--- /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 1
+#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 "i586-mingw32msvc-gcc -std=gnu99"
+#define __GMP_CFLAGS "-m32 -O2 -pedantic -fomit-frame-pointer -mtune=pentium -march=pentium -mno-cygwin"
+
+/* 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
+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.
+
+\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
+ software manuals; it can be used for any textual work, regardless
+ of subject matter or whether it is published as a printed book.
+ We recommend this License principally for works whose purpose is
+ instruction or reference.
+
+ 1. APPLICABILITY AND DEFINITIONS
+
+ This License applies to any manual or other work, in any medium,
+ that contains a notice placed by the copyright holder saying it
+ can be distributed under the terms of this License. Such a notice
+ grants a world-wide, royalty-free license, unlimited in duration,
+ to use that work under the conditions stated herein. The
+ "Document", below, refers to any such manual or work. Any member
+ of the public is a licensee, and is addressed as "you". You
+ accept the license if you copy, modify or distribute the work in a
+ way requiring permission under copyright law.
+
+ A "Modified Version" of the Document means any work containing the
+ Document or a portion of it, either copied verbatim, or with
+ modifications and/or translated into another language.
+
+ A "Secondary Section" is a named appendix or a front-matter section
+ of the Document that deals exclusively with the relationship of the
+ publishers or authors of the Document to the Document's overall
+ subject (or to related matters) and contains nothing that could
+ fall directly within that overall subject. (Thus, if the Document
+ is in part a textbook of mathematics, a Secondary Section may not
+ explain any mathematics.) The relationship could be a matter of
+ historical connection with the subject or with related matters, or
+ of legal, commercial, philosophical, ethical or political position
+ regarding them.
+
+ The "Invariant Sections" are certain Secondary Sections whose
+ titles are designated, as being those of Invariant Sections, in
+ the notice that says that the Document is released under this
+ License. If a section does not fit the above definition of
+ Secondary then it is not allowed to be designated as Invariant.
+ The Document may contain zero Invariant Sections. If the Document
+ does not identify any Invariant Sections then there are none.
+
+ The "Cover Texts" are certain short passages of text that are
+ listed, as Front-Cover Texts or Back-Cover Texts, in the notice
+ that says that the Document is released under this License. A
+ Front-Cover Text may be at most 5 words, and a Back-Cover Text may
+ be at most 25 words.
+
+ A "Transparent" copy of the Document means a machine-readable copy,
+ represented in a format whose specification is available to the
+ general public, that is suitable for revising the document
+ straightforwardly with generic text editors or (for images
+ composed of pixels) generic paint programs or (for drawings) some
+ widely available drawing editor, and that is suitable for input to
+ text formatters or for automatic translation to a variety of
+ formats suitable for input to text formatters. A copy made in an
+ otherwise Transparent file format whose markup, or absence of
+ markup, has been arranged to thwart or discourage subsequent
+ modification by readers is not Transparent. An image format is
+ not Transparent if used for any substantial amount of text. A
+ copy that is not "Transparent" is called "Opaque".
+
+ Examples of suitable formats for Transparent copies include plain
+ ASCII without markup, Texinfo input format, LaTeX input format,
+ SGML or XML using a publicly available DTD, and
+ standard-conforming simple HTML, PostScript or PDF designed for
+ human modification. Examples of transparent image formats include
+ PNG, XCF and JPG. Opaque formats include proprietary formats that
+ can be read and edited only by proprietary word processors, SGML or
+ XML for which the DTD and/or processing tools are not generally
+ available, and the machine-generated HTML, PostScript or PDF
+ produced by some word processors for output purposes only.
+
+ The "Title Page" means, for a printed book, the title page itself,
+ plus such following pages as are needed to hold, legibly, the
+ material this License requires to appear in the title page. For
+ works in formats which do not have any title page as such, "Title
+ Page" means the text near the most prominent appearance of the
+ work's title, preceding the beginning of the body of the text.
+
+ The "publisher" means any person or entity that distributes copies
+ of the Document to the public.
+
+ A section "Entitled XYZ" means a named subunit of the Document
+ whose title either is precisely XYZ or contains XYZ in parentheses
+ following text that translates XYZ in another language. (Here XYZ
+ stands for a specific section name mentioned below, such as
+ "Acknowledgements", "Dedications", "Endorsements", or "History".)
+ To "Preserve the Title" of such a section when you modify the
+ Document means that it remains a section "Entitled XYZ" according
+ to this definition.
+
+ The Document may include Warranty Disclaimers next to the notice
+ which states that this License applies to the Document. These
+ Warranty Disclaimers are considered to be included by reference in
+ this License, but only as regards disclaiming warranties: any other
+ implication that these Warranty Disclaimers may have is void and
+ has no effect on the meaning of this License.
+
+ 2. VERBATIM COPYING
+
+ You may copy and distribute the Document in any medium, either
+ commercially or noncommercially, provided that this License, the
+ copyright notices, and the license notice saying this License
+ applies to the Document are reproduced in all copies, and that you
+ add no other conditions whatsoever to those of this License. You
+ may not use technical measures to obstruct or control the reading
+ or further copying of the copies you make or distribute. However,
+ you may accept compensation in exchange for copies. If you
+ distribute a large enough number of copies you must also follow
+ the conditions in section 3.
+
+ You may also lend copies, under the same conditions stated above,
+ and you may publicly display copies.
+
+ 3. COPYING IN QUANTITY
+
+ If you publish printed copies (or copies in media that commonly
+ have printed covers) of the Document, numbering more than 100, and
+ the Document's license notice requires Cover Texts, you must
+ enclose the copies in covers that carry, clearly and legibly, all
+ these Cover Texts: Front-Cover Texts on the front cover, and
+ Back-Cover Texts on the back cover. Both covers must also clearly
+ and legibly identify you as the publisher of these copies. The
+ front cover must present the full title with all words of the
+ title equally prominent and visible. You may add other material
+ on the covers in addition. Copying with changes limited to the
+ covers, as long as they preserve the title of the Document and
+ satisfy these conditions, can be treated as verbatim copying in
+ other respects.
+
+ If the required texts for either cover are too voluminous to fit
+ legibly, you should put the first ones listed (as many as fit
+ reasonably) on the actual cover, and continue the rest onto
+ adjacent pages.
+
+ If you publish or distribute Opaque copies of the Document
+ numbering more than 100, you must either include a
+ machine-readable Transparent copy along with each Opaque copy, or
+ state in or with each Opaque copy a computer-network location from
+ which the general network-using public has access to download
+ using public-standard network protocols a complete Transparent
+ copy of the Document, free of added material. If you use the
+ latter option, you must take reasonably prudent steps, when you
+ begin distribution of Opaque copies in quantity, to ensure that
+ this Transparent copy will remain thus accessible at the stated
+ location until at least one year after the last time you
+ distribute an Opaque copy (directly or through your agents or
+ retailers) of that edition to the public.
+
+ It is requested, but not required, that you contact the authors of
+ the Document well before redistributing any large number of
+ copies, to give them a chance to provide you with an updated
+ version of the Document.
+
+ 4. MODIFICATIONS
+
+ You may copy and distribute a Modified Version of the Document
+ under the conditions of sections 2 and 3 above, provided that you
+ release the Modified Version under precisely this License, with
+ the Modified Version filling the role of the Document, thus
+ licensing distribution and modification of the Modified Version to
+ whoever possesses a copy of it. In addition, you must do these
+ things in the Modified Version:
+
+ A. Use in the Title Page (and on the covers, if any) a title
+ distinct from that of the Document, and from those of
+ previous versions (which should, if there were any, be listed
+ in the History section of the Document). You may use the
+ same title as a previous version if the original publisher of
+ that version gives permission.
+
+ B. List on the Title Page, as authors, one or more persons or
+ entities responsible for authorship of the modifications in
+ the Modified Version, together with at least five of the
+ principal authors of the Document (all of its principal
+ authors, if it has fewer than five), unless they release you
+ from this requirement.
+
+ C. State on the Title page the name of the publisher of the
+ Modified Version, as the publisher.
+
+ D. Preserve all the copyright notices of the Document.
+
+ E. Add an appropriate copyright notice for your modifications
+ adjacent to the other copyright notices.
+
+ F. Include, immediately after the copyright notices, a license
+ notice giving the public permission to use the Modified
+ Version under the terms of this License, in the form shown in
+ the Addendum below.
+
+ G. Preserve in that license notice the full lists of Invariant
+ Sections and required Cover Texts given in the Document's
+ license notice.
+
+ H. Include an unaltered copy of this License.
+
+ I. Preserve the section Entitled "History", Preserve its Title,
+ and add to it an item stating at least the title, year, new
+ authors, and publisher of the Modified Version as given on
+ the Title Page. If there is no section Entitled "History" in
+ the Document, create one stating the title, year, authors,
+ and publisher of the Document as given on its Title Page,
+ then add an item describing the Modified Version as stated in
+ the previous sentence.
+
+ J. Preserve the network location, if any, given in the Document
+ for public access to a Transparent copy of the Document, and
+ likewise the network locations given in the Document for
+ previous versions it was based on. These may be placed in
+ the "History" section. You may omit a network location for a
+ work that was published at least four years before the
+ Document itself, or if the original publisher of the version
+ it refers to gives permission.
+
+ K. For any section Entitled "Acknowledgements" or "Dedications",
+ Preserve the Title of the section, and preserve in the
+ section all the substance and tone of each of the contributor
+ acknowledgements and/or dedications given therein.
+
+ L. Preserve all the Invariant Sections of the Document,
+ unaltered in their text and in their titles. Section numbers
+ or the equivalent are not considered part of the section
+ titles.
+
+ M. Delete any section Entitled "Endorsements". Such a section
+ may not be included in the Modified Version.
+
+ N. Do not retitle any existing section to be Entitled
+ "Endorsements" or to conflict in title with any Invariant
+ Section.
+
+ O. Preserve any Warranty Disclaimers.
+
+ If the Modified Version includes new front-matter sections or
+ appendices that qualify as Secondary Sections and contain no
+ material copied from the Document, you may at your option
+ designate some or all of these sections as invariant. To do this,
+ add their titles to the list of Invariant Sections in the Modified
+ Version's license notice. These titles must be distinct from any
+ other section titles.
+
+ You may add a section Entitled "Endorsements", provided it contains
+ nