU_MP_VERSION_MINOR
- -- Macro: __GNU_MP_VERSION_PATCHLEVEL
- The major and minor GMP version, and patch level, respectively, as
- integers. For GMP i.j, these numbers will be i, j, and 0,
- respectively. For GMP i.j.k, these numbers will be i, j, and k,
- respectively.
-
- -- Global Constant: const char * const gmp_version
- The GMP version number, as a null-terminated string, in the form
- "i.j.k". This release is "5.0.1". Note that the format "i.j" was
- used when k was zero was used before version 4.3.0.
-
- -- Macro: __GMP_CC
- -- Macro: __GMP_CFLAGS
- The compiler and compiler flags, respectively, used when compiling
- GMP, as strings.
-
-\1f
-File: gmp.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: GMP Basics
-
-3.9 Compatibility with older versions
-=====================================
-
-This version of GMP is upwardly binary compatible with all 4.x and 3.x
-versions, and upwardly compatible at the source level with all 2.x
-versions, with the following exceptions.
-
- * `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
- consistency with other `mpn' functions.
-
- * `mpf_get_prec' counted precision slightly differently in GMP 3.0
- and 3.0.1, but in 3.1 reverted to the 2.x style.
-
- There are a number of compatibility issues between GMP 1 and GMP 2
-that of course also apply when porting applications from GMP 1 to GMP
-4. Please see the GMP 2 manual for details.
-
- The Berkeley MP compatibility library (*note BSD Compatible
-Functions::) is source and binary compatible with the standard `libmp'.
-
-\1f
-File: gmp.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: GMP Basics
-
-3.10 Demonstration programs
-===========================
-
-The `demos' subdirectory has some sample programs using GMP. These
-aren't built or installed, but there's a `Makefile' with rules for them.
-For instance,
-
- make pexpr
- ./pexpr 68^975+10
-
-The following programs are provided
-
- * `pexpr' is an expression evaluator, the program used on the GMP
- web page.
-
- * The `calc' subdirectory has a similar but simpler evaluator using
- `lex' and `yacc'.
-
- * The `expr' subdirectory is yet another expression evaluator, a
- library designed for ease of use within a C program. See
- `demos/expr/README' for more information.
-
- * `factorize' is a Pollard-Rho factorization program.
-
- * `isprime' is a command-line interface to the `mpz_probab_prime_p'
- function.
-
- * `primes' counts or lists primes in an interval, using a sieve.
-
- * `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic
- class numbers.
-
- * The `perl' subdirectory is a comprehensive perl interface to GMP.
- See `demos/perl/INSTALL' for more information. Documentation is
- in POD format in `demos/perl/GMP.pm'.
-
- As an aside, consideration has been given at various times to some
-sort of expression evaluation within the main GMP library. Going
-beyond something minimal quickly leads to matters like user-defined
-functions, looping, fixnums for control variables, etc, which are
-considered outside the scope of GMP (much closer to language
-interpreters or compilers, *Note Language Bindings::.) Something
-simple for program input convenience may yet be a possibility, a
-combination of the `expr' demo and the `pexpr' tree back-end perhaps.
-But for now the above evaluators are offered as illustrations.
-
-\1f
-File: gmp.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: GMP Basics
-
-3.11 Efficiency
-===============
-
-Small Operands
- On small operands, the time for function call overheads and memory
- allocation can be significant in comparison to actual calculation.
- This is unavoidable in a general purpose variable precision
- library, although GMP attempts to be as efficient as it can on
- both large and small operands.
-
-Static Linking
- On some CPUs, in particular the x86s, the static `libgmp.a' should
- be used for maximum speed, since the PIC code in the shared
- `libgmp.so' will have a small overhead on each function call and
- global data address. For many programs this will be
- insignificant, but for long calculations there's a gain to be had.
-
-Initializing and Clearing
- Avoid excessive initializing and clearing of variables, since this
- can be quite time consuming, especially in comparison to otherwise
- fast operations like addition.
-
- A language interpreter might want to keep a free list or stack of
- initialized variables ready for use. It should be possible to
- integrate something like that with a garbage collector too.
-
-Reallocations
- An `mpz_t' or `mpq_t' variable used to hold successively increasing
- values will have its memory repeatedly `realloc'ed, which could be
- quite slow or could fragment memory, depending on the C library.
- If an application can estimate the final size then `mpz_init2' or
- `mpz_realloc2' can be called to allocate the necessary space from
- the beginning (*note Initializing Integers::).
-
- It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
- is too small, since all functions will do a further reallocation
- if necessary. Badly overestimating memory required will waste
- space though.
-
-`2exp' Functions
- It's up to an application to call functions like `mpz_mul_2exp'
- when appropriate. General purpose functions like `mpz_mul' make
- no attempt to identify powers of two or other special forms,
- because such inputs will usually be very rare and testing every
- time would be wasteful.
-
-`ui' and `si' Functions
- The `ui' functions and the small number of `si' functions exist for
- convenience and should be used where applicable. But if for
- example an `mpz_t' contains a value that fits in an `unsigned
- long' there's no need extract it and call a `ui' function, just
- use the regular `mpz' function.
-
-In-Place Operations
- `mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
- `mpf_neg' are fast when used for in-place operations like
- `mpz_abs(x,x)', since in the current implementation only a single
- field of `x' needs changing. On suitable compilers (GCC for
- instance) this is inlined too.
-
- `mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
- from an in-place operation like `mpz_add_ui(x,x,y)', since usually
- only one or two limbs of `x' will need to be changed. The same
- applies to the full precision `mpz_add' etc if `y' is small. If
- `y' is big then cache locality may be helped, but that's all.
-
- `mpz_mul' is currently the opposite, a separate destination is
- slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
- only one limb, make a temporary copy of `x' before forming the
- result. Normally that copying will only be a tiny fraction of the
- time for the multiply, so this is not a particularly important
- consideration.
-
- `mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
- attempt to recognise a copy of something to itself, so a call like
- `mpz_set(x,x)' will be wasteful. Naturally that would never be
- written deliberately, but if it might arise from two pointers to
- the same object then a test to avoid it might be desirable.
-
- if (x != y)
- mpz_set (x, y);
-
- Note that it's never worth introducing extra `mpz_set' calls just
- to get in-place operations. If a result should go to a particular
- variable then just direct it there and let GMP take care of data
- movement.
-
-Divisibility Testing (Small Integers)
- `mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
- functions for testing whether an `mpz_t' is divisible by an
- individual small integer. They use an algorithm which is faster
- than `mpz_tdiv_ui', but which gives no useful information about
- the actual remainder, only whether it's zero (or a particular
- value).
-
- However when testing divisibility by several small integers, it's
- best to take a remainder modulo their product, to save
- multi-precision operations. For instance to test whether a number
- is divisible by any of 23, 29 or 31 take a remainder modulo
- 23*29*31 = 20677 and then test that.
-
- The division functions like `mpz_tdiv_q_ui' which give a quotient
- as well as a remainder are generally a little slower than the
- remainder-only functions like `mpz_tdiv_ui'. If the quotient is
- only rarely wanted then it's probably best to just take a
- remainder and then go back and calculate the quotient if and when
- it's wanted (`mpz_divexact_ui' can be used if the remainder is
- zero).
-
-Rational Arithmetic
- The `mpq' functions operate on `mpq_t' values with no common
- factors in the numerator and denominator. Common factors are
- checked-for and cast out as necessary. In general, cancelling
- factors every time is the best approach since it minimizes the
- sizes for subsequent operations.
-
- However, applications that know something about the factorization
- of the values they're working with might be able to avoid some of
- the GCDs used for canonicalization, or swap them for divisions.
- For example when multiplying by a prime it's enough to check for
- factors of it in the denominator instead of doing a full GCD. Or
- when forming a big product it might be known that very little
- cancellation will be possible, and so canonicalization can be left
- to the end.
-
- The `mpq_numref' and `mpq_denref' macros give access to the
- numerator and denominator to do things outside the scope of the
- supplied `mpq' functions. *Note Applying Integer Functions::.
-
- The canonical form for rationals allows mixed-type `mpq_t' and
- integer additions or subtractions to be done directly with
- multiples of the denominator. This will be somewhat faster than
- `mpq_add'. For example,
-
- /* mpq increment */
- mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
-
- /* mpq += unsigned long */
- mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
-
- /* mpq -= mpz */
- mpz_submul (mpq_numref(q), mpq_denref(q), z);
-
-Number Sequences
- Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
- designed for calculating isolated values. If a range of values is
- wanted it's probably best to call to get a starting point and
- iterate from there.
-
-Text Input/Output
- Hexadecimal or octal are suggested for input or output in text
- form. Power-of-2 bases like these can be converted much more
- efficiently than other bases, like decimal. For big numbers
- there's usually nothing of particular interest to be seen in the
- digits, so the base doesn't matter much.
-
- Maybe we can hope octal will one day become the normal base for
- everyday use, as proposed by King Charles XII of Sweden and later
- reformers.
-
-\1f
-File: gmp.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: GMP Basics
-
-3.12 Debugging
-==============
-
-Stack Overflow
- Depending on the system, a segmentation violation or bus error
- might be the only indication of stack overflow. See
- `--enable-alloca' choices in *Note Build Options::, for how to
- address this.
-
- In new enough versions of GCC, `-fstack-check' may be able to
- ensure an overflow is recognised by the system before too much
- damage is done, or `-fstack-limit-symbol' or
- `-fstack-limit-register' may be able to add checking if the system
- itself doesn't do any (*note Options for Code Generation:
- (gcc)Code Gen Options.). These options must be added to the
- `CFLAGS' used in the GMP build (*note Build Options::), adding
- them just to an application will have no effect. Note also
- they're a slowdown, adding overhead to each function call and each
- stack allocation.
-
-Heap Problems
- The most likely cause of application problems with GMP is heap
- corruption. Failing to `init' GMP variables will have
- unpredictable effects, and corruption arising elsewhere in a
- program may well affect GMP. Initializing GMP variables more than
- once or failing to clear them will cause memory leaks.
-
- In all such cases a `malloc' debugger is recommended. On a GNU or
- BSD system the standard C library `malloc' has some diagnostic
- facilities, see *Note Allocation Debugging: (libc)Allocation
- Debugging, or `man 3 malloc'. Other possibilities, in no
- particular order, include
-
- `http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/'
- `http://dmalloc.com/'
- `http://www.perens.com/FreeSoftware/' (electric fence)
- `http://packages.debian.org/stable/devel/fda'
- `http://www.gnupdate.org/components/leakbug/'
- `http://people.redhat.com/~otaylor/memprof/'
- `http://www.cbmamiga.demon.co.uk/mpatrol/'
-
- The GMP default allocation routines in `memory.c' also have a
- simple sentinel scheme which can be enabled with `#define DEBUG'
- in that file. This is mainly designed for detecting buffer
- overruns during GMP development, but might find other uses.
-
-Stack Backtraces
- On some systems the compiler options GMP uses by default can
- interfere with debugging. In particular on x86 and 68k systems
- `-fomit-frame-pointer' is used and this generally inhibits stack
- backtracing. Recompiling without such options may help while
- debugging, though the usual caveats about it potentially moving a
- memory problem or hiding a compiler bug will apply.
-
-GDB, the GNU Debugger
- A sample `.gdbinit' is included in the distribution, showing how
- to call some undocumented dump functions to print GMP variables
- from within GDB. Note that these functions shouldn't be used in
- final application code since they're undocumented and may be
- subject to incompatible changes in future versions of GMP.
-
-Source File Paths
- GMP has multiple source files with the same name, in different
- directories. For example `mpz', `mpq' and `mpf' each have an
- `init.c'. If the debugger can't already determine the right one
- it may help to build with absolute paths on each C file. One way
- to do that is to use a separate object directory with an absolute
- path to the source directory.
-
- cd /my/build/dir
- /my/source/dir/gmp-5.0.1/configure
-
- This works via `VPATH', and might require GNU `make'. Alternately
- it might be possible to change the `.c.lo' rules appropriately.
-
-Assertion Checking
- The build option `--enable-assert' is available to add some
- consistency checks to the library (see *Note Build Options::).
- These are likely to be of limited value to most applications.
- Assertion failures are just as likely to indicate memory
- corruption as a library or compiler bug.
-
- Applications using the low-level `mpn' functions, however, will
- benefit from `--enable-assert' since it adds checks on the
- parameters of most such functions, many of which have subtle
- restrictions on their usage. Note however that only the generic C
- code has checks, not the assembly code, so CPU `none' should be
- used for maximum checking.
-
-Temporary Memory Checking
- The build option `--enable-alloca=debug' arranges that each block
- of temporary memory in GMP is allocated with a separate call to
- `malloc' (or the allocation function set with
- `mp_set_memory_functions').
-
- This can help a malloc debugger detect accesses outside the
- intended bounds, or detect memory not released. In a normal
- build, on the other hand, temporary memory is allocated in blocks
- which GMP divides up for its own use, or may be allocated with a
- compiler builtin `alloca' which will go nowhere near any malloc
- debugger hooks.
-
-Maximum Debuggability
- To summarize the above, a GMP build for maximum debuggability
- would be
-
- ./configure --disable-shared --enable-assert \
- --enable-alloca=debug --host=none CFLAGS=-g
-
- For C++, add `--enable-cxx CXXFLAGS=-g'.
-
-Checker
- The GCC checker (`http://savannah.nongnu.org/projects/checker/')
- can be used with GMP. It contains a stub library which means GMP
- applications compiled with checker can use a normal GMP build.
-
- A build of GMP with checking within GMP itself can be made. This
- will run very very slowly. On GNU/Linux for example,
-
- ./configure --host=none-pc-linux-gnu CC=checkergcc
-
- `--host=none' must be used, since the GMP assembly code doesn't
- support the checking scheme. The GMP C++ features cannot be used,
- since current versions of checker (0.9.9.1) don't yet support the
- standard C++ library.
-
-Valgrind
- The valgrind program (`http://valgrind.org/') is a memory checker
- for x86s. It translates and emulates machine instructions to do
- strong checks for uninitialized data (at the level of individual
- bits), memory accesses through bad pointers, and memory leaks.
-
- Recent versions of Valgrind are getting support for MMX and
- SSE/SSE2 instructions, for past versions GMP will need to be
- configured not to use those, ie. for an x86 without them (for
- instance plain `i486').
-
-Other Problems
- Any suspected bug in GMP itself should be isolated to make sure
- it's not an application problem, see *Note Reporting Bugs::.
-
-\1f
-File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics
-
-3.13 Profiling
-==============
-
-Running a program under a profiler is a good way to find where it's
-spending most time and where improvements can be best sought. The
-profiling choices for a GMP build are as follows.
-
-`--disable-profiling'
- The default is to add nothing special for profiling.
-
- It should be possible to just compile the mainline of a program
- with `-p' and use `prof' to get a profile consisting of
- timer-based sampling of the program counter. Most of the GMP
- assembly code has the necessary symbol information.
-
- This approach has the advantage of minimizing interference with
- normal program operation, but on most systems the resolution of
- the sampling is quite low (10 milliseconds for instance),
- requiring long runs to get accurate information.
-
-`--enable-profiling=prof'
- Build with support for the system `prof', which means `-p' added
- to the `CFLAGS'.
-
- This provides call counting in addition to program counter
- sampling, which allows the most frequently called routines to be
- identified, and an average time spent in each routine to be
- determined.
-
- The x86 assembly code has support for this option, but on other
- processors the assembly routines will be as if compiled without
- `-p' and therefore won't appear in the call counts.
-
- On some systems, such as GNU/Linux, `-p' in fact means `-pg' and in
- this case `--enable-profiling=gprof' described below should be used
- instead.
-
-`--enable-profiling=gprof'
- Build with support for `gprof', which means `-pg' added to the
- `CFLAGS'.
-
- This provides call graph construction in addition to call counting
- and program counter sampling, which makes it possible to count
- calls coming from different locations. For example the number of
- calls to `mpn_mul' from `mpz_mul' versus the number from
- `mpf_mul'. The program counter sampling is still flat though, so
- only a total time in `mpn_mul' would be accumulated, not a
- separate amount for each call site.
-
- The x86 assembly code has support for this option, but on other
- processors the assembly routines will be as if compiled without
- `-pg' and therefore not be included in the call counts.
-
- On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
- incompatible, so the latter is omitted from the default flags in
- that case, which might result in poorer code generation.
-
- Incidentally, it should be possible to use the `gprof' program
- with a plain `--enable-profiling=prof' build. But in that case
- only the `gprof -p' flat profile and call counts can be expected
- to be valid, not the `gprof -q' call graph.
-
-`--enable-profiling=instrument'
- Build with the GCC option `-finstrument-functions' added to the
- `CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
- Options.).
-
- This inserts special instrumenting calls at the start and end of
- each function, allowing exact timing and full call graph
- construction.
-
- This instrumenting is not normally a standard system feature and
- will require support from an external library, such as
-
- `http://sourceforge.net/projects/fnccheck/'
-
- This should be included in `LIBS' during the GMP configure so that
- test programs will link. For example,
-
- ./configure --enable-profiling=instrument LIBS=-lfc
-
- On a GNU system the C library provides dummy instrumenting
- functions, so programs compiled with this option will link. In
- this case it's only necessary to ensure the correct library is
- added when linking an application.
-
- The x86 assembly code supports this option, but on other
- processors the assembly routines will be as if compiled without
- `-finstrument-functions' meaning time spent in them will
- effectively be attributed to their caller.
-
-\1f
-File: gmp.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: GMP Basics
-
-3.14 Autoconf
-=============
-
-Autoconf based applications can easily check whether GMP is installed.
-The only thing to be noted is that GMP library symbols from version 3
-onwards have prefixes like `__gmpz'. The following therefore would be
-a simple test,
-
- AC_CHECK_LIB(gmp, __gmpz_init)
-
- This just uses the default `AC_CHECK_LIB' actions for found or not
-found, but an application that must have GMP would want to generate an
-error if not found. For example,
-
- AC_CHECK_LIB(gmp, __gmpz_init, ,
- [AC_MSG_ERROR([GNU MP not found, see http://gmplib.org/])])
-
- If functions added in some particular version of GMP are required,
-then one of those can be used when checking. For example `mpz_mul_si'
-was added in GMP 3.1,
-
- AC_CHECK_LIB(gmp, __gmpz_mul_si, ,
- [AC_MSG_ERROR(
- [GNU MP not found, or not 3.1 or up, see http://gmplib.org/])])
-
- An alternative would be to test the version number in `gmp.h' using
-say `AC_EGREP_CPP'. That would make it possible to test the exact
-version, if some particular sub-minor release is known to be necessary.
-
- In general it's recommended that applications should simply demand a
-new enough GMP rather than trying to provide supplements for features
-not available in past versions.
-
- Occasionally an application will need or want to know the size of a
-type at configuration or preprocessing time, not just with `sizeof' in
-the code. This can be done in the normal way with `mp_limb_t' etc, but
-GMP 4.0 or up is best for this, since prior versions needed certain
-`-D' defines on systems using a `long long' limb. The following would
-suit Autoconf 2.50 or up,
-
- AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
-
-\1f
-File: gmp.info, Node: Emacs, Prev: Autoconf, Up: GMP Basics
-
-3.15 Emacs
-==========
-
-<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
-C functions while editing (*note Info Documentation Lookup: (emacs)Info
-Lookup.).
-
- The GMP manual can be included in such lookups by putting the
-following in your `.emacs',
-
- (eval-after-load "info-look"
- '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
- (setcar (nthcdr 3 mode-value)
- (cons '("(gmp)Function Index" nil "^ -.* " "\\>")
- (nth 3 mode-value)))))
-
-\1f
-File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: GMP Basics, Up: Top
-
-4 Reporting Bugs
-****************
-
-If you think you have found a bug in the GMP library, please
-investigate it and report it. We have made this library available to
-you, and it is not too much to ask you to report the bugs you find.
-
- Before you report a bug, check it's not already addressed in *Note
-Known Build Problems::, or perhaps *Note Notes for Particular
-Systems::. You may also want to check `http://gmplib.org/' for patches
-for this release.
-
- Please include the following in any report,
-
- * The GMP version number, and if pre-packaged or patched then say so.
-
- * A test program that makes it possible for us to reproduce the bug.
- Include instructions on how to run the program.
-
- * A description of what is wrong. If the results are incorrect, in
- what way. If you get a crash, say so.
-
- * If you get a crash, include a stack backtrace from the debugger if
- it's informative (`where' in `gdb', or `$C' in `adb').
-
- * Please do not send core dumps, executables or `strace's.
-
- * The configuration options you used when building GMP, if any.
-
- * The name of the compiler and its version. For `gcc', get the
- version with `gcc -v', otherwise perhaps `what `which cc`', or
- similar.
-
- * The output from running `uname -a'.
-
- * The output from running `./config.guess', and from running
- `./configfsf.guess' (might be the same).
-
- * If the bug is related to `configure', then the compressed contents
- of `config.log'.
-
- * If the bug is related to an `asm' file not assembling, then the
- contents of `config.m4' and the offending line or lines from the
- temporary `mpn/tmp-<file>.s'.
-
- Please make an effort to produce a self-contained report, with
-something definite that can be tested or debugged. Vague queries or
-piecemeal messages are difficult to act on and don't help the
-development effort.
-
- It is not uncommon that an observed problem is actually due to a bug
-in the compiler; the GMP code tends to explore interesting corners in
-compilers.
-
- If your bug report is good, we will do our best to help you get a
-corrected version of the library; if the bug report is poor, we won't
-do anything about it (except maybe ask you to send a better report).
-
- Send your report to: <gmp-bugs@gmplib.org>.
-
- If you think something in this manual is unclear, or downright
-incorrect, or if the language needs to be improved, please send a note
-to the same address.
-
-\1f
-File: gmp.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
-
-5 Integer Functions
-*******************
-
-This chapter describes the GMP functions for performing integer
-arithmetic. These functions start with the prefix `mpz_'.
-
- GMP integers are stored in objects of type `mpz_t'.
-
-* Menu:
-
-* Initializing Integers::
-* Assigning Integers::
-* Simultaneous Integer Init & Assign::
-* Converting Integers::
-* Integer Arithmetic::
-* Integer Division::
-* Integer Exponentiation::
-* Integer Roots::
-* Number Theoretic Functions::
-* Integer Comparisons::
-* Integer Logic and Bit Fiddling::
-* I/O of Integers::
-* Integer Random Numbers::
-* Integer Import and Export::
-* Miscellaneous Integer Functions::
-* Integer Special Functions::
-
-\1f
-File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions
-
-5.1 Initialization Functions
-============================
-
-The functions for integer arithmetic assume that all integer objects are
-initialized. You do that by calling the function `mpz_init'. For
-example,
-
- {
- mpz_t integ;
- mpz_init (integ);
- ...
- mpz_add (integ, ...);
- ...
- mpz_sub (integ, ...);
-
- /* Unless the program is about to exit, do ... */
- mpz_clear (integ);
- }
-
- As you can see, you can store new values any number of times, once an
-object is initialized.
-
- -- Function: void mpz_init (mpz_t X)
- Initialize X, and set its value to 0.
-
- -- Function: void mpz_inits (mpz_t X, ...)
- Initialize a NULL-terminated list of `mpz_t' variables, and set
- their values to 0.
-
- -- Function: void mpz_init2 (mpz_t X, mp_bitcnt_t N)
- Initialize X, with space for N-bit numbers, and set its value to 0.
- Calling this function instead of `mpz_init' or `mpz_inits' is never
- necessary; reallocation is handled automatically by GMP when
- needed.
-
- N is only the initial space, X will grow automatically in the
- normal way, if necessary, for subsequent values stored.
- `mpz_init2' makes it possible to avoid such reallocations if a
- maximum size is known in advance.
-
- -- Function: void mpz_clear (mpz_t X)
- Free the space occupied by X. Call this function for all `mpz_t'
- variables when you are done with them.
-
- -- Function: void mpz_clears (mpz_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpz_t'
- variables.
-
- -- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N)
- Change the space allocated for X to N bits. The value in X is
- preserved if it fits, or is set to 0 if not.
-
- Calling this function is never necessary; reallocation is handled
- automatically by GMP when needed. But this function can be used
- to increase the space for a variable in order to avoid repeated
- automatic reallocations, or to decrease it to give memory back to
- the heap.
-
-\1f
-File: gmp.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions
-
-5.2 Assignment Functions
-========================
-
-These functions assign new values to already initialized integers
-(*note Initializing Integers::).
-
- -- Function: void mpz_set (mpz_t ROP, mpz_t OP)
- -- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
- -- Function: void mpz_set_si (mpz_t ROP, signed long int OP)
- -- Function: void mpz_set_d (mpz_t ROP, double OP)
- -- Function: void mpz_set_q (mpz_t ROP, mpq_t OP)
- -- Function: void mpz_set_f (mpz_t ROP, mpf_t OP)
- Set the value of ROP from OP.
-
- `mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
- integer.
-
- -- Function: int mpz_set_str (mpz_t ROP, char *STR, int BASE)
- Set the value of ROP from STR, a null-terminated C string in base
- BASE. White space is allowed in the string, and is simply ignored.
-
- The BASE may vary from 2 to 62, or if BASE is 0, then the leading
- characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
- for binary, `0' for octal, or decimal otherwise.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value. For bases 37 to 62, upper-case
- letter represent the usual 10..35 while lower-case letter
- represent 36..61.
-
- This function returns 0 if the entire string is a valid number in
- base BASE. Otherwise it returns -1.
-
- -- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
- Swap the values ROP1 and ROP2 efficiently.
-
-\1f
-File: gmp.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions
-
-5.3 Combined Initialization and Assignment Functions
-====================================================
-
-For convenience, GMP provides a parallel series of initialize-and-set
-functions which initialize the output and then store the value there.
-These functions' names have the form `mpz_init_set...'
-
- Here is an example of using one:
-
- {
- mpz_t pie;
- mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
- ...
- mpz_sub (pie, ...);
- ...
- mpz_clear (pie);
- }
-
-Once the integer has been initialized by any of the `mpz_init_set...'
-functions, it can be used as the source or destination operand for the
-ordinary integer functions. Don't use an initialize-and-set function
-on a variable already initialized!
-
- -- Function: void mpz_init_set (mpz_t ROP, mpz_t OP)
- -- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
- -- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
- -- Function: void mpz_init_set_d (mpz_t ROP, double OP)
- Initialize ROP with limb space and set the initial numeric value
- from OP.
-
- -- Function: int mpz_init_set_str (mpz_t ROP, char *STR, int BASE)
- Initialize ROP and set its value like `mpz_set_str' (see its
- documentation above for details).
-
- If the string is a correct base BASE number, the function returns
- 0; if an error occurs it returns -1. ROP is initialized even if
- an error occurs. (I.e., you have to call `mpz_clear' for it.)
-
-\1f
-File: gmp.info, Node: Converting Integers, Next: Integer Arithmetic, Prev: Simultaneous Integer Init & Assign, Up: Integer Functions
-
-5.4 Conversion Functions
-========================
-
-This section describes functions for converting GMP integers to
-standard C types. Functions for converting _to_ GMP integers are
-described in *Note Assigning Integers:: and *Note I/O of Integers::.
-
- -- Function: unsigned long int mpz_get_ui (mpz_t OP)
- Return the value of OP as an `unsigned long'.
-
- If OP is too big to fit an `unsigned long' then just the least
- significant bits that do fit are returned. The sign of OP is
- ignored, only the absolute value is used.
-
- -- Function: signed long int mpz_get_si (mpz_t OP)
- If OP fits into a `signed long int' return the value of OP.
- Otherwise return the least significant part of OP, with the same
- sign as OP.
-
- If OP is too big to fit in a `signed long int', the returned
- result is probably not very useful. To find out if the value will
- fit, use the function `mpz_fits_slong_p'.
-
- -- Function: double mpz_get_d (mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent from the conversion is too big, the result is
- system dependent. An infinity is returned where available. A
- hardware overflow trap may or may not occur.
-
- -- Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero), and returning the exponent separately.
-
- The return value is in the range 0.5<=abs(D)<1 and the exponent is
- stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
- is zero, the return is 0.0 and 0 is stored to `*EXP'.
-
- This is similar to the standard C `frexp' function (*note
- Normalization Functions: (libc)Normalization Functions.).
-
- -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
- Convert OP to a string of digits in base BASE. The base argument
- may vary from 2 to 62 or from -2 to -36.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of storage large
- enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
- The two extra bytes are for a possible minus sign, and the
- null-terminator.
-
- A pointer to the result string is returned, being either the
- allocated block, or the given STR.
-
-\1f
-File: gmp.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
-
-5.5 Arithmetic Functions
-========================
-
- -- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 + OP2.
-
- -- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
- -- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
- OP2)
- Set ROP to OP1 - OP2.
-
- -- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2)
- -- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 times OP2.
-
- -- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long
- int OP2)
- Set ROP to ROP + OP1 times OP2.
-
- -- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long
- int OP2)
- Set ROP to ROP - OP1 times OP2.
-
- -- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2. This operation can also be
- defined as a left shift by OP2 bits.
-
- -- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
- Set ROP to -OP.
-
- -- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
- Set ROP to the absolute value of OP.
-
-\1f
-File: gmp.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
-
-5.6 Division Functions
-======================
-
-Division is undefined if the divisor is zero. Passing a zero divisor
-to the division or modulo functions (including the modular powering
-functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
-division by zero. This lets a program handle arithmetic exceptions in
-these functions the same way as for normal C `int' arithmetic.
-
- -- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_ui (mpz_t N,
- unsigned long int D)
- -- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- -- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_ui (mpz_t N,
- unsigned long int D)
- -- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- -- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
- -- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
- mpz_t N, unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_ui (mpz_t N,
- unsigned long int D)
- -- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N, mp_bitcnt_t B)
- -- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N, mp_bitcnt_t B)
-
- Divide N by D, forming a quotient Q and/or remainder R. For the
- `2exp' functions, D=2^B. The rounding is in three styles, each
- suiting different applications.
-
- * `cdiv' rounds Q up towards +infinity, and R will have the
- opposite sign to D. The `c' stands for "ceil".
-
- * `fdiv' rounds Q down towards -infinity, and R will have the
- same sign as D. The `f' stands for "floor".
-
- * `tdiv' rounds Q towards zero, and R will have the same sign
- as N. The `t' stands for "truncate".
-
- In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
- 0<=abs(R)<abs(D).
-
- The `q' functions calculate only the quotient, the `r' functions
- only the remainder, and the `qr' functions calculate both. Note
- that for `qr' the same variable cannot be passed for both Q and R,
- or results will be unpredictable.
-
- For the `ui' variants the return value is the remainder, and in
- fact returning the remainder is all the `div_ui' functions do. For
- `tdiv' and `cdiv' the remainder can be negative, so for those the
- return value is the absolute value of the remainder.
-
- For the `2exp' variants the divisor is 2^B. These functions are
- implemented as right shifts and bit masks, but of course they
- round the same as the other functions.
-
- For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
- simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
- effectively an arithmetic right shift treating N as twos complement
- the same as the bitwise logical functions do, whereas
- `mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
-
- -- Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
- -- Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N,
- unsigned long int D)
- Set R to N `mod' D. The sign of the divisor is ignored; the
- result is always non-negative.
-
- `mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
- remainder as well as setting R. See `mpz_fdiv_ui' above if only
- the return value is wanted.
-
- -- Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
- -- Function: void mpz_divexact_ui (mpz_t Q, mpz_t N, unsigned long D)
- Set Q to N/D. These functions produce correct results only when
- it is known in advance that D divides N.
-
- These routines are much faster than the other division functions,
- and are the best choice when exact division is known to occur, for
- example reducing a rational to lowest terms.
-
- -- Function: int mpz_divisible_p (mpz_t N, mpz_t D)
- -- Function: int mpz_divisible_ui_p (mpz_t N, unsigned long int D)
- -- Function: int mpz_divisible_2exp_p (mpz_t N, mp_bitcnt_t B)
- Return non-zero if N is exactly divisible by D, or in the case of
- `mpz_divisible_2exp_p' by 2^B.
-
- N is divisible by D if there exists an integer Q satisfying N =
- Q*D. Unlike the other division functions, D=0 is accepted and
- following the rule it can be seen that only 0 is considered
- divisible by 0.
-
- -- Function: int mpz_congruent_p (mpz_t N, mpz_t C, mpz_t D)
- -- Function: int mpz_congruent_ui_p (mpz_t N, unsigned long int C,
- unsigned long int D)
- -- Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, mp_bitcnt_t B)
- Return non-zero if N is congruent to C modulo D, or in the case of
- `mpz_congruent_2exp_p' modulo 2^B.
-
- N is congruent to C mod D if there exists an integer Q satisfying
- N = C + Q*D. Unlike the other division functions, D=0 is accepted
- and following the rule it can be seen that N and C are considered
- congruent mod 0 only when exactly equal.
-
-\1f
-File: gmp.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
-
-5.7 Exponentiation Functions
-============================
-
- -- Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t
- MOD)
- -- Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long
- int EXP, mpz_t MOD)
- Set ROP to (BASE raised to EXP) modulo MOD.
-
- Negative EXP is supported if an inverse BASE^-1 mod MOD exists
- (see `mpz_invert' in *Note Number Theoretic Functions::). If an
- inverse doesn't exist then a divide by zero is raised.
-
- -- Function: void mpz_powm_sec (mpz_t ROP, mpz_t BASE, mpz_t EXP,
- mpz_t MOD)
- Set ROP to (BASE raised to EXP) modulo MOD.
-
- It is required that EXP > 0 and that MOD is odd.
-
- This function is designed to take the same time and have the same
- cache access patterns for any two same-size arguments, assuming
- that function arguments are placed at the same position and that
- the machine state is identical upon function entry. This function
- is intended for cryptographic purposes, where resilience to
- side-channel attacks is desired.
-
- -- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
- EXP)
- -- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
- unsigned long int EXP)
- Set ROP to BASE raised to EXP. The case 0^0 yields 1.
-
-\1f
-File: gmp.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions
-
-5.8 Root Extraction Functions
-=============================
-
- -- Function: int mpz_root (mpz_t ROP, mpz_t OP, unsigned long int N)
- Set ROP to the truncated integer part of the Nth root of OP.
- Return non-zero if the computation was exact, i.e., if OP is ROP
- to the Nth power.
-
- -- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, mpz_t U,
- unsigned long int N)
- Set ROOT to the truncated integer part of the Nth root of U. Set
- REM to the remainder, U-ROOT**N.
-
- -- Function: void mpz_sqrt (mpz_t ROP, mpz_t OP)
- Set ROP to the truncated integer part of the square root of OP.
-
- -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, mpz_t OP)
- Set ROP1 to the truncated integer part of the square root of OP,
- like `mpz_sqrt'. Set ROP2 to the remainder OP-ROP1*ROP1, which
- will be zero if OP is a perfect square.
-
- If ROP1 and ROP2 are the same variable, the results are undefined.
-
- -- Function: int mpz_perfect_power_p (mpz_t OP)
- Return non-zero if OP is a perfect power, i.e., if there exist
- integers A and B, with B>1, such that OP equals A raised to the
- power B.
-
- Under this definition both 0 and 1 are considered to be perfect
- powers. Negative values of OP are accepted, but of course can
- only be odd perfect powers.
-
- -- Function: int mpz_perfect_square_p (mpz_t OP)
- Return non-zero if OP is a perfect square, i.e., if the square
- root of OP is an integer. Under this definition both 0 and 1 are
- considered to be perfect squares.
-
-\1f
-File: gmp.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
-
-5.9 Number Theoretic Functions
-==============================
-
- -- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
- Determine whether N is prime. Return 2 if N is definitely prime,
- return 1 if N is probably prime (without being certain), or return
- 0 if N is definitely composite.
-
- This function does some trial divisions, then some Miller-Rabin
- probabilistic primality tests. REPS controls how many such tests
- are done, 5 to 10 is a reasonable number, more will reduce the
- chances of a composite being returned as "probably prime".
-
- Miller-Rabin and similar tests can be more properly called
- compositeness tests. Numbers which fail are known to be composite
- but those which pass might be prime or might be composite. Only a
- few composites pass, hence those which pass are considered
- probably prime.
-
- -- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
- Set ROP to the next prime greater than OP.
-
- This function uses a probabilistic algorithm to identify primes.
- For practical purposes it's adequate, the chance of a composite
- passing will be extremely small.
-
- -- Function: void mpz_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to the greatest common divisor of OP1 and OP2. The result
- is always positive even if one or both input operands are negative.
-
- -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
- unsigned long int OP2)
- Compute the greatest common divisor of OP1 and OP2. If ROP is not
- `NULL', store the result there.
-
- If the result is small enough to fit in an `unsigned long int', it
- is returned. If the result does not fit, 0 is returned, and the
- result is equal to the argument OP1. Note that the result will
- always fit if OP2 is non-zero.
-
- -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A,
- mpz_t B)
- Set G to the greatest common divisor of A and B, and in addition
- set S and T to coefficients satisfying A*S + B*T = G. The value
- in G is always positive, even if one or both of A and B are
- negative. The values in S and T are chosen such that abs(S) <=
- abs(B) and abs(T) <= abs(A).
-
- If T is `NULL' then that value is not computed.
-
- -- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- -- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2)
- Set ROP to the least common multiple of OP1 and OP2. ROP is
- always positive, irrespective of the signs of OP1 and OP2. ROP
- will be zero if either OP1 or OP2 is zero.
-
- -- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Compute the inverse of OP1 modulo OP2 and put the result in ROP.
- If the inverse exists, the return value is non-zero and ROP will
- satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return
- value is zero and ROP is undefined.
-
- -- Function: int mpz_jacobi (mpz_t A, mpz_t B)
- Calculate the Jacobi symbol (A/B). This is defined only for B odd.
-
- -- Function: int mpz_legendre (mpz_t A, mpz_t P)
- Calculate the Legendre symbol (A/P). This is defined only for P
- an odd positive prime, and for such P it's identical to the Jacobi
- symbol.
-
- -- Function: int mpz_kronecker (mpz_t A, mpz_t B)
- -- Function: int mpz_kronecker_si (mpz_t A, long B)
- -- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
- -- Function: int mpz_si_kronecker (long A, mpz_t B)
- -- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
- Calculate the Jacobi symbol (A/B) with the Kronecker extension
- (a/2)=(2/a) when a odd, or (a/2)=0 when a even.
-
- When B is odd the Jacobi symbol and Kronecker symbol are
- identical, so `mpz_kronecker_ui' etc can be used for mixed
- precision Jacobi symbols too.
-
- For more information see Henri Cohen section 1.4.2 (*note
- References::), or any number theory textbook. See also the
- example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
-
- -- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, mpz_t OP, mpz_t F)
- Remove all occurrences of the factor F from OP and store the
- result in ROP. The return value is how many such occurrences were
- removed.
-
- -- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
- Set ROP to OP!, the factorial of OP.
-
- -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
- -- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
- unsigned long int K)
- Compute the binomial coefficient N over K and store the result in
- ROP. Negative values of N are supported by `mpz_bin_ui', using
- the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
- section 1.2.6 part G.
-
- -- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
- -- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
- int N)
- `mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
- `mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
-
- These functions are designed for calculating isolated Fibonacci
- numbers. When a sequence of values is wanted it's best to start
- with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
- similar.
-
- -- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
- -- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
- long int N)
- `mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
- `mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
-
- These functions are designed for calculating isolated Lucas
- numbers. When a sequence of values is wanted it's best to start
- with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
- or similar.
-
- The Fibonacci numbers and Lucas numbers are related sequences, so
- it's never necessary to call both `mpz_fib2_ui' and
- `mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
- can be found in *Note Lucas Numbers Algorithm::, the reverse is
- straightforward too.
-
-\1f
-File: gmp.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
-
-5.10 Comparison Functions
-=========================
-
- -- Function: int mpz_cmp (mpz_t OP1, mpz_t OP2)
- -- Function: int mpz_cmp_d (mpz_t OP1, double OP2)
- -- Macro: int mpz_cmp_si (mpz_t OP1, signed long int OP2)
- -- Macro: int mpz_cmp_ui (mpz_t OP1, unsigned long int OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, or a negative value if OP1 < OP2.
-
- `mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their
- arguments more than once. `mpz_cmp_d' can be called with an
- infinity, but results are undefined for a NaN.
-
- -- Function: int mpz_cmpabs (mpz_t OP1, mpz_t OP2)
- -- Function: int mpz_cmpabs_d (mpz_t OP1, double OP2)
- -- Function: int mpz_cmpabs_ui (mpz_t OP1, unsigned long int OP2)
- Compare the absolute values of OP1 and OP2. Return a positive
- value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
- negative value if abs(OP1) < abs(OP2).
-
- `mpz_cmpabs_d' can be called with an infinity, but results are
- undefined for a NaN.
-
- -- Macro: int mpz_sgn (mpz_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates
- its argument multiple times.
-
-\1f
-File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
-
-5.11 Logical and Bit Manipulation Functions
-===========================================
-
-These functions behave as if twos complement arithmetic were used
-(although sign-magnitude is the actual implementation). The least
-significant bit is number 0.
-
- -- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise-and OP2.
-
- -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise inclusive-or OP2.
-
- -- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
- Set ROP to OP1 bitwise exclusive-or OP2.
-
- -- Function: void mpz_com (mpz_t ROP, mpz_t OP)
- Set ROP to the one's complement of OP.
-
- -- Function: mp_bitcnt_t mpz_popcount (mpz_t OP)
- If OP>=0, return the population count of OP, which is the number
- of 1 bits in the binary representation. If OP<0, the number of 1s
- is infinite, and the return value is the largest possible
- `mp_bitcnt_t'.
-
- -- Function: mp_bitcnt_t mpz_hamdist (mpz_t OP1, mpz_t OP2)
- If OP1 and OP2 are both >=0 or both <0, return the hamming
- distance between the two operands, which is the number of bit
- positions where OP1 and OP2 have different bit values. If one
- operand is >=0 and the other <0 then the number of bits different
- is infinite, and the return value is the largest possible
- `mp_bitcnt_t'.
-
- -- Function: mp_bitcnt_t mpz_scan0 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
- -- Function: mp_bitcnt_t mpz_scan1 (mpz_t OP, mp_bitcnt_t STARTING_BIT)
- Scan OP, starting from bit STARTING_BIT, towards more significant
- bits, until the first 0 or 1 bit (respectively) is found. Return
- the index of the found bit.
-
- If the bit at STARTING_BIT is already what's sought, then
- STARTING_BIT is returned.
-
- If there's no bit found, then the largest possible `mp_bitcnt_t' is
- returned. This will happen in `mpz_scan0' past the end of a
- negative number, or `mpz_scan1' past the end of a nonnegative
- number.
-
- -- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Set bit BIT_INDEX in ROP.
-
- -- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Clear bit BIT_INDEX in ROP.
-
- -- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
- Complement bit BIT_INDEX in ROP.
-
- -- Function: int mpz_tstbit (mpz_t OP, mp_bitcnt_t BIT_INDEX)
- Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
-
-\1f
-File: gmp.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
-
-5.12 Input and Output Functions
-===============================
-
-Functions that perform input from a stdio stream, and functions that
-output to a stdio stream. Passing a `NULL' pointer for a STREAM
-argument to any of these functions will make them read from `stdin' and
-write to `stdout', respectively.
-
- When using any of these functions, it is a good idea to include
-`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
-prototypes for these functions.
-
- -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, mpz_t OP)
- Output OP on stdio stream STREAM, as a string of digits in base
- BASE. The base argument may vary from 2 to 62 or from -2 to -36.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- -- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
- Input a possibly white-space preceded string in base BASE from
- stdio stream STREAM, and put the read integer in ROP.
-
- The BASE may vary from 2 to 62, or if BASE is 0, then the leading
- characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
- for binary, `0' for octal, or decimal otherwise.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value. For bases 37 to 62, upper-case
- letter represent the usual 10..35 while lower-case letter
- represent 36..61.
-
- Return the number of bytes read, or if an error occurred, return 0.
-
- -- Function: size_t mpz_out_raw (FILE *STREAM, mpz_t OP)
- Output OP on stdio stream STREAM, in raw binary format. The
- integer is written in a portable format, with 4 bytes of size
- information, and that many bytes of limbs. Both the size and the
- limbs are written in decreasing significance order (i.e., in
- big-endian).
-
- The output can be read with `mpz_inp_raw'.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- The output of this can not be read by `mpz_inp_raw' from GMP 1,
- because of changes necessary for compatibility between 32-bit and
- 64-bit machines.
-
- -- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
- Input from stdio stream STREAM in the format written by
- `mpz_out_raw', and put the result in ROP. Return the number of
- bytes read, or if an error occurred, return 0.
-
- This routine can read the output from `mpz_out_raw' also from GMP
- 1, in spite of changes necessary for compatibility between 32-bit
- and 64-bit machines.
-
-\1f
-File: gmp.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
-
-5.13 Random Number Functions
-============================
-
-The random number functions of GMP come in two groups; older function
-that rely on a global state, and newer functions that accept a state
-parameter that is read and modified. Please see the *Note Random
-Number Functions:: for more information on how to use and not to use
-random number functions.
-
- -- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t N)
- Generate a uniformly distributed random integer in the range 0 to
- 2^N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
- mpz_t N)
- Generate a uniform random integer in the range 0 to N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t N)
- Generate a random integer with long strings of zeros and ones in
- the binary representation. Useful for testing functions and
- algorithms, since this kind of random numbers have proven to be
- more likely to trigger corner-case bugs. The random number will
- be in the range 0 to 2^N-1, inclusive.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
- Generate a random integer of at most MAX_SIZE limbs. The generated
- random number doesn't satisfy any particular requirements of
- randomness. Negative random numbers are generated when MAX_SIZE
- is negative.
-
- This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
- instead.
-
- -- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
- Generate a random integer of at most MAX_SIZE limbs, with long
- strings of zeros and ones in the binary representation. Useful
- for testing functions and algorithms, since this kind of random
- numbers have proven to be more likely to trigger corner-case bugs.
- Negative random numbers are generated when MAX_SIZE is negative.
-
- This function is obsolete. Use `mpz_rrandomb' instead.
-
-\1f
-File: gmp.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions
-
-5.14 Integer Import and Export
-==============================
-
-`mpz_t' variables can be converted to and from arbitrary words of binary
-data with the following functions.
-
- -- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
- size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
- Set ROP from an array of word data at OP.
-
- The parameters specify the format of the data. COUNT many words
- are read, each SIZE bytes. ORDER can be 1 for most significant
- word first or -1 for least significant first. Within each word
- ENDIAN can be 1 for most significant byte first, -1 for least
- significant first, or 0 for the native endianness of the host CPU.
- The most significant NAILS bits of each word are skipped, this
- can be 0 to use the full words.
-
- There is no sign taken from the data, ROP will simply be a positive
- integer. An application can handle any sign itself, and apply it
- for instance with `mpz_neg'.
-
- There are no data alignment restrictions on OP, any address is
- allowed.
-
- Here's an example converting an array of `unsigned long' data, most
- significant element first, and host byte order within each value.
-
- unsigned long a[20];
- /* Initialize Z and A */
- mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
-
- This example assumes the full `sizeof' bytes are used for data in
- the given type, which is usually true, and certainly true for
- `unsigned long' everywhere we know of. However on Cray vector
- systems it may be noted that `short' and `int' are always stored
- in 8 bytes (and with `sizeof' indicating that) but use only 32 or
- 46 bits. The NAILS feature can account for this, by passing for
- instance `8*sizeof(int)-INT_BIT'.
-
- -- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
- size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
- Fill ROP with word data from OP.
-
- The parameters specify the format of the data produced. Each word
- will be SIZE bytes and ORDER can be 1 for most significant word
- first or -1 for least significant first. Within each word ENDIAN
- can be 1 for most significant byte first, -1 for least significant
- first, or 0 for the native endianness of the host CPU. The most
- significant NAILS bits of each word are unused and set to zero,
- this can be 0 to produce full words.
-
- The number of words produced is written to `*COUNTP', or COUNTP
- can be `NULL' to discard the count. ROP must have enough space
- for the data, or if ROP is `NULL' then a result array of the
- necessary size is allocated using the current GMP allocation
- function (*note Custom Allocation::). In either case the return
- value is the destination used, either ROP or the allocated block.
-
- If OP is non-zero then the most significant word produced will be
- non-zero. If OP is zero then the count returned will be zero and
- nothing written to ROP. If ROP is `NULL' in this case, no block
- is allocated, just `NULL' is returned.
-
- The sign of OP is ignored, just the absolute value is exported. An
- application can use `mpz_sgn' to get the sign and handle it as
- desired. (*note Integer Comparisons::)
-
- There are no data alignment restrictions on ROP, any address is
- allowed.
-
- When an application is allocating space itself the required size
- can be determined with a calculation like the following. Since
- `mpz_sizeinbase' always returns at least 1, `count' here will be
- at least one, which avoids any portability problems with
- `malloc(0)', though if `z' is zero no space at all is actually
- needed (or written).
-
- numb = 8*size - nail;
- count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
- p = malloc (count * size);
-
-\1f
-File: gmp.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
-
-5.15 Miscellaneous Functions
-============================
-
- -- Function: int mpz_fits_ulong_p (mpz_t OP)
- -- Function: int mpz_fits_slong_p (mpz_t OP)
- -- Function: int mpz_fits_uint_p (mpz_t OP)
- -- Function: int mpz_fits_sint_p (mpz_t OP)
- -- Function: int mpz_fits_ushort_p (mpz_t OP)
- -- Function: int mpz_fits_sshort_p (mpz_t OP)
- Return non-zero iff the value of OP fits in an `unsigned long int',
- `signed long int', `unsigned int', `signed int', `unsigned short
- int', or `signed short int', respectively. Otherwise, return zero.
-
- -- Macro: int mpz_odd_p (mpz_t OP)
- -- Macro: int mpz_even_p (mpz_t OP)
- Determine whether OP is odd or even, respectively. Return
- non-zero if yes, zero if no. These macros evaluate their argument
- more than once.
-
- -- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
- Return the size of OP measured in number of digits in the given
- BASE. BASE can vary from 2 to 62. The sign of OP is ignored,
- just the absolute value is used. The result will be either exact
- or 1 too big. If BASE is a power of 2, the result is always
- exact. If OP is zero the return value is always 1.
-
- This function can be used to determine the space required when
- converting OP to a string. The right amount of allocation is
- normally two more than the value returned by `mpz_sizeinbase', one
- extra for a minus sign and one for the null-terminator.
-
- It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
- the most significant 1 bit in OP, counting from 1. (Unlike the
- bitwise functions which start from 0, *Note Logical and Bit
- Manipulation Functions: Integer Logic and Bit Fiddling.)
-
-\1f
-File: gmp.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
-
-5.16 Special Functions
-======================
-
-The functions in this section are for various special purposes. Most
-applications will not need them.
-
- -- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_size_t
- ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
- This is a special type of initialization. *Fixed* space of
- FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in
- INTEGER_ARRAY. There is no way to free the storage allocated by
- this function. Don't call `mpz_clear'!
-
- The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For
- example,
-
- mpz_t arr[20000];
- mpz_array_init (arr[0], 20000, 512);
-
- This function is only intended for programs that create a large
- number of integers and need to reduce memory usage by avoiding the
- overheads of allocating and reallocating lots of small blocks. In
- normal programs this function is not recommended.
-
- The space allocated to each integer by this function will not be
- automatically increased, unlike the normal `mpz_init', so an
- application must ensure it is sufficient for any value stored.
- The following space requirements apply to various routines,
-
- * `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
- `mpz_set_ui' need room for the value they store.
-
- * `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
- for the larger of the two operands, plus an extra
- `mp_bits_per_limb'.
-
- * `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
- of the number of bits in their operands, but each rounded up
- to a multiple of `mp_bits_per_limb'.
-
- * `mpz_swap' can be used between two array variables, but not
- between an array and a normal variable.
-
- For other functions, or if in doubt, the suggestion is to
- calculate in a regular `mpz_init' variable and copy the result to
- an array variable with `mpz_set'.
-
- -- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
- Change the space for INTEGER to NEW_ALLOC limbs. The value in
- INTEGER is preserved if it fits, or is set to 0 if not. The return
- value is not useful to applications and should be ignored.
-
- `mpz_realloc2' is the preferred way to accomplish allocation
- changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
- except that `_mpz_realloc' takes its size in limbs.
-
- -- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
- Return limb number N from OP. The sign of OP is ignored, just the
- absolute value is used. The least significant limb is number 0.
-
- `mpz_size' can be used to find how many limbs make up OP.
- `mpz_getlimbn' returns zero if N is outside the range 0 to
- `mpz_size(OP)-1'.
-
- -- Function: size_t mpz_size (mpz_t OP)
- Return the size of OP measured in number of limbs. If OP is zero,
- the returned value will be zero.
-
-\1f
-File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
-
-6 Rational Number Functions
-***************************
-
-This chapter describes the GMP functions for performing arithmetic on
-rational numbers. These functions start with the prefix `mpq_'.
-
- Rational numbers are stored in objects of type `mpq_t'.
-
- All rational arithmetic functions assume operands have a canonical
-form, and canonicalize their result. The canonical from means that the
-denominator and the numerator have no common factors, and that the
-denominator is positive. Zero has the unique representation 0/1.
-
- Pure assignment functions do not canonicalize the assigned variable.
-It is the responsibility of the user to canonicalize the assigned
-variable before any arithmetic operations are performed on that
-variable.
-
- -- Function: void mpq_canonicalize (mpq_t OP)
- Remove any factors that are common to the numerator and
- denominator of OP, and make the denominator positive.
-
-* Menu:
-
-* Initializing Rationals::
-* Rational Conversions::
-* Rational Arithmetic::
-* Comparing Rationals::
-* Applying Integer Functions::
-* I/O of Rationals::
-
-\1f
-File: gmp.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
-
-6.1 Initialization and Assignment Functions
-===========================================
-
- -- Function: void mpq_init (mpq_t X)
- Initialize X and set it to 0/1. Each variable should normally
- only be initialized once, or at least cleared out (using the
- function `mpq_clear') between each initialization.
-
- -- Function: void mpq_inits (mpq_t X, ...)
- Initialize a NULL-terminated list of `mpq_t' variables, and set
- their values to 0/1.
-
- -- Function: void mpq_clear (mpq_t X)
- Free the space occupied by X. Make sure to call this function for
- all `mpq_t' variables when you are done with them.
-
- -- Function: void mpq_clears (mpq_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpq_t'
- variables.
-
- -- Function: void mpq_set (mpq_t ROP, mpq_t OP)
- -- Function: void mpq_set_z (mpq_t ROP, mpz_t OP)
- Assign ROP from OP.
-
- -- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
- unsigned long int OP2)
- -- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
- long int OP2)
- Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
- common factors, ROP has to be passed to `mpq_canonicalize' before
- any operations are performed on ROP.
-
- -- Function: int mpq_set_str (mpq_t ROP, char *STR, int BASE)
- Set ROP from a null-terminated string STR in the given BASE.
-
- The string can be an integer like "41" or a fraction like
- "41/152". The fraction must be in canonical form (*note Rational
- Number Functions::), or if not then `mpq_canonicalize' must be
- called.
-
- The numerator and optional denominator are parsed the same as in
- `mpz_set_str' (*note Assigning Integers::). White space is
- allowed in the string, and is simply ignored. The BASE can vary
- from 2 to 62, or if BASE is 0 then the leading characters are
- used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
- octal, or decimal otherwise. Note that this is done separately
- for the numerator and denominator, so for instance `0xEF/100' is
- 239/100, whereas `0xEF/0x100' is 239/256.
-
- The return value is 0 if the entire string is a valid number, or
- -1 if not.
-
- -- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
- Swap the values ROP1 and ROP2 efficiently.
-
-\1f
-File: gmp.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
-
-6.2 Conversion Functions
-========================
-
- -- Function: double mpq_get_d (mpq_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent from the conversion is too big or too small to fit
- a `double' then the result is system dependent. For too big an
- infinity is returned when available. For too small 0.0 is
- normally returned. Hardware overflow, underflow and denorm traps
- may or may not occur.
-
- -- Function: void mpq_set_d (mpq_t ROP, double OP)
- -- Function: void mpq_set_f (mpq_t ROP, mpf_t OP)
- Set ROP to the value of OP. There is no rounding, this conversion
- is exact.
-
- -- Function: char * mpq_get_str (char *STR, int BASE, mpq_t OP)
- Convert OP to a string of digits in base BASE. The base may vary
- from 2 to 36. The string will be of the form `num/den', or if the
- denominator is 1 then just `num'.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of storage large
- enough for the result, that being
-
- mpz_sizeinbase (mpq_numref(OP), BASE)
- + mpz_sizeinbase (mpq_denref(OP), BASE) + 3
-
- The three extra bytes are for a possible minus sign, possible
- slash, and the null-terminator.
-
- A pointer to the result string is returned, being either the
- allocated block, or the given STR.
-
-\1f
-File: gmp.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
-
-6.3 Arithmetic Functions
-========================
-
- -- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
- Set SUM to ADDEND1 + ADDEND2.
-
- -- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
- SUBTRAHEND)
- Set DIFFERENCE to MINUEND - SUBTRAHEND.
-
- -- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
- MULTIPLICAND)
- Set PRODUCT to MULTIPLIER times MULTIPLICAND.
-
- -- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2.
-
- -- Function: void mpq_div (mpq_t QUOTIENT, mpq_t DIVIDEND, mpq_t
- DIVISOR)
- Set QUOTIENT to DIVIDEND/DIVISOR.
-
- -- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 divided by 2 raised to OP2.
-
- -- Function: void mpq_neg (mpq_t NEGATED_OPERAND, mpq_t OPERAND)
- Set NEGATED_OPERAND to -OPERAND.
-
- -- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
- Set ROP to the absolute value of OP.
-
- -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
- Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
- this routine will divide by zero.
-
-\1f
-File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
-
-6.4 Comparison Functions
-========================
-
- -- Function: int mpq_cmp (mpq_t OP1, mpq_t OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- To determine if two rationals are equal, `mpq_equal' is faster than
- `mpq_cmp'.
-
- -- Macro: int mpq_cmp_ui (mpq_t OP1, unsigned long int NUM2, unsigned
- long int DEN2)
- -- Macro: int mpq_cmp_si (mpq_t OP1, long int NUM2, unsigned long int
- DEN2)
- Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
- NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
- NUM2/DEN2.
-
- NUM2 and DEN2 are allowed to have common factors.
-
- These functions are implemented as a macros and evaluate their
- arguments multiple times.
-
- -- Macro: int mpq_sgn (mpq_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates its
- arguments multiple times.
-
- -- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
- Return non-zero if OP1 and OP2 are equal, zero if they are
- non-equal. Although `mpq_cmp' can be used for the same purpose,
- this function is much faster.
-
-\1f
-File: gmp.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
-
-6.5 Applying Integer Functions to Rationals
-===========================================
-
-The set of `mpq' functions is quite small. In particular, there are few
-functions for either input or output. The following functions give
-direct access to the numerator and denominator of an `mpq_t'.
-
- Note that if an assignment to the numerator and/or denominator could
-take an `mpq_t' out of the canonical form described at the start of
-this chapter (*note Rational Number Functions::) then
-`mpq_canonicalize' must be called before any other `mpq' functions are
-applied to that `mpq_t'.
-
- -- Macro: mpz_t mpq_numref (mpq_t OP)
- -- Macro: mpz_t mpq_denref (mpq_t OP)
- Return a reference to the numerator and denominator of OP,
- respectively. The `mpz' functions can be used on the result of
- these macros.
-
- -- Function: void mpq_get_num (mpz_t NUMERATOR, mpq_t RATIONAL)
- -- Function: void mpq_get_den (mpz_t DENOMINATOR, mpq_t RATIONAL)
- -- Function: void mpq_set_num (mpq_t RATIONAL, mpz_t NUMERATOR)
- -- Function: void mpq_set_den (mpq_t RATIONAL, mpz_t DENOMINATOR)
- Get or set the numerator or denominator of a rational. These
- functions are equivalent to calling `mpz_set' with an appropriate
- `mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
- `mpq_denref' is recommended instead of these functions.
-
-\1f
-File: gmp.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
-
-6.6 Input and Output Functions
-==============================
-
-When using any of these functions, it's a good idea to include `stdio.h'
-before `gmp.h', since that will allow `gmp.h' to define prototypes for
-these functions.
-
- Passing a `NULL' pointer for a STREAM argument to any of these
-functions will make them read from `stdin' and write to `stdout',
-respectively.
-
- -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
- Output OP on stdio stream STREAM, as a string of digits in base
- BASE. The base may vary from 2 to 36. Output is in the form
- `num/den' or if the denominator is 1 then just `num'.
-
- Return the number of bytes written, or if an error occurred,
- return 0.
-
- -- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
- Read a string of digits from STREAM and convert them to a rational
- in ROP. Any initial white-space characters are read and
- discarded. Return the number of characters read (including white
- space), or 0 if a rational could not be read.
-
- The input can be a fraction like `17/63' or just an integer like
- `123'. Reading stops at the first character not in this form, and
- white space is not permitted within the string. If the input
- might not be in canonical form, then `mpq_canonicalize' must be
- called (*note Rational Number Functions::).
-
- The BASE can be between 2 and 36, or can be 0 in which case the
- leading characters of the string determine the base, `0x' or `0X'
- for hexadecimal, `0' for octal, or decimal otherwise. The leading
- characters are examined separately for the numerator and
- denominator of a fraction, so for instance `0x10/11' is 16/11,
- whereas `0x10/0x11' is 16/17.
-
-\1f
-File: gmp.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
-
-7 Floating-point Functions
-**************************
-
-GMP floating point numbers are stored in objects of type `mpf_t' and
-functions operating on them have an `mpf_' prefix.
-
- The mantissa of each float has a user-selectable precision, limited
-only by available memory. Each variable has its own precision, and
-that can be increased or decreased at any time.
-
- The exponent of each float is a fixed precision, one machine word on
-most systems. In the current implementation the exponent is a count of
-limbs, so for example on a 32-bit system this means a range of roughly
-2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
-greater. Note however `mpf_get_str' can only return an exponent which
-fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
-bigger than a `long'.
-
- Each variable keeps a size for the mantissa data actually in use.
-This means that if a float is exactly represented in only a few bits
-then only those bits will be used in a calculation, even if the
-selected precision is high.
-
- All calculations are performed to the precision of the destination
-variable. Each function is defined to calculate with "infinite
-precision" followed by a truncation to the destination precision, but
-of course the work done is only what's needed to determine a result
-under that definition.
-
- The precision selected for a variable is a minimum value, GMP may
-increase it a little to facilitate efficient calculation. Currently
-this means rounding up to a whole limb, and then sometimes having a
-further partial limb, depending on the high limb of the mantissa. But
-applications shouldn't be concerned by such details.
-
- The mantissa in stored in binary, as might be imagined from the fact
-precisions are expressed in bits. One consequence of this is that
-decimal fractions like 0.1 cannot be represented exactly. The same is
-true of plain IEEE `double' floats. This makes both highly unsuitable
-for calculations involving money or other values that should be exact
-decimal fractions. (Suitably scaled integers, or perhaps rationals,
-are better choices.)
-
- `mpf' functions and variables have no special notion of infinity or
-not-a-number, and applications must take care not to overflow the
-exponent or results will be unpredictable. This might change in a
-future release.
-
- Note that the `mpf' functions are _not_ intended as a smooth
-extension to IEEE P754 arithmetic. In particular results obtained on
-one computer often differ from the results on a computer with a
-different word size.
-
-* Menu:
-
-* Initializing Floats::
-* Assigning Floats::
-* Simultaneous Float Init & Assign::
-* Converting Floats::
-* Float Arithmetic::
-* Float Comparison::
-* I/O of Floats::
-* Miscellaneous Float Functions::
-
-\1f
-File: gmp.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
-
-7.1 Initialization Functions
-============================
-
- -- Function: void mpf_set_default_prec (mp_bitcnt_t PREC)
- Set the default precision to be *at least* PREC bits. All
- subsequent calls to `mpf_init' will use this precision, but
- previously initialized variables are unaffected.
-
- -- Function: mp_bitcnt_t mpf_get_default_prec (void)
- Return the default precision actually used.
-
- An `mpf_t' object must be initialized before storing the first value
-in it. The functions `mpf_init' and `mpf_init2' are used for that
-purpose.
-
- -- Function: void mpf_init (mpf_t X)
- Initialize X to 0. Normally, a variable should be initialized
- once only or at least be cleared, using `mpf_clear', between
- initializations. The precision of X is undefined unless a default
- precision has already been established by a call to
- `mpf_set_default_prec'.
-
- -- Function: void mpf_init2 (mpf_t X, mp_bitcnt_t PREC)
- Initialize X to 0 and set its precision to be *at least* PREC
- bits. Normally, a variable should be initialized once only or at
- least be cleared, using `mpf_clear', between initializations.
-
- -- Function: void mpf_inits (mpf_t X, ...)
- Initialize a NULL-terminated list of `mpf_t' variables, and set
- their values to 0. The precision of the initialized variables is
- undefined unless a default precision has already been established
- by a call to `mpf_set_default_prec'.
-
- -- Function: void mpf_clear (mpf_t X)
- Free the space occupied by X. Make sure to call this function for
- all `mpf_t' variables when you are done with them.
-
- -- Function: void mpf_clears (mpf_t X, ...)
- Free the space occupied by a NULL-terminated list of `mpf_t'
- variables.
-
- Here is an example on how to initialize floating-point variables:
- {
- mpf_t x, y;
- mpf_init (x); /* use default precision */
- mpf_init2 (y, 256); /* precision _at least_ 256 bits */
- ...
- /* Unless the program is about to exit, do ... */
- mpf_clear (x);
- mpf_clear (y);
- }
-
- The following three functions are useful for changing the precision
-during a calculation. A typical use would be for adjusting the
-precision gradually in iterative algorithms like Newton-Raphson, making
-the computation precision closely match the actual accurate part of the
-numbers.
-
- -- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP)
- Return the current precision of OP, in bits.
-
- -- Function: void mpf_set_prec (mpf_t ROP, mp_bitcnt_t PREC)
- Set the precision of ROP to be *at least* PREC bits. The value in
- ROP will be truncated to the new precision.
-
- This function requires a call to `realloc', and so should not be
- used in a tight loop.
-
- -- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC)
- Set the precision of ROP to be *at least* PREC bits, without
- changing the memory allocated.
-
- PREC must be no more than the allocated precision for ROP, that
- being the precision when ROP was initialized, or in the most recent
- `mpf_set_prec'.
-
- The value in ROP is unchanged, and in particular if it had a higher
- precision than PREC it will retain that higher precision. New
- values written to ROP will use the new PREC.
-
- Before calling `mpf_clear' or the full `mpf_set_prec', another
- `mpf_set_prec_raw' call must be made to restore ROP to its original
- allocated precision. Failing to do so will have unpredictable
- results.
-
- `mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
- original allocated precision. After `mpf_set_prec_raw' it
- reflects the PREC value set.
-
- `mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
- at different precisions during a calculation, perhaps to gradually
- increase precision in an iteration, or just to use various
- different precisions for different purposes during a calculation.
-
-\1f
-File: gmp.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
-
-7.2 Assignment Functions
-========================
-
-These functions assign new values to already initialized floats (*note
-Initializing Floats::).
-
- -- Function: void mpf_set (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
- -- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
- -- Function: void mpf_set_d (mpf_t ROP, double OP)
- -- Function: void mpf_set_z (mpf_t ROP, mpz_t OP)
- -- Function: void mpf_set_q (mpf_t ROP, mpq_t OP)
- Set the value of ROP from OP.
-
- -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
- Set the value of ROP from the string in STR. The string is of the
- form `M@N' or, if the base is 10 or less, alternatively `MeN'.
- `M' is the mantissa and `N' is the exponent. The mantissa is
- always in the specified base. The exponent is either in the
- specified base or, if BASE is negative, in decimal. The decimal
- point expected is taken from the current locale, on systems
- providing `localeconv'.
-
- The argument BASE may be in the ranges 2 to 62, or -62 to -2.
- Negative values are used to specify that the exponent is in
- decimal.
-
- For bases up to 36, case is ignored; upper-case and lower-case
- letters have the same value; for bases 37 to 62, upper-case letter
- represent the usual 10..35 while lower-case letter represent
- 36..61.
-
- Unlike the corresponding `mpz' function, the base will not be
- determined from the leading characters of the string if BASE is 0.
- This is so that numbers like `0.23' are not interpreted as octal.
-
- White space is allowed in the string, and is simply ignored.
- [This is not really true; white-space is ignored in the beginning
- of the string and within the mantissa, but not in other places,
- such as after a minus sign or in the exponent. We are considering
- changing the definition of this function, making it fail when
- there is any white-space in the input, since that makes a lot of
- sense. Please tell us your opinion about this change. Do you
- really want it to accept "3 14" as meaning 314 as it does now?]
-
- This function returns 0 if the entire string is a valid number in
- base BASE. Otherwise it returns -1.
-
- -- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
- Swap ROP1 and ROP2 efficiently. Both the values and the
- precisions of the two variables are swapped.
-
-\1f
-File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
-
-7.3 Combined Initialization and Assignment Functions
-====================================================
-
-For convenience, GMP provides a parallel series of initialize-and-set
-functions which initialize the output and then store the value there.
-These functions' names have the form `mpf_init_set...'
-
- Once the float has been initialized by any of the `mpf_init_set...'
-functions, it can be used as the source or destination operand for the
-ordinary float functions. Don't use an initialize-and-set function on
-a variable already initialized!
-
- -- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
- -- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
- -- Function: void mpf_init_set_d (mpf_t ROP, double OP)
- Initialize ROP and set its value from OP.
-
- The precision of ROP will be taken from the active default
- precision, as set by `mpf_set_default_prec'.
-
- -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
- Initialize ROP and set its value from the string in STR. See
- `mpf_set_str' above for details on the assignment operation.
-
- Note that ROP is initialized even if an error occurs. (I.e., you
- have to call `mpf_clear' for it.)
-
- The precision of ROP will be taken from the active default
- precision, as set by `mpf_set_default_prec'.
-
-\1f
-File: gmp.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
-
-7.4 Conversion Functions
-========================
-
- -- Function: double mpf_get_d (mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero).
-
- If the exponent in OP is too big or too small to fit a `double'
- then the result is system dependent. For too big an infinity is
- returned when available. For too small 0.0 is normally returned.
- Hardware overflow, underflow and denorm traps may or may not occur.
-
- -- Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
- towards zero), and with an exponent returned separately.
-
- The return value is in the range 0.5<=abs(D)<1 and the exponent is
- stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
- is zero, the return is 0.0 and 0 is stored to `*EXP'.
-
- This is similar to the standard C `frexp' function (*note
- Normalization Functions: (libc)Normalization Functions.).
-
- -- Function: long mpf_get_si (mpf_t OP)
- -- Function: unsigned long mpf_get_ui (mpf_t OP)
- Convert OP to a `long' or `unsigned long', truncating any fraction
- part. If OP is too big for the return type, the result is
- undefined.
-
- See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
- Miscellaneous Float Functions::).
-
- -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
- BASE, size_t N_DIGITS, mpf_t OP)
- Convert OP to a string of digits in base BASE. The base argument
- may vary from 2 to 62 or from -2 to -36. Up to N_DIGITS digits
- will be generated. Trailing zeros are not returned. No more
- digits than can be accurately represented by OP are ever
- generated. If N_DIGITS is 0 then that accurate maximum number of
- digits are generated.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- If STR is `NULL', the result string is allocated using the current
- allocation function (*note Custom Allocation::). The block will be
- `strlen(str)+1' bytes, that being exactly enough for the string and
- null-terminator.
-
- If STR is not `NULL', it should point to a block of N_DIGITS + 2
- bytes, that being enough for the mantissa, a possible minus sign,
- and a null-terminator. When N_DIGITS is 0 to get all significant
- digits, an application won't be able to know the space required,
- and STR should be `NULL' in that case.
-
- The generated string is a fraction, with an implicit radix point
- immediately to the left of the first digit. The applicable
- exponent is written through the EXPPTR pointer. For example, the
- number 3.1416 would be returned as string "31416" and exponent 1.
-
- When OP is zero, an empty string is produced and the exponent
- returned is 0.
-
- A pointer to the result string is returned, being either the
- allocated block or the given STR.
-
-\1f
-File: gmp.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
-
-7.5 Arithmetic Functions
-========================
-
- -- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 + OP2.
-
- -- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
- OP2)
- -- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 - OP2.
-
- -- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 times OP2.
-
- Division is undefined if the divisor is zero, and passing a zero
-divisor to the divide functions will make these functions intentionally
-divide by zero. This lets the user handle arithmetic exceptions in
-these functions in the same manner as other arithmetic exceptions.
-
- -- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- -- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
- OP2)
- -- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1/OP2.
-
- -- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
- Set ROP to the square root of OP.
-
- -- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
- OP2)
- Set ROP to OP1 raised to the power OP2.
-
- -- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
- Set ROP to -OP.
-
- -- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
- Set ROP to the absolute value of OP.
-
- -- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 times 2 raised to OP2.
-
- -- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, mp_bitcnt_t OP2)
- Set ROP to OP1 divided by 2 raised to OP2.
-
-\1f
-File: gmp.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
-
-7.6 Comparison Functions
-========================
-
- -- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
- -- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
- -- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
- -- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- `mpf_cmp_d' can be called with an infinity, but results are
- undefined for a NaN.
-
- -- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, mp_bitcnt_t op3)
- Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
- zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
-
- Caution 1: All version of GMP up to version 4.2.4 compared just
- whole limbs, meaning sometimes more than OP3 bits, sometimes fewer.
-
- Caution 2: This function will consider XXX11...111 and XX100...000
- different, even if ... is replaced by a semi-infinite number of
- bits. Such numbers are really just one ulp off, and should be
- considered equal.
-
- -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
- Compute the relative difference between OP1 and OP2 and store the
- result in ROP. This is abs(OP1-OP2)/OP1.
-
- -- Macro: int mpf_sgn (mpf_t OP)
- Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
-
- This function is actually implemented as a macro. It evaluates
- its arguments multiple times.
-
-\1f
-File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
-
-7.7 Input and Output Functions
-==============================
-
-Functions that perform input from a stdio stream, and functions that
-output to a stdio stream. Passing a `NULL' pointer for a STREAM
-argument to any of these functions will make them read from `stdin' and
-write to `stdout', respectively.
-
- When using any of these functions, it is a good idea to include
-`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
-prototypes for these functions.
-
- -- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
- N_DIGITS, mpf_t OP)
- Print OP to STREAM, as a string of digits. Return the number of
- bytes written, or if an error occurred, return 0.
-
- The mantissa is prefixed with an `0.' and is in the given BASE,
- which may vary from 2 to 62 or from -2 to -36. An exponent is
- then printed, separated by an `e', or if the base is greater than
- 10 then by an `@'. The exponent is always in decimal. The
- decimal point follows the current locale, on systems providing
- `localeconv'.
-
- For BASE in the range 2..36, digits and lower-case letters are
- used; for -2..-36, digits and upper-case letters are used; for
- 37..62, digits, upper-case letters, and lower-case letters (in
- that significance order) are used.
-
- Up to N_DIGITS will be printed from the mantissa, except that no
- more digits than are accurately representable by OP will be
- printed. N_DIGITS can be 0 to select that accurate maximum.
-
- -- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
- Read a string in base BASE from STREAM, and put the read float in
- ROP. The string is of the form `M@N' or, if the base is 10 or
- less, alternatively `MeN'. `M' is the mantissa and `N' is the
- exponent. The mantissa is always in the specified base. The
- exponent is either in the specified base or, if BASE is negative,
- in decimal. The decimal point expected is taken from the current
- locale, on systems providing `localeconv'.
-
- The argument BASE may be in the ranges 2 to 36, or -36 to -2.
- Negative values are used to specify that the exponent is in
- decimal.
-
- Unlike the corresponding `mpz' function, the base will not be
- determined from the leading characters of the string if BASE is 0.
- This is so that numbers like `0.23' are not interpreted as octal.
-
- Return the number of bytes read, or if an error occurred, return 0.
-
-\1f
-File: gmp.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
-
-7.8 Miscellaneous Functions
-===========================
-
- -- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
- -- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
- Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
- next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
- to the integer towards zero.
-
- -- Function: int mpf_integer_p (mpf_t OP)
- Return non-zero if OP is an integer.
-
- -- Function: int mpf_fits_ulong_p (mpf_t OP)
- -- Function: int mpf_fits_slong_p (mpf_t OP)
- -- Function: int mpf_fits_uint_p (mpf_t OP)
- -- Function: int mpf_fits_sint_p (mpf_t OP)
- -- Function: int mpf_fits_ushort_p (mpf_t OP)
- -- Function: int mpf_fits_sshort_p (mpf_t OP)
- Return non-zero if OP would fit in the respective C data type, when
- truncated to an integer.
-
- -- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
- mp_bitcnt_t NBITS)
- Generate a uniformly distributed random float in ROP, such that 0
- <= ROP < 1, with NBITS significant bits in the mantissa.
-
- The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
- before invoking this function.
-
- -- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
- EXP)
- Generate a random float of at most MAX_SIZE limbs, with long
- strings of zeros and ones in the binary representation. The
- exponent of the number is in the interval -EXP to EXP (in limbs).
- This function is useful for testing functions and algorithms,
- since these kind of random numbers have proven to be more likely
- to trigger corner-case bugs. Negative random numbers are
- generated when MAX_SIZE is negative.
-
-\1f
-File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
-
-8 Low-level Functions
-*********************
-
-This chapter describes low-level GMP functions, used to implement the
-high-level GMP functions, but also intended for time-critical user code.
-
- These functions start with the prefix `mpn_'.
-
- The `mpn' functions are designed to be as fast as possible, *not* to
-provide a coherent calling interface. The different functions have
-somewhat similar interfaces, but there are variations that make them
-hard to use. These functions do as little as possible apart from the
-real multiple precision computation, so that no time is spent on things
-that not all callers need.
-
- A source operand is specified by a pointer to the least significant
-limb and a limb count. A destination operand is specified by just a
-pointer. It is the responsibility of the caller to ensure that the
-destination has enough space for storing the result.
-
- With this way of specifying operands, it is possible to perform
-computations on subranges of an argument, and store the result into a
-subrange of a destination.
-
- A common requirement for all functions is that each source area
-needs at least one limb. No size argument may be zero. Unless
-otherwise stated, in-place operations are allowed where source and
-destination are the same, but not where they only partly overlap.
-
- The `mpn' functions are the base for the implementation of the
-`mpz_', `mpf_', and `mpq_' functions.
-
- This example adds the number beginning at S1P and the number
-beginning at S2P and writes the sum at DESTP. All areas have N limbs.
-
- cy = mpn_add_n (destp, s1p, s2p, n)
-
- It should be noted that the `mpn' functions make no attempt to
-identify high or low zero limbs on their operands, or other special
-forms. On random data such cases will be unlikely and it'd be wasteful
-for every function to check every time. An application knowing
-something about its data can take steps to trim or perhaps split its
-calculations.
-
-
-In the notation used below, a source operand is identified by the
-pointer to the least significant limb, and the limb count in braces.
-For example, {S1P, S1N}.
-
- -- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Add {S1P, N} and {S2P, N}, and write the N least significant limbs
- of the result to RP. Return carry, either 0 or 1.
-
- This is the lowest-level function for addition. It is the
- preferred function for addition, since it is written in assembly
- for most CPUs. For addition of a variable to itself (i.e., S1P
- equals S2P) use `mpn_lshift' with a count of 1 for optimal speed.
-
- -- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Add {S1P, N} and S2LIMB, and write the N least significant limbs
- of the result to RP. Return carry, either 0 or 1.
-
- -- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
- limbs of the result to RP. Return carry, either 0 or 1.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Subtract {S2P, N} from {S1P, N}, and write the N least significant
- limbs of the result to RP. Return borrow, either 0 or 1.
-
- This is the lowest-level function for subtraction. It is the
- preferred function for subtraction, since it is written in
- assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Subtract S2LIMB from {S1P, N}, and write the N least significant
- limbs of the result to RP. Return borrow, either 0 or 1.
-
- -- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
- significant limbs of the result to RP. Return borrow, either 0 or
- 1.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: void mpn_neg (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N)
- Perform the negation of {SP, N}, and write the result to {RP, N}.
- Return carry-out.
-
- -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
- RP.
-
- The destination has to have space for 2*N limbs, even if the
- product's most significant limb is zero. No overlap is permitted
- between the destination and either source.
-
- If the two input operands are the same, use `mpn_sqr'.
-
- -- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
- Multiply {S1P, S1N} and {S2P, S2N}, and write the (S1N+S2N)-limb
- result to RP. Return the most significant limb of the result.
-
- The destination has to have space for S1N + S2N limbs, even if the
- product's most significant limb is zero. No overlap is permitted
- between the destination and either source.
-
- This function requires that S1N is greater than or equal to S2N.
-
- -- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Compute the square of {S1P, N} and write the 2*N-limb result to RP.
-
- The destination has to have space for 2*N limbs, even if the
- result's most significant limb is zero. No overlap is permitted
- between the destination and the source.
-
- -- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} by S2LIMB, and write the N least significant
- limbs of the product to RP. Return the most significant limb of
- the product. {S1P, N} and {RP, N} are allowed to overlap provided
- RP <= S1P.
-
- This is a low-level function that is a building block for general
- multiplication as well as other operations in GMP. It is written
- in assembly for most CPUs.
-
- Don't call this function if S2LIMB is a power of 2; use
- `mpn_lshift' with a count equal to the logarithm of S2LIMB
- instead, for optimal speed.
-
- -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
- *S1P, mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} and S2LIMB, and add the N least significant
- limbs of the product to {RP, N} and write the result to RP.
- Return the most significant limb of the product, plus carry-out
- from the addition.
-
- This is a low-level function that is a building block for general
- multiplication as well as other operations in GMP. It is written
- in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
- *S1P, mp_size_t N, mp_limb_t S2LIMB)
- Multiply {S1P, N} and S2LIMB, and subtract the N least significant
- limbs of the product from {RP, N} and write the result to RP.
- Return the most significant limb of the product, plus borrow-out
- from the subtraction.
-
- This is a low-level function that is a building block for general
- multiplication and division as well as other operations in GMP.
- It is written in assembly for most CPUs.
-
- -- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
- QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
- mp_size_t DN)
- Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
- and the remainder at {RP, DN}. The quotient is rounded towards 0.
-
- No overlap is permitted between arguments, except that NP might
- equal RP. The dividend size NN must be greater than or equal to
- divisor size DN. The most significant limb of the divisor must be
- non-zero. The QXN operand must be zero.
-
- -- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
- mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
- mp_size_t S3N)
- [This function is obsolete. Please call `mpn_tdiv_qr' instead for
- best performance.]
-
- Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
- with the exception of the most significant limb, which is
- returned. The remainder replaces the dividend at RS2P; it will be
- S3N limbs long (i.e., as many limbs as the divisor).
-
- In addition to an integer quotient, QXN fraction limbs are
- developed, and stored after the integral limbs. For most usages,
- QXN will be zero.
-
- It is required that RS2N is greater than or equal to S3N. It is
- required that the most significant bit of the divisor is set.
-
- If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
- that special case, no overlap between arguments is permitted.
-
- Return the most significant limb of the quotient, either 0 or 1.
-
- The area at R1P needs to be RS2N - S3N + QXN limbs large.
-
- -- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
- mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
- -- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
- mp_size_t S2N, mp_limb_t S3LIMB)
- Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P.
- Return the remainder.
-
- The integer quotient is written to {R1P+QXN, S2N} and in addition
- QXN fraction limbs are developed and written to {R1P, QXN}.
- Either or both S2N and QXN can be zero. For most usages, QXN will
- be zero.
-
- `mpn_divmod_1' exists for upward source compatibility and is
- simply a macro calling `mpn_divrem_1' with a QXN of 0.
-
- The areas at R1P and S2P have to be identical or completely
- separate, not partially overlapping.
-
- -- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
- mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
- [This function is obsolete. Please call `mpn_tdiv_qr' instead for
- best performance.]
-
- -- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
- mp_size_t N)
- -- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t
- *SP, mp_size_t N, mp_limb_t CARRY)
- Divide {SP, N} by 3, expecting it to divide exactly, and writing
- the result to {RP, N}. If 3 divides exactly, the return value is
- zero and the result is the quotient. If not, the return value is
- non-zero and the result won't be anything useful.
-
- `mpn_divexact_by3c' takes an initial carry parameter, which can be
- the return value from a previous call, so a large calculation can
- be done piece by piece from low to high. `mpn_divexact_by3' is
- simply a macro calling `mpn_divexact_by3c' with a 0 carry
- parameter.
-
- These routines use a multiply-by-inverse and will be faster than
- `mpn_divrem_1' on CPUs with fast multiplication but slow division.
-
- The source a, result q, size n, initial carry i, and return value
- c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
- c is always 0, 1 or 2, and the initial carry i must also be 0, 1
- or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
- When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b
- == 1 mod 3 (when `mp_bits_per_limb' is even, which is always so
- currently).
-
- -- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
- mp_limb_t S2LIMB)
- Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
- zero.
-
- -- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N, unsigned int COUNT)
- Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
- The bits shifted out at the left are returned in the least
- significant COUNT bits of the return value (the rest of the return
- value is zero).
-
- COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
- {SP, N} and {RP, N} may overlap, provided RP >= SP.
-
- This function is written in assembly for most CPUs.
-
- -- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N, unsigned int COUNT)
- Shift {SP, N} right by COUNT bits, and write the result to {RP,
- N}. The bits shifted out at the right are returned in the most
- significant COUNT bits of the return value (the rest of the return
- value is zero).
-
- COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
- {SP, N} and {RP, N} may overlap, provided RP <= SP.
-
- This function is written in assembly for most CPUs.
-
- -- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
- mp_size_t N)
- Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
- S2, 0 if they are equal, or a negative value if S1 < S2.
-
- -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *XP,
- mp_size_t XN, mp_limb_t *YP, mp_size_t YN)
- Set {RP, RETVAL} to the greatest common divisor of {XP, XN} and
- {YP, YN}. The result can be up to YN limbs, the return value is
- the actual number produced. Both source operands are destroyed.
-
- {XP, XN} must have at least as many bits as {YP, YN}. {YP, YN}
- must be odd. Both operands must have non-zero most significant
- limbs. No overlap is permitted between {XP, XN} and {YP, YN}.
-
- -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *XP, mp_size_t XN,
- mp_limb_t YLIMB)
- Return the greatest common divisor of {XP, XN} and YLIMB. Both
- operands must be non-zero.
-
- -- Function: mp_size_t mpn_gcdext (mp_limb_t *GP, mp_limb_t *SP,
- mp_size_t *SN, mp_limb_t *XP, mp_size_t XN, mp_limb_t *YP,
- mp_size_t YN)
- Let U be defined by {XP, XN} and let V be defined by {YP, YN}.
-
- Compute the greatest common divisor G of U and V. Compute a
- cofactor S such that G = US + VT. The second cofactor T is not
- computed but can easily be obtained from (G - U*S) / V (the
- division will be exact). It is required that U >= V > 0.
-
- S satisfies S = 1 or abs(S) < V / (2 G). S = 0 if and only if V
- divides U (i.e., G = V).
-
- Store G at GP and let the return value define its limb count.
- Store S at SP and let |*SN| define its limb count. S can be
- negative; when this happens *SN will be negative. The areas at GP
- and SP should each have room for XN+1 limbs.
-
- The areas {XP, XN+1} and {YP, YN+1} are destroyed (i.e. the input
- operands plus an extra limb past the end of each).
-
- Compatibility note: GMP 4.3.0 and 4.3.1 defined S less strictly.
- Earlier as well as later GMP releases define S as described here.
-
- -- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
- const mp_limb_t *SP, mp_size_t N)
- Compute the square root of {SP, N} and put the result at {R1P,
- ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space
- for N limbs, but the return value indicates how many are produced.
-
- The most significant limb of {SP, N} must be non-zero. The areas
- {R1P, ceil(N/2)} and {SP, N} must be completely separate. The
- areas {R2P, N} and {SP, N} must be either identical or completely
- separate.
-
- If the remainder is not wanted then R2P can be `NULL', and in this
- case the return value is zero or non-zero according to whether the
- remainder would have been zero or non-zero.
-
- A return value of zero indicates a perfect square. See also
- `mpz_perfect_square_p'.
-
- -- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
- mp_limb_t *S1P, mp_size_t S1N)
- Convert {S1P, S1N} to a raw unsigned char array at STR in base
- BASE, and return the number of characters produced. There may be
- leading zeros in the string. The string is not in ASCII; to
- convert it to printable format, add the ASCII codes for `0' or
- `A', depending on the base and range. BASE can vary from 2 to 256.
-
- The most significant limb of the input {S1P, S1N} must be
- non-zero. The input {S1P, S1N} is clobbered, except when BASE is
- a power of 2, in which case it's unchanged.
-
- The area at STR has to have space for the largest possible number
- represented by a S1N long limb array, plus one extra character.
-
- -- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
- *STR, size_t STRSIZE, int BASE)
- Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
-
- STR[0] is the most significant byte and STR[STRSIZE-1] is the
- least significant. Each byte should be a value in the range 0 to
- BASE-1, not an ASCII character. BASE can vary from 2 to 256.
-
- The return value is the number of limbs written to RP. If the most
- significant input byte is non-zero then the high limb at RP will be
- non-zero, and only that exact number of limbs will be required
- there.
-
- If the most significant input byte is zero then there may be high
- zero limbs written to RP and included in the return value.
-
- STRSIZE must be at least 1, and no overlap is permitted between
- {STR,STRSIZE} and the result at RP.
-
- -- Function: mp_bitcnt_t mpn_scan0 (const mp_limb_t *S1P, mp_bitcnt_t
- BIT)
- Scan S1P from bit position BIT for the next clear bit.
-
- It is required that there be a clear bit within the area at S1P at
- or beyond bit position BIT, so that the function has something to
- return.
-
- -- Function: mp_bitcnt_t mpn_scan1 (const mp_limb_t *S1P, mp_bitcnt_t
- BIT)
- Scan S1P from bit position BIT for the next set bit.
-
- It is required that there be a set bit within the area at S1P at or
- beyond bit position BIT, so that the function has something to
- return.
-
- -- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
- -- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
- Generate a random number of length R1N and store it at R1P. The
- most significant limb is always non-zero. `mpn_random' generates
- uniformly distributed limb data, `mpn_random2' generates long
- strings of zeros and ones in the binary representation.
-
- `mpn_random2' is intended for testing the correctness of the `mpn'
- routines.
-
- -- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t
- N)
- Count the number of set bits in {S1P, N}.
-
- -- Function: mp_bitcnt_t mpn_hamdist (const mp_limb_t *S1P, const
- mp_limb_t *S2P, mp_size_t N)
- Compute the hamming distance between {S1P, N} and {S2P, N}, which
- is the number of bit positions where the two operands have
- different bit values.
-
- -- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
- N)
- Return non-zero iff {S1P, N} is a perfect square.
-
- -- Function: void mpn_and_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
- write the result to {RP, N}.
-
- -- Function: void mpn_ior_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
- and write the result to {RP, N}.
-
- -- Function: void mpn_xor_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
- and write the result to {RP, N}.
-
- -- Function: void mpn_andn_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and the bitwise
- complement of {S2P, N}, and write the result to {RP, N}.
-
- -- Function: void mpn_iorn_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and the
- bitwise complement of {S2P, N}, and write the result to {RP, N}.
-
- -- Function: void mpn_nand_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical and of {S1P, N} and {S2P, N}, and
- write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_nior_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
- and write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_xnor_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
- Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
- and write the bitwise complement of the result to {RP, N}.
-
- -- Function: void mpn_com (mp_limb_t *RP, const mp_limb_t *SP,
- mp_size_t N)
- Perform the bitwise complement of {SP, N}, and write the result to
- {RP, N}.
-
- -- Function: void mpn_copyi (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Copy from {S1P, N} to {RP, N}, increasingly.
-
- -- Function: void mpn_copyd (mp_limb_t *RP, const mp_limb_t *S1P,
- mp_size_t N)
- Copy from {S1P, N} to {RP, N}, decreasingly.
-
- -- Function: void mpn_zero (mp_limb_t *RP, mp_size_t N)
- Zero {RP, N}.
-
-
-8.1 Nails
-=========
-
-*Everything in this section is highly experimental and may disappear or
-be subject to incompatible changes in a future version of GMP.*
-
- Nails are an experimental feature whereby a few bits are left unused
-at the top of each `mp_limb_t'. This can significantly improve carry
-handling on some processors.
-
- All the `mpn' functions accepting limb data will expect the nail
-bits to be zero on entry, and will return data with the nails similarly
-all zero. This applies both to limb vectors and to single limb
-arguments.
-
- Nails can be enabled by configuring with `--enable-nails'. By
-default the number of bits will be chosen according to what suits the
-host processor, but a particular number can be selected with
-`--enable-nails=N'.
-
- At the mpn level, a nail build is neither source nor binary
-compatible with a non-nail build, strictly speaking. But programs
-acting on limbs only through the mpn functions are likely to work
-equally well with either build, and judicious use of the definitions
-below should make any program compatible with either build, at the
-source level.
-
- For the higher level routines, meaning `mpz' etc, a nail build
-should be fully source and binary compatible with a non-nail build.
-
- -- Macro: GMP_NAIL_BITS
- -- Macro: GMP_NUMB_BITS
- -- Macro: GMP_LIMB_BITS
- `GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are
- not in use. `GMP_NUMB_BITS' is the number of data bits in a limb.
- `GMP_LIMB_BITS' is the total number of bits in an `mp_limb_t'. In
- all cases
-
- GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
-
- -- Macro: GMP_NAIL_MASK
- -- Macro: GMP_NUMB_MASK
- Bit masks for the nail and number parts of a limb.
- `GMP_NAIL_MASK' is 0 when nails are not in use.
-
- `GMP_NAIL_MASK' is not often needed, since the nail part can be
- obtained with `x >> GMP_NUMB_BITS', and that means one less large
- constant, which can help various RISC chips.
-
- -- Macro: GMP_NUMB_MAX
- The maximum value that can be stored in the number part of a limb.
- This is the same as `GMP_NUMB_MASK', but can be used for clarity
- when doing comparisons rather than bit-wise operations.
-
- The term "nails" comes from finger or toe nails, which are at the
-ends of a limb (arm or leg). "numb" is short for number, but is also
-how the developers felt after trying for a long time to come up with
-sensible names for these things.
-
- In the future (the distant future most likely) a non-zero nail might
-be permitted, giving non-unique representations for numbers in a limb
-vector. This would help vector processors since carries would only
-ever need to propagate one or two limbs.
-
-\1f
-File: gmp.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
-
-9 Random Number Functions
-*************************
-
-Sequences of pseudo-random numbers in GMP are generated using a
-variable of type `gmp_randstate_t', which holds an algorithm selection
-and a current state. Such a variable must be initialized by a call to
-one of the `gmp_randinit' functions, and can be seeded with one of the
-`gmp_randseed' functions.
-
- The functions actually generating random numbers are described in
-*Note Integer Random Numbers::, and *Note Miscellaneous Float
-Functions::.
-
- The older style random number functions don't accept a
-`gmp_randstate_t' parameter but instead share a global variable of that
-type. They use a default algorithm and are currently not seeded
-(though perhaps that will change in the future). The new functions
-accepting a `gmp_randstate_t' are recommended for applications that
-care about randomness.
-
-* Menu:
-
-* Random State Initialization::
-* Random State Seeding::
-* Random State Miscellaneous::
-
-\1f
-File: gmp.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
-
-9.1 Random State Initialization
-===============================
-
- -- Function: void gmp_randinit_default (gmp_randstate_t STATE)
- Initialize STATE with a default algorithm. This will be a
- compromise between speed and randomness, and is recommended for
- applications with no special requirements. Currently this is
- `gmp_randinit_mt'.
-
- -- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
- Initialize STATE for a Mersenne Twister algorithm. This algorithm
- is fast and has good randomness properties.
-
- -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t
- A, unsigned long C, mp_bitcnt_t M2EXP)
- Initialize STATE with a linear congruential algorithm X = (A*X +
- C) mod 2^M2EXP.
-
- The low bits of X in this algorithm are not very random. The least
- significant bit will have a period no more than 2, and the second
- bit no more than 4, etc. For this reason only the high half of
- each X is actually used.
-
- When a random number of more than M2EXP/2 bits is to be generated,
- multiple iterations of the recurrence are used and the results
- concatenated.
-
- -- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
- mp_bitcnt_t SIZE)
- Initialize STATE for a linear congruential algorithm as per
- `gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
- chosen so that SIZE bits (or more) of each X will be used, ie.
- M2EXP/2 >= SIZE.
-
- If successful the return value is non-zero. If SIZE is bigger
- than the table data provides then the return value is zero. The
- maximum SIZE currently supported is 128.
-
- -- Function: void gmp_randinit_set (gmp_randstate_t ROP,
- gmp_randstate_t OP)
- Initialize ROP with a copy of the algorithm and state from OP.
-
- -- Function: void gmp_randinit (gmp_randstate_t STATE,
- gmp_randalg_t ALG, ...)
- *This function is obsolete.*
-
- Initialize STATE with an algorithm selected by ALG. The only
- choice is `GMP_RAND_ALG_LC', which is `gmp_randinit_lc_2exp_size'
- described above. A third parameter of type `unsigned long' is
- required, this is the SIZE for that function.
- `GMP_RAND_ALG_DEFAULT' or 0 are the same as `GMP_RAND_ALG_LC'.
-
- `gmp_randinit' sets bits in the global variable `gmp_errno' to
- indicate an error. `GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is
- unsupported, or `GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter
- is too big. It may be noted this error reporting is not thread
- safe (a good reason to use `gmp_randinit_lc_2exp_size' instead).
-
- -- Function: void gmp_randclear (gmp_randstate_t STATE)
- Free all memory occupied by STATE.
-
-\1f
-File: gmp.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
-
-9.2 Random State Seeding
-========================
-
- -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
- -- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
- unsigned long int SEED)
- Set an initial seed value into STATE.
-
- The size of a seed determines how many different sequences of
- random numbers that it's possible to generate. The "quality" of
- the seed is the randomness of a given seed compared to the
- previous seed used, and this affects the randomness of separate
- number sequences. The method for choosing a seed is critical if
- the generated numbers are to be used for important applications,
- such as generating cryptographic keys.
-
- Traditionally the system time has been used to seed, but care
- needs to be taken with this. If an application seeds often and
- the resolution of the system clock is low, then the same sequence
- of numbers might be repeated. Also, the system time is quite easy
- to guess, so if unpredictability is required then it should
- definitely not be the only source for the seed value. On some
- systems there's a special device `/dev/random' which provides
- random data better suited for use as a seed.
-
-\1f
-File: gmp.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
-
-9.3 Random State Miscellaneous
-==============================
-
- -- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
- unsigned long N)
- Return a uniformly distributed random number of N bits, ie. in the
- range 0 to 2^N-1 inclusive. N must be less than or equal to the
- number of bits in an `unsigned long'.
-
- -- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE,
- unsigned long N)
- Return a uniformly distributed random number in the range 0 to
- N-1, inclusive.
-
-\1f
-File: gmp.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
-
-10 Formatted Output
-*******************
-
-* Menu:
-
-* Formatted Output Strings::
-* Formatted Output Functions::
-* C++ Formatted Output::
-
-\1f
-File: gmp.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output
-
-10.1 Format Strings
-===================
-
-`gmp_printf' and friends accept format strings similar to the standard C
-`printf' (*note Formatted Output: (libc)Formatted Output.). A format
-specification is of the form
-
- % [flags] [width] [.[precision]] [type] conv
-
- GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
-respectively, `M' for `mp_limb_t', and `N' for an `mp_limb_t' array.
-`Z', `Q', `M' and `N' behave like integers. `Q' will print a `/' and a
-denominator, if needed. `F' behaves like a float. For example,
-
- mpz_t z;
- gmp_printf ("%s is an mpz %Zd\n", "here", z);
-
- mpq_t q;
- gmp_printf ("a hex rational: %#40Qx\n", q);
-
- mpf_t f;
- int n;
- gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
-
- mp_limb_t l;
- gmp_printf ("limb %Mu\n", l);
-
- const mp_limb_t *ptr;
- mp_size_t size;
- gmp_printf ("limb array %Nx\n", ptr, size);
-
- For `N' the limbs are expected least significant first, as per the
-`mpn' functions (*note Low-level Functions::). A negative size can be
-given to print the value as a negative.
-
- All the standard C `printf' types behave the same as the C library
-`printf', and can be freely intermixed with the GMP extensions. In the
-current implementation the standard parts of the format string are
-simply handed to `printf' and only the GMP extensions handled directly.
-
- The flags accepted are as follows. GLIBC style ' is only for the
-standard C types (not the GMP types), and only if the C library
-supports it.
-
- 0 pad with zeros (rather than spaces)
- # show the base with `0x', `0X' or `0'
- + always show a sign
- (space) show a space or a `-' sign
- ' group digits, GLIBC style (not GMP types)
-
- The optional width and precision can be given as a number within the
-format string, or as a `*' to take an extra parameter of type `int', the
-same as the standard `printf'.
-
- The standard types accepted are as follows. `h' and `l' are
-portable, the rest will depend on the compiler (or include files) for
-the type and the C library for the output.
-
- h short
- hh char
- j intmax_t or uintmax_t
- l long or wchar_t
- ll long long
- L long double
- q quad_t or u_quad_t
- t ptrdiff_t
- z size_t
-
-The GMP types are
-
- F mpf_t, float conversions
- Q mpq_t, integer conversions
- M mp_limb_t, integer conversions
- N mp_limb_t array, integer conversions
- Z mpz_t, integer conversions
-
- The conversions accepted are as follows. `a' and `A' are always
-supported for `mpf_t' but depend on the C library for standard C float
-types. `m' and `p' depend on the C library.
-
- a A hex floats, C99 style
- c character
- d decimal integer
- e E scientific format float
- f fixed point float
- i same as d
- g G fixed or scientific float
- m `strerror' string, GLIBC style
- n store characters written so far
- o octal integer
- p pointer
- s string
- u unsigned integer
- x X hex integer
-
- `o', `x' and `X' are unsigned for the standard C types, but for
-types `Z', `Q' and `N' they are signed. `u' is not meaningful for `Z',
-`Q' and `N'.
-
- `M' is a proxy for the C library `l' or `L', according to the size
-of `mp_limb_t'. Unsigned conversions will be usual, but a signed
-conversion can be used and will interpret the value as a twos complement
-negative.
-
- `n' can be used with any type, even the GMP types.
-
- Other types or conversions that might be accepted by the C library
-`printf' cannot be used through `gmp_printf', this includes for
-instance extensions registered with GLIBC `register_printf_function'.
-Also currently there's no support for POSIX `$' style numbered arguments
-(perhaps this will be added in the future).
-
- The precision field has it's usual meaning for integer `Z' and float
-`F' types, but is currently undefined for `Q' and should not be used
-with that.
-
- `mpf_t' conversions only ever generate as many digits as can be
-accurately represented by the operand, the same as `mpf_get_str' does.
-Zeros will be used if necessary to pad to the requested precision. This
-happens even for an `f' conversion of an `mpf_t' which is an integer,
-for instance 2^1024 in an `mpf_t' of 128 bits precision will only
-produce about 40 digits, then pad with zeros to the decimal point. An
-empty precision field like `%.Fe' or `%.Ff' can be used to specifically
-request just the significant digits.
-
- The decimal point character (or string) is taken from the current
-locale settings on systems which provide `localeconv' (*note Locales
-and Internationalization: (libc)Locales.). The C library will normally
-do the same for standard float output.
-
- The format string is only interpreted as plain `char's, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-\1f
-File: gmp.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output
-
-10.2 Functions
-==============
-
-Each of the following functions is similar to the corresponding C
-library function. The basic `printf' forms take a variable argument
-list. The `vprintf' forms take an argument pointer, see *Note Variadic
-Functions: (libc)Variadic Functions, or `man 3 va_start'.
-
- It should be emphasised that if a format string is invalid, or the
-arguments don't match what the format specifies, then the behaviour of
-any of these functions will be unpredictable. GCC format string
-checking is not available, since it doesn't recognise the GMP
-extensions.
-
- The file based functions `gmp_printf' and `gmp_fprintf' will return
--1 to indicate a write error. Output is not "atomic", so partial
-output may be produced if a write error occurs. All the functions can
-return -1 if the C library `printf' variant in use returns -1, but this
-shouldn't normally occur.
-
- -- Function: int gmp_printf (const char *FMT, ...)
- -- Function: int gmp_vprintf (const char *FMT, va_list AP)
- Print to the standard output `stdout'. Return the number of
- characters written, or -1 if an error occurred.
-
- -- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
- -- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
- Print to the stream FP. Return the number of characters written,
- or -1 if an error occurred.
-
- -- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
- -- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
- Form a null-terminated string in BUF. Return the number of
- characters written, excluding the terminating null.
-
- No overlap is permitted between the space at BUF and the string
- FMT.
-
- These functions are not recommended, since there's no protection
- against exceeding the space available at BUF.
-
- -- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char
- *FMT, ...)
- -- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
- *FMT, va_list AP)
- Form a null-terminated string in BUF. No more than SIZE bytes
- will be written. To get the full output, SIZE must be enough for
- the string and null-terminator.
-
- The return value is the total number of characters which ought to
- have been produced, excluding the terminating null. If RETVAL >=
- SIZE then the actual output has been truncated to the first SIZE-1
- characters, and a null appended.
-
- No overlap is permitted between the region {BUF,SIZE} and the FMT
- string.
-
- Notice the return value is in ISO C99 `snprintf' style. This is
- so even if the C library `vsnprintf' is the older GLIBC 2.0.x
- style.
-
- -- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
- -- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP)
- Form a null-terminated string in a block of memory obtained from
- the current memory allocation function (*note Custom
- Allocation::). The block will be the size of the string and
- null-terminator. The address of the block in stored to *PP. The
- return value is the number of characters produced, excluding the
- null-terminator.
-
- Unlike the C library `asprintf', `gmp_asprintf' doesn't return -1
- if there's no more memory available, it lets the current allocation
- function handle that.
-
- -- Function: int gmp_obstack_printf (struct obstack *OB, const char
- *FMT, ...)
- -- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
- *FMT, va_list AP)
- Append to the current object in OB. The return value is the
- number of characters written. A null-terminator is not written.
-
- FMT cannot be within the current object in OB, since that object
- might move as it grows.
-
- These functions are available only when the C library provides the
- obstack feature, which probably means only on GNU systems, see
- *Note Obstacks: (libc)Obstacks.
-
-\1f
-File: gmp.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
-
-10.3 C++ Formatted Output
-=========================
-
-The following functions are provided in `libgmpxx' (*note Headers and
-Libraries::), which is built if C++ support is enabled (*note Build
-Options::). Prototypes are available from `<gmp.h>'.
-
- -- Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- In hex or octal, OP is printed as a signed number, the same as for
- decimal. This is unlike the standard `operator<<' routines on
- `int' etc, which instead give twos complement.
-
- -- Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- Output will be a fraction like `5/9', or if the denominator is 1
- then just a plain integer like `123'.
-
- In hex or octal, OP is printed as a signed value, the same as for
- decimal. If `ios::showbase' is set then a base indicator is shown
- on both the numerator and denominator (if the denominator is
- required).
-
- -- Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
- Print OP to STREAM, using its `ios' formatting settings.
- `ios::width' is reset to 0 after output, the same as the standard
- `ostream operator<<' routines do.
-
- The decimal point follows the standard library float `operator<<',
- which on recent systems means the `std::locale' imbued on STREAM.
-
- Hex and octal are supported, unlike the standard `operator<<' on
- `double'. The mantissa will be in hex or octal, the exponent will
- be in decimal. For hex the exponent delimiter is an `@'. This is
- as per `mpf_out_str'.
-
- `ios::showbase' is supported, and will put a base on the mantissa,
- for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
- This last form is slightly strange, but at least differentiates
- itself from decimal.
-
- These operators mean that GMP types can be printed in the usual C++
-way, for example,
-
- mpz_t z;
- int n;
- ...
- cout << "iteration " << n << " value " << z << "\n";
-
- But note that `ostream' output (and `istream' input, *note C++
-Formatted Input::) is the only overloading available for the GMP types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *Note C++ Class Interface::.
-
-\1f
-File: gmp.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
-
-11 Formatted Input
-******************
-
-* Menu:
-
-* Formatted Input Strings::
-* Formatted Input Functions::
-* C++ Formatted Input::
-
-\1f
-File: gmp.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input
-
-11.1 Formatted Input Strings
-============================
-
-`gmp_scanf' and friends accept format strings similar to the standard C
-`scanf' (*note Formatted Input: (libc)Formatted Input.). A format
-specification is of the form
-
- % [flags] [width] [type] conv
-
- GMP adds types `Z', `Q' and `F' for `mpz_t', `mpq_t' and `mpf_t'
-respectively. `Z' and `Q' behave like integers. `Q' will read a `/'
-and a denominator, if present. `F' behaves like a float.
-
- GMP variables don't require an `&' when passed to `gmp_scanf', since
-they're already "call-by-reference". For example,
-
- /* to read say "a(5) = 1234" */
- int n;
- mpz_t z;
- gmp_scanf ("a(%d) = %Zd\n", &n, z);
-
- mpq_t q1, q2;
- gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
-
- /* to read say "topleft (1.55,-2.66)" */
- mpf_t x, y;
- char buf[32];
- gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
-
- All the standard C `scanf' types behave the same as in the C library
-`scanf', and can be freely intermixed with the GMP extensions. In the
-current implementation the standard parts of the format string are
-simply handed to `scanf' and only the GMP extensions handled directly.
-
- The flags accepted are as follows. `a' and `'' will depend on
-support from the C library, and `'' cannot be used with GMP types.
-
- * read but don't store
- a allocate a buffer (string conversions)
- ' grouped digits, GLIBC style (not GMP
- types)
-
- The standard types accepted are as follows. `h' and `l' are
-portable, the rest will depend on the compiler (or include files) for
-the type and the C library for the input.
-
- h short
- hh char
- j intmax_t or uintmax_t
- l long int, double or wchar_t
- ll long long
- L long double
- q quad_t or u_quad_t
- t ptrdiff_t
- z size_t
-
-The GMP types are
-
- F mpf_t, float conversions
- Q mpq_t, integer conversions
- Z mpz_t, integer conversions
-
- The conversions accepted are as follows. `p' and `[' will depend on
-support from the C library, the rest are standard.
-
- c character or characters
- d decimal integer
- e E f g G float
- i integer with base indicator
- n characters read so far
- o octal integer
- p pointer
- s string of non-whitespace characters
- u decimal integer
- x X hex integer
- [ string of characters in a set
-
- `e', `E', `f', `g' and `G' are identical, they all read either fixed
-point or scientific format, and either upper or lower case `e' for the
-exponent in scientific format.
-
- C99 style hex float format (`printf %a', *note Formatted Output
-Strings::) is always accepted for `mpf_t', but for the standard float
-types it will depend on the C library.
-
- `x' and `X' are identical, both accept both upper and lower case
-hexadecimal.
-
- `o', `u', `x' and `X' all read positive or negative values. For the
-standard C types these are described as "unsigned" conversions, but
-that merely affects certain overflow handling, negatives are still
-allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of
-Integers.). For GMP types there are no overflows, so `d' and `u' are
-identical.
-
- `Q' type reads the numerator and (optional) denominator as given.
-If the value might not be in canonical form then `mpq_canonicalize'
-must be called before using it in any calculations (*note Rational
-Number Functions::).
-
- `Qi' will read a base specification separately for the numerator and
-denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11'
-would be 16/17.
-
- `n' can be used with any of the types above, even the GMP types.
-`*' to suppress assignment is allowed, though in that case it would do
-nothing at all.
-
- Other conversions or types that might be accepted by the C library
-`scanf' cannot be used through `gmp_scanf'.
-
- Whitespace is read and discarded before a field, except for `c' and
-`[' conversions.
-
- For float conversions, the decimal point character (or string)
-expected is taken from the current locale settings on systems which
-provide `localeconv' (*note Locales and Internationalization:
-(libc)Locales.). The C library will normally do the same for standard
-float input.
-
- The format string is only interpreted as plain `char's, multibyte
-characters are not recognised. Perhaps this will change in the future.
-
-\1f
-File: gmp.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input
-
-11.2 Formatted Input Functions
-==============================
-
-Each of the following functions is similar to the corresponding C
-library function. The plain `scanf' forms take a variable argument
-list. The `vscanf' forms take an argument pointer, see *Note Variadic
-Functions: (libc)Variadic Functions, or `man 3 va_start'.
-
- It should be emphasised that if a format string is invalid, or the
-arguments don't match what the format specifies, then the behaviour of
-any of these functions will be unpredictable. GCC format string
-checking is not available, since it doesn't recognise the GMP
-extensions.
-
- No overlap is permitted between the FMT string and any of the results
-produced.
-
- -- Function: int gmp_scanf (const char *FMT, ...)
- -- Function: int gmp_vscanf (const char *FMT, va_list AP)
- Read from the standard input `stdin'.
-
- -- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
- -- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
- Read from the stream FP.
-
- -- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
- -- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
- AP)
- Read from a null-terminated string S.
-
- The return value from each of these functions is the same as the
-standard C99 `scanf', namely the number of fields successfully parsed
-and stored. `%n' fields and fields read but suppressed by `*' don't
-count towards the return value.
-
- If end of input (or a file error) is reached before a character for
-a field or a literal, and if no previous non-suppressed fields have
-matched, then the return value is `EOF' instead of 0. A whitespace
-character in the format string is only an optional match and doesn't
-induce an `EOF' in this fashion. Leading whitespace read and discarded
-for a field don't count as characters for that field.
-
- For the GMP types, input parsing follows C99 rules, namely one
-character of lookahead is used and characters are read while they
-continue to meet the format requirements. If this doesn't provide a
-complete number then the function terminates, with that field not
-stored nor counted towards the return value. For instance with `mpf_t'
-an input `1.23e-XYZ' would be read up to the `X' and that character
-pushed back since it's not a digit. The string `1.23e-' would then be
-considered invalid since an `e' must be followed by at least one digit.
-
- For the standard C types, in the current implementation GMP calls
-the C library `scanf' functions, which might have looser rules about
-what constitutes a valid input.
-
- Note that `gmp_sscanf' is the same as `gmp_fscanf' and only does one
-character of lookahead when parsing. Although clearly it could look at
-its entire input, it is deliberately made identical to `gmp_fscanf',
-the same way C99 `sscanf' is the same as `fscanf'.
-
-\1f
-File: gmp.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
-
-11.3 C++ Formatted Input
-========================
-
-The following functions are provided in `libgmpxx' (*note Headers and
-Libraries::), which is built only if C++ support is enabled (*note
-Build Options::). Prototypes are available from `<gmp.h>'.
-
- -- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
-
- -- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
- An integer like `123' will be read, or a fraction like `5/9'. No
- whitespace is allowed around the `/'. If the fraction is not in
- canonical form then `mpq_canonicalize' must be called (*note
- Rational Number Functions::) before operating on it.
-
- As per integer input, an `0' or `0x' base indicator is read when
- none of `ios::dec', `ios::oct' or `ios::hex' are set. This is
- done separately for numerator and denominator, so that for instance
- `0x10/11' is 16/11 and `0x10/0x11' is 16/17.
-
- -- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
- Read ROP from STREAM, using its `ios' formatting settings.
-
- Hex or octal floats are not supported, but might be in the future,
- or perhaps it's best to accept only what the standard float
- `operator>>' does.
-
- Note that digit grouping specified by the `istream' locale is
-currently not accepted. Perhaps this will change in the future.
-
-
- These operators mean that GMP types can be read in the usual C++
-way, for example,
-
- mpz_t z;
- ...
- cin >> z;
-
- But note that `istream' input (and `ostream' output, *note C++
-Formatted Output::) is the only overloading available for the GMP types
-and that for instance using `+' with an `mpz_t' will have unpredictable
-results. For classes with overloading, see *Note C++ Class Interface::.
-
-\1f
-File: gmp.info, Node: C++ Class Interface, Next: BSD Compatible Functions, Prev: Formatted Input, Up: Top
-
-12 C++ Class Interface
-**********************
-
-This chapter describes the C++ class based interface to GMP.
-
- All GMP C language types and functions can be used in C++ programs,
-since `gmp.h' has `extern "C"' qualifiers, but the class interface
-offers overloaded functions and operators which may be more convenient.
-
- Due to the implementation of this interface, a reasonably recent C++
-compiler is required, one supporting namespaces, partial specialization
-of templates and member templates. For GCC this means version 2.91 or
-later.
-
- *Everything described in this chapter is to be considered preliminary
-and might be subject to incompatible changes if some unforeseen
-difficulty reveals itself.*
-
-* Menu:
-
-* C++ Interface General::
-* C++ Interface Integers::
-* C++ Interface Rationals::
-* C++ Interface Floats::
-* C++ Interface Random Numbers::
-* C++ Interface Limitations::
-
-\1f
-File: gmp.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface
-
-12.1 C++ Interface General
-==========================
-
-All the C++ classes and functions are available with
-
- #include <gmpxx.h>
-
- Programs should be linked with the `libgmpxx' and `libgmp'
-libraries. For example,
-
- g++ mycxxprog.cc -lgmpxx -lgmp
-
-The classes defined are
-
- -- Class: mpz_class
- -- Class: mpq_class
- -- Class: mpf_class
-
- The standard operators and various standard functions are overloaded
-to allow arithmetic with these classes. For example,
-
- int
- main (void)
- {
- mpz_class a, b, c;
-
- a = 1234;
- b = "-5678";
- c = a+b;
- cout << "sum is " << c << "\n";
- cout << "absolute value is " << abs(c) << "\n";
-
- return 0;
- }
-
- An important feature of the implementation is that an expression like
-`a=b+c' results in a single call to the corresponding `mpz_add',
-without using a temporary for the `b+c' part. Expressions which by
-their nature imply intermediate values, like `a=b*c+d*e', still use
-temporaries though.
-
- The classes can be freely intermixed in expressions, as can the
-classes and the standard types `long', `unsigned long' and `double'.
-Smaller types like `int' or `float' can also be intermixed, since C++
-will promote them.
-
- Note that `bool' is not accepted directly, but must be explicitly
-cast to an `int' first. This is because C++ will automatically convert
-any pointer to a `bool', so if GMP accepted `bool' it would make all
-sorts of invalid class and pointer combinations compile but almost
-certainly not do anything sensible.
-
- Conversions back from the classes to standard C++ types aren't done
-automatically, instead member functions like `get_si' are provided (see
-the following sections for details).
-
- Also there are no automatic conversions from the classes to the
-corresponding GMP C types, instead a reference to the underlying C
-object can be obtained with the following functions,
-
- -- Function: mpz_t mpz_class::get_mpz_t ()
- -- Function: mpq_t mpq_class::get_mpq_t ()
- -- Function: mpf_t mpf_class::get_mpf_t ()
-
- These can be used to call a C function which doesn't have a C++ class
-interface. For example to set `a' to the GCD of `b' and `c',
-
- mpz_class a, b, c;
- ...
- mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
-
- In the other direction, a class can be initialized from the
-corresponding GMP C type, or assigned to if an explicit constructor is
-used. In both cases this makes a copy of the value, it doesn't create
-any sort of association. For example,
-
- mpz_t z;
- // ... init and calculate z ...
- mpz_class x(z);
- mpz_class y;
- y = mpz_class (z);
-
- There are no namespace setups in `gmpxx.h', all types and functions
-are simply put into the global namespace. This is what `gmp.h' has
-done in the past, and continues to do for compatibility. The extras
-provided by `gmpxx.h' follow GMP naming conventions and are unlikely to
-clash with anything.
-
-\1f
-File: gmp.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface
-
-12.2 C++ Interface Integers
-===========================
-
- -- Function: void mpz_class::mpz_class (type N)
- Construct an `mpz_class'. All the standard C++ types may be used,
- except `long long' and `long double', and all the GMP C++ classes
- can be used. Any necessary conversion follows the corresponding C
- function, for example `double' follows `mpz_set_d' (*note
- Assigning Integers::).
-
- -- Function: void mpz_class::mpz_class (mpz_t Z)
- Construct an `mpz_class' from an `mpz_t'. The value in Z is
- copied into the new `mpz_class', there won't be any permanent
- association between it and Z.
-
- -- Function: void mpz_class::mpz_class (const char *S)
- -- Function: void mpz_class::mpz_class (const char *S, int BASE = 0)
- -- Function: void mpz_class::mpz_class (const string& S)
- -- Function: void mpz_class::mpz_class (const string& S, int BASE = 0)
- Construct an `mpz_class' converted from a string using
- `mpz_set_str' (*note Assigning Integers::).
-
- If the string is not a valid integer, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: mpz_class operator/ (mpz_class A, mpz_class D)
- -- Function: mpz_class operator% (mpz_class A, mpz_class D)
- Divisions involving `mpz_class' round towards zero, as per the
- `mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::).
- This is the same as the C99 `/' and `%' operators.
-
- The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called
- directly if desired. For example,
-
- mpz_class q, a, d;
- ...
- mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
-
- -- Function: mpz_class abs (mpz_class OP1)
- -- Function: int cmp (mpz_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpz_class OP2)
- -- Function: bool mpz_class::fits_sint_p (void)
- -- Function: bool mpz_class::fits_slong_p (void)
- -- Function: bool mpz_class::fits_sshort_p (void)
- -- Function: bool mpz_class::fits_uint_p (void)
- -- Function: bool mpz_class::fits_ulong_p (void)
- -- Function: bool mpz_class::fits_ushort_p (void)
- -- Function: double mpz_class::get_d (void)
- -- Function: long mpz_class::get_si (void)
- -- Function: string mpz_class::get_str (int BASE = 10)
- -- Function: unsigned long mpz_class::get_ui (void)
- -- Function: int mpz_class::set_str (const char *STR, int BASE)
- -- Function: int mpz_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpz_class OP)
- -- Function: mpz_class sqrt (mpz_class OP)
- These functions provide a C++ class interface to the corresponding
- GMP C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
-
-
- Overloaded operators for combinations of `mpz_class' and `double'
-are provided for completeness, but it should be noted that if the given
-`double' is not an integer then the way any rounding is done is
-currently unspecified. The rounding might take place at the start, in
-the middle, or at the end of the operation, and it might change in the
-future.
-
- Conversions between `mpz_class' and `double', however, are defined
-to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
-And comparisons are always made exactly, as per `mpz_cmp_d'.
-
-\1f
-File: gmp.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface
-
-12.3 C++ Interface Rationals
-============================
-
-In all the following constructors, if a fraction is given then it
-should be in canonical form, or if not then `mpq_class::canonicalize'
-called.
-
- -- Function: void mpq_class::mpq_class (type OP)
- -- Function: void mpq_class::mpq_class (integer NUM, integer DEN)
- Construct an `mpq_class'. The initial value can be a single value
- of any type, or a pair of integers (`mpz_class' or standard C++
- integer types) representing a fraction, except that `long long'
- and `long double' are not supported. For example,
-
- mpq_class q (99);
- mpq_class q (1.75);
- mpq_class q (1, 3);
-
- -- Function: void mpq_class::mpq_class (mpq_t Q)
- Construct an `mpq_class' from an `mpq_t'. The value in Q is
- copied into the new `mpq_class', there won't be any permanent
- association between it and Q.
-
- -- Function: void mpq_class::mpq_class (const char *S)
- -- Function: void mpq_class::mpq_class (const char *S, int BASE = 0)
- -- Function: void mpq_class::mpq_class (const string& S)
- -- Function: void mpq_class::mpq_class (const string& S, int BASE = 0)
- Construct an `mpq_class' converted from a string using
- `mpq_set_str' (*note Initializing Rationals::).
-
- If the string is not a valid rational, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: void mpq_class::canonicalize ()
- Put an `mpq_class' into canonical form, as per *Note Rational
- Number Functions::. All arithmetic operators require their
- operands in canonical form, and will return results in canonical
- form.
-
- -- Function: mpq_class abs (mpq_class OP)
- -- Function: int cmp (mpq_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpq_class OP2)
- -- Function: double mpq_class::get_d (void)
- -- Function: string mpq_class::get_str (int BASE = 10)
- -- Function: int mpq_class::set_str (const char *STR, int BASE)
- -- Function: int mpq_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpq_class OP)
- These functions provide a C++ class interface to the corresponding
- GMP C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
-
- -- Function: mpz_class& mpq_class::get_num ()
- -- Function: mpz_class& mpq_class::get_den ()
- Get a reference to an `mpz_class' which is the numerator or
- denominator of an `mpq_class'. This can be used both for read and
- write access. If the object returned is modified, it modifies the
- original `mpq_class'.
-
- If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
-
- -- Function: mpz_t mpq_class::get_num_mpz_t ()
- -- Function: mpz_t mpq_class::get_den_mpz_t ()
- Get a reference to the underlying `mpz_t' numerator or denominator
- of an `mpq_class'. This can be passed to C functions expecting an
- `mpz_t'. Any modifications made to the `mpz_t' will modify the
- original `mpq_class'.
-
- If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
-
- -- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
- Read ROP from STREAM, using its `ios' formatting settings, the
- same as `mpq_t operator>>' (*note C++ Formatted Input::).
-
- If the ROP read might not be in canonical form then
- `mpq_class::canonicalize' must be called.
-
-\1f
-File: gmp.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
-
-12.4 C++ Interface Floats
-=========================
-
-When an expression requires the use of temporary intermediate
-`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
-same precision as the destination `f'. Explicit constructors can be
-used if this doesn't suit.
-
- -- Function: mpf_class::mpf_class (type OP)
- -- Function: mpf_class::mpf_class (type OP, unsigned long PREC)
- Construct an `mpf_class'. Any standard C++ type can be used,
- except `long long' and `long double', and any of the GMP C++
- classes can be used.
-
- If PREC is given, the initial precision is that value, in bits. If
- PREC is not given, then the initial precision is determined by the
- type of OP given. An `mpz_class', `mpq_class', or C++ builtin
- type will give the default `mpf' precision (*note Initializing
- Floats::). An `mpf_class' or expression will give the precision
- of that value. The precision of a binary expression is the higher
- of the two operands.
-
- mpf_class f(1.5); // default precision
- mpf_class f(1.5, 500); // 500 bits (at least)
- mpf_class f(x); // precision of x
- mpf_class f(abs(x)); // precision of x
- mpf_class f(-g, 1000); // 1000 bits (at least)
- mpf_class f(x+y); // greater of precisions of x and y
-
- -- Function: void mpf_class::mpf_class (const char *S)
- -- Function: void mpf_class::mpf_class (const char *S, unsigned long
- PREC, int BASE = 0)
- -- Function: void mpf_class::mpf_class (const string& S)
- -- Function: void mpf_class::mpf_class (const string& S, unsigned long
- PREC, int BASE = 0)
- Construct an `mpf_class' converted from a string using
- `mpf_set_str' (*note Assigning Floats::). If PREC is given, the
- initial precision is that value, in bits. If not, the default
- `mpf' precision (*note Initializing Floats::) is used.
-
- If the string is not a valid float, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
-
- -- Function: mpf_class& mpf_class::operator= (type OP)
- Convert and store the given OP value to an `mpf_class' object. The
- same types are accepted as for the constructors above.
-
- Note that `operator=' only stores a new value, it doesn't copy or
- change the precision of the destination, instead the value is
- truncated if necessary. This is the same as `mpf_set' etc. Note
- in particular this means for `mpf_class' a copy constructor is not
- the same as a default constructor plus assignment.
-
- mpf_class x (y); // x created with precision of y
-
- mpf_class x; // x created with default precision
- x = y; // value truncated to that precision
-
- Applications using templated code may need to be careful about the
- assumptions the code makes in this area, when working with
- `mpf_class' values of various different or non-default precisions.
- For instance implementations of the standard `complex' template
- have been seen in both styles above, though of course `complex' is
- normally only actually specified for use with the builtin float
- types.
-
- -- Function: mpf_class abs (mpf_class OP)
- -- Function: mpf_class ceil (mpf_class OP)
- -- Function: int cmp (mpf_class OP1, type OP2)
- -- Function: int cmp (type OP1, mpf_class OP2)
- -- Function: bool mpf_class::fits_sint_p (void)
- -- Function: bool mpf_class::fits_slong_p (void)
- -- Function: bool mpf_class::fits_sshort_p (void)
- -- Function: bool mpf_class::fits_uint_p (void)
- -- Function: bool mpf_class::fits_ulong_p (void)
- -- Function: bool mpf_class::fits_ushort_p (void)
- -- Function: mpf_class floor (mpf_class OP)
- -- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
- -- Function: double mpf_class::get_d (void)
- -- Function: long mpf_class::get_si (void)
- -- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
- size_t DIGITS = 0)
- -- Function: unsigned long mpf_class::get_ui (void)
- -- Function: int mpf_class::set_str (const char *STR, int BASE)
- -- Function: int mpf_class::set_str (const string& STR, int BASE)
- -- Function: int sgn (mpf_class OP)
- -- Function: mpf_class sqrt (mpf_class OP)
- -- Function: mpf_class trunc (mpf_class OP)
- These functions provide a C++ class interface to the corresponding
- GMP C routines.
-
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
-
- The accuracy provided by `hypot' is not currently guaranteed.
-
- -- Function: mp_bitcnt_t mpf_class::get_prec ()
- -- Function: void mpf_class::set_prec (mp_bitcnt_t PREC)
- -- Function: void mpf_class::set_prec_raw (mp_bitcnt_t PREC)
- Get or set the current precision of an `mpf_class'.
-
- The restrictions described for `mpf_set_prec_raw' (*note
- Initializing Floats::) apply to `mpf_class::set_prec_raw'. Note
- in particular that the `mpf_class' must be restored to it's
- allocated precision before being destroyed. This must be done by
- application code, there's no automatic mechanism for it.
-
-\1f
-File: gmp.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
-
-12.5 C++ Interface Random Numbers
-=================================
-
- -- Class: gmp_randclass
- The C++ class interface to the GMP random number functions uses
- `gmp_randclass' to hold an algorithm selection and current state,
- as per `gmp_randstate_t'.
-
- -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
- (gmp_randstate_t, ...), ...)
- Construct a `gmp_randclass', using a call to the given RANDINIT
- function (*note Random State Initialization::). The arguments
- expected are the same as RANDINIT, but with `mpz_class' instead of
- `mpz_t'. For example,
-
- gmp_randclass r1 (gmp_randinit_default);
- gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
- gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
- gmp_randclass r4 (gmp_randinit_mt);
-
- `gmp_randinit_lc_2exp_size' will fail if the size requested is too
- big, an `std::length_error' exception is thrown in that case.
-
- -- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
- Construct a `gmp_randclass' using the same parameters as
- `gmp_randinit' (*note Random State Initialization::). This
- function is obsolete and the above RANDINIT style should be
- preferred.
-
- -- Function: void gmp_randclass::seed (unsigned long int S)
- -- Function: void gmp_randclass::seed (mpz_class S)
- Seed a random number generator. See *note Random Number
- Functions::, for how to choose a good seed.
-
- -- Function: mpz_class gmp_randclass::get_z_bits (unsigned long BITS)
- -- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
- Generate a random integer with a specified number of bits.
-
- -- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
- Generate a random integer in the range 0 to N-1 inclusive.
-
- -- Function: mpf_class gmp_randclass::get_f ()
- -- Function: mpf_class gmp_randclass::get_f (unsigned long PREC)
- Generate a random float F in the range 0 <= F < 1. F will be to
- PREC bits precision, or if PREC is not given then to the precision
- of the destination. For example,
-
- gmp_randclass r;
- ...
- mpf_class f (0, 512); // 512 bits precision
- f = r.get_f(); // random number, 512 bits
-
-\1f
-File: gmp.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface
-
-12.6 C++ Interface Limitations
-==============================
-
-`mpq_class' and Templated Reading
- A generic piece of template code probably won't know that
- `mpq_class' requires a `canonicalize' call if inputs read with
- `operator>>' might be non-canonical. This can lead to incorrect
- results.
-
- `operator>>' behaves as it does for reasons of efficiency. A
- canonicalize can be quite time consuming on large operands, and is
- best avoided if it's not necessary.
-
- But this potential difficulty reduces the usefulness of
- `mpq_class'. Perhaps a mechanism to tell `operator>>' what to do
- will be adopted in the future, maybe a preprocessor define, a
- global flag, or an `ios' flag pressed into service. Or maybe, at
- the risk of inconsistency, the `mpq_class' `operator>>' could
- canonicalize and leave `mpq_t' `operator>>' not doing so, for use
- on those occasions when that's acceptable. Send feedback or
- alternate ideas to <gmp-bugs@gmplib.org>.
-
-Subclassing
- Subclassing the GMP C++ classes works, but is not currently
- recommended.
-
- Expressions involving subclasses resolve correctly (or seem to),
- but in normal C++ fashion the subclass doesn't inherit
- constructors and assignments. There's many of those in the GMP
- classes, and a good way to reestablish them in a subclass is not
- yet provided.
-
-Templated Expressions
- A subtle difficulty exists when using expressions together with
- application-defined template functions. Consider the following,
- with `T' intended to be some numeric type,
-
- template <class T>
- T fun (const T &, const T &);
-
- When used with, say, plain `mpz_class' variables, it works fine:
- `T' is resolved as `mpz_class'.
-
- mpz_class f(1), g(2);
- fun (f, g); // Good
-
- But when one of the arguments is an expression, it doesn't work.
-
- mpz_class f(1), g(2), h(3);
- fun (f, g+h); // Bad
-
- This is because `g+h' ends up being a certain expression template
- type internal to `gmpxx.h', which the C++ template resolution
- rules are unable to automatically convert to `mpz_class'. The
- workaround is simply to add an explicit cast.
-
- mpz_class f(1), g(2), h(3);
- fun (f, mpz_class(g+h)); // Good
-
- Similarly, within `fun' it may be necessary to cast an expression
- to type `T' when calling a templated `fun2'.
-
- template <class T>
- void fun (T f, T g)
- {
- fun2 (f, f+g); // Bad
- }
-
- template <class T>
- void fun (T f, T g)
- {
- fun2 (f, T(f+g)); // Good
- }
-
-\1f
-File: gmp.info, Node: BSD Compatible Functions, Next: Custom Allocation, Prev: C++ Class Interface, Up: Top
-
-13 Berkeley MP Compatible Functions
-***********************************
-
-These functions are intended to be fully compatible with the Berkeley MP
-library which is available on many BSD derived U*ix systems. The
-`--enable-mpbsd' option must be used when building GNU MP to make these
-available (*note Installing GMP::).
-
- The original Berkeley MP library has a usage restriction: you cannot
-use the same variable as both source and destination in a single
-function call. The compatible functions in GNU MP do not share this
-restriction--inputs and outputs may overlap.
-
- It is not recommended that new programs are written using these
-functions. Apart from the incomplete set of functions, the interface
-for initializing `MINT' objects is more error prone, and the `pow'
-function collides with `pow' in `libm.a'.
-
- Include the header `mp.h' to get the definition of the necessary
-types and functions. If you are on a BSD derived system, make sure to
-include GNU `mp.h' if you are going to link the GNU `libmp.a' to your
-program. This means that you probably need to give the `-I<dir>'
-option to the compiler, where `<dir>' is the directory where you have
-GNU `mp.h'.
-
- -- Function: MINT * itom (signed short int INITIAL_VALUE)
- Allocate an integer consisting of a `MINT' object and dynamic limb
- space. Initialize the integer to INITIAL_VALUE. Return a pointer
- to the `MINT' object.
-
- -- Function: MINT * xtom (char *INITIAL_VALUE)
- Allocate an integer consisting of a `MINT' object and dynamic limb
- space. Initialize the integer from INITIAL_VALUE, a hexadecimal,
- null-terminated C string. Return a pointer to the `MINT' object.
-
- -- Function: void move (MINT *SRC, MINT *DEST)
- Set DEST to SRC by copying. Both variables must be previously
- initialized.
-
- -- Function: void madd (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Add SRC_1 and SRC_2 and put the sum in DESTINATION.
-
- -- Function: void msub (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Subtract SRC_2 from SRC_1 and put the difference in DESTINATION.
-
- -- Function: void mult (MINT *SRC_1, MINT *SRC_2, MINT *DESTINATION)
- Multiply SRC_1 and SRC_2 and put the product in DESTINATION.
-
- -- Function: void mdiv (MINT *DIVIDEND, MINT *DIVISOR, MINT *QUOTIENT,
- MINT *REMAINDER)
- -- Function: void sdiv (MINT *DIVIDEND, signed short int DIVISOR, MINT
- *QUOTIENT, signed short int *REMAINDER)
- Set QUOTIENT to DIVIDEND/DIVISOR, and REMAINDER to DIVIDEND mod
- DIVISOR. The quotient is rounded towards zero; the remainder has
- the same sign as the dividend unless it is zero.
-
- Some implementations of these functions work differently--or not
- at all--for negative arguments.
-
- -- Function: void msqrt (MINT *OP, MINT *ROOT, MINT *REMAINDER)
- Set ROOT to the truncated integer part of the square root of OP,
- like `mpz_sqrt'. Set REMAINDER to OP-ROOT*ROOT, i.e. zero if OP
- is a perfect square.
-
- If ROOT and REMAINDER are the same variable, the results are
- undefined.
-
- -- Function: void pow (MINT *BASE, MINT *EXP, MINT *MOD, MINT *DEST)
- Set DEST to (BASE raised to EXP) modulo MOD.
-
- Note that the name `pow' clashes with `pow' from the standard C
- math library (*note Exponentiation and Logarithms: (libc)Exponents
- and Logarithms.). An application will only be able to use one or
- the other.
-
- -- Function: void rpow (MINT *BASE, signed short int EXP, MINT *DEST)
- Set DEST to BASE raised to EXP.
-
- -- Function: void gcd (MINT *OP1, MINT *OP2, MINT *RES)
- Set RES to the greatest common divisor of OP1 and OP2.
-
- -- Function: int mcmp (MINT *OP1, MINT *OP2)
- Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
- if OP1 = OP2, and a negative value if OP1 < OP2.
-
- -- Function: void min (MINT *DEST)
- Input a decimal string from `stdin', and put the read integer in
- DEST. SPC and TAB are allowed in the number string, and are
- ignored.
-
- -- Function: void mout (MINT *SRC)
- Output SRC to `stdout', as a decimal string. Also output a
- newline.
-
- -- Function: char * mtox (MINT *OP)
- Convert OP to a hexadecimal string, and return a pointer to the
- string. The returned string is allocated using the default memory
- allocation function, `malloc' by default. It will be
- `strlen(str)+1' bytes, that being exactly enough for the string
- and null-terminator.
-
- -- Function: void mfree (MINT *OP)
- De-allocate, the space used by OP. *This function should only be
- passed a value returned by `itom' or `xtom'.*
-
-\1f
-File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: BSD Compatible Functions, Up: Top
-
-14 Custom Allocation
-********************
-
-By default GMP uses `malloc', `realloc' and `free' for memory
-allocation, and if they fail GMP prints a message to the standard error
-output and terminates the program.
-
- Alternate functions can be specified, to allocate memory in a
-different way or to have a different error action on running out of
-memory.
-
- This feature is available in the Berkeley compatibility library
-(*note BSD Compatible Functions::) as well as the main GMP library.
-
- -- Function: void mp_set_memory_functions (
- void *(*ALLOC_FUNC_PTR) (size_t),
- void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
- void (*FREE_FUNC_PTR) (void *, size_t))
- Replace the current allocation functions from the arguments. If
- an argument is `NULL', the corresponding default function is used.
-
- These functions will be used for all memory allocation done by
- GMP, apart from temporary space from `alloca' if that function is
- available and GMP is configured to use it (*note Build Options::).
-
- *Be sure to call `mp_set_memory_functions' only when there are no
- active GMP objects allocated using the previous memory functions!
- Usually that means calling it before any other GMP function.*
-
- The functions supplied should fit the following declarations:
-
- -- Function: void * allocate_function (size_t ALLOC_SIZE)
- Return a pointer to newly allocated space with at least ALLOC_SIZE
- bytes.
-
- -- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
- size_t NEW_SIZE)
- Resize a previously allocated block PTR of OLD_SIZE bytes to be
- NEW_SIZE bytes.
-
- The block may be moved if necessary or if desired, and in that
- case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
- the new location. The return value is a pointer to the resized
- block, that being the new location if moved or just PTR if not.
-
- PTR is never `NULL', it's always a previously allocated block.
- NEW_SIZE may be bigger or smaller than OLD_SIZE.
-
- -- Function: void free_function (void *PTR, size_t SIZE)
- De-allocate the space pointed to by PTR.
-
- PTR is never `NULL', it's always a previously allocated block of
- SIZE bytes.
-
- A "byte" here means the unit used by the `sizeof' operator.
-
- The OLD_SIZE parameters to REALLOCATE_FUNCTION and FREE_FUNCTION are
-passed for convenience, but of course can be ignored if not needed.
-The default functions using `malloc' and friends for instance don't use
-them.
-
- No error return is allowed from any of these functions, if they
-return then they must have performed the specified operation. In
-particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
-return `NULL'.
-
- Getting a different fatal error action is a good use for custom
-allocation functions, for example giving a graphical dialog rather than
-the default print to `stderr'. How much is possible when genuinely out
-of memory is another question though.
-
- There's currently no defined way for the allocation functions to
-recover from an error such as out of memory, they must terminate
-program execution. A `longjmp' or throwing a C++ exception will have
-undefined results. This may change in the future.
-
- GMP may use allocated blocks to hold pointers to other allocated
-blocks. This will limit the assumptions a conservative garbage
-collection scheme can make.
-
- Since the default GMP allocation uses `malloc' and friends, those
-functions will be linked in even if the first thing a program does is an
-`mp_set_memory_functions'. It's necessary to change the GMP sources if
-this is a problem.
-
-
- -- Function: void mp_get_memory_functions (
- void *(**ALLOC_FUNC_PTR) (size_t),
- void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t),
- void (**FREE_FUNC_PTR) (void *, size_t))
- Get the current allocation functions, storing function pointers to
- the locations given by the arguments. If an argument is `NULL',
- that function pointer is not stored.
-
- For example, to get just the current free function,
-
- void (*freefunc) (void *, size_t);
-
- mp_get_memory_functions (NULL, NULL, &freefunc);
-
-\1f
-File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
-
-15 Language Bindings
-********************
-
-The following packages and projects offer access to GMP from languages
-other than C, though perhaps with varying levels of functionality and
-efficiency.
-
-
-C++
- * GMP C++ class interface, *note C++ Class Interface::
- Straightforward interface, expression templates to eliminate
- temporaries.
-
- * ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
- Linear algebra and polynomials using templates.
-
- * Arithmos `http://www.win.ua.ac.be/~cant/arithmos/'
- Rationals with infinities and square roots.
-
- * CLN `http://www.ginac.de/CLN/'
- High level classes for arithmetic.
-
- * LiDIA `http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/'
- A C++ library for computational number theory.
-
- * Linbox `http://www.linalg.org/'
- Sparse vectors and matrices.
-
- * NTL `http://www.shoup.net/ntl/'
- A C++ number theory library.
-
-Fortran
- * Omni F77 `http://phase.hpcc.jp/Omni/home.html'
- Arbitrary precision floats.
-
-Haskell
- * Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
-
-Java
- * Kaffe `http://www.kaffe.org/'
-
- * Kissme `http://kissme.sourceforge.net/'
-
-Lisp
- * GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html'
-
- * Librep `http://librep.sourceforge.net/'
-
- * XEmacs (21.5.18 beta and up) `http://www.xemacs.org'
- Optional big integers, rationals and floats using GMP.
-
-M4
- * GNU m4 betas `http://www.seindal.dk/rene/gnu/'
- Optionally provides an arbitrary precision `mpeval'.
-
-ML
- * MLton compiler `http://mlton.org/'
-
-Objective Caml
- * MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en'
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
- Optionally using GMP.
-
-Oz
- * Mozart `http://www.mozart-oz.org/'
-
-Pascal
- * GNU Pascal Compiler `http://www.gnu-pascal.de/'
- GMP unit.
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
- For Free Pascal, optionally using GMP.
-
-Perl
- * GMP module, see `demos/perl' in the GMP sources (*note
- Demonstration Programs::).
-
- * Math::GMP `http://www.cpan.org/'
- Compatible with Math::BigInt, but not as many functions as
- the GMP module above.
-
- * Math::BigInt::GMP `http://www.cpan.org/'
- Plug Math::GMP into normal Math::BigInt operations.
-
-Pike
- * mpz module in the standard distribution,
- `http://pike.ida.liu.se/'
-
-Prolog
- * SWI Prolog `http://www.swi-prolog.org/'
- Arbitrary precision floats.
-
-Python
- * mpz module in the standard distribution,
- `http://www.python.org/'
-
- * GMPY `http://gmpy.sourceforge.net/'
-
-Scheme
- * GNU Guile (upcoming 1.8)
- `http://www.gnu.org/software/guile/guile.html'
-
- * RScheme `http://www.rscheme.org/'
-
- * STklos `http://www.stklos.org/'
-
-Smalltalk
- * GNU Smalltalk
- `http://www.smalltalk.org/versions/GNUSmalltalk.html'
-
-Other
- * Axiom `http://savannah.nongnu.org/projects/axiom'
- Computer algebra using GCL.
-
- * DrGenius `http://drgenius.seul.org/'
- Geometry system and mathematical programming language.
-
- * GiNaC `http://www.ginac.de/'
- C++ computer algebra using CLN.
-
- * GOO `http://www.googoogaga.org/'
- Dynamic object oriented language.
-
- * Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
- Macsyma computer algebra using GCL.
-
- * Q `http://q-lang.sourceforge.net/'
- Equational programming system.
-
- * Regina `http://regina.sourceforge.net/'
- Topological calculator.
-
- * Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
- Yet another computer algebra system.
-
-
-\1f
-File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
-
-16 Algorithms
-*************
-
-This chapter is an introduction to some of the algorithms used for
-various GMP operations. The code is likely to be hard to understand
-without knowing something about the algorithms.
-
- Some GMP internals are mentioned, but applications that expect to be
-compatible with future GMP releases should take care to use only the
-documented functions.
-
-* Menu:
-
-* Multiplication Algorithms::
-* Division Algorithms::
-* Greatest Common Divisor Algorithms::
-* Powering Algorithms::
-* Root Extraction Algorithms::
-* Radix Conversion Algorithms::
-* Other Algorithms::
-* Assembly Coding::
-
-\1f
-File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
-
-16.1 Multiplication
-===================
-
-NxN limb multiplications and squares are done using one of five
-algorithms, as the size N increases.
-
- Algorithm Threshold
- Basecase (none)
- Karatsuba `MUL_TOOM22_THRESHOLD'
- Toom-3 `MUL_TOOM33_THRESHOLD'
- Toom-4 `MUL_TOOM44_THRESHOLD'
- FFT `MUL_FFT_THRESHOLD'
-
- Similarly for squaring, with the `SQR' thresholds.
-
- NxM multiplications of operands with different sizes above
-`MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired
-algorithms or directly with FFT, depending on operand size (*note
-Unbalanced Multiplication::).
-
-* Menu:
-
-* Basecase Multiplication::
-* Karatsuba Multiplication::
-* Toom 3-Way Multiplication::
-* Toom 4-Way Multiplication::
-* FFT Multiplication::
-* Other Multiplication::
-* Unbalanced Multiplication::
-
-\1f
-File: gmp.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
-
-16.1.1 Basecase Multiplication
-------------------------------
-
-Basecase NxM multiplication is a straightforward rectangular set of
-cross-products, the same as long multiplication done by hand and for
-that reason sometimes known as the schoolbook or grammar school method.
-This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M
-(*note References::), and the `mpn/generic/mul_basecase.c' code.
-
- Assembly implementations of `mpn_mul_basecase' are essentially the
-same as the generic C code, but have all the usual assembly tricks and
-obscurities introduced for speed.
-
- A square can be done in roughly half the time of a multiply, by
-using the fact that the cross products above and below the diagonal are
-the same. A triangle of products below the diagonal is formed, doubled
-(left shift by one bit), and then the products on the diagonal added.
-This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembly
-implementations take essentially the same approach.
-
- u0 u1 u2 u3 u4
- +---+---+---+---+---+
- u0 | d | | | | |
- +---+---+---+---+---+
- u1 | | d | | | |
- +---+---+---+---+---+
- u2 | | | d | | |
- +---+---+---+---+---+
- u3 | | | | d | |
- +---+---+---+---+---+
- u4 | | | | | d |
- +---+---+---+---+---+
-
- In practice squaring isn't a full 2x faster than multiplying, it's
-usually around 1.5x. Less than 1.5x probably indicates
-`mpn_sqr_basecase' wants improving on that CPU.
-
- On some CPUs `mpn_mul_basecase' can be faster than the generic C
-`mpn_sqr_basecase' on some small sizes. `SQR_BASECASE_THRESHOLD' is
-the size at which to use `mpn_sqr_basecase', this will be zero if that
-routine should be used always.
-
-\1f
-File: gmp.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
-
-16.1.2 Karatsuba Multiplication
--------------------------------
-
-The Karatsuba multiplication algorithm is described in Knuth section
-4.3.3 part A, and various other textbooks. A brief description is
-given here.
-
- The inputs x and y are treated as each split into two parts of equal
-length (or the most significant part one limb shorter if N is odd).
-
- high low
- +----------+----------+
- | x1 | x0 |
- +----------+----------+
-
- +----------+----------+
- | y1 | y0 |
- +----------+----------+
-
- Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
-(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
-y=y1*b+y0, and the following holds,
-
- x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
-
- This formula means doing only three multiplies of (N/2)x(N/2) limbs,
-whereas a basecase multiply of NxN limbs is equivalent to four
-multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
-positions where the three products must be added.
-
- high low
- +--------+--------+ +--------+--------+
- | x1*y1 | | x0*y0 |
- +--------+--------+ +--------+--------+
- +--------+--------+
- add | x1*y1 |
- +--------+--------+
- +--------+--------+
- add | x0*y0 |
- +--------+--------+
- +--------+--------+
- sub | (x1-x0)*(y1-y0) |
- +--------+--------+
-
- The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
-and the sign used to choose to add or subtract. Notice the sum
-high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
-additions, rather than 6*k, but in GMP extra function call overheads
-outweigh the saving.
-
- Squaring is similar to multiplying, but with x=y the formula reduces
-to an equivalent with three squares,
-
- x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
-
- The final result is accumulated from those three squares the same
-way as for the three multiplies above. The middle term (x1-x0)^2 is now
-always positive.
-
- A similar formula for both multiplying and squaring can be
-constructed with a middle term (x1+x0)*(y1+y0). But those sums can
-exceed k limbs, leading to more carry handling and additions than the
-form above.
-
- Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
-the exponent being log(3)/log(2), representing 3 multiplies each 1/2
-the size of the inputs. This is a big improvement over the basecase
-multiply at O(N^2) and the advantage soon overcomes the extra additions
-Karatsuba performs. `MUL_TOOM22_THRESHOLD' can be as little as 10
-limbs. The `SQR' threshold is usually about twice the `MUL'.
-
- The basecase algorithm will take a time of the form M(N) = a*N^2 +
-b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which
-expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4
-for a means per-crossproduct speedups in the basecase code will
-increase the threshold since they benefit M(N) more than K(N). And
-conversely the 3/2 for b means linear style speedups of b will increase
-the threshold since they benefit K(N) more than M(N). The latter can
-be seen for instance when adding an optimized `mpn_sqr_diagonal' to
-`mpn_sqr_basecase'. Of course all speedups reduce total time, and in
-that sense the algorithm thresholds are merely of academic interest.
-
-\1f
-File: gmp.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
-
-16.1.3 Toom 3-Way Multiplication
---------------------------------
-
-The Karatsuba formula is the simplest case of a general approach to
-splitting inputs that leads to both Toom and FFT algorithms. A
-description of Toom can be found in Knuth section 4.3.3, with an
-example 3-way calculation after Theorem A. The 3-way form used in GMP
-is described here.
-
- The operands are each considered split into 3 pieces of equal length
-(or the most significant part 1 or 2 limbs shorter than the other two).
-
- high low
- +----------+----------+----------+
- | x2 | x1 | x0 |
- +----------+----------+----------+
-
- +----------+----------+----------+
- | y2 | y1 | y0 |
- +----------+----------+----------+
-
-These parts are treated as the coefficients of two polynomials
-
- X(t) = x2*t^2 + x1*t + x0
- Y(t) = y2*t^2 + y1*t + y0
-
- Let b equal the power of 2 which is the size of the x0, x1, y0 and
-y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb).
-With this x=X(b) and y=Y(b).
-
- Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
-
- W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
-
- The w[i] are going to be determined, and when they are they'll give
-the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
-coefficients will be roughly b^2 each, and the final W(b) will be an
-addition like,
-
- high low
- +-------+-------+
- | w4 |
- +-------+-------+
- +--------+-------+
- | w3 |
- +--------+-------+
- +--------+-------+
- | w2 |
- +--------+-------+
- +--------+-------+
- | w1 |
- +--------+-------+
- +-------+-------+
- | w0 |
- +-------+-------+
-
- The w[i] coefficients could be formed by a simple set of cross
-products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
-this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
-equivalent merely to a basecase multiply. Instead the following
-approach is used.
-
- X(t) and Y(t) are evaluated and multiplied at 5 points, giving
-values of W(t) at those points. In GMP the following points are used,
-
- Point Value
- t=0 x0 * y0, which gives w0 immediately
- t=1 (x2+x1+x0) * (y2+y1+y0)
- t=-1 (x2-x1+x0) * (y2-y1+y0)
- t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
- t=inf x2 * y2, which gives w4 immediately
-
- At t=-1 the values can be negative and that's handled using the
-absolute values and tracking the sign separately. At t=inf the value
-is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
-it's much easier to think of as simply x2*y2 giving w4 immediately
-(much like x0*y0 at t=0 gives w0 immediately).
-
- Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
-linear combination of the w[i] coefficients, and the value of those
-combinations has just been calculated.
-
- W(0) = w0
- W(1) = w4 + w3 + w2 + w1 + w0
- W(-1) = w4 - w3 + w2 - w1 + w0
- W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
- W(inf) = w4
-
- This is a set of five equations in five unknowns, and some
-elementary linear algebra quickly isolates each w[i]. This involves
-adding or subtracting one W(t) value from another, and a couple of
-divisions by powers of 2 and one division by 3, the latter using the
-special `mpn_divexact_by3' (*note Exact Division::).
-
- The conversion of W(t) values to the coefficients is interpolation.
-A polynomial of degree 4 like W(t) is uniquely determined by values
-known at 5 different points. The points are arbitrary and can be
-chosen to make the linear equations come out with a convenient set of
-steps for quickly isolating the w[i].
-
- Squaring follows the same procedure as multiplication, but there's
-only one X(t) and it's evaluated at the 5 points, and those values
-squared to give values of W(t). The interpolation is then identical,
-and in fact the same `toom3_interpolate' subroutine is used for both
-squaring and multiplying.
-
- Toom-3 is asymptotically O(N^1.465), the exponent being
-log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
-size each. This is an improvement over Karatsuba at O(N^1.585), though
-Toom does more work in the evaluation and interpolation and so it only
-realizes its advantage above a certain size.
-
- Near the crossover between Toom-3 and Karatsuba there's generally a
-range of sizes where the difference between the two is small.
-`MUL_TOOM33_THRESHOLD' is a somewhat arbitrary point in that range and
-successive runs of the tune program can give different values due to
-small variations in measuring. A graph of time versus size for the two
-shows the effect, see `tune/README'.
-
- At the fairly small sizes where the Toom-3 thresholds occur it's
-worth remembering that the asymptotic behaviour for Karatsuba and
-Toom-3 can't be expected to make accurate predictions, due of course to
-the big influence of all sorts of overheads, and the fact that only a
-few recursions of each are being performed. Even at large sizes
-there's a good chance machine dependent effects like cache architecture
-will mean actual performance deviates from what might be predicted.
-
- The formula given for the Karatsuba algorithm (*note Karatsuba
-Multiplication::) has an equivalent for Toom-3 involving only five
-multiplies, but this would be complicated and unenlightening.
-
- An alternate view of Toom-3 can be found in Zuras (*note
-References::), using a vector to represent the x and y splits and a
-matrix multiplication for the evaluation and interpolation stages. The
-matrix inverses are not meant to be actually used, and they have
-elements with values much greater than in fact arise in the
-interpolation steps. The diagram shown for the 3-way is attractive,
-but again doesn't have to be implemented that way and for example with
-a bit of rearrangement just one division by 6 can be done.
-
-\1f
-File: gmp.info, Node: Toom 4-Way Multiplication, Next: FFT Multiplication, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
-
-16.1.4 Toom 4-Way Multiplication
---------------------------------
-
-Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
-respectively. Toom-4 analogously splits the operands into 4
-coefficients. Using the notation from the section on Toom-3
-multiplication, we form two polynomials:
-
- X(t) = x3*t^3 + x2*t^2 + x1*t + x0
- Y(t) = y3*t^3 + y2*t^2 + y1*t + y0
-
- X(t) and Y(t) are evaluated and multiplied at 7 points, giving
-values of W(t) at those points. In GMP the following points are used,
-
- Point Value
- t=0 x0 * y0, which gives w0 immediately
- t=1/2 (x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)
- t=-1/2 (-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)
- t=1 (x3+x2+x1+x0) * (y3+y2+y1+y0)
- t=-1 (-x3+x2-x1+x0) * (-y3+y2-y1+y0)
- t=2 (8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)
- t=inf x3 * y3, which gives w6 immediately
-
- The number of additions and subtractions for Toom-4 is much larger
-than for Toom-3. But several subexpressions occur multiple times, for
-example x2+x0, occurs for both t=1 and t=-1.
-
- Toom-4 is asymptotically O(N^1.404), the exponent being
-log(7)/log(4), representing 7 recursive multiplies of 1/4 the original
-size each.
-
-\1f
-File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
-
-16.1.5 FFT Multiplication
--------------------------
-
-At large to very large sizes a Fermat style FFT multiplication is used,
-following Scho"nhage and Strassen (*note References::). Descriptions
-of FFTs in various forms can be found in many textbooks, for instance
-Knuth section 4.3.3 part C or Lipson chapter IX. A brief description
-of the form used in GMP is given here.
-
- The multiplication done is x*y mod 2^N+1, for a given N. A full
-product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x
-and y with high zero limbs. The modular product is the native form for
-the algorithm, so padding to get a full product is unavoidable.
-
- The algorithm follows a split, evaluate, pointwise multiply,
-interpolate and combine similar to that described above for Karatsuba
-and Toom-3. A k parameter controls the split, with an FFT-k splitting
-into 2^k pieces of M=N/2^k bits each. N must be a multiple of
-(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
-bit shifts in the split and combine stages.
-
- The evaluations, pointwise multiplications, and interpolation, are
-all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
-2^k and of `mp_bits_per_limb'. The results of interpolation will be
-the following negacyclic convolution of the input pieces, and the
-choice of N' ensures these sums aren't truncated.
-
- ---
- \ b
- w[n] = / (-1) * x[i] * y[j]
- ---
- i+j==b*2^k+n
- b=0,1
-
- The points used for the evaluation are g^i for i=0 to 2^k-1 where
-g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
-necessary cancellations at the interpolation stage, and it's also a
-power of 2 so the fast Fourier transforms used for the evaluation and
-interpolation do only shifts, adds and negations.
-
- The pointwise multiplications are done modulo 2^N'+1 and either
-recurse into a further FFT or use a plain multiplication (Toom-3,
-Karatsuba or basecase), whichever is optimal at the size N'. The
-interpolation is an inverse fast Fourier transform. The resulting set
-of sums of x[i]*y[j] are added at appropriate offsets to give the final
-result.
-
- Squaring is the same, but x is the only input so it's one transform
-at the evaluate stage and the pointwise multiplies are squares. The
-interpolation is the same.
-
- For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm,
-the exponent representing 2^k recursed modular multiplies each
-1/2^(k-1) the size of the original. Each successive k is an asymptotic
-improvement, but overheads mean each is only faster at bigger and
-bigger sizes. In the code, `MUL_FFT_TABLE' and `SQR_FFT_TABLE' are the
-thresholds where each k is used. Each new k effectively swaps some
-multiplying for some shifts, adds and overheads.
-
- A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
-plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
-A k=4 FFT at O(N^1.333) can be expected to be the first faster than
-Toom-3 at O(N^1.465). In practice this is what's found, with
-`MUL_FFT_MODF_THRESHOLD' and `SQR_FFT_MODF_THRESHOLD' being between 300
-and 1000 limbs, depending on the CPU. So far it's been found that only
-very large FFTs recurse into pointwise multiplies above these sizes.
-
- When an FFT is to give a full product, the change of N to 2N doesn't
-alter the theoretical complexity for a given k, but for the purposes of
-considering where an FFT might be first used it can be assumed that the
-FFT is recursing into a normal multiply and that on that basis it's
-doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
-making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
-first FFT faster than Toom-3. In practice `MUL_FFT_THRESHOLD' and
-`SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere
-between 3000 and 10000 limbs.
-
- The way N is split into 2^k pieces and then 2M+k+3 is rounded up to
-a multiple of 2^k and `mp_bits_per_limb' means that when
-2^k>=mp_bits_per_limb the effective N is a multiple of 2^(2k-1) bits.
-The +k+3 means some values of N just under such a multiple will be
-rounded to the next. The complexity calculations above assume that a
-favourable size is used, meaning one which isn't padded through
-rounding, and it's also assumed that the extra +k+3 bits are negligible
-at typical FFT sizes.
-
- The practical effect of the 2^(2k-1) constraint is to introduce a
-step-effect into measured speeds. For example k=8 will round N up to a
-multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
-groups of sizes for which `mpn_mul_n' runs at the same speed. Or for
-k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice
-it's been found each k is used at quite small multiples of its size
-constraint and so the step effect is quite noticeable in a time versus
-size graph.
-
- The threshold determinations currently measure at the mid-points of
-size steps, but this is sub-optimal since at the start of a new step it
-can happen that it's better to go back to the previous k for a while.
-Something more sophisticated for `MUL_FFT_TABLE' and `SQR_FFT_TABLE'
-will be needed.
-
-\1f
-File: gmp.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
-
-16.1.6 Other Multiplication
----------------------------
-
-The Toom algorithms described above (*note Toom 3-Way Multiplication::,
-*note Toom 4-Way Multiplication::) generalizes to split into an
-arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C.
-This is not currently used. The notes here are merely for interest.
-
- In general a split into r+1 pieces is made, and evaluations and
-pointwise multiplications done at 2*r+1 points. A 4-way split does 7
-pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
-algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
-multiplications count towards big-O complexity, but the time spent in
-the evaluate and interpolate stages grows with r and has a significant
-practical impact, with the asymptotic advantage of each r realized only
-at bigger and bigger sizes. The overheads grow as O(N*r), whereas in
-an r=2^k FFT they grow only as O(N*log(r)).
-
- Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
-uses -r,...,0,...,r and the latter saves some small multiplies in the
-evaluate stage (or rather trades them for additions), and has a further
-saving of nearly half the interpolate steps. The idea is to separate
-odd and even final coefficients and then perform algorithm C steps C7
-and C8 on them separately. The divisors at step C7 become j^2 and the
-multipliers at C8 become 2*t*j-j^2.
-
- Splitting odd and even parts through positive and negative points
-can be thought of as using -1 as a square root of unity. If a 4th root
-of unity was available then a further split and speedup would be
-possible, but no such root exists for plain integers. Going to complex
-integers with i=sqrt(-1) doesn't help, essentially because in Cartesian
-form it takes three real multiplies to do a complex multiply. The
-existence of 2^k'th roots of unity in a suitable ring or field lets the
-fast Fourier transform keep splitting and get to O(N*log(r)).
-
- Floating point FFTs use complex numbers approximating Nth roots of
-unity. Some processors have special support for such FFTs. But these
-are not used in GMP since it's very difficult to guarantee an exact
-result (to some number of bits). An occasional difference of 1 in the
-last bit might not matter to a typical signal processing algorithm, but
-is of course of vital importance to GMP.
-
-\1f
-File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
-
-16.1.7 Unbalanced Multiplication
---------------------------------
-
-Multiplication of operands with different sizes, both below
-`MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication
-(*note Basecase Multiplication::).
-
- For really large operands, we invoke FFT directly.
-
- For operands between these sizes, we use Toom inspired algorithms
-suggested by Alberto Zanoni and Marco Bodrato. The idea is to split
-the operands into polynomials of different degree. GMP currently
-splits the smaller operand onto 2 coefficients, i.e., a polynomial of
-degree 1, but the larger operand can be split into 2, 3, or 4
-coefficients, i.e., a polynomial of degree 1 to 3.
-
-\1f
-File: gmp.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
-
-16.2 Division Algorithms
-========================
-
-* Menu:
-
-* Single Limb Division::
-* Basecase Division::
-* Divide and Conquer Division::
-* Block-Wise Barrett Division::
-* Exact Division::
-* Exact Remainder::
-* Small Quotient Division::
-
-\1f
-File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
-
-16.2.1 Single Limb Division
----------------------------
-
-Nx1 division is implemented using repeated 2x1 divisions from high to
-low, either with a hardware divide instruction or a multiplication by
-inverse, whichever is best on a given CPU.
-
- The multiply by inverse follows "Improved division by invariant
-integers" by Mo"ller and Granlund (*note References::) and is
-implemented as `udiv_qrnnd_preinv' in `gmp-impl.h'. The idea is to
-have a fixed-point approximation to 1/d (see `invert_limb') and then
-multiply by the high limb (plus one bit) of the dividend to get a
-quotient q. With d normalized (high bit set), q is no more than 1 too
-small. Subtracting q*d from the dividend gives a remainder, and
-reveals whether q or q-1 is correct.
-
- The result is a division done with two multiplications and four or
-five arithmetic operations. On CPUs with low latency multipliers this
-can be much faster than a hardware divide, though the cost of
-calculating the inverse at the start may mean it's only better on
-inputs bigger than say 4 or 5 limbs.
-
- When a divisor must be normalized, either for the generic C
-`__udiv_qrnnd_c' or the multiply by inverse, the division performed is
-actually a*2^k by d*2^k where a is the dividend and k is the power
-necessary to have the high bit of d*2^k set. The bit shifts for the
-dividend are usually accomplished "on the fly" meaning by extracting
-the appropriate bits at each step. Done this way the quotient limbs
-come out aligned ready to store. When only the remainder is wanted, an
-alternative is to take the dividend limbs unshifted and calculate r = a
-mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
-help on CPUs with poor bit shifts or few registers.
-
- The multiply by inverse can be done two limbs at a time. The
-calculation is basically the same, but the inverse is two limbs and the
-divisor treated as if padded with a low zero limb. This means more
-work, since the inverse will need a 2x2 multiply, but the four 1x1s to
-do that are independent and can therefore be done partly or wholly in
-parallel. Likewise for a 2x1 calculating q*d. The net effect is to
-process two limbs with roughly the same two multiplies worth of latency
-that one limb at a time gives. This extends to 3 or 4 limbs at a time,
-though the extra work to apply the inverse will almost certainly soon
-reach the limits of multiplier throughput.
-
- A similar approach in reverse can be taken to process just half a
-limb at a time if the divisor is only a half limb. In this case the
-1x1 multiply for the inverse effectively becomes two (1/2)x1 for each
-limb, which can be a saving on CPUs with a fast half limb multiply, or
-in fact if the only multiply is a half limb, and especially if it's not
-pipelined.
-
-\1f
-File: gmp.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
-
-16.2.2 Basecase Division
-------------------------
-
-Basecase NxM division is like long division done by hand, but in base
-2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and
-`mpn/generic/sb_divrem_mn.c'.
-
- Briefly stated, while the dividend remains larger than the divisor,
-a high quotient limb is formed and the Nx1 product q*d subtracted at
-the top end of the dividend. With a normalized divisor (most
-significant bit set), each quotient limb can be formed with a 2x1
-division and a 1x1 multiplication plus some subtractions. The 2x1
-division is by the high limb of the divisor and is done either with a
-hardware divide or a multiply by inverse (the same as in *Note Single
-Limb Division::) whichever is faster. Such a quotient is sometimes one
-too big, requiring an addback of the divisor, but that happens rarely.
-
- With Q=N-M being the number of quotient limbs, this is an O(Q*M)
-algorithm and will run at a speed similar to a basecase QxM
-multiplication, differing in fact only in the extra multiply and divide
-for each of the Q quotient limbs.
-
-\1f
-File: gmp.info, Node: Divide and Conquer Division, Next: Block-Wise Barrett Division, Prev: Basecase Division, Up: Division Algorithms
-
-16.2.3 Divide and Conquer Division
-----------------------------------
-
-For divisors larger than `DC_DIV_QR_THRESHOLD', division is done by
-dividing. Or to be precise by a recursive divide and conquer algorithm
-based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
-(*note References::).
-
- The algorithm consists essentially of recognising that a 2NxN
-division can be done with the basecase division algorithm (*note
-Basecase Division::), but using N/2 limbs as a base, not just a single
-limb. This way the multiplications that arise are (N/2)x(N/2) and can
-take advantage of Karatsuba and higher multiplication algorithms (*note
-Multiplication Algorithms::). The two "digits" of the quotient are
-formed by recursive Nx(N/2) divisions.
-
- If the (N/2)x(N/2) multiplies are done with a basecase multiplication
-then the work is about the same as a basecase division, but with more
-function call overheads and with some subtractions separated from the
-multiplies. These overheads mean that it's only when N/2 is above
-`MUL_TOOM22_THRESHOLD' that divide and conquer is of use.
-
- `DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be
-somewhere above twice `MUL_TOOM22_THRESHOLD', but how much above
-depends on the CPU. An optimized `mpn_mul_basecase' can lower
-`DC_DIV_QR_THRESHOLD' a little by offering a ready-made advantage over
-repeated `mpn_submul_1' calls.
-
- Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
-the time for an NxN multiplication done with FFTs. The actual time is
-a sum over multiplications of the recursed sizes, as can be seen near
-the end of section 2.2 of Burnikel and Ziegler. For example, within
-the Toom-3 range, divide and conquer is 2.63*M(N). With higher
-algorithms the M(N) term improves and the multiplier tends to log(N).
-In practice, at moderate to large sizes, a 2NxN division is about 2 to
-4 times slower than an NxN multiplication.
-
-\1f
-File: gmp.info, Node: Block-Wise Barrett Division, Next: Exact Division, Prev: Divide and Conquer Division, Up: Division Algorithms
-
-16.2.4 Block-Wise Barrett Division
-----------------------------------
-
-For the largest divisions, a block-wise Barrett division algorithm is
-used. Here, the divisor is inverted to a precision determined by the
-relative size of the dividend and divisor. Blocks of quotient limbs
-are then generated by multiplying blocks from the dividend by the
-inverse.
-
- Our block-wise algorithm computes a smaller inverse than in the
-plain Barrett algorithm. For a 2n/n division, the inverse will be just
-ceil(n/2) limbs.
-
-\1f
-File: gmp.info, Node: Exact Division, Next: Exact Remainder, Prev: Block-Wise Barrett Division, Up: Division Algorithms
-
-16.2.5 Exact Division
----------------------
-
-A so-called exact division is when the dividend is known to be an exact
-multiple of the divisor. Jebelean's exact division algorithm uses this
-knowledge to make some significant optimizations (*note References::).
-
- The idea can be illustrated in decimal for example with 368154
-divided by 543. Because the low digit of the dividend is 4, the low
-digit of the quotient must be 8. This is arrived at from 4*7 mod 10,
-using the fact 7 is the modular inverse of 3 (the low digit of the
-divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from
-the dividend leaving 363810. Notice the low digit has become zero.
-
- The procedure is repeated at the second digit, with the next
-quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving
-325800. And finally at the third digit with quotient digit 6 (8*7 mod
-10), subtracting 6*543=3258 leaving 0. So the quotient is 678.
-
- Notice however that the multiplies and subtractions don't need to
-extend past the low three digits of the dividend, since that's enough
-to determine the three quotient digits. For the last quotient digit no
-subtraction is needed at all. On a 2NxN division like this one, only
-about half the work of a normal basecase division is necessary.
-
- For an NxM exact division producing Q=N-M quotient limbs, the saving
-over a normal basecase division is in two parts. Firstly, each of the
-Q quotient limbs needs only one multiply, not a 2x1 divide and
-multiply. Secondly, the crossproducts are reduced when Q>M to
-Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are
-complementary. If Q is big then many divisions are saved, or if Q is
-small then the crossproducts reduce to a small number.
-
- The modular inverse used is calculated efficiently by `binvert_limb'
-in `gmp-impl.h'. This does four multiplies for a 32-bit limb, or six
-for a 64-bit limb. `tune/modlinv.c' has some alternate implementations
-that might suit processors better at bit twiddling than multiplying.
-
- The sub-quadratic exact division described by Jebelean in "Exact
-Division with Karatsuba Complexity" is not currently implemented. It
-uses a rearrangement similar to the divide and conquer for normal
-division (*note Divide and Conquer Division::), but operating from low
-to high. A further possibility not currently implemented is
-"Bidirectional Exact Integer Division" by Krandick and Jebelean which
-forms quotient limbs from both the high and low ends of the dividend,
-and can halve once more the number of crossproducts needed in a 2NxN
-division.
-
- A special case exact division by 3 exists in `mpn_divexact_by3',
-supporting Toom-3 multiplication and `mpq' canonicalizations. It forms
-quotient digits with a multiply by the modular inverse of 3 (which is
-`0xAA..AAB') and uses two comparisons to determine a borrow for the next
-limb. The multiplications don't need to be on the dependent chain, as
-long as the effect of the borrows is applied, which can help chips with
-pipelined multipliers.
-
-\1f
-File: gmp.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
-
-16.2.6 Exact Remainder
-----------------------
-
-If the exact division algorithm is done with a full subtraction at each
-stage and the dividend isn't a multiple of the divisor, then low zero
-limbs are produced but with a remainder in the high limbs. For
-dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
-remainder r is of the form
-
- a = q*d + r*b^n
-
- n represents the number of zero limbs produced by the subtractions,
-that being the number of limbs produced for q. r will be in the range
-0<=r<d and can be viewed as a remainder, but one shifted up by a factor
-of b^n.
-
- Carrying out full subtractions at each stage means the same number
-of cross products must be done as a normal division, but there's still
-some single limb divisions saved. When d is a single limb some
-simplifications arise, providing good speedups on a number of
-processors.
-
- `mpn_divexact_by3', `mpn_modexact_1_odd' and the `mpn_redc_X'
-functions differ subtly in how they return r, leading to some negations
-in the above formula, but all are essentially the same.
-
- Clearly r is zero when a is a multiple of d, and this leads to
-divisibility or congruence tests which are potentially more efficient
-than a normal division.
-
- The factor of b^n on r can be ignored in a GCD when d is odd, hence
-the use of `mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui'
-etc (*note Greatest Common Divisor Algorithms::).
-
- Montgomery's REDC method for modular multiplications uses operands
-of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
-uses the factor of b^n in the exact remainder to reach a product in the
-same form (x*y)*b^-n (*note Modular Powering Algorithm::).
-
- Notice that r generally gives no useful information about the
-ordinary remainder a mod d since b^n mod d could be anything. If
-however b^n == 1 mod d, then r is the negative of the ordinary
-remainder. This occurs whenever d is a factor of b^n-1, as for example
-with 3 in `mpn_divexact_by3'. For a 32 or 64 bit limb other such
-factors include 5, 17 and 257, but no particular use has been found for
-this.
-
-\1f
-File: gmp.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
-
-16.2.7 Small Quotient Division
-------------------------------
-
-An NxM division where the number of quotient limbs Q=N-M is small can
-be optimized somewhat.
-
- An ordinary basecase division normalizes the divisor by shifting it
-to make the high bit set, shifting the dividend accordingly, and
-shifting the remainder back down at the end of the calculation. This
-is wasteful if only a few quotient limbs are to be formed. Instead a
-division of just the top 2*Q limbs of the dividend by the top Q limbs
-of the divisor can be used to form a trial quotient. This requires
-only those limbs normalized, not the whole of the divisor and dividend.
-
- A multiply and subtract then applies the trial quotient to the M-Q
-unused limbs of the divisor and N-Q dividend limbs (which includes Q
-limbs remaining from the trial quotient division). The starting trial
-quotient can be 1 or 2 too big, but all cases of 2 too big and most
-cases of 1 too big are detected by first comparing the most significant
-limbs that will arise from the subtraction. An addback is done if the
-quotient still turns out to be 1 too big.
-
- This whole procedure is essentially the same as one step of the
-basecase algorithm done in a Q limb base, though with the trial
-quotient test done only with the high limbs, not an entire Q limb
-"digit" product. The correctness of this weaker test can be
-established by following the argument of Knuth section 4.3.1 exercise
-20 but with the v2*q>b*r+u2 condition appropriately relaxed.
-
-\1f
-File: gmp.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
-
-16.3 Greatest Common Divisor
-============================
-
-* Menu:
-
-* Binary GCD::
-* Lehmer's Algorithm::
-* Subquadratic GCD::
-* Extended GCD::
-* Jacobi Symbol::
-
-\1f
-File: gmp.info, Node: Binary GCD, Next: Lehmer's Algorithm, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
-
-16.3.1 Binary GCD
------------------
-
-At small sizes GMP uses an O(N^2) binary style GCD. This is described
-in many textbooks, for example Knuth section 4.5.2 algorithm B. It
-simply consists of successively reducing odd operands a and b using
-
- a,b = abs(a-b),min(a,b)
- strip factors of 2 from a
-
- The Euclidean GCD algorithm, as per Knuth algorithms E and A,
-repeatedly computes the quotient q = floor(a/b) and replaces a,b by v,
-u - q v. The binary algorithm has so far been found to be faster than
-the Euclidean algorithm everywhere. One reason the binary method does
-well is that the implied quotient at each step is usually small, so
-often only one or two subtractions are needed to get the same effect as
-a division. Quotients 1, 2 and 3 for example occur 67.7% of the time,
-see Knuth section 4.5.3 Theorem E.
-
- When the implied quotient is large, meaning b is much smaller than
-a, then a division is worthwhile. This is the basis for the initial a
-mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
-and 1x1 cases). But after that initial reduction, big quotients occur
-too rarely to make it worth checking for them.
-
-
- The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
-described above. For two N-bit operands, the algorithm takes about
-0.68 iterations per bit. For optimum performance some attention needs
-to be paid to the way the factors of 2 are stripped from a.
-
- Firstly it may be noted that in twos complement the number of low
-zero bits on a-b is the same as b-a, so counting or testing can begin on
-a-b without waiting for abs(a-b) to be determined.
-
- A loop stripping low zero bits tends not to branch predict well,
-since the condition is data dependent. But on average there's only a
-few low zeros, so an option is to strip one or two bits arithmetically
-then loop for more (as done for AMD K6). Or use a lookup table to get
-a count for several bits then loop for more (as done for AMD K7). An
-alternative approach is to keep just one of a or b odd and iterate
-
- a,b = abs(a-b), min(a,b)
- a = a/2 if even
- b = b/2 if even
-
- This requires about 1.25 iterations per bit, but stripping of a
-single bit at each step avoids any branching. Repeating the bit strip
-reduces to about 0.9 iterations per bit, which may be a worthwhile
-tradeoff.
-
- Generally with the above approaches a speed of perhaps 6 cycles per
-bit can be achieved, which is still not terribly fast with for instance
-a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
-means it's not usually advantageous to combine a set of divisibility
-tests into a GCD.
-
- Currently, the binary algorithm is used for GCD only when N < 3.
-
-\1f
-File: gmp.info, Node: Lehmer's Algorithm, Next: Subquadratic GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.2 Lehmer's algorithm
--------------------------
-
-Lehmer's improvement of the Euclidean algorithms is based on the
-observation that the initial part of the quotient sequence depends only
-on the most significant parts of the inputs. The variant of Lehmer's
-algorithm used in GMP splits off the most significant two limbs, as
-suggested, e.g., in "A Double-Digit Lehmer-Euclid Algorithm" by
-Jebelean (*note References::). The quotients of two double-limb inputs
-are collected as a 2 by 2 matrix with single-limb elements. This is
-done by the function `mpn_hgcd2'. The resulting matrix is applied to
-the inputs using `mpn_mul_1' and `mpn_submul_1'. Each iteration usually
-reduces the inputs by almost one limb. In the rare case of a large
-quotient, no progress can be made by examining just the most
-significant two limbs, and the quotient is computing using plain
-division.
-
- The resulting algorithm is asymptotically O(N^2), just as the
-Euclidean algorithm and the binary algorithm. The quadratic part of the
-work are the calls to `mpn_mul_1' and `mpn_submul_1'. For small sizes,
-the linear work is also significant. There are roughly N calls to the
-`mpn_hgcd2' function. This function uses a couple of important
-optimizations:
-
- * It uses the same relaxed notion of correctness as `mpn_hgcd' (see
- next section). This means that when called with the most
- significant two limbs of two large numbers, the returned matrix
- does not always correspond exactly to the initial quotient
- sequence for the two large numbers; the final quotient may
- sometimes be one off.
-
- * It takes advantage of the fact the quotients are usually small.
- The division operator is not used, since the corresponding
- assembler instruction is very slow on most architectures. (This
- code could probably be improved further, it uses many branches
- that are unfriendly to prediction).
-
- * It switches from double-limb calculations to single-limb
- calculations half-way through, when the input numbers have been
- reduced in size from two limbs to one and a half.
-
-
-\1f
-File: gmp.info, Node: Subquadratic GCD, Next: Extended GCD, Prev: Lehmer's Algorithm, Up: Greatest Common Divisor Algorithms
-
-16.3.3 Subquadratic GCD
------------------------
-
-For inputs larger than `GCD_DC_THRESHOLD', GCD is computed via the HGCD
-(Half GCD) function, as a generalization to Lehmer's algorithm.
-
- Let the inputs a,b be of size N limbs each. Put S = floor(N/2) + 1.
-Then HGCD(a,b) returns a transformation matrix T with non-negative
-elements, and reduced numbers (c;d) = T^-1 (a;b). The reduced numbers
-c,d must be larger than S limbs, while their difference abs(c-d) must
-fit in S limbs. The matrix elements will also be of size roughly N/2.
-
- The HGCD base case uses Lehmer's algorithm, but with the above stop
-condition that returns reduced numbers and the corresponding
-transformation matrix half-way through. For inputs larger than
-`HGCD_THRESHOLD', HGCD is computed recursively, using the divide and
-conquer algorithm in "On Scho"nhage's algorithm and subquadratic
-integer GCD computation" by Mo"ller (*note References::). The recursive
-algorithm consists of these main steps.
-
- * Call HGCD recursively, on the most significant N/2 limbs. Apply the
- resulting matrix T_1 to the full numbers, reducing them to a size
- just above 3N/2.
-
- * Perform a small number of division or subtraction steps to reduce
- the numbers to size below 3N/2. This is essential mainly for the
- unlikely case of large quotients.
-
- * Call HGCD recursively, on the most significant N/2 limbs of the
- reduced numbers. Apply the resulting matrix T_2 to the full
- numbers, reducing them to a size just above N/2.
-
- * Compute T = T_1 T_2.
-
- * Perform a small number of division and subtraction steps to
- satisfy the requirements, and return.
-
- GCD is then implemented as a loop around HGCD, similarly to Lehmer's
-algorithm. Where Lehmer repeatedly chops off the top two limbs, calls
-`mpn_hgcd2', and applies the resulting matrix to the full numbers, the
-subquadratic GCD chops off the most significant third of the limbs (the
-proportion is a tuning parameter, and 1/3 seems to be more efficient
-than, e.g, 1/2), calls `mpn_hgcd', and applies the resulting matrix.
-Once the input numbers are reduced to size below `GCD_DC_THRESHOLD',
-Lehmer's algorithm is used for the rest of the work.
-
- The asymptotic running time of both HGCD and GCD is O(M(N)*log(N)),
-where M(N) is the time for multiplying two N-limb numbers.
-
-\1f
-File: gmp.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Subquadratic GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.4 Extended GCD
--------------------
-
-The extended GCD function, or GCDEXT, calculates gcd(a,b) and also
-cofactors x and y satisfying a*x+b*y=gcd(a,b). All the algorithms used
-for plain GCD are extended to handle this case. The binary algorithm is
-used only for single-limb GCDEXT. Lehmer's algorithm is used for sizes
-up to `GCDEXT_DC_THRESHOLD'. Above this threshold, GCDEXT is
-implemented as a loop around HGCD, but with more book-keeping to keep
-track of the cofactors. This gives the same asymptotic running time as
-for GCD and HGCD, O(M(N)*log(N))
-
- One difference to plain GCD is that while the inputs a and b are
-reduced as the algorithm proceeds, the cofactors x and y grow in size.
-This makes the tuning of the chopping-point more difficult. The current
-code chops off the most significant half of the inputs for the call to
-HGCD in the first iteration, and the most significant two thirds for
-the remaining calls. This strategy could surely be improved. Also the
-stop condition for the loop, where Lehmer's algorithm is invoked once
-the inputs are reduced below `GCDEXT_DC_THRESHOLD', could maybe be
-improved by taking into account the current size of the cofactors.
-
-\1f
-File: gmp.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
-
-16.3.5 Jacobi Symbol
---------------------
-
-`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
-simple binary algorithm similar to that described for the GCDs (*note
-Binary GCD::). They're not very fast when both inputs are large.
-Lehmer's multi-step improvement or a binary based multi-step algorithm
-is likely to be better.
-
- When one operand fits a single limb, and that includes
-`mpz_kronecker_ui' and friends, an initial reduction is done with
-either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
-algorithm on a single limb. The binary algorithm is well suited to a
-single limb, and the whole calculation in this case is quite efficient.
-
- In all the routines sign changes for the result are accumulated
-using some bit twiddling, avoiding table lookups or conditional jumps.
-
+++ /dev/null
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
-
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
-
- Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
-2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010 Free
-Software Foundation, Inc.
-
- Permission is granted to copy, distribute and/or modify this
-document under the terms of the GNU Free Documentation License, Version
-1.3 or any later version published by the Free Software Foundation;
-with no Invariant Sections, with the Front-Cover Texts being "A GNU
-Manual", and with the Back-Cover Texts being "You have freedom to copy
-and modify this GNU Manual, like GNU software". A copy of the license
-is included in *Note GNU Free Documentation License::.
-
-INFO-DIR-SECTION GNU libraries
-START-INFO-DIR-ENTRY
-* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
-END-INFO-DIR-ENTRY
-
-\1f
-File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
-
-16.4 Powering Algorithms
-========================
-
-* Menu:
-
-* Normal Powering Algorithm::
-* Modular Powering Algorithm::
-
-\1f
-File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
-
-16.4.1 Normal Powering
-----------------------
-
-Normal `mpz' or `mpf' powering uses a simple binary algorithm,
-successively squaring and then multiplying by the base when a 1 bit is
-seen in the exponent, as per Knuth section 4.6.3. The "left to right"
-variant described there is used rather than algorithm A, since it's
-just as easy and can be done with somewhat less temporary memory.
-
-\1f
-File: gmp.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
-
-16.4.2 Modular Powering
------------------------
-
-Modular powering is implemented using a 2^k-ary sliding window
-algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
-(*note References::). k is chosen according to the size of the
-exponent. Larger exponents use larger values of k, the choice being
-made to minimize the average number of multiplications that must
-supplement the squaring.
-
- The modular multiplies and squares use either a simple division or
-the REDC method by Montgomery (*note References::). REDC is a little
-faster, essentially saving N single limb divisions in a fashion similar
-to an exact remainder (*note Exact Remainder::).
-
-\1f
-File: gmp.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
-
-16.5 Root Extraction Algorithms
-===============================
-
-* Menu:
-
-* Square Root Algorithm::
-* Nth Root Algorithm::
-* Perfect Square Algorithm::
-* Perfect Power Algorithm::
-
-\1f
-File: gmp.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
-
-16.5.1 Square Root
-------------------
-
-Square roots are taken using the "Karatsuba Square Root" algorithm by
-Paul Zimmermann (*note References::).
-
- An input n is split into four parts of k bits each, so with b=2^k we
-have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
-that either the high or second highest bit is set. In GMP, k is kept
-on a limb boundary and the input is left shifted (by an even number of
-bits) to normalize.
-
- The square root of the high two parts is taken, by recursive
-application of the algorithm (bottoming out in a one-limb Newton's
-method),
-
- s1,r1 = sqrtrem (a3*b + a2)
-
- This is an approximation to the desired root and is extended by a
-division to give s,r,
-
- q,u = divrem (r1*b + a1, 2*s1)
- s = s1*b + q
- r = u*b + a0 - q^2
-
- The normalization requirement on a3 means at this point s is either
-correct or 1 too big. r is negative in the latter case, so
-
- if r < 0 then
- r = r + 2*s - 1
- s = s - 1
-
- The algorithm is expressed in a divide and conquer form, but as
-noted in the paper it can also be viewed as a discrete variant of
-Newton's method, or as a variation on the schoolboy method (no longer
-taught) for square roots two digits at a time.
-
- If the remainder r is not required then usually only a few high limbs
-of r and u need to be calculated to determine whether an adjustment to
-s is required. This optimization is not currently implemented.
-
- In the Karatsuba multiplication range this algorithm is
-O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
-limbs. In the FFT multiplication range this grows to a bound of
-O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
-Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
-
- The algorithm does all its calculations in integers and the resulting
-`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
-precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.
-
-\1f
-File: gmp.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
-
-16.5.2 Nth Root
----------------
-
-Integer Nth roots are taken using Newton's method with the following
-iteration, where A is the input and n is the root to be taken.
-
- 1 A
- a[i+1] = - * ( --------- + (n-1)*a[i] )
- n a[i]^(n-1)
-
- The initial approximation a[1] is generated bitwise by successively
-powering a trial root with or without new 1 bits, aiming to be just
-above the true root. The iteration converges quadratically when
-started from a good approximation. When n is large more initial bits
-are needed to get good convergence. The current implementation is not
-particularly well optimized.
-
-\1f
-File: gmp.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
-
-16.5.3 Perfect Square
----------------------
-
-A significant fraction of non-squares can be quickly identified by
-checking whether the input is a quadratic residue modulo small integers.
-
- `mpz_perfect_square_p' first tests the input mod 256, which means
-just examining the low byte. Only 44 different values occur for
-squares mod 256, so 82.8% of inputs can be immediately identified as
-non-squares.
-
- On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
-for a total 99.25% of inputs identified as non-squares. On a 64-bit
-system 97 is tested too, for a total 99.62%.
-
- These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
-for 64-bits), and such a remainder can be quickly taken just using
-additions (see `mpn_mod_34lsub1').
-
- When nails are in use moduli are instead selected by the `gen-psqr.c'
-program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
-could be done with nails using some extra bit shifts, but this is not
-currently implemented.
-
- In any case each modulus is applied to the `mpn_mod_34lsub1' or
-`mpn_mod_1' remainder and a table lookup identifies non-squares. By
-using a "modexact" style calculation, and suitably permuted tables,
-just one multiply each is required, see the code for details. Moduli
-are also combined to save operations, so long as the lookup tables
-don't become too big. `gen-psqr.c' does all the pre-calculations.
-
- A square root must still be taken for any value that passes these
-tests, to verify it's really a square and not one of the small fraction
-of non-squares that get through (ie. a pseudo-square to all the tested
-bases).
-
- Clearly more residue tests could be done, `mpz_perfect_square_p' only
-uses a compact and efficient set. Big inputs would probably benefit
-from more residue testing, small inputs might be better off with less.
-The assumed distribution of squares versus non-squares in the input
-would affect such considerations.
-
-\1f
-File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
-
-16.5.4 Perfect Power
---------------------
-
-Detecting perfect powers is required by some factorization algorithms.
-Currently `mpz_perfect_power_p' is implemented using repeated Nth root
-extractions, though naturally only prime roots need to be considered.
-(*Note Nth Root Algorithm::.)
-
- If a prime divisor p with multiplicity e can be found, then only
-roots which are divisors of e need to be considered, much reducing the
-work necessary. To this end divisibility by a set of small primes is
-checked.
-
-\1f
-File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
-
-16.6 Radix Conversion
-=====================
-
-Radix conversions are less important than other algorithms. A program
-dominated by conversions should probably use a different data
-representation.
-
-* Menu:
-
-* Binary to Radix::
-* Radix to Binary::
-
-\1f
-File: gmp.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
-
-16.6.1 Binary to Radix
-----------------------
-
-Conversions from binary to a power-of-2 radix use a simple and fast
-O(N) bit extraction algorithm.
-
- Conversions from binary to other radices use one of two algorithms.
-Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
-Repeated divisions by b^n are made, where b is the radix and n is the
-biggest power that fits in a limb. But instead of simply using the
-remainder r from such divisions, an extra divide step is done to give a
-fractional limb representing r/b^n. The digits of r can then be
-extracted using multiplications by b rather than divisions. Special
-case code is provided for decimal, allowing multiplications by 10 to
-optimize to shifts and adds.
-
- Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
-used. For an input t, powers b^(n*2^i) of the radix are calculated,
-until a power between t and sqrt(t) is reached. t is then divided by
-that largest power, giving a quotient which is the digits above that
-power, and a remainder which is those below. These two parts are in
-turn divided by the second highest power, and so on recursively. When
-a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
-limbs, the basecase algorithm described above is used.
-
- The advantage of this algorithm is that big divisions can make use
-of the sub-quadratic divide and conquer division (*note Divide and
-Conquer Division::), and big divisions tend to have less overheads than
-lots of separate single limb divisions anyway. But in any case the
-cost of calculating the powers b^(n*2^i) must first be overcome.
-
- `GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
-the same basic thing, the point where it becomes worth doing a big
-division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
-includes the cost of calculating the radix power required, whereas
-`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
-case when recursing.
-
- Since the base case produces digits from least to most significant
-but they want to be stored from most to least, it's necessary to
-calculate in advance how many digits there will be, or at least be sure
-not to underestimate that. For GMP the number of input bits is
-multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
-The result is either correct or one too big.
-
- Examining some of the high bits of the input could increase the
-chance of getting the exact number of digits, but an exact result every
-time would not be practical, since in general the difference between
-numbers 100... and 99... is only in the last few bits and the work to
-identify 99... might well be almost as much as a full conversion.
-
- `mpf_get_str' doesn't currently use the algorithm described here, it
-multiplies or divides by a power of b to move the radix point to the
-just above the highest non-zero digit (or at worst one above that
-location), then multiplies by b^n to bring out digits. This is O(N^2)
-and is certainly not optimal.
-
- The r/b^n scheme described above for using multiplications to bring
-out digits might be useful for more than a single limb. Some brief
-experiments with it on the base case when recursing didn't give a
-noticeable improvement, but perhaps that was only due to the
-implementation. Something similar would work for the sub-quadratic
-divisions too, though there would be the cost of calculating a bigger
-radix power.
-
- Another possible improvement for the sub-quadratic part would be to
-arrange for radix powers that balanced the sizes of quotient and
-remainder produced, ie. the highest power would be an b^(n*k)
-approximately equal to sqrt(t), not restricted to a 2^i factor. That
-ought to smooth out a graph of times against sizes, but may or may not
-be a net speedup.
-
-\1f
-File: gmp.info, Node: Radix to Binary, Prev: Binary to Radix, Up: Radix Conversion Algorithms
-
-16.6.2 Radix to Binary
-----------------------
-
-*This section needs to be rewritten, it currently describes the
-algorithms used before GMP 4.3.*
-
- Conversions from a power-of-2 radix into binary use a simple and fast
-O(N) bitwise concatenation algorithm.
-
- Conversions from other radices use one of two algorithms. Sizes
-below `SET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method. Groups
-of n digits are converted to limbs, where n is the biggest power of the
-base b which will fit in a limb, then those groups are accumulated into
-the result by multiplying by b^n and adding. This saves
-multi-precision operations, as per Knuth section 4.4 part E (*note
-References::). Some special case code is provided for decimal, giving
-the compiler a chance to optimize multiplications by 10.
-
- Above `SET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
-used. First groups of n digits are converted into limbs. Then adjacent
-limbs are combined into limb pairs with x*b^n+y, where x and y are the
-limbs. Adjacent limb pairs are combined into quads similarly with
-x*b^(2n)+y. This continues until a single block remains, that being
-the result.
-
- The advantage of this method is that the multiplications for each x
-are big blocks, allowing Karatsuba and higher algorithms to be used.
-But the cost of calculating the powers b^(n*2^i) must be overcome.
-`SET_STR_PRECOMPUTE_THRESHOLD' usually ends up quite big, around 5000
-digits, and on some processors much bigger still.
-
- `SET_STR_PRECOMPUTE_THRESHOLD' is based on the input digits (and
-tuned for decimal), though it might be better based on a limb count, so
-as to be independent of the base. But that sort of count isn't used by
-the base case and so would need some sort of initial calculation or
-estimate.
-
- The main reason `SET_STR_PRECOMPUTE_THRESHOLD' is so much bigger
-than the corresponding `GET_STR_PRECOMPUTE_THRESHOLD' is that
-`mpn_mul_1' is much faster than `mpn_divrem_1' (often by a factor of 5,
-or more).
-
-\1f
-File: gmp.info, Node: Other Algorithms, Next: Assembly Coding, Prev: Radix Conversion Algorithms, Up: Algorithms
-
-16.7 Other Algorithms
-=====================
-
-* Menu:
-
-* Prime Testing Algorithm::
-* Factorial Algorithm::
-* Binomial Coefficients Algorithm::
-* Fibonacci Numbers Algorithm::
-* Lucas Numbers Algorithm::
-* Random Number Algorithms::
-
-\1f
-File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
-
-16.7.1 Prime Testing
---------------------
-
-The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
-Functions::) first does some trial division by small factors and then
-uses the Miller-Rabin probabilistic primality testing algorithm, as
-described in Knuth section 4.5.4 algorithm P (*note References::).
-
- For an odd input n, and with n = q*2^k+1 where q is odd, this
-algorithm selects a random base x and tests whether x^q mod n is 1 or
--1, or an x^(q*2^j) mod n is 1, for 1<=j<=k. If so then n is probably
-prime, if not then n is definitely composite.
-
- Any prime n will pass the test, but some composites do too. Such
-composites are known as strong pseudoprimes to base x. No n is a
-strong pseudoprime to more than 1/4 of all bases (see Knuth exercise
-22), hence with x chosen at random there's no more than a 1/4 chance a
-"probable prime" will in fact be composite.
-
- In fact strong pseudoprimes are quite rare, making the test much more
-powerful than this analysis would suggest, but 1/4 is all that's proven
-for an arbitrary n.
-
-\1f
-File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
-
-16.7.2 Factorial
-----------------
-
-Factorials are calculated by a combination of removal of twos,
-powering, and binary splitting. The procedure can be best illustrated
-with an example,
-
- 23! = 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18.19.20.21.22.23
-
-has factors of two removed,
-
- 23! = 2^19.1.1.3.1.5.3.7.1.9.5.11.3.13.7.15.1.17.9.19.5.21.11.23
-
-and the resulting terms collected up according to their multiplicity,
-
- 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)
-
- Each sequence such as 13.15.17.19.21.23 is evaluated by splitting
-into every second term, as for instance (13.17.21).(15.19.23), and the
-same recursively on each half. This is implemented iteratively using
-some bit twiddling.
-
- Such splitting is more efficient than repeated Nx1 multiplies since
-it forms big multiplies, allowing Karatsuba and higher algorithms to be
-used. And even below the Karatsuba threshold a big block of work can
-be more efficient for the basecase algorithm.
-
- Splitting into subsequences of every second term keeps the resulting
-products more nearly equal in size than would the simpler approach of
-say taking the first half and second half of the sequence. Nearly
-equal products are more efficient for the current multiply
-implementation.
-
-\1f
-File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
-
-16.7.3 Binomial Coefficients
-----------------------------
-
-Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
-using C(n,k) = C(n,n-k) if necessary, and then evaluating the following
-product simply from i=2 to i=k.
-
- k (n-k+i)
- C(n,k) = (n-k+1) * prod -------
- i=2 i
-
- It's easy to show that each denominator i will divide the product so
-far, so the exact division algorithm is used (*note Exact Division::).
-
- The numerators n-k+i and denominators i are first accumulated into
-as many fit a limb, to save multi-precision operations, though for
-`mpz_bin_ui' this applies only to the divisors, since n is an `mpz_t'
-and n-k+i in general won't fit in a limb at all.
-
-\1f
-File: gmp.info, Node: Fibonacci Numbers Algorithm, Next: Lucas Numbers Algorithm, Prev: Binomial Coefficients Algorithm, Up: Other Algorithms
-
-16.7.4 Fibonacci Numbers
-------------------------
-
-The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
-calculating isolated F[n] or F[n],F[n-1] values efficiently.
-
- For small n, a table of single limb values in `__gmp_fib_table' is
-used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb up
-to F[93]. For convenience the table starts at F[-1].
-
- Beyond the table, values are generated with a binary powering
-algorithm, calculating a pair F[n] and F[n-1] working from high to low
-across the bits of n. The formulas used are
-
- F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k
- F[2k-1] = F[k]^2 + F[k-1]^2
-
- F[2k] = F[2k+1] - F[2k-1]
-
- At each step, k is the high b bits of n. If the next bit of n is 0
-then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used,
-and the process repeated until all bits of n are incorporated. Notice
-these formulas require just two squares per bit of n.
-
- It'd be possible to handle the first few n above the single limb
-table with simple additions, using the defining Fibonacci recurrence
-F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to
-be faster for only about 10 or 20 values of n, and including a block of
-code for just those doesn't seem worthwhile. If they really mattered
-it'd be better to extend the data table.
-
- Using a table avoids lots of calculations on small numbers, and
-makes small n go fast. A bigger table would make more small n go fast,
-it's just a question of balancing size against desired speed. For GMP
-the code is kept compact, with the emphasis primarily on a good
-powering algorithm.
-
- `mpz_fib2_ui' returns both F[n] and F[n-1], but `mpz_fib_ui' is only
-interested in F[n]. In this case the last step of the algorithm can
-become one multiply instead of two squares. One of the following two
-formulas is used, according as n is odd or even.
-
- F[2k] = F[k]*(F[k]+2F[k-1])
-
- F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
-
- F[2k+1] here is the same as above, just rearranged to be a multiply.
-For interest, the 2*(-1)^k term both here and above can be applied
-just to the low limb of the calculation, without a carry or borrow into
-further limbs, which saves some code size. See comments with
-`mpz_fib_ui' and the internal `mpn_fib2_ui' for how this is done.
-
-\1f
-File: gmp.info, Node: Lucas Numbers Algorithm, Next: Random Number Algorithms, Prev: Fibonacci Numbers Algorithm, Up: Other Algorithms
-
-16.7.5 Lucas Numbers
---------------------
-
-`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
-Fibonacci numbers with the following simple formulas.
-
- L[k] = F[k] + 2*F[k-1]
- L[k-1] = 2*F[k] - F[k-1]
-
- `mpz_lucnum_ui' is only interested in L[n], and some work can be
-saved. Trailing zero bits on n can be handled with a single square
-each.
-
- L[2k] = L[k]^2 - 2*(-1)^k
-
- And the lowest 1 bit can be handled with one multiply of a pair of
-Fibonacci numbers, similar to what `mpz_fib_ui' does.
-
- L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
-
-\1f
-File: gmp.info, Node: Random Number Algorithms, Prev: Lucas Numbers Algorithm, Up: Other Algorithms
-
-16.7.6 Random Numbers
----------------------
-
-For the `urandomb' functions, random numbers are generated simply by
-concatenating bits produced by the generator. As long as the generator
-has good randomness properties this will produce well-distributed N bit
-numbers.
-
- For the `urandomm' functions, random numbers in a range 0<=R<N are
-generated by taking values R of ceil(log2(N)) bits each until one
-satisfies R<N. This will normally require only one or two attempts,
-but the attempts are limited in case the generator is somehow
-degenerate and produces only 1 bits or similar.
-
- The Mersenne Twister generator is by Matsumoto and Nishimura (*note
-References::). It has a non-repeating period of 2^19937-1, which is a
-Mersenne prime, hence the name of the generator. The state is 624
-words of 32-bits each, which is iterated with one XOR and shift for each
-32-bit word generated, making the algorithm very fast. Randomness
-properties are also very good and this is the default algorithm used by
-GMP.
-
- Linear congruential generators are described in many text books, for
-instance Knuth volume 2 (*note References::). With a modulus M and
-parameters A and C, a integer state S is iterated by the formula S <-
-A*S+C mod M. At each step the new state is a linear function of the
-previous, mod M, hence the name of the generator.
-
- In GMP only moduli of the form 2^N are supported, and the current
-implementation is not as well optimized as it could be. Overheads are
-significant when N is small, and when N is large clearly the multiply
-at each step will become slow. This is not a big concern, since the
-Mersenne Twister generator is better in every respect and is therefore
-recommended for all normal applications.
-
- For both generators the current state can be deduced by observing
-enough output and applying some linear algebra (over GF(2) in the case
-of the Mersenne Twister). This generally means raw output is
-unsuitable for cryptographic applications without further hashing or
-the like.
-
-\1f
-File: gmp.info, Node: Assembly Coding, Prev: Other Algorithms, Up: Algorithms
-
-16.8 Assembly Coding
-====================
-
-The assembly subroutines in GMP are the most significant source of
-speed at small to moderate sizes. At larger sizes algorithm selection
-becomes more important, but of course speedups in low level routines
-will still speed up everything proportionally.
-
- Carry handling and widening multiplies that are important for GMP
-can't be easily expressed in C. GCC `asm' blocks help a lot and are
-provided in `longlong.h', but hand coding low level routines invariably
-offers a speedup over generic C by a factor of anything from 2 to 10.
-
-* Menu:
-
-* Assembly Code Organisation::
-* Assembly Basics::
-* Assembly Carry Propagation::
-* Assembly Cache Handling::
-* Assembly Functional Units::
-* Assembly Floating Point::
-* Assembly SIMD Instructions::
-* Assembly Software Pipelining::
-* Assembly Loop Unrolling::
-* Assembly Writing Guide::
-
-\1f
-File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding
-
-16.8.1 Code Organisation
-------------------------
-
-The various `mpn' subdirectories contain machine-dependent code, written
-in C or assembly. The `mpn/generic' subdirectory contains default code,
-used when there's no machine-specific version of a particular file.
-
- Each `mpn' subdirectory is for an ISA family. Generally 32-bit and
-64-bit variants in a family cannot share code and have separate
-directories. Within a family further subdirectories may exist for CPU
-variants.
-
- In each directory a `nails' subdirectory may exist, holding code with
-nails support for that CPU variant. A `NAILS_SUPPORT' directive in each
-file indicates the nails values the code handles. Nails code only
-exists where it's faster, or promises to be faster, than plain code.
-There's no effort put into nails if they're not going to enhance a
-given CPU.
-
-\1f
-File: gmp.info, Node: Assembly Basics, Next: Assembly Carry Propagation, Prev: Assembly Code Organisation, Up: Assembly Coding
-
-16.8.2 Assembly Basics
-----------------------
-
-`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
-overall GMP performance. All multiplications and divisions come down to
-repeated calls to these. `mpn_add_n', `mpn_sub_n', `mpn_lshift' and
-`mpn_rshift' are next most important.
-
- On some CPUs assembly versions of the internal functions
-`mpn_mul_basecase' and `mpn_sqr_basecase' give significant speedups,
-mainly through avoiding function call overheads. They can also
-potentially make better use of a wide superscalar processor, as can
-bigger primitives like `mpn_addmul_2' or `mpn_addmul_4'.
-
- The restrictions on overlaps between sources and destinations (*note
-Low-level Functions::) are designed to facilitate a variety of
-implementations. For example, knowing `mpn_add_n' won't have partly
-overlapping sources and destination means reading can be done far ahead
-of writing on superscalar processors, and loops can be vectorized on a
-vector processor, depending on the carry handling.
-
-\1f
-File: gmp.info, Node: Assembly Carry Propagation, Next: Assembly Cache Handling, Prev: Assembly Basics, Up: Assembly Coding
-
-16.8.3 Carry Propagation
-------------------------
-
-The problem that presents most challenges in GMP is propagating carries
-from one limb to the next. In functions like `mpn_addmul_1' and
-`mpn_add_n', carries are the only dependencies between limb operations.
-
- On processors with carry flags, a straightforward CISC style `adc' is
-generally best. AMD K6 `mpn_addmul_1' however is an example of an
-unusual set of circumstances where a branch works out better.
-
- On RISC processors generally an add and compare for overflow is
-used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some
-carry propagation schemes require 4 instructions, meaning at least 4
-cycles per limb, but other schemes may use just 1 or 2. On wide
-superscalar processors performance may be completely determined by the
-number of dependent instructions between carry-in and carry-out for
-each limb.
-
- On vector processors good use can be made of the fact that a carry
-bit only very rarely propagates more than one limb. When adding a
-single bit to a limb, there's only a carry out if that limb was
-`0xFF...FF' which on random data will be only 1 in 2^mp_bits_per_limb.
-`mpn/cray/add_n.c' is an example of this, it adds all limbs in
-parallel, adds one set of carry bits in parallel and then only rarely
-needs to fall through to a loop propagating further carries.
-
- On the x86s, GCC (as of version 2.95.2) doesn't generate
-particularly good code for the RISC style idioms that are necessary to
-handle carry bits in C. Often conditional jumps are generated where
-`adc' or `sbb' forms would be better. And so unfortunately almost any
-loop involving carry bits needs to be coded in assembly for best
-results.
-
-\1f
-File: gmp.info, Node: Assembly Cache Handling, Next: Assembly Functional Units, Prev: Assembly Carry Propagation, Up: Assembly Coding
-
-16.8.4 Cache Handling
----------------------
-
-GMP aims to perform well both on operands that fit entirely in L1 cache
-and those which don't.
-
- Basic routines like `mpn_add_n' or `mpn_lshift' are often used on
-large operands, so L2 and main memory performance is important for them.
-`mpn_mul_1' and `mpn_addmul_1' are mostly used for multiply and square
-basecases, so L1 performance matters most for them, unless assembly
-versions of `mpn_mul_basecase' and `mpn_sqr_basecase' exist, in which
-case the remaining uses are mostly for larger operands.
-
- For L2 or main memory operands, memory access times will almost
-certainly be more than the calculation time. The aim therefore is to
-maximize memory throughput, by starting a load of the next cache line
-while processing the contents of the previous one. Clearly this is
-only possible if the chip has a lock-up free cache or some sort of
-prefetch instruction. Most current chips have both these features.
-
- Prefetching sources combines well with loop unrolling, since a
-prefetch can be initiated once per unrolled loop (or more than once if
-the loop covers more than one cache line).
-
- On CPUs without write-allocate caches, prefetching destinations will
-ensure individual stores don't go further down the cache hierarchy,
-limiting bandwidth. Of course for calculations which are slow anyway,
-like `mpn_divrem_1', write-throughs might be fine.
-
- The distance ahead to prefetch will be determined by memory latency
-versus throughput. The aim of course is to have data arriving
-continuously, at peak throughput. Some CPUs have limits on the number
-of fetches or prefetches in progress.
-
- If a special prefetch instruction doesn't exist then a plain load
-can be used, but in that case care must be taken not to attempt to read
-past the end of an operand, since that might produce a segmentation
-violation.
-
- Some CPUs or systems have hardware that detects sequential memory
-accesses and initiates suitable cache movements automatically, making
-life easy.
-
-\1f
-File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding
-
-16.8.5 Functional Units
------------------------
-
-When choosing an approach for an assembly loop, consideration is given
-to what operations can execute simultaneously and what throughput can
-thereby be achieved. In some cases an algorithm can be tweaked to
-accommodate available resources.
-
- Loop control will generally require a counter and pointer updates,
-costing as much as 5 instructions, plus any delays a branch introduces.
-CPU addressing modes might reduce pointer updates, perhaps by allowing
-just one updating pointer and others expressed as offsets from it, or
-on CISC chips with all addressing done with the loop counter as a
-scaled index.
-
- The final loop control cost can be amortised by processing several
-limbs in each iteration (*note Assembly Loop Unrolling::). This at
-least ensures loop control isn't a big fraction the work done.
-
- Memory throughput is always a limit. If perhaps only one load or
-one store can be done per cycle then 3 cycles/limb will the top speed
-for "binary" operations like `mpn_add_n', and any code achieving that
-is optimal.
-
- Integer resources can be freed up by having the loop counter in a
-float register, or by pressing the float units into use for some
-multiplying, perhaps doing every second limb on the float side (*note
-Assembly Floating Point::).
-
- Float resources can be freed up by doing carry propagation on the
-integer side, or even by doing integer to float conversions in integers
-using bit twiddling.
-
-\1f
-File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding
-
-16.8.6 Floating Point
----------------------
-
-Floating point arithmetic is used in GMP for multiplications on CPUs
-with poor integer multipliers. It's mostly useful for `mpn_mul_1',
-`mpn_addmul_1' and `mpn_submul_1' on 64-bit machines, and
-`mpn_mul_basecase' on both 32-bit and 64-bit machines.
-
- With IEEE 53-bit double precision floats, integer multiplications
-producing up to 53 bits will give exact results. Breaking a 64x64
-multiplication into eight 16x32->48 bit pieces is convenient. With
-some care though six 21x32->53 bit products can be used, if one of the
-lower two 21-bit pieces also uses the sign bit.
-
- For the `mpn_mul_1' family of functions on a 64-bit machine, the
-invariant single limb is split at the start, into 3 or 4 pieces.
-Inside the loop, the bignum operand is split into 32-bit pieces. Fast
-conversion of these unsigned 32-bit pieces to floating point is highly
-machine-dependent. In some cases, reading the data into the integer
-unit, zero-extending to 64-bits, then transferring to the floating
-point unit back via memory is the only option.
-
- Converting partial products back to 64-bit limbs is usually best
-done as a signed conversion. Since all values are smaller than 2^53,
-signed and unsigned are the same, but most processors lack unsigned
-conversions.
-
-
-
- Here is a diagram showing 16x32 bit products for an `mpn_mul_1' or
-`mpn_addmul_1' with a 64-bit limb. The single limb operand V is split
-into four 16-bit parts. The multi-limb operand U is split in the loop
-into two 32-bit parts.
-
- +---+---+---+---+
- |v48|v32|v16|v00| V operand
- +---+---+---+---+
-
- +-------+---+---+
- x | u32 | u00 | U operand (one limb)
- +---------------+
-
- ---------------------------------
-
- +-----------+
- | u00 x v00 | p00 48-bit products
- +-----------+
- +-----------+
- | u00 x v16 | p16
- +-----------+
- +-----------+
- | u00 x v32 | p32
- +-----------+
- +-----------+
- | u00 x v48 | p48
- +-----------+
- +-----------+
- | u32 x v00 | r32
- +-----------+
- +-----------+
- | u32 x v16 | r48
- +-----------+
- +-----------+
- | u32 x v32 | r64
- +-----------+
- +-----------+
- | u32 x v48 | r80
- +-----------+
-
- p32 and r32 can be summed using floating-point addition, and
-likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from
-the previous iteration.
-
- For each loop then, four 49-bit quantities are transferred to the
-integer unit, aligned as follows,
-
- |-----64bits----|-----64bits----|
- +------------+
- | p00 + r64' | i00
- +------------+
- +------------+
- | p16 + r80' | i16
- +------------+
- +------------+
- | p32 + r32 | i32
- +------------+
- +------------+
- | p48 + r48 | i48
- +------------+
-
- The challenge then is to sum these efficiently and add in a carry
-limb, generating a low 64-bit result limb and a high 33-bit carry limb
-(i48 extends 33 bits into the high half).
-
-\1f
-File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding
-
-16.8.7 SIMD Instructions
-------------------------
-
-The single-instruction multiple-data support in current microprocessors
-is aimed at signal processing algorithms where each data point can be
-treated more or less independently. There's generally not much support
-for propagating the sort of carries that arise in GMP.
-
- SIMD multiplications of say four 16x16 bit multiplies only do as much
-work as one 32x32 from GMP's point of view, and need some shifts and
-adds besides. But of course if say the SIMD form is fully pipelined
-and uses less instruction decoding then it may still be worthwhile.
-
- On the x86 chips, MMX has so far found a use in `mpn_rshift' and
-`mpn_lshift', and is used in a special case for 16-bit multipliers in
-the P55 `mpn_mul_1'. SSE2 is used for Pentium 4 `mpn_mul_1',
-`mpn_addmul_1', and `mpn_submul_1'.
-
-\1f
-File: gmp.info, Node: Assembly Software Pipelining, Next: Assembly Loop Unrolling, Prev: Assembly SIMD Instructions, Up: Assembly Coding
-
-16.8.8 Software Pipelining
---------------------------
-
-Software pipelining consists of scheduling instructions around the
-branch point in a loop. For example a loop might issue a load not for
-use in the present iteration but the next, thereby allowing extra
-cycles for the data to arrive from memory.
-
- Naturally this is wanted only when doing things like loads or
-multiplies that take several cycles to complete, and only where a CPU
-has multiple functional units so that other work can be done in the
-meantime.
-
- A pipeline with several stages will have a data value in progress at
-each stage and each loop iteration moves them along one stage. This is
-like juggling.
-
- If the latency of some instruction is greater than the loop time
-then it will be necessary to unroll, so one register has a result ready
-to use while another (or multiple others) are still in progress.
-(*note Assembly Loop Unrolling::).
-
-\1f
-File: gmp.info, Node: Assembly Loop Unrolling, Next: Assembly Writing Guide, Prev: Assembly Software Pipelining, Up: Assembly Coding
-
-16.8.9 Loop Unrolling
----------------------
-
-Loop unrolling consists of replicating code so that several limbs are
-processed in each loop. At a minimum this reduces loop overheads by a
-corresponding factor, but it can also allow better register usage, for
-example alternately using one register combination and then another.
-Judicious use of `m4' macros can help avoid lots of duplication in the
-source code.
-
- Any amount of unrolling can be handled with a loop counter that's
-decremented by N each time, stopping when the remaining count is less
-than the further N the loop will process. Or by subtracting N at the
-start, the termination condition becomes when the counter C is less
-than 0 (and the count of remaining limbs is C+N).
-
- Alternately for a power of 2 unroll the loop count and remainder can
-be established with a shift and mask. This is convenient if also
-making a computed jump into the middle of a large loop.
-
- The limbs not a multiple of the unrolling can be handled in various
-ways, for example
-
- * A simple loop at the end (or the start) to process the excess.
- Care will be wanted that it isn't too much slower than the
- unrolled part.
-
- * A set of binary tests, for example after an 8-limb unrolling, test
- for 4 more limbs to process, then a further 2 more or not, and
- finally 1 more or not. This will probably take more code space
- than a simple loop.
-
- * A `switch' statement, providing separate code for each possible
- excess, for example an 8-limb unrolling would have separate code
- for 0 remaining, 1 remaining, etc, up to 7 remaining. This might
- take a lot of code, but may be the best way to optimize all cases
- in combination with a deep pipelined loop.
-
- * A computed jump into the middle of the loop, thus making the first
- iteration handle the excess. This should make times smoothly
- increase with size, which is attractive, but setups for the jump
- and adjustments for pointers can be tricky and could become quite
- difficult in combination with deep pipelining.
-
-\1f
-File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding
-
-16.8.10 Writing Guide
----------------------
-
-This is a guide to writing software pipelined loops for processing limb
-vectors in assembly.
-
- First determine the algorithm and which instructions are needed.
-Code it without unrolling or scheduling, to make sure it works. On a
-3-operand CPU try to write each new value to a new register, this will
-greatly simplify later steps.
-
- Then note for each instruction the functional unit and/or issue port
-requirements. If an instruction can use either of two units, like U0
-or U1 then make a category "U0/U1". Count the total using each unit
-(or combined unit), and count all instructions.
-
- Figure out from those counts the best possible loop time. The goal
-will be to find a perfect schedule where instruction latencies are
-completely hidden. The total instruction count might be the limiting
-factor, or perhaps a particular functional unit. It might be possible
-to tweak the instructions to help the limiting factor.
-
- Suppose the loop time is N, then make N issue buckets, with the
-final loop branch at the end of the last. Now fill the buckets with
-dummy instructions using the functional units desired. Run this to
-make sure the intended speed is reached.
-
- Now replace the dummy instructions with the real instructions from
-the slow but correct loop you started with. The first will typically
-be a load instruction. Then the instruction using that value is placed
-in a bucket an appropriate distance down. Run the loop again, to check
-it still runs at target speed.
-
- Keep placing instructions, frequently measuring the loop. After a
-few you will need to wrap around from the last bucket back to the top
-of the loop. If you used the new-register for new-value strategy above
-then there will be no register conflicts. If not then take care not to
-clobber something already in use. Changing registers at this time is
-very error prone.
-
- The loop will overlap two or more of the original loop iterations,
-and the computation of one vector element result will be started in one
-iteration of the new loop, and completed one or several iterations
-later.
-
- The final step is to create feed-in and wind-down code for the loop.
-A good way to do this is to make a copy (or copies) of the loop at the
-start and delete those instructions which don't have valid antecedents,
-and at the end replicate and delete those whose results are unwanted
-(including any further loads).
-
- The loop will have a minimum number of limbs loaded and processed,
-so the feed-in code must test if the request size is smaller and skip
-either to a suitable part of the wind-down or to special code for small
-sizes.
-
-\1f
-File: gmp.info, Node: Internals, Next: Contributors, Prev: Algorithms, Up: Top
-
-17 Internals
-************
-
-*This chapter is provided only for informational purposes and the
-various internals described here may change in future GMP releases.
-Applications expecting to be compatible with future releases should use
-only the documented interfaces described in previous chapters.*
-
-* Menu:
-
-* Integer Internals::
-* Rational Internals::
-* Float Internals::
-* Raw Output Internals::
-* C++ Interface Internals::
-
-\1f
-File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
-
-17.1 Integer Internals
-======================
-
-`mpz_t' variables represent integers using sign and magnitude, in space
-dynamically allocated and reallocated. The fields are as follows.
-
-`_mp_size'
- The number of limbs, or the negative of that when representing a
- negative integer. Zero is represented by `_mp_size' set to zero,
- in which case the `_mp_d' data is unused.
-
-`_mp_d'
- A pointer to an array of limbs which is the magnitude. These are
- stored "little endian" as per the `mpn' functions, so `_mp_d[0]'
- is the least significant limb and `_mp_d[ABS(_mp_size)-1]' is the
- most significant. Whenever `_mp_size' is non-zero, the most
- significant limb is non-zero.
-
- Currently there's always at least one limb allocated, so for
- instance `mpz_set_ui' never needs to reallocate, and `mpz_get_ui'
- can fetch `_mp_d[0]' unconditionally (though its value is then
- only wanted if `_mp_size' is non-zero).
-
-`_mp_alloc'
- `_mp_alloc' is the number of limbs currently allocated at `_mp_d',
- and naturally `_mp_alloc >= ABS(_mp_size)'. When an `mpz' routine
- is about to (or might be about to) increase `_mp_size', it checks
- `_mp_alloc' to see whether there's enough space, and reallocates
- if not. `MPZ_REALLOC' is generally used for this.
-
- The various bitwise logical functions like `mpz_and' behave as if
-negative values were twos complement. But sign and magnitude is always
-used internally, and necessary adjustments are made during the
-calculations. Sometimes this isn't pretty, but sign and magnitude are
-best for other routines.
-
- Some internal temporary variables are setup with `MPZ_TMP_INIT' and
-these have `_mp_d' space obtained from `TMP_ALLOC' rather than the
-memory allocation functions. Care is taken to ensure that these are
-big enough that no reallocation is necessary (since it would have
-unpredictable consequences).
-
- `_mp_size' and `_mp_alloc' are `int', although `mp_size_t' is
-usually a `long'. This is done to make the fields just 32 bits on some
-64 bits systems, thereby saving a few bytes of data space but still
-providing plenty of range.
-
-\1f
-File: gmp.info, Node: Rational Internals, Next: Float Internals, Prev: Integer Internals, Up: Internals
-
-17.2 Rational Internals
-=======================
-
-`mpq_t' variables represent rationals using an `mpz_t' numerator and
-denominator (*note Integer Internals::).
-
- The canonical form adopted is denominator positive (and non-zero),
-no common factors between numerator and denominator, and zero uniquely
-represented as 0/1.
-
- It's believed that casting out common factors at each stage of a
-calculation is best in general. A GCD is an O(N^2) operation so it's
-better to do a few small ones immediately than to delay and have to do
-a big one later. Knowing the numerator and denominator have no common
-factors can be used for example in `mpq_mul' to make only two cross
-GCDs necessary, not four.
-
- This general approach to common factors is badly sub-optimal in the
-presence of simple factorizations or little prospect for cancellation,
-but GMP has no way to know when this will occur. As per *Note
-Efficiency::, that's left to applications. The `mpq_t' framework might
-still suit, with `mpq_numref' and `mpq_denref' for direct access to the
-numerator and denominator, or of course `mpz_t' variables can be used
-directly.
-
-\1f
-File: gmp.info, Node: Float Internals, Next: Raw Output Internals, Prev: Rational Internals, Up: Internals
-
-17.3 Float Internals
-====================
-
-Efficient calculation is the primary aim of GMP floats and the use of
-whole limbs and simple rounding facilitates this.
-
- `mpf_t' floats have a variable precision mantissa and a single
-machine word signed exponent. The mantissa is represented using sign
-and magnitude.
-
- most least
- significant significant
- limb limb
-
- _mp_d
- |---- _mp_exp ---> |
- _____ _____ _____ _____ _____
- |_____|_____|_____|_____|_____|
- . <------------ radix point
-
- <-------- _mp_size --------->
-
-The fields are as follows.
-
-`_mp_size'
- The number of limbs currently in use, or the negative of that when
- representing a negative value. Zero is represented by `_mp_size'
- and `_mp_exp' both set to zero, and in that case the `_mp_d' data
- is unused. (In the future `_mp_exp' might be undefined when
- representing zero.)
-
-`_mp_prec'
- The precision of the mantissa, in limbs. In any calculation the
- aim is to produce `_mp_prec' limbs of result (the most significant
- being non-zero).
-
-`_mp_d'
- A pointer to the array of limbs which is the absolute value of the
- mantissa. These are stored "little endian" as per the `mpn'
- functions, so `_mp_d[0]' is the least significant limb and
- `_mp_d[ABS(_mp_size)-1]' the most significant.
-
- The most significant limb is always non-zero, but there are no
- other restrictions on its value, in particular the highest 1 bit
- can be anywhere within the limb.
-
- `_mp_prec+1' limbs are allocated to `_mp_d', the extra limb being
- for convenience (see below). There are no reallocations during a
- calculation, only in a change of precision with `mpf_set_prec'.
-
-`_mp_exp'
- The exponent, in limbs, determining the location of the implied
- radix point. Zero means the radix point is just above the most
- significant limb. Positive values mean a radix point offset
- towards the lower limbs and hence a value >= 1, as for example in
- the diagram above. Negative exponents mean a radix point further
- above the highest limb.
-
- Naturally the exponent can be any value, it doesn't have to fall
- within the limbs as the diagram shows, it can be a long way above
- or a long way below. Limbs other than those included in the
- `{_mp_d,_mp_size}' data are treated as zero.
-
- The `_mp_size' and `_mp_prec' fields are `int', although the
-`mp_size_t' type is usually a `long'. The `_mp_exp' field is usually
-`long'. This is done to make some fields just 32 bits on some 64 bits
-systems, thereby saving a few bytes of data space but still providing
-plenty of precision and a very large range.
-
-
-The following various points should be noted.
-
-Low Zeros
- The least significant limbs `_mp_d[0]' etc can be zero, though
- such low zeros can always be ignored. Routines likely to produce
- low zeros check and avoid them to save time in subsequent
- calculations, but for most routines they're quite unlikely and
- aren't checked.
-
-Mantissa Size Range
- The `_mp_size' count of limbs in use can be less than `_mp_prec' if
- the value can be represented in less. This means low precision
- values or small integers stored in a high precision `mpf_t' can
- still be operated on efficiently.
-
- `_mp_size' can also be greater than `_mp_prec'. Firstly a value is
- allowed to use all of the `_mp_prec+1' limbs available at `_mp_d',
- and secondly when `mpf_set_prec_raw' lowers `_mp_prec' it leaves
- `_mp_size' unchanged and so the size can be arbitrarily bigger than
- `_mp_prec'.
-
-Rounding
- All rounding is done on limb boundaries. Calculating `_mp_prec'
- limbs with the high non-zero will ensure the application requested
- minimum precision is obtained.
-
- The use of simple "trunc" rounding towards zero is efficient,
- since there's no need to examine extra limbs and increment or
- decrement.
-
-Bit Shifts
- Since the exponent is in limbs, there are no bit shifts in basic
- operations like `mpf_add' and `mpf_mul'. When differing exponents
- are encountered all that's needed is to adjust pointers to line up
- the relevant limbs.
-
- Of course `mpf_mul_2exp' and `mpf_div_2exp' will require bit
- shifts, but the choice is between an exponent in limbs which
- requires shifts there, or one in bits which requires them almost
- everywhere else.
-
-Use of `_mp_prec+1' Limbs
- The extra limb on `_mp_d' (`_mp_prec+1' rather than just
- `_mp_prec') helps when an `mpf' routine might get a carry from its
- operation. `mpf_add' for instance will do an `mpn_add' of
- `_mp_prec' limbs. If there's no carry then that's the result, but
- if there is a carry then it's stored in the extra limb of space and
- `_mp_size' becomes `_mp_prec+1'.
-
- Whenever `_mp_prec+1' limbs are held in a variable, the low limb
- is not needed for the intended precision, only the `_mp_prec' high
- limbs. But zeroing it out or moving the rest down is unnecessary.
- Subsequent routines reading the value will simply take the high
- limbs they need, and this will be `_mp_prec' if their target has
- that same precision. This is no more than a pointer adjustment,
- and must be checked anyway since the destination precision can be
- different from the sources.
-
- Copy functions like `mpf_set' will retain a full `_mp_prec+1' limbs
- if available. This ensures that a variable which has `_mp_size'
- equal to `_mp_prec+1' will get its full exact value copied.
- Strictly speaking this is unnecessary since only `_mp_prec' limbs
- are needed for the application's requested precision, but it's
- considered that an `mpf_set' from one variable into another of the
- same precision ought to produce an exact copy.
-
-Application Precisions
- `__GMPF_BITS_TO_PREC' converts an application requested precision
- to an `_mp_prec'. The value in bits is rounded up to a whole limb
- then an extra limb is added since the most significant limb of
- `_mp_d' is only non-zero and therefore might contain only one bit.
-
- `__GMPF_PREC_TO_BITS' does the reverse conversion, and removes the
- extra limb from `_mp_prec' before converting to bits. The net
- effect of reading back with `mpf_get_prec' is simply the precision
- rounded up to a multiple of `mp_bits_per_limb'.
-
- Note that the extra limb added here for the high only being
- non-zero is in addition to the extra limb allocated to `_mp_d'.
- For example with a 32-bit limb, an application request for 250
- bits will be rounded up to 8 limbs, then an extra added for the
- high being only non-zero, giving an `_mp_prec' of 9. `_mp_d' then
- gets 10 limbs allocated. Reading back with `mpf_get_prec' will
- take `_mp_prec' subtract 1 limb and multiply by 32, giving 256
- bits.
-
- Strictly speaking, the fact the high limb has at least one bit
- means that a float with, say, 3 limbs of 32-bits each will be
- holding at least 65 bits, but for the purposes of `mpf_t' it's
- considered simply to be 64 bits, a nice multiple of the limb size.
-
-\1f
-File: gmp.info, Node: Raw Output Internals, Next: C++ Interface Internals, Prev: Float Internals, Up: Internals
-
-17.4 Raw Output Internals
-=========================
-
-`mpz_out_raw' uses the following format.
-
- +------+------------------------+
- | size | data bytes |
- +------+------------------------+
-
- The size is 4 bytes written most significant byte first, being the
-number of subsequent data bytes, or the twos complement negative of
-that when a negative integer is represented. The data bytes are the
-absolute value of the integer, written most significant byte first.
-
- The most significant data byte is always non-zero, so the output is
-the same on all systems, irrespective of limb size.
-
- In GMP 1, leading zero bytes were written to pad the data bytes to a
-multiple of the limb size. `mpz_inp_raw' will still accept this, for
-compatibility.
-
- The use of "big endian" for both the size and data fields is
-deliberate, it makes the data easy to read in a hex dump of a file.
-Unfortunately it also means that the limb data must be reversed when
-reading or writing, so neither a big endian nor little endian system
-can just read and write `_mp_d'.
-
-\1f
-File: gmp.info, Node: C++ Interface Internals, Prev: Raw Output Internals, Up: Internals
-
-17.5 C++ Interface Internals
-============================
-
-A system of expression templates is used to ensure something like
-`a=b+c' turns into a simple call to `mpz_add' etc. For `mpf_class' the
-scheme also ensures the precision of the final destination is used for
-any temporaries within a statement like `f=w*x+y*z'. These are
-important features which a naive implementation cannot provide.
-
- A simplified description of the scheme follows. The true scheme is
-complicated by the fact that expressions have different return types.
-For detailed information, refer to the source code.
-
- To perform an operation, say, addition, we first define a "function
-object" evaluating it,
-
- struct __gmp_binary_plus
- {
- static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
- };
-
-And an "additive expression" object,
-
- __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
- operator+(const mpf_class &f, const mpf_class &g)
- {
- return __gmp_expr
- <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
- }
-
- The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
-to encapsulate any possible kind of expression into a single template
-type. In fact even `mpf_class' etc are `typedef' specializations of
-`__gmp_expr'.
-
- Next we define assignment of `__gmp_expr' to `mpf_class'.
-
- template <class T>
- mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
- {
- expr.eval(this->get_mpf_t(), this->precision());
- return *this;
- }
-
- template <class Op>
- void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
- (mpf_t f, mp_bitcnt_t precision)
- {
- Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
- }
-
- where `expr.val1' and `expr.val2' are references to the expression's
-operands (here `expr' is the `__gmp_binary_expr' stored within the
-`__gmp_expr').
-
- This way, the expression is actually evaluated only at the time of
-assignment, when the required precision (that of `f') is known.
-Furthermore the target `mpf_t' is now available, thus we can call
-`mpf_add' directly with `f' as the output argument.
-
- Compound expressions are handled by defining operators taking
-subexpressions as their arguments, like this:
-
- template <class T, class U>
- __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
- {
- return __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
- (expr1, expr2);
- }
-
- And the corresponding specializations of `__gmp_expr::eval':
-
- template <class T, class U, class Op>
- void __gmp_expr
- <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
- (mpf_t f, mp_bitcnt_t precision)
- {
- // declare two temporaries
- mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
- Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
- }
-
- The expression is thus recursively evaluated to any level of
-complexity and all subexpressions are evaluated to the precision of `f'.
-
-\1f
-File: gmp.info, Node: Contributors, Next: References, Prev: Internals, Up: Top
-
-Appendix A Contributors
-***********************
-
-Torbjo"rn Granlund wrote the original GMP library and is still the main
-developer. Code not explicitly attributed to others, was contributed by
-Torbjo"rn. Several other individuals and organizations have contributed
-GMP. Here is a list in chronological order on first contribution:
-
- Gunnar Sjo"din and Hans Riesel helped with mathematical problems in
-early versions of the library.
-
- Richard Stallman helped with the interface design and revised the
-first version of this manual.
-
- Brian Beuning and Doug Lea helped with testing of early versions of
-the library and made creative suggestions.
-
- John Amanatides of York University in Canada contributed the function
-`mpz_probab_prime_p'.
-
- Paul Zimmermann wrote the REDC-based mpz_powm code, the
-Scho"nhage-Strassen FFT multiply code, and the Karatsuba square root
-code. He also improved the Toom3 code for GMP 4.2. Paul sparked the
-development of GMP 2, with his comparisons between bignum packages.
-The ECMNET project Paul is organizing was a driving force behind many
-of the optimizations in GMP 3. Paul also wrote the new GMP 4.3 nth
-root code (with Torbjo"rn).
-
- Ken Weber (Kent State University, Universidade Federal do Rio Grande
-do Sul) contributed now defunct versions of `mpz_gcd', `mpz_divexact',
-`mpn_gcd', and `mpn_bdivmod', partially supported by CNPq (Brazil)
-grant 301314194-2.
-
- Per Bothner of Cygnus Support helped to set up GMP to use Cygnus'
-configure. He has also made valuable suggestions and tested numerous
-intermediary releases.
-
- Joachim Hollman was involved in the design of the `mpf' interface,
-and in the `mpz' design revisions for version 2.
-
- Bennet Yee contributed the initial versions of `mpz_jacobi' and
-`mpz_legendre'.
-
- Andreas Schwab contributed the files `mpn/m68k/lshift.S' and
-`mpn/m68k/rshift.S' (now in `.asm' form).
-
- Robert Harley of Inria, France and David Seal of ARM, England,
-suggested clever improvements for population count. Robert also wrote
-highly optimized Karatsuba and 3-way Toom multiplication functions for
-GMP 3, and contributed the ARM assembly code.
-
- Torsten Ekedahl of the Mathematical department of Stockholm
-University provided significant inspiration during several phases of
-the GMP development. His mathematical expertise helped improve several
-algorithms.
-
- Linus Nordberg wrote the new configure system based on autoconf and
-implemented the new random functions.
-
- Kevin Ryde worked on a large number of things: optimized x86 code,
-m4 asm macros, parameter tuning, speed measuring, the configure system,
-function inlining, divisibility tests, bit scanning, Jacobi symbols,
-Fibonacci and Lucas number functions, printf and scanf functions, perl
-interface, demo expression parser, the algorithms chapter in the
-manual, `gmpasm-mode.el', and various miscellaneous improvements
-elsewhere.
-
- Kent Boortz made the Mac OS 9 port.
-
- Steve Root helped write the optimized alpha 21264 assembly code.
-
- Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
-`istream' input routines.
-
- Jason Moxham rewrote `mpz_fac_ui'.
-
- Pedro Gimeno implemented the Mersenne Twister and made other random
-number improvements.
-
- Niels Mo"ller wrote the sub-quadratic GCD and extended GCD code, the
-quadratic Hensel division code, and (with Torbjo"rn) the new divide and
-conquer division code for GMP 4.3. Niels also helped implement the new
-Toom multiply code for GMP 4.3 and implemented helper functions to
-simplify Toom evaluations for GMP 5.0. He wrote the original version
-of mpn_mulmod_bnm1.
-
- Alberto Zanoni and Marco Bodrato suggested the unbalanced multiply
-strategy, and found the optimal strategies for evaluation and
-interpolation in Toom multiplication.
-
- Marco Bodrato helped implement the new Toom multiply code for GMP
-4.3 and implemented most of the new Toom multiply and squaring code for
-5.0. He is the main author of the current mpn_mulmod_bnm1 and
-mpn_mullo_n. Marco also wrote the functions mpn_invert and
-mpn_invertappr.
-
- David Harvey suggested the internal function `mpn_bdiv_dbm1',
-implementing division relevant to Toom multiplication. He also worked
-on fast assembly sequences, in particular on a fast AMD64
-`mpn_mul_basecase'.
-
- Martin Boij wrote `mpn_perfect_power_p'.
-
- (This list is chronological, not ordered after significance. If you
-have contributed to GMP but are not listed above, please tell
-<gmp-devel@gmplib.org> about the omission!)
-
- The development of floating point functions of GNU MP 2, were
-supported in part by the ESPRIT-BRA (Basic Research Activities) 6846
-project POSSO (POlynomial System SOlving).
-
- The development of GMP 2, 3, and 4 was supported in part by the IDA
-Center for Computing Sciences.
-
- Thanks go to Hans Thorsen for donating an SGI system for the GMP
-test system environment.
-
-\1f
-File: gmp.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
-
-Appendix B References
-*********************
-
-B.1 Books
-=========
-
- * Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study
- in Analytic Number Theory and Computational Complexity", Wiley,
- 1998.
-
- * Richard Crandall and Carl Pomerance, "Prime Numbers: A
- Computational Perspective", 2nd edition, Springer-Verlag, 2005.
- `http://math.dartmouth.edu/~carlp/'
-
- * Henri Cohen, "A Course in Computational Algebraic Number Theory",
- Graduate Texts in Mathematics number 138, Springer-Verlag, 1993.
- `http://www.math.u-bordeaux.fr/~cohen/'
-
- * Donald E. Knuth, "The Art of Computer Programming", volume 2,
- "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998.
- `http://www-cs-faculty.stanford.edu/~knuth/taocp.html'
-
- * John D. Lipson, "Elements of Algebra and Algebraic Computing", The
- Benjamin Cummings Publishing Company Inc, 1981.
-
- * Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone,
- "Handbook of Applied Cryptography",
- `http://www.cacr.math.uwaterloo.ca/hac/'
-
- * Richard M. Stallman and the GCC Developer Community, "Using the
- GNU Compiler Collection", Free Software Foundation, 2008,
- available online `http://gcc.gnu.org/onlinedocs/', and in the GCC
- package `ftp://ftp.gnu.org/gnu/gcc/'
-
-B.2 Papers
-==========
-
- * Yves Bertot, Nicolas Magaud and Paul Zimmermann, "A Proof of GMP
- Square Root", Journal of Automated Reasoning, volume 29, 2002, pp.
- 225-252. Also available online as INRIA Research Report 4475,
- June 2001, `http://www.inria.fr/rrrt/rr-4475.html'
-
- * Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division",
- Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022,
- `http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022'
-
- * Torbjo"rn Granlund and Peter L. Montgomery, "Division by Invariant
- Integers using Multiplication", in Proceedings of the SIGPLAN
- PLDI'94 Conference, June 1994. Also available
- `ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz' (and .psl.gz).
-
- * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
- invariant integers", to appear.
-
- * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large
- and small", to appear.
-
- * Tudor Jebelean, "An algorithm for exact division", Journal of
- Symbolic Computation, volume 15, 1993, pp. 169-180. Research
- report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz'
-
- * Tudor Jebelean, "Exact Division with Karatsuba Complexity -
- Extended Abstract", RISC-Linz technical report 96-31,
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz'
-
- * Tudor Jebelean, "Practical Integer Division with Karatsuba
- Complexity", ISSAC 97, pp. 339-341. Technical report available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz'
-
- * Tudor Jebelean, "A Generalization of the Binary GCD Algorithm",
- ISSAC 93, pp. 111-116. Technical report version available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz'
-
- * Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for
- Finding the GCD of Long Integers", Journal of Symbolic
- Computation, volume 19, 1995, pp. 145-157. Technical report
- version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz'
-
- * Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer
- Division", Journal of Symbolic Computation, volume 21, 1996, pp.
- 441-455. Early technical report version also available
- `ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz'
-
- * Makoto Matsumoto and Takuji Nishimura, "Mersenne Twister: A
- 623-dimensionally equidistributed uniform pseudorandom number
- generator", ACM Transactions on Modelling and Computer Simulation,
- volume 8, January 1998, pp. 3-30. Available online
- `http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.ps.gz'
- (or .pdf)
-
- * R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
- Proceedings of the 13th Annual IEEE Symposium on Switching and
- Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast
- Modular Transforms", Journal of Computer and System Sciences,
- volume 8, number 3, June 1974, pp. 366-386.
-
- * Niels Mo"ller, "On Scho"nhage's algorithm and subquadratic integer
- GCD computation", in Mathematics of Computation, volume 77,
- January 2008, pp. 589-607.
-
- * Peter L. Montgomery, "Modular Multiplication Without Trial
- Division", in Mathematics of Computation, volume 44, number 170,
- April 1985.
-
- * Arnold Scho"nhage and Volker Strassen, "Schnelle Multiplikation
- grosser Zahlen", Computing 7, 1971, pp. 281-292.
-
- * Kenneth Weber, "The accelerated integer GCD algorithm", ACM
- Transactions on Mathematical Software, volume 21, number 1, March
- 1995, pp. 111-122.
-
- * Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
- 3805, November 1999, `http://www.inria.fr/rrrt/rr-3805.html'
-
- * Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
- Implementations",
- `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
-
- * Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11:
- IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271.
- Reprinted as "More on Multiplying and Squaring Large Integers",
- IEEE Transactions on Computers, volume 43, number 8, August 1994,
- pp. 899-908.
-
-\1f
-File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
-
-Appendix C GNU Free Documentation License
-*****************************************
-
- Version 1.3, 3 November 2008
-
- Copyright (C) 2000, 2001, 2002, 2007, 2008 Free Software Foundation, Inc.
- `http://fsf.org/'
-
- Everyone is permitted to copy and distribute verbatim copies
- of this license document, but changing it is not allowed.
-
- 0. PREAMBLE
-
- The purpose of this License is to make a manual, textbook, or other
- functional and useful document "free" in the sense of freedom: to
- assure everyone the effective freedom to copy and redistribute it,
- with or without modifying it, either commercially or
- noncommercially. Secondarily, this License preserves for the
- author and publisher a way to get credit for their work, while not
- being considered responsible for modifications made by others.
-
- This License is a kind of "copyleft", which means that derivative
- works of the document must themselves be free in the same sense.
- It complements the GNU General Public License, which is a copyleft
- license designed for free software.
-
- We have designed this License in order to use it for manuals for
- free software, because free software needs free documentation: a
- free program should come with manuals providing the same freedoms
- that the software does. But this License is not limited to
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- We recommend this License principally for works whose purpose is
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-
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- 6. COLLECTIONS OF DOCUMENTS
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- You may make a collection consisting of the Document and other
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- translation of this License, and all the license notices in the
- Document, and any Warranty Disclaimers, provided that you also
- include the original English version of this License and the
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- "Dedications", or "History", the requirement (section 4) to
- Preserve its Title (section 1) will typically require changing the
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- 9. TERMINATION
-
- You may not copy, modify, sublicense, or distribute the Document
- except as expressly provided under this License. Any attempt
- otherwise to copy, modify, sublicense, or distribute it is void,
- and will automatically terminate your rights under this License.
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- copyright holder fails to notify you of the violation by some
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- the same material does not give you any rights to use it.
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- 10. FUTURE REVISIONS OF THIS LICENSE
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- the GNU Free Documentation License from time to time. Such new
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-ADDENDUM: How to use this License for your documents
-====================================================
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-To use this License in a document you have written, include a copy of
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- Copyright (C) YEAR YOUR NAME.
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- or any later version published by the Free Software Foundation;
- with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
- Texts. A copy of the license is included in the section entitled ``GNU
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-Texts, replace the "with...Texts." line with this:
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-
-\1f
-File: gmp.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top
-
-Concept Index
-*************
-
-\0\b[index\0\b]
-* Menu:
-
-* #include: Headers and Libraries.
- (line 6)
-* --build: Build Options. (line 52)
-* --disable-fft: Build Options. (line 317)
-* --disable-shared: Build Options. (line 45)
-* --disable-static: Build Options. (line 45)
-* --enable-alloca: Build Options. (line 278)
-* --enable-assert: Build Options. (line 327)
-* --enable-cxx: Build Options. (line 230)
-* --enable-fat: Build Options. (line 164)
-* --enable-mpbsd: Build Options. (line 322)
-* --enable-profiling <1>: Profiling. (line 6)
-* --enable-profiling: Build Options. (line 331)
-* --exec-prefix: Build Options. (line 32)
-* --host: Build Options. (line 66)
-* --prefix: Build Options. (line 32)
-* -finstrument-functions: Profiling. (line 66)
-* 2exp functions: Efficiency. (line 43)
-* 68000: Notes for Particular Systems.
- (line 80)
-* 80x86: Notes for Particular Systems.
- (line 126)
-* ABI <1>: Build Options. (line 171)
-* ABI: ABI and ISA. (line 6)
-* About this manual: Introduction to GMP. (line 58)
-* AC_CHECK_LIB: Autoconf. (line 11)
-* AIX <1>: ABI and ISA. (line 184)
-* AIX <2>: Notes for Particular Systems.
- (line 7)
-* AIX: ABI and ISA. (line 169)
-* Algorithms: Algorithms. (line 6)
-* alloca: Build Options. (line 278)
-* Allocation of memory: Custom Allocation. (line 6)
-* AMD64: ABI and ISA. (line 44)
-* Anonymous FTP of latest version: Introduction to GMP. (line 38)
-* Application Binary Interface: ABI and ISA. (line 6)
-* Arithmetic functions <1>: Float Arithmetic. (line 6)
-* Arithmetic functions <2>: Integer Arithmetic. (line 6)
-* Arithmetic functions: Rational Arithmetic. (line 6)
-* ARM: Notes for Particular Systems.
- (line 20)
-* Assembly cache handling: Assembly Cache Handling.
- (line 6)
-* Assembly carry propagation: Assembly Carry Propagation.
- (line 6)
-* Assembly code organisation: Assembly Code Organisation.
- (line 6)
-* Assembly coding: Assembly Coding. (line 6)
-* Assembly floating Point: Assembly Floating Point.
- (line 6)
-* Assembly loop unrolling: Assembly Loop Unrolling.
- (line 6)
-* Assembly SIMD: Assembly SIMD Instructions.
- (line 6)
-* Assembly software pipelining: Assembly Software Pipelining.
- (line 6)
-* Assembly writing guide: Assembly Writing Guide.
- (line 6)
-* Assertion checking <1>: Debugging. (line 79)
-* Assertion checking: Build Options. (line 327)
-* Assignment functions <1>: Assigning Floats. (line 6)
-* Assignment functions <2>: Initializing Rationals.
- (line 6)
-* Assignment functions <3>: Simultaneous Integer Init & Assign.
- (line 6)
-* Assignment functions <4>: Simultaneous Float Init & Assign.
- (line 6)
-* Assignment functions: Assigning Integers. (line 6)
-* Autoconf: Autoconf. (line 6)
-* Basics: GMP Basics. (line 6)
-* Berkeley MP compatible functions <1>: Build Options. (line 322)
-* Berkeley MP compatible functions: BSD Compatible Functions.
- (line 6)
-* Binomial coefficient algorithm: Binomial Coefficients Algorithm.
- (line 6)
-* Binomial coefficient functions: Number Theoretic Functions.
- (line 100)
-* Binutils strip: Known Build Problems.
- (line 28)
-* Bit manipulation functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Bit scanning functions: Integer Logic and Bit Fiddling.
- (line 38)
-* Bit shift left: Integer Arithmetic. (line 35)
-* Bit shift right: Integer Division. (line 53)
-* Bits per limb: Useful Macros and Constants.
- (line 7)
-* BSD MP compatible functions <1>: Build Options. (line 322)
-* BSD MP compatible functions: BSD Compatible Functions.
- (line 6)
-* Bug reporting: Reporting Bugs. (line 6)
-* Build directory: Build Options. (line 19)
-* Build notes for binary packaging: Notes for Package Builds.
- (line 6)
-* Build notes for particular systems: Notes for Particular Systems.
- (line 6)
-* Build options: Build Options. (line 6)
-* Build problems known: Known Build Problems.
- (line 6)
-* Build system: Build Options. (line 52)
-* Building GMP: Installing GMP. (line 6)
-* Bus error: Debugging. (line 7)
-* C compiler: Build Options. (line 182)
-* C++ compiler: Build Options. (line 254)
-* C++ interface: C++ Class Interface. (line 6)
-* C++ interface internals: C++ Interface Internals.
- (line 6)
-* C++ istream input: C++ Formatted Input. (line 6)
-* C++ ostream output: C++ Formatted Output.
- (line 6)
-* C++ support: Build Options. (line 230)
-* CC: Build Options. (line 182)
-* CC_FOR_BUILD: Build Options. (line 217)
-* CFLAGS: Build Options. (line 182)
-* Checker: Debugging. (line 115)
-* checkergcc: Debugging. (line 122)
-* Code organisation: Assembly Code Organisation.
- (line 6)
-* Compaq C++: Notes for Particular Systems.
- (line 25)
-* Comparison functions <1>: Integer Comparisons. (line 6)
-* Comparison functions <2>: Comparing Rationals. (line 6)
-* Comparison functions: Float Comparison. (line 6)
-* Compatibility with older versions: Compatibility with older versions.
- (line 6)
-* Conditions for copying GNU MP: Copying. (line 6)
-* Configuring GMP: Installing GMP. (line 6)
-* Congruence algorithm: Exact Remainder. (line 29)
-* Congruence functions: Integer Division. (line 124)
-* Constants: Useful Macros and Constants.
- (line 6)
-* Contributors: Contributors. (line 6)
-* Conventions for parameters: Parameter Conventions.
- (line 6)
-* Conventions for variables: Variable Conventions.
- (line 6)
-* Conversion functions <1>: Converting Integers. (line 6)
-* Conversion functions <2>: Converting Floats. (line 6)
-* Conversion functions: Rational Conversions.
- (line 6)
-* Copying conditions: Copying. (line 6)
-* CPPFLAGS: Build Options. (line 208)
-* CPU types <1>: Introduction to GMP. (line 24)
-* CPU types: Build Options. (line 108)
-* Cross compiling: Build Options. (line 66)
-* Custom allocation: Custom Allocation. (line 6)
-* CXX: Build Options. (line 254)
-* CXXFLAGS: Build Options. (line 254)
-* Cygwin: Notes for Particular Systems.
- (line 43)
-* Darwin: Known Build Problems.
- (line 51)
-* Debugging: Debugging. (line 6)
-* Demonstration programs: Demonstration Programs.
- (line 6)
-* Digits in an integer: Miscellaneous Integer Functions.
- (line 23)
-* Divisibility algorithm: Exact Remainder. (line 29)
-* Divisibility functions: Integer Division. (line 124)
-* Divisibility testing: Efficiency. (line 91)
-* Division algorithms: Division Algorithms. (line 6)
-* Division functions <1>: Rational Arithmetic. (line 22)
-* Division functions <2>: Integer Division. (line 6)
-* Division functions: Float Arithmetic. (line 33)
-* DJGPP <1>: Notes for Particular Systems.
- (line 43)
-* DJGPP: Known Build Problems.
- (line 18)
-* DLLs: Notes for Particular Systems.
- (line 56)
-* DocBook: Build Options. (line 354)
-* Documentation formats: Build Options. (line 347)
-* Documentation license: GNU Free Documentation License.
- (line 6)
-* DVI: Build Options. (line 350)
-* Efficiency: Efficiency. (line 6)
-* Emacs: Emacs. (line 6)
-* Exact division functions: Integer Division. (line 102)
-* Exact remainder: Exact Remainder. (line 6)
-* Example programs: Demonstration Programs.
- (line 6)
-* Exec prefix: Build Options. (line 32)
-* Execution profiling <1>: Profiling. (line 6)
-* Execution profiling: Build Options. (line 331)
-* Exponentiation functions <1>: Integer Exponentiation.
- (line 6)
-* Exponentiation functions: Float Arithmetic. (line 41)
-* Export: Integer Import and Export.
- (line 45)
-* Expression parsing demo: Demonstration Programs.
- (line 18)
-* Extended GCD: Number Theoretic Functions.
- (line 45)
-* Factor removal functions: Number Theoretic Functions.
- (line 90)
-* Factorial algorithm: Factorial Algorithm. (line 6)
-* Factorial functions: Number Theoretic Functions.
- (line 95)
-* Factorization demo: Demonstration Programs.
- (line 25)
-* Fast Fourier Transform: FFT Multiplication. (line 6)
-* Fat binary: Build Options. (line 164)
-* FFT multiplication <1>: FFT Multiplication. (line 6)
-* FFT multiplication: Build Options. (line 317)
-* Fibonacci number algorithm: Fibonacci Numbers Algorithm.
- (line 6)
-* Fibonacci sequence functions: Number Theoretic Functions.
- (line 108)
-* Float arithmetic functions: Float Arithmetic. (line 6)
-* Float assignment functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Float assignment functions: Assigning Floats. (line 6)
-* Float comparison functions: Float Comparison. (line 6)
-* Float conversion functions: Converting Floats. (line 6)
-* Float functions: Floating-point Functions.
- (line 6)
-* Float initialization functions <1>: Simultaneous Float Init & Assign.
- (line 6)
-* Float initialization functions: Initializing Floats. (line 6)
-* Float input and output functions: I/O of Floats. (line 6)
-* Float internals: Float Internals. (line 6)
-* Float miscellaneous functions: Miscellaneous Float Functions.
- (line 6)
-* Float random number functions: Miscellaneous Float Functions.
- (line 27)
-* Float rounding functions: Miscellaneous Float Functions.
- (line 9)
-* Float sign tests: Float Comparison. (line 33)
-* Floating point mode: Notes for Particular Systems.
- (line 34)
-* Floating-point functions: Floating-point Functions.
- (line 6)
-* Floating-point number: Nomenclature and Types.
- (line 21)
-* fnccheck: Profiling. (line 77)
-* Formatted input: Formatted Input. (line 6)
-* Formatted output: Formatted Output. (line 6)
-* Free Documentation License: GNU Free Documentation License.
- (line 6)
-* frexp <1>: Converting Floats. (line 23)
-* frexp: Converting Integers. (line 42)
-* FTP of latest version: Introduction to GMP. (line 38)
-* Function classes: Function Classes. (line 6)
-* FunctionCheck: Profiling. (line 77)
-* GCC Checker: Debugging. (line 115)
-* GCD algorithms: Greatest Common Divisor Algorithms.
- (line 6)
-* GCD extended: Number Theoretic Functions.
- (line 45)
-* GCD functions: Number Theoretic Functions.
- (line 30)
-* GDB: Debugging. (line 58)
-* Generic C: Build Options. (line 153)
-* GMP Perl module: Demonstration Programs.
- (line 35)
-* GMP version number: Useful Macros and Constants.
- (line 12)
-* gmp.h: Headers and Libraries.
- (line 6)
-* gmpxx.h: C++ Interface General.
- (line 8)
-* GNU Debugger: Debugging. (line 58)
-* GNU Free Documentation License: GNU Free Documentation License.
- (line 6)
-* GNU strip: Known Build Problems.
- (line 28)
-* gprof: Profiling. (line 41)
-* Greatest common divisor algorithms: Greatest Common Divisor Algorithms.
- (line 6)
-* Greatest common divisor functions: Number Theoretic Functions.
- (line 30)
-* Hardware floating point mode: Notes for Particular Systems.
- (line 34)
-* Headers: Headers and Libraries.
- (line 6)
-* Heap problems: Debugging. (line 24)
-* Home page: Introduction to GMP. (line 34)
-* Host system: Build Options. (line 66)
-* HP-UX: ABI and ISA. (line 107)
-* HPPA: ABI and ISA. (line 68)
-* I/O functions <1>: I/O of Integers. (line 6)
-* I/O functions <2>: I/O of Rationals. (line 6)
-* I/O functions: I/O of Floats. (line 6)
-* i386: Notes for Particular Systems.
- (line 126)
-* IA-64: ABI and ISA. (line 107)
-* Import: Integer Import and Export.
- (line 11)
-* In-place operations: Efficiency. (line 57)
-* Include files: Headers and Libraries.
- (line 6)
-* info-lookup-symbol: Emacs. (line 6)
-* Initialization functions <1>: Initializing Integers.
- (line 6)
-* Initialization functions <2>: Initializing Rationals.
- (line 6)
-* Initialization functions <3>: Random State Initialization.
- (line 6)
-* Initialization functions <4>: Simultaneous Float Init & Assign.
- (line 6)
-* Initialization functions <5>: Simultaneous Integer Init & Assign.
- (line 6)
-* Initialization functions: Initializing Floats. (line 6)
-* Initializing and clearing: Efficiency. (line 21)
-* Input functions <1>: I/O of Integers. (line 6)
-* Input functions <2>: I/O of Rationals. (line 6)
-* Input functions <3>: I/O of Floats. (line 6)
-* Input functions: Formatted Input Functions.
- (line 6)
-* Install prefix: Build Options. (line 32)
-* Installing GMP: Installing GMP. (line 6)
-* Instruction Set Architecture: ABI and ISA. (line 6)
-* instrument-functions: Profiling. (line 66)
-* Integer: Nomenclature and Types.
- (line 6)
-* Integer arithmetic functions: Integer Arithmetic. (line 6)
-* Integer assignment functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
-* Integer assignment functions: Assigning Integers. (line 6)
-* Integer bit manipulation functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Integer comparison functions: Integer Comparisons. (line 6)
-* Integer conversion functions: Converting Integers. (line 6)
-* Integer division functions: Integer Division. (line 6)
-* Integer exponentiation functions: Integer Exponentiation.
- (line 6)
-* Integer export: Integer Import and Export.
- (line 45)
-* Integer functions: Integer Functions. (line 6)
-* Integer import: Integer Import and Export.
- (line 11)
-* Integer initialization functions <1>: Simultaneous Integer Init & Assign.
- (line 6)
-* Integer initialization functions: Initializing Integers.
- (line 6)
-* Integer input and output functions: I/O of Integers. (line 6)
-* Integer internals: Integer Internals. (line 6)
-* Integer logical functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Integer miscellaneous functions: Miscellaneous Integer Functions.
- (line 6)
-* Integer random number functions: Integer Random Numbers.
- (line 6)
-* Integer root functions: Integer Roots. (line 6)
-* Integer sign tests: Integer Comparisons. (line 28)
-* Integer special functions: Integer Special Functions.
- (line 6)
-* Interix: Notes for Particular Systems.
- (line 51)
-* Internals: Internals. (line 6)
-* Introduction: Introduction to GMP. (line 6)
-* Inverse modulo functions: Number Theoretic Functions.
- (line 60)
-* IRIX <1>: Known Build Problems.
- (line 38)
-* IRIX: ABI and ISA. (line 132)
-* ISA: ABI and ISA. (line 6)
-* istream input: C++ Formatted Input. (line 6)
-* Jacobi symbol algorithm: Jacobi Symbol. (line 6)
-* Jacobi symbol functions: Number Theoretic Functions.
- (line 66)
-* Karatsuba multiplication: Karatsuba Multiplication.
- (line 6)
-* Karatsuba square root algorithm: Square Root Algorithm.
- (line 6)
-* Kronecker symbol functions: Number Theoretic Functions.
- (line 78)
-* Language bindings: Language Bindings. (line 6)
-* Latest version of GMP: Introduction to GMP. (line 38)
-* LCM functions: Number Theoretic Functions.
- (line 55)
-* Least common multiple functions: Number Theoretic Functions.
- (line 55)
-* Legendre symbol functions: Number Theoretic Functions.
- (line 69)
-* libgmp: Headers and Libraries.
- (line 22)
-* libgmpxx: Headers and Libraries.
- (line 27)
-* Libraries: Headers and Libraries.
- (line 22)
-* Libtool: Headers and Libraries.
- (line 33)
-* Libtool versioning: Notes for Package Builds.
- (line 9)
-* License conditions: Copying. (line 6)
-* Limb: Nomenclature and Types.
- (line 31)
-* Limb size: Useful Macros and Constants.
- (line 7)
-* Linear congruential algorithm: Random Number Algorithms.
- (line 25)
-* Linear congruential random numbers: Random State Initialization.
- (line 32)
-* Linking: Headers and Libraries.
- (line 22)
-* Logical functions: Integer Logic and Bit Fiddling.
- (line 6)
-* Low-level functions: Low-level Functions. (line 6)
-* Lucas number algorithm: Lucas Numbers Algorithm.
- (line 6)
-* Lucas number functions: Number Theoretic Functions.
- (line 119)
-* MacOS X: Known Build Problems.
- (line 51)
-* Mailing lists: Introduction to GMP. (line 45)
-* Malloc debugger: Debugging. (line 30)
-* Malloc problems: Debugging. (line 24)
-* Memory allocation: Custom Allocation. (line 6)
-* Memory management: Memory Management. (line 6)
-* Mersenne twister algorithm: Random Number Algorithms.
- (line 17)
-* Mersenne twister random numbers: Random State Initialization.
- (line 13)
-* MINGW: Notes for Particular Systems.
- (line 43)
-* MIPS: ABI and ISA. (line 132)
-* Miscellaneous float functions: Miscellaneous Float Functions.
- (line 6)
-* Miscellaneous integer functions: Miscellaneous Integer Functions.
- (line 6)
-* MMX: Notes for Particular Systems.
- (line 132)
-* Modular inverse functions: Number Theoretic Functions.
- (line 60)
-* Most significant bit: Miscellaneous Integer Functions.
- (line 34)
-* mp.h: BSD Compatible Functions.
- (line 21)
-* MPN_PATH: Build Options. (line 335)
-* MS Windows: Notes for Particular Systems.
- (line 56)
-* MS-DOS: Notes for Particular Systems.
- (line 43)
-* Multi-threading: Reentrancy. (line 6)
-* Multiplication algorithms: Multiplication Algorithms.
- (line 6)
-* Nails: Low-level Functions. (line 478)
-* Native compilation: Build Options. (line 52)
-* NeXT: Known Build Problems.
- (line 57)
-* Next prime function: Number Theoretic Functions.
- (line 23)
-* Nomenclature: Nomenclature and Types.
- (line 6)
-* Non-Unix systems: Build Options. (line 11)
-* Nth root algorithm: Nth Root Algorithm. (line 6)
-* Number sequences: Efficiency. (line 147)
-* Number theoretic functions: Number Theoretic Functions.
- (line 6)
-* Numerator and denominator: Applying Integer Functions.
- (line 6)
-* obstack output: Formatted Output Functions.
- (line 81)
-* OpenBSD: Notes for Particular Systems.
- (line 86)
-* Optimizing performance: Performance optimization.
- (line 6)
-* ostream output: C++ Formatted Output.
- (line 6)
-* Other languages: Language Bindings. (line 6)
-* Output functions <1>: I/O of Floats. (line 6)
-* Output functions <2>: I/O of Rationals. (line 6)
-* Output functions <3>: Formatted Output Functions.
- (line 6)
-* Output functions: I/O of Integers. (line 6)
-* Packaged builds: Notes for Package Builds.
- (line 6)
-* Parameter conventions: Parameter Conventions.
- (line 6)
-* Parsing expressions demo: Demonstration Programs.
- (line 21)
-* Particular systems: Notes for Particular Systems.
- (line 6)
-* Past GMP versions: Compatibility with older versions.
- (line 6)
-* PDF: Build Options. (line 350)
-* Perfect power algorithm: Perfect Power Algorithm.
- (line 6)
-* Perfect power functions: Integer Roots. (line 27)
-* Perfect square algorithm: Perfect Square Algorithm.
- (line 6)
-* Perfect square functions: Integer Roots. (line 36)
-* perl: Demonstration Programs.
- (line 35)
-* Perl module: Demonstration Programs.
- (line 35)
-* Postscript: Build Options. (line 350)
-* Power/PowerPC <1>: Known Build Problems.
- (line 63)
-* Power/PowerPC: Notes for Particular Systems.
- (line 92)
-* Powering algorithms: Powering Algorithms. (line 6)
-* Powering functions <1>: Float Arithmetic. (line 41)
-* Powering functions: Integer Exponentiation.
- (line 6)
-* PowerPC: ABI and ISA. (line 167)
-* Precision of floats: Floating-point Functions.
- (line 6)
-* Precision of hardware floating point: Notes for Particular Systems.
- (line 34)
-* Prefix: Build Options. (line 32)
-* Prime testing algorithms: Prime Testing Algorithm.
- (line 6)
-* Prime testing functions: Number Theoretic Functions.
- (line 7)
-* printf formatted output: Formatted Output. (line 6)
-* Probable prime testing functions: Number Theoretic Functions.
- (line 7)
-* prof: Profiling. (line 24)
-* Profiling: Profiling. (line 6)
-* Radix conversion algorithms: Radix Conversion Algorithms.
- (line 6)
-* Random number algorithms: Random Number Algorithms.
- (line 6)
-* Random number functions <1>: Integer Random Numbers.
- (line 6)
-* Random number functions <2>: Miscellaneous Float Functions.
- (line 27)
-* Random number functions: Random Number Functions.
- (line 6)
-* Random number seeding: Random State Seeding.
- (line 6)
-* Random number state: Random State Initialization.
- (line 6)
-* Random state: Nomenclature and Types.
- (line 46)
-* Rational arithmetic: Efficiency. (line 113)
-* Rational arithmetic functions: Rational Arithmetic. (line 6)
-* Rational assignment functions: Initializing Rationals.
- (line 6)
-* Rational comparison functions: Comparing Rationals. (line 6)
-* Rational conversion functions: Rational Conversions.
- (line 6)
-* Rational initialization functions: Initializing Rationals.
- (line 6)
-* Rational input and output functions: I/O of Rationals. (line 6)
-* Rational internals: Rational Internals. (line 6)
-* Rational number: Nomenclature and Types.
- (line 16)
-* Rational number functions: Rational Number Functions.
- (line 6)
-* Rational numerator and denominator: Applying Integer Functions.
- (line 6)
-* Rational sign tests: Comparing Rationals. (line 27)
-* Raw output internals: Raw Output Internals.
- (line 6)
-* Reallocations: Efficiency. (line 30)
-* Reentrancy: Reentrancy. (line 6)
-* References: References. (line 6)
-* Remove factor functions: Number Theoretic Functions.
- (line 90)
-* Reporting bugs: Reporting Bugs. (line 6)
-* Root extraction algorithm: Nth Root Algorithm. (line 6)
-* Root extraction algorithms: Root Extraction Algorithms.
- (line 6)
-* Root extraction functions <1>: Float Arithmetic. (line 37)
-* Root extraction functions: Integer Roots. (line 6)
-* Root testing functions: Integer Roots. (line 36)
-* Rounding functions: Miscellaneous Float Functions.
- (line 9)
-* Sample programs: Demonstration Programs.
- (line 6)
-* Scan bit functions: Integer Logic and Bit Fiddling.
- (line 38)
-* scanf formatted input: Formatted Input. (line 6)
-* SCO: Known Build Problems.
- (line 38)
-* Seeding random numbers: Random State Seeding.
- (line 6)
-* Segmentation violation: Debugging. (line 7)
-* Sequent Symmetry: Known Build Problems.
- (line 68)
-* Services for Unix: Notes for Particular Systems.
- (line 51)
-* Shared library versioning: Notes for Package Builds.
- (line 9)
-* Sign tests <1>: Float Comparison. (line 33)
-* Sign tests <2>: Integer Comparisons. (line 28)
-* Sign tests: Comparing Rationals. (line 27)
-* Size in digits: Miscellaneous Integer Functions.
- (line 23)
-* Small operands: Efficiency. (line 7)
-* Solaris <1>: ABI and ISA. (line 201)
-* Solaris: Known Build Problems.
- (line 78)
-* Sparc: Notes for Particular Systems.
- (line 108)
-* Sparc V9: ABI and ISA. (line 201)
-* Special integer functions: Integer Special Functions.
- (line 6)
-* Square root algorithm: Square Root Algorithm.
- (line 6)
-* SSE2: Notes for Particular Systems.
- (line 132)
-* Stack backtrace: Debugging. (line 50)
-* Stack overflow <1>: Debugging. (line 7)
-* Stack overflow: Build Options. (line 278)
-* Static linking: Efficiency. (line 14)
-* stdarg.h: Headers and Libraries.
- (line 17)
-* stdio.h: Headers and Libraries.
- (line 11)
-* Stripped libraries: Known Build Problems.
- (line 28)
-* Sun: ABI and ISA. (line 201)
-* SunOS: Notes for Particular Systems.
- (line 120)
-* Systems: Notes for Particular Systems.
- (line 6)
-* Temporary memory: Build Options. (line 278)
-* Texinfo: Build Options. (line 347)
-* Text input/output: Efficiency. (line 153)
-* Thread safety: Reentrancy. (line 6)
-* Toom multiplication <1>: Other Multiplication.
- (line 6)
-* Toom multiplication <2>: Toom 4-Way Multiplication.
- (line 6)
-* Toom multiplication: Toom 3-Way Multiplication.
- (line 6)
-* Types: Nomenclature and Types.
- (line 6)
-* ui and si functions: Efficiency. (line 50)
-* Unbalanced multiplication: Unbalanced Multiplication.
- (line 6)
-* Upward compatibility: Compatibility with older versions.
- (line 6)
-* Useful macros and constants: Useful Macros and Constants.
- (line 6)
-* User-defined precision: Floating-point Functions.
- (line 6)
-* Valgrind: Debugging. (line 130)
-* Variable conventions: Variable Conventions.
- (line 6)
-* Version number: Useful Macros and Constants.
- (line 12)
-* Web page: Introduction to GMP. (line 34)
-* Windows: Notes for Particular Systems.
- (line 56)
-* x86: Notes for Particular Systems.
- (line 126)
-* x87: Notes for Particular Systems.
- (line 34)
-* XML: Build Options. (line 354)
-
-\1f
-File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top
-
-Function and Type Index
-***********************
-
-\0\b[index\0\b]
-* Menu:
-
-* __GMP_CC: Useful Macros and Constants.
- (line 23)
-* __GMP_CFLAGS: Useful Macros and Constants.
- (line 24)
-* __GNU_MP_VERSION: Useful Macros and Constants.
- (line 10)
-* __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
- (line 11)
-* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
- (line 12)
-* _mpz_realloc: Integer Special Functions.
- (line 51)
-* abs <1>: C++ Interface Rationals.
- (line 43)
-* abs <2>: C++ Interface Integers.
- (line 42)
-* abs: C++ Interface Floats.
- (line 70)
-* ceil: C++ Interface Floats.
- (line 71)
-* cmp <1>: C++ Interface Floats.
- (line 72)
-* cmp <2>: C++ Interface Rationals.
- (line 44)
-* cmp <3>: C++ Interface Integers.
- (line 44)
-* cmp: C++ Interface Rationals.
- (line 45)
-* floor: C++ Interface Floats.
- (line 80)
-* gcd: BSD Compatible Functions.
- (line 82)
-* gmp_asprintf: Formatted Output Functions.
- (line 65)
-* gmp_errno: Random State Initialization.
- (line 55)
-* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization.
- (line 55)
-* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization.
- (line 55)
-* gmp_fprintf: Formatted Output Functions.
- (line 29)
-* gmp_fscanf: Formatted Input Functions.
- (line 25)
-* GMP_LIMB_BITS: Low-level Functions. (line 508)
-* GMP_NAIL_BITS: Low-level Functions. (line 506)
-* GMP_NAIL_MASK: Low-level Functions. (line 516)
-* GMP_NUMB_BITS: Low-level Functions. (line 507)
-* GMP_NUMB_MASK: Low-level Functions. (line 517)
-* GMP_NUMB_MAX: Low-level Functions. (line 525)
-* gmp_obstack_printf: Formatted Output Functions.
- (line 79)
-* gmp_obstack_vprintf: Formatted Output Functions.
- (line 81)
-* gmp_printf: Formatted Output Functions.
- (line 24)
-* GMP_RAND_ALG_DEFAULT: Random State Initialization.
- (line 49)
-* GMP_RAND_ALG_LC: Random State Initialization.
- (line 49)
-* gmp_randclass: C++ Interface Random Numbers.
- (line 7)
-* gmp_randclass::get_f: C++ Interface Random Numbers.
- (line 45)
-* gmp_randclass::get_z_bits: C++ Interface Random Numbers.
- (line 39)
-* gmp_randclass::get_z_range: C++ Interface Random Numbers.
- (line 42)
-* gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
- (line 13)
-* gmp_randclass::seed: C++ Interface Random Numbers.
- (line 33)
-* gmp_randclear: Random State Initialization.
- (line 62)
-* gmp_randinit: Random State Initialization.
- (line 47)
-* gmp_randinit_default: Random State Initialization.
- (line 7)
-* gmp_randinit_lc_2exp: Random State Initialization.
- (line 18)
-* gmp_randinit_lc_2exp_size: Random State Initialization.
- (line 32)
-* gmp_randinit_mt: Random State Initialization.
- (line 13)
-* gmp_randinit_set: Random State Initialization.
- (line 43)
-* gmp_randseed: Random State Seeding.
- (line 7)
-* gmp_randseed_ui: Random State Seeding.
- (line 9)
-* gmp_randstate_t: Nomenclature and Types.
- (line 46)
-* gmp_scanf: Formatted Input Functions.
- (line 21)
-* gmp_snprintf: Formatted Output Functions.
- (line 46)
-* gmp_sprintf: Formatted Output Functions.
- (line 34)
-* gmp_sscanf: Formatted Input Functions.
- (line 29)
-* gmp_urandomb_ui: Random State Miscellaneous.
- (line 8)
-* gmp_urandomm_ui: Random State Miscellaneous.
- (line 14)
-* gmp_vasprintf: Formatted Output Functions.
- (line 66)
-* gmp_version: Useful Macros and Constants.
- (line 18)
-* gmp_vfprintf: Formatted Output Functions.
- (line 30)
-* gmp_vfscanf: Formatted Input Functions.
- (line 26)
-* gmp_vprintf: Formatted Output Functions.
- (line 25)
-* gmp_vscanf: Formatted Input Functions.
- (line 22)
-* gmp_vsnprintf: Formatted Output Functions.
- (line 48)
-* gmp_vsprintf: Formatted Output Functions.
- (line 35)
-* gmp_vsscanf: Formatted Input Functions.
- (line 31)
-* hypot: C++ Interface Floats.
- (line 81)
-* itom: BSD Compatible Functions.
- (line 29)
-* madd: BSD Compatible Functions.
- (line 43)
-* mcmp: BSD Compatible Functions.
- (line 85)
-* mdiv: BSD Compatible Functions.
- (line 53)
-* mfree: BSD Compatible Functions.
- (line 105)
-* min: BSD Compatible Functions.
- (line 89)
-* MINT: BSD Compatible Functions.
- (line 21)
-* mout: BSD Compatible Functions.
- (line 94)
-* move: BSD Compatible Functions.
- (line 39)
-* mp_bitcnt_t: Nomenclature and Types.
- (line 42)
-* mp_bits_per_limb: Useful Macros and Constants.
- (line 7)
-* mp_exp_t: Nomenclature and Types.
- (line 27)
-* mp_get_memory_functions: Custom Allocation. (line 93)
-* mp_limb_t: Nomenclature and Types.
- (line 31)
-* mp_set_memory_functions: Custom Allocation. (line 21)
-* mp_size_t: Nomenclature and Types.
- (line 37)
-* mpf_abs: Float Arithmetic. (line 47)
-* mpf_add: Float Arithmetic. (line 7)
-* mpf_add_ui: Float Arithmetic. (line 9)
-* mpf_ceil: Miscellaneous Float Functions.
- (line 7)
-* mpf_class: C++ Interface General.
- (line 20)
-* mpf_class::fits_sint_p: C++ Interface Floats.
- (line 74)
-* mpf_class::fits_slong_p: C++ Interface Floats.
- (line 75)
-* mpf_class::fits_sshort_p: C++ Interface Floats.
- (line 76)
-* mpf_class::fits_uint_p: C++ Interface Floats.
- (line 77)
-* mpf_class::fits_ulong_p: C++ Interface Floats.
- (line 78)
-* mpf_class::fits_ushort_p: C++ Interface Floats.
- (line 79)
-* mpf_class::get_d: C++ Interface Floats.
- (line 82)
-* mpf_class::get_mpf_t: C++ Interface General.
- (line 66)
-* mpf_class::get_prec: C++ Interface Floats.
- (line 100)
-* mpf_class::get_si: C++ Interface Floats.
- (line 83)
-* mpf_class::get_str: C++ Interface Floats.
- (line 85)
-* mpf_class::get_ui: C++ Interface Floats.
- (line 86)
-* mpf_class::mpf_class: C++ Interface Floats.
- (line 38)
-* mpf_class::operator=: C++ Interface Floats.
- (line 47)
-* mpf_class::set_prec: C++ Interface Floats.
- (line 101)
-* mpf_class::set_prec_raw: C++ Interface Floats.
- (line 102)
-* mpf_class::set_str: C++ Interface Floats.
- (line 88)
-* mpf_clear: Initializing Floats. (line 37)
-* mpf_clears: Initializing Floats. (line 41)
-* mpf_cmp: Float Comparison. (line 7)
-* mpf_cmp_d: Float Comparison. (line 8)
-* mpf_cmp_si: Float Comparison. (line 10)
-* mpf_cmp_ui: Float Comparison. (line 9)
-* mpf_div: Float Arithmetic. (line 29)
-* mpf_div_2exp: Float Arithmetic. (line 53)
-* mpf_div_ui: Float Arithmetic. (line 33)
-* mpf_eq: Float Comparison. (line 17)
-* mpf_fits_sint_p: Miscellaneous Float Functions.
- (line 20)
-* mpf_fits_slong_p: Miscellaneous Float Functions.
- (line 18)
-* mpf_fits_sshort_p: Miscellaneous Float Functions.
- (line 22)
-* mpf_fits_uint_p: Miscellaneous Float Functions.
- (line 19)
-* mpf_fits_ulong_p: Miscellaneous Float Functions.
- (line 17)
-* mpf_fits_ushort_p: Miscellaneous Float Functions.
- (line 21)
-* mpf_floor: Miscellaneous Float Functions.
- (line 8)
-* mpf_get_d: Converting Floats. (line 7)
-* mpf_get_d_2exp: Converting Floats. (line 16)
-* mpf_get_default_prec: Initializing Floats. (line 12)
-* mpf_get_prec: Initializing Floats. (line 62)
-* mpf_get_si: Converting Floats. (line 27)
-* mpf_get_str: Converting Floats. (line 37)
-* mpf_get_ui: Converting Floats. (line 28)
-* mpf_init: Initializing Floats. (line 19)
-* mpf_init2: Initializing Floats. (line 26)
-* mpf_init_set: Simultaneous Float Init & Assign.
- (line 16)
-* mpf_init_set_d: Simultaneous Float Init & Assign.
- (line 19)
-* mpf_init_set_si: Simultaneous Float Init & Assign.
- (line 18)
-* mpf_init_set_str: Simultaneous Float Init & Assign.
- (line 25)
-* mpf_init_set_ui: Simultaneous Float Init & Assign.
- (line 17)
-* mpf_inits: Initializing Floats. (line 31)
-* mpf_inp_str: I/O of Floats. (line 37)
-* mpf_integer_p: Miscellaneous Float Functions.
- (line 14)
-* mpf_mul: Float Arithmetic. (line 19)
-* mpf_mul_2exp: Float Arithmetic. (line 50)
-* mpf_mul_ui: Float Arithmetic. (line 21)
-* mpf_neg: Float Arithmetic. (line 44)
-* mpf_out_str: I/O of Floats. (line 17)
-* mpf_pow_ui: Float Arithmetic. (line 41)
-* mpf_random2: Miscellaneous Float Functions.
- (line 36)
-* mpf_reldiff: Float Comparison. (line 29)
-* mpf_set: Assigning Floats. (line 10)
-* mpf_set_d: Assigning Floats. (line 13)
-* mpf_set_default_prec: Initializing Floats. (line 7)
-* mpf_set_prec: Initializing Floats. (line 65)
-* mpf_set_prec_raw: Initializing Floats. (line 72)
-* mpf_set_q: Assigning Floats. (line 15)
-* mpf_set_si: Assigning Floats. (line 12)
-* mpf_set_str: Assigning Floats. (line 18)
-* mpf_set_ui: Assigning Floats. (line 11)
-* mpf_set_z: Assigning Floats. (line 14)
-* mpf_sgn: Float Comparison. (line 33)
-* mpf_sqrt: Float Arithmetic. (line 36)
-* mpf_sqrt_ui: Float Arithmetic. (line 37)
-* mpf_sub: Float Arithmetic. (line 12)
-* mpf_sub_ui: Float Arithmetic. (line 16)
-* mpf_swap: Assigning Floats. (line 52)
-* mpf_t: Nomenclature and Types.
- (line 21)
-* mpf_trunc: Miscellaneous Float Functions.
- (line 9)
-* mpf_ui_div: Float Arithmetic. (line 31)
-* mpf_ui_sub: Float Arithmetic. (line 14)
-* mpf_urandomb: Miscellaneous Float Functions.
- (line 27)
-* mpn_add: Low-level Functions. (line 69)
-* mpn_add_1: Low-level Functions. (line 64)
-* mpn_add_n: Low-level Functions. (line 54)
-* mpn_addmul_1: Low-level Functions. (line 148)
-* mpn_and_n: Low-level Functions. (line 420)
-* mpn_andn_n: Low-level Functions. (line 435)
-* mpn_cmp: Low-level Functions. (line 284)
-* mpn_com: Low-level Functions. (line 460)
-* mpn_copyd: Low-level Functions. (line 469)
-* mpn_copyi: Low-level Functions. (line 465)
-* mpn_divexact_by3: Low-level Functions. (line 229)
-* mpn_divexact_by3c: Low-level Functions. (line 231)
-* mpn_divmod: Low-level Functions. (line 224)
-* mpn_divmod_1: Low-level Functions. (line 208)
-* mpn_divrem: Low-level Functions. (line 182)
-* mpn_divrem_1: Low-level Functions. (line 206)
-* mpn_gcd: Low-level Functions. (line 289)
-* mpn_gcd_1: Low-level Functions. (line 299)
-* mpn_gcdext: Low-level Functions. (line 305)
-* mpn_get_str: Low-level Functions. (line 346)
-* mpn_hamdist: Low-level Functions. (line 410)
-* mpn_ior_n: Low-level Functions. (line 425)
-* mpn_iorn_n: Low-level Functions. (line 440)
-* mpn_lshift: Low-level Functions. (line 260)
-* mpn_mod_1: Low-level Functions. (line 255)
-* mpn_mul: Low-level Functions. (line 114)
-* mpn_mul_1: Low-level Functions. (line 133)
-* mpn_mul_n: Low-level Functions. (line 103)
-* mpn_nand_n: Low-level Functions. (line 445)
-* mpn_neg: Low-level Functions. (line 98)
-* mpn_nior_n: Low-level Functions. (line 450)
-* mpn_perfect_square_p: Low-level Functions. (line 416)
-* mpn_popcount: Low-level Functions. (line 406)
-* mpn_random: Low-level Functions. (line 395)
-* mpn_random2: Low-level Functions. (line 396)
-* mpn_rshift: Low-level Functions. (line 272)
-* mpn_scan0: Low-level Functions. (line 380)
-* mpn_scan1: Low-level Functions. (line 388)
-* mpn_set_str: Low-level Functions. (line 361)
-* mpn_sqr: Low-level Functions. (line 125)
-* mpn_sqrtrem: Low-level Functions. (line 328)
-* mpn_sub: Low-level Functions. (line 90)
-* mpn_sub_1: Low-level Functions. (line 85)
-* mpn_sub_n: Low-level Functions. (line 76)
-* mpn_submul_1: Low-level Functions. (line 159)
-* mpn_tdiv_qr: Low-level Functions. (line 171)
-* mpn_xnor_n: Low-level Functions. (line 455)
-* mpn_xor_n: Low-level Functions. (line 430)
-* mpn_zero: Low-level Functions. (line 472)
-* mpq_abs: Rational Arithmetic. (line 31)
-* mpq_add: Rational Arithmetic. (line 7)
-* mpq_canonicalize: Rational Number Functions.
- (line 22)
-* mpq_class: C++ Interface General.
- (line 19)
-* mpq_class::canonicalize: C++ Interface Rationals.
- (line 37)
-* mpq_class::get_d: C++ Interface Rationals.
- (line 46)
-* mpq_class::get_den: C++ Interface Rationals.
- (line 58)
-* mpq_class::get_den_mpz_t: C++ Interface Rationals.
- (line 68)
-* mpq_class::get_mpq_t: C++ Interface General.
- (line 65)
-* mpq_class::get_num: C++ Interface Rationals.
- (line 57)
-* mpq_class::get_num_mpz_t: C++ Interface Rationals.
- (line 67)
-* mpq_class::get_str: C++ Interface Rationals.
- (line 47)
-* mpq_class::mpq_class: C++ Interface Rationals.
- (line 22)
-* mpq_class::set_str: C++ Interface Rationals.
- (line 49)
-* mpq_clear: Initializing Rationals.
- (line 16)
-* mpq_clears: Initializing Rationals.
- (line 20)
-* mpq_cmp: Comparing Rationals. (line 7)
-* mpq_cmp_si: Comparing Rationals. (line 17)
-* mpq_cmp_ui: Comparing Rationals. (line 15)
-* mpq_denref: Applying Integer Functions.
- (line 18)
-* mpq_div: Rational Arithmetic. (line 22)
-* mpq_div_2exp: Rational Arithmetic. (line 25)
-* mpq_equal: Comparing Rationals. (line 33)
-* mpq_get_d: Rational Conversions.
- (line 7)
-* mpq_get_den: Applying Integer Functions.
- (line 24)
-* mpq_get_num: Applying Integer Functions.
- (line 23)
-* mpq_get_str: Rational Conversions.
- (line 22)
-* mpq_init: Initializing Rationals.
- (line 7)
-* mpq_inits: Initializing Rationals.
- (line 12)
-* mpq_inp_str: I/O of Rationals. (line 23)
-* mpq_inv: Rational Arithmetic. (line 34)
-* mpq_mul: Rational Arithmetic. (line 15)
-* mpq_mul_2exp: Rational Arithmetic. (line 18)
-* mpq_neg: Rational Arithmetic. (line 28)
-* mpq_numref: Applying Integer Functions.
- (line 17)
-* mpq_out_str: I/O of Rationals. (line 15)
-* mpq_set: Initializing Rationals.
- (line 24)
-* mpq_set_d: Rational Conversions.
- (line 17)
-* mpq_set_den: Applying Integer Functions.
- (line 26)
-* mpq_set_f: Rational Conversions.
- (line 18)
-* mpq_set_num: Applying Integer Functions.
- (line 25)
-* mpq_set_si: Initializing Rationals.
- (line 31)
-* mpq_set_str: Initializing Rationals.
- (line 36)
-* mpq_set_ui: Initializing Rationals.
- (line 29)
-* mpq_set_z: Initializing Rationals.
- (line 25)
-* mpq_sgn: Comparing Rationals. (line 27)
-* mpq_sub: Rational Arithmetic. (line 11)
-* mpq_swap: Initializing Rationals.
- (line 56)
-* mpq_t: Nomenclature and Types.
- (line 16)
-* mpz_abs: Integer Arithmetic. (line 42)
-* mpz_add: Integer Arithmetic. (line 7)
-* mpz_add_ui: Integer Arithmetic. (line 9)
-* mpz_addmul: Integer Arithmetic. (line 25)
-* mpz_addmul_ui: Integer Arithmetic. (line 27)
-* mpz_and: Integer Logic and Bit Fiddling.
- (line 11)
-* mpz_array_init: Integer Special Functions.
- (line 11)
-* mpz_bin_ui: Number Theoretic Functions.
- (line 98)
-* mpz_bin_uiui: Number Theoretic Functions.
- (line 100)
-* mpz_cdiv_q: Integer Division. (line 13)
-* mpz_cdiv_q_2exp: Integer Division. (line 24)
-* mpz_cdiv_q_ui: Integer Division. (line 17)
-* mpz_cdiv_qr: Integer Division. (line 15)
-* mpz_cdiv_qr_ui: Integer Division. (line 21)
-* mpz_cdiv_r: Integer Division. (line 14)
-* mpz_cdiv_r_2exp: Integer Division. (line 25)
-* mpz_cdiv_r_ui: Integer Division. (line 19)
-* mpz_cdiv_ui: Integer Division. (line 23)
-* mpz_class: C++ Interface General.
- (line 18)
-* mpz_class::fits_sint_p: C++ Interface Integers.
- (line 45)
-* mpz_class::fits_slong_p: C++ Interface Integers.
- (line 46)
-* mpz_class::fits_sshort_p: C++ Interface Integers.
- (line 47)
-* mpz_class::fits_uint_p: C++ Interface Integers.
- (line 48)
-* mpz_class::fits_ulong_p: C++ Interface Integers.
- (line 49)
-* mpz_class::fits_ushort_p: C++ Interface Integers.
- (line 50)
-* mpz_class::get_d: C++ Interface Integers.
- (line 51)
-* mpz_class::get_mpz_t: C++ Interface General.
- (line 64)
-* mpz_class::get_si: C++ Interface Integers.
- (line 52)
-* mpz_class::get_str: C++ Interface Integers.
- (line 53)
-* mpz_class::get_ui: C++ Interface Integers.
- (line 54)
-* mpz_class::mpz_class: C++ Interface Integers.
- (line 7)
-* mpz_class::set_str: C++ Interface Integers.
- (line 56)
-* mpz_clear: Initializing Integers.
- (line 44)
-* mpz_clears: Initializing Integers.
- (line 48)
-* mpz_clrbit: Integer Logic and Bit Fiddling.
- (line 54)
-* mpz_cmp: Integer Comparisons. (line 7)
-* mpz_cmp_d: Integer Comparisons. (line 8)
-* mpz_cmp_si: Integer Comparisons. (line 9)
-* mpz_cmp_ui: Integer Comparisons. (line 10)
-* mpz_cmpabs: Integer Comparisons. (line 18)
-* mpz_cmpabs_d: Integer Comparisons. (line 19)
-* mpz_cmpabs_ui: Integer Comparisons. (line 20)
-* mpz_com: Integer Logic and Bit Fiddling.
- (line 20)
-* mpz_combit: Integer Logic and Bit Fiddling.
- (line 57)
-* mpz_congruent_2exp_p: Integer Division. (line 124)
-* mpz_congruent_p: Integer Division. (line 121)
-* mpz_congruent_ui_p: Integer Division. (line 123)
-* mpz_divexact: Integer Division. (line 101)
-* mpz_divexact_ui: Integer Division. (line 102)
-* mpz_divisible_2exp_p: Integer Division. (line 112)
-* mpz_divisible_p: Integer Division. (line 110)
-* mpz_divisible_ui_p: Integer Division. (line 111)
-* mpz_even_p: Miscellaneous Integer Functions.
- (line 18)
-* mpz_export: Integer Import and Export.
- (line 45)
-* mpz_fac_ui: Number Theoretic Functions.
- (line 95)
-* mpz_fdiv_q: Integer Division. (line 27)
-* mpz_fdiv_q_2exp: Integer Division. (line 38)
-* mpz_fdiv_q_ui: Integer Division. (line 31)
-* mpz_fdiv_qr: Integer Division. (line 29)
-* mpz_fdiv_qr_ui: Integer Division. (line 35)
-* mpz_fdiv_r: Integer Division. (line 28)
-* mpz_fdiv_r_2exp: Integer Division. (line 39)
-* mpz_fdiv_r_ui: Integer Division. (line 33)
-* mpz_fdiv_ui: Integer Division. (line 37)
-* mpz_fib2_ui: Number Theoretic Functions.
- (line 108)
-* mpz_fib_ui: Number Theoretic Functions.
- (line 106)
-* mpz_fits_sint_p: Miscellaneous Integer Functions.
- (line 10)
-* mpz_fits_slong_p: Miscellaneous Integer Functions.
- (line 8)
-* mpz_fits_sshort_p: Miscellaneous Integer Functions.
- (line 12)
-* mpz_fits_uint_p: Miscellaneous Integer Functions.
- (line 9)
-* mpz_fits_ulong_p: Miscellaneous Integer Functions.
- (line 7)
-* mpz_fits_ushort_p: Miscellaneous Integer Functions.
- (line 11)
-* mpz_gcd: Number Theoretic Functions.
- (line 30)
-* mpz_gcd_ui: Number Theoretic Functions.
- (line 35)
-* mpz_gcdext: Number Theoretic Functions.
- (line 45)
-* mpz_get_d: Converting Integers. (line 27)
-* mpz_get_d_2exp: Converting Integers. (line 35)
-* mpz_get_si: Converting Integers. (line 18)
-* mpz_get_str: Converting Integers. (line 46)
-* mpz_get_ui: Converting Integers. (line 11)
-* mpz_getlimbn: Integer Special Functions.
- (line 60)
-* mpz_hamdist: Integer Logic and Bit Fiddling.
- (line 29)
-* mpz_import: Integer Import and Export.
- (line 11)
-* mpz_init: Initializing Integers.
- (line 26)
-* mpz_init2: Initializing Integers.
- (line 33)
-* mpz_init_set: Simultaneous Integer Init & Assign.
- (line 27)
-* mpz_init_set_d: Simultaneous Integer Init & Assign.
- (line 30)
-* mpz_init_set_si: Simultaneous Integer Init & Assign.
- (line 29)
-* mpz_init_set_str: Simultaneous Integer Init & Assign.
- (line 34)
-* mpz_init_set_ui: Simultaneous Integer Init & Assign.
- (line 28)
-* mpz_inits: Initializing Integers.
- (line 29)
-* mpz_inp_raw: I/O of Integers. (line 59)
-* mpz_inp_str: I/O of Integers. (line 28)
-* mpz_invert: Number Theoretic Functions.
- (line 60)
-* mpz_ior: Integer Logic and Bit Fiddling.
- (line 14)
-* mpz_jacobi: Number Theoretic Functions.
- (line 66)
-* mpz_kronecker: Number Theoretic Functions.
- (line 74)
-* mpz_kronecker_si: Number Theoretic Functions.
- (line 75)
-* mpz_kronecker_ui: Number Theoretic Functions.
- (line 76)
-* mpz_lcm: Number Theoretic Functions.
- (line 54)
-* mpz_lcm_ui: Number Theoretic Functions.
- (line 55)
-* mpz_legendre: Number Theoretic Functions.
- (line 69)
-* mpz_lucnum2_ui: Number Theoretic Functions.
- (line 119)
-* mpz_lucnum_ui: Number Theoretic Functions.
- (line 117)
-* mpz_mod: Integer Division. (line 91)
-* mpz_mod_ui: Integer Division. (line 93)
-* mpz_mul: Integer Arithmetic. (line 19)
-* mpz_mul_2exp: Integer Arithmetic. (line 35)
-* mpz_mul_si: Integer Arithmetic. (line 20)
-* mpz_mul_ui: Integer Arithmetic. (line 22)
-* mpz_neg: Integer Arithmetic. (line 39)
-* mpz_nextprime: Number Theoretic Functions.
- (line 23)
-* mpz_odd_p: Miscellaneous Integer Functions.
- (line 17)
-* mpz_out_raw: I/O of Integers. (line 43)
-* mpz_out_str: I/O of Integers. (line 16)
-* mpz_perfect_power_p: Integer Roots. (line 27)
-* mpz_perfect_square_p: Integer Roots. (line 36)
-* mpz_popcount: Integer Logic and Bit Fiddling.
- (line 23)
-* mpz_pow_ui: Integer Exponentiation.
- (line 31)
-* mpz_powm: Integer Exponentiation.
- (line 8)
-* mpz_powm_sec: Integer Exponentiation.
- (line 18)
-* mpz_powm_ui: Integer Exponentiation.
- (line 10)
-* mpz_probab_prime_p: Number Theoretic Functions.
- (line 7)
-* mpz_random: Integer Random Numbers.
- (line 42)
-* mpz_random2: Integer Random Numbers.
- (line 51)
-* mpz_realloc2: Initializing Integers.
- (line 52)
-* mpz_remove: Number Theoretic Functions.
- (line 90)
-* mpz_root: Integer Roots. (line 7)
-* mpz_rootrem: Integer Roots. (line 13)
-* mpz_rrandomb: Integer Random Numbers.
- (line 31)
-* mpz_scan0: Integer Logic and Bit Fiddling.
- (line 37)
-* mpz_scan1: Integer Logic and Bit Fiddling.
- (line 38)
-* mpz_set: Assigning Integers. (line 10)
-* mpz_set_d: Assigning Integers. (line 13)
-* mpz_set_f: Assigning Integers. (line 15)
-* mpz_set_q: Assigning Integers. (line 14)
-* mpz_set_si: Assigning Integers. (line 12)
-* mpz_set_str: Assigning Integers. (line 21)
-* mpz_set_ui: Assigning Integers. (line 11)
-* mpz_setbit: Integer Logic and Bit Fiddling.
- (line 51)
-* mpz_sgn: Integer Comparisons. (line 28)
-* mpz_si_kronecker: Number Theoretic Functions.
- (line 77)
-* mpz_size: Integer Special Functions.
- (line 68)
-* mpz_sizeinbase: Miscellaneous Integer Functions.
- (line 23)
-* mpz_sqrt: Integer Roots. (line 17)
-* mpz_sqrtrem: Integer Roots. (line 20)
-* mpz_sub: Integer Arithmetic. (line 12)
-* mpz_sub_ui: Integer Arithmetic. (line 14)
-* mpz_submul: Integer Arithmetic. (line 30)
-* mpz_submul_ui: Integer Arithmetic. (line 32)
-* mpz_swap: Assigning Integers. (line 37)
-* mpz_t: Nomenclature and Types.
- (line 6)
-* mpz_tdiv_q: Integer Division. (line 41)
-* mpz_tdiv_q_2exp: Integer Division. (line 52)
-* mpz_tdiv_q_ui: Integer Division. (line 45)
-* mpz_tdiv_qr: Integer Division. (line 43)
-* mpz_tdiv_qr_ui: Integer Division. (line 49)
-* mpz_tdiv_r: Integer Division. (line 42)
-* mpz_tdiv_r_2exp: Integer Division. (line 53)
-* mpz_tdiv_r_ui: Integer Division. (line 47)
-* mpz_tdiv_ui: Integer Division. (line 51)
-* mpz_tstbit: Integer Logic and Bit Fiddling.
- (line 60)
-* mpz_ui_kronecker: Number Theoretic Functions.
- (line 78)
-* mpz_ui_pow_ui: Integer Exponentiation.
- (line 33)
-* mpz_ui_sub: Integer Arithmetic. (line 16)
-* mpz_urandomb: Integer Random Numbers.
- (line 14)
-* mpz_urandomm: Integer Random Numbers.
- (line 23)
-* mpz_xor: Integer Logic and Bit Fiddling.
- (line 17)
-* msqrt: BSD Compatible Functions.
- (line 63)
-* msub: BSD Compatible Functions.
- (line 46)
-* mtox: BSD Compatible Functions.
- (line 98)
-* mult: BSD Compatible Functions.
- (line 49)
-* operator%: C++ Interface Integers.
- (line 30)
-* operator/: C++ Interface Integers.
- (line 29)
-* operator<<: C++ Formatted Output.
- (line 20)
-* operator>> <1>: C++ Formatted Input. (line 11)
-* operator>>: C++ Interface Rationals.
- (line 77)
-* pow: BSD Compatible Functions.
- (line 71)
-* rpow: BSD Compatible Functions.
- (line 79)
-* sdiv: BSD Compatible Functions.
- (line 55)
-* sgn <1>: C++ Interface Rationals.
- (line 50)
-* sgn <2>: C++ Interface Integers.
- (line 57)
-* sgn: C++ Interface Floats.
- (line 89)
-* sqrt <1>: C++ Interface Integers.
- (line 58)
-* sqrt: C++ Interface Floats.
- (line 90)
-* trunc: C++ Interface Floats.
- (line 91)
-* xtom: BSD Compatible Functions.
- (line 34)
-
-
case `uname -m` in
x86_64)
# No cp commands, we want to use static linking instead.
- export CC="$CC -I../../../../misc/builddeps/linux64/d0_blind_id/include"
- export CC="$CC -L../../../../misc/builddeps/linux64/d0_blind_id/lib"
- export CC="$CC -Wl,-rpath,../../../../misc/builddeps/linux64/d0_blind_id/lib"
- export CC="$CC -I../../../../misc/builddeps/linux64/gmp/include"
- export CC="$CC -L../../../../misc/builddeps/linux64/gmp/lib"
- export CC="$CC -Wl,-rpath,../../../../misc/builddeps/linux64/gmp/lib"
- MAKEFLAGS="$MAKEFLAGS DP_LINK_CRYPTO=shared DP_LINK_CRYPTO_RIJNDAEL=shared LIB_CRYPTO=../../../../misc/builddeps/linux64/d0_blind_id/lib/libd0_blind_id.a LIB_CRYPTO+=../../../../misc/builddeps/linux64/gmp/lib/libgmp.a LIB_CRYPTO_RIJNDAEL=../../../../misc/builddeps/linux64/d0_blind_id/lib/libd0_rijndael.a"
+ MAKEFLAGS="$MAKEFLAGS DP_LINK_CRYPTO=shared DP_LINK_CRYPTO_RIJNDAEL=shared"
+ export CC="$CC -I../../../../"
+ export CC="$CC -L../../../../d0_blind_id/.libs"
+ export CC="$CC -Wl,-rpath,./d0_blind_id/.libs"
;;
*86)
# No cp commands, we want to use static linking instead.
- export CC="$CC -I../../../../misc/builddeps/linux32/d0_blind_id/include"
- export CC="$CC -L../../../../misc/builddeps/linux32/d0_blind_id/lib"
- export CC="$CC -Wl,-rpath,../../../../misc/builddeps/linux32/d0_blind_id/lib"
- export CC="$CC -I../../../../misc/builddeps/linux32/gmp/include"
- export CC="$CC -L../../../../misc/builddeps/linux32/gmp/lib"
- export CC="$CC -Wl,-rpath,../../../../misc/builddeps/linux32/gmp/lib"
- MAKEFLAGS="$MAKEFLAGS DP_LINK_CRYPTO=shared DP_LINK_CRYPTO_RIJNDAEL=shared LIB_CRYPTO=../../../../misc/builddeps/linux32/d0_blind_id/lib/libd0_blind_id.a LIB_CRYPTO+=../../../../misc/builddeps/linux32/gmp/lib/libgmp.a LIB_CRYPTO_RIJNDAEL=../../../../misc/builddeps/linux32/d0_blind_id/lib/libd0_rijndael.a"
+ export CC="$CC -I../../../../"
+ export CC="$CC -L../../../../d0_blind_id/.libs"
+ export CC="$CC -Wl,-rpath,./d0_blind_id/.libs"
+ MAKEFLAGS="$MAKEFLAGS DP_LINK_CRYPTO=shared DP_LINK_CRYPTO_RIJNDAEL=shared"
;;
*)
compiled0=true
projects / xonotic / xonotic.git / commitdiff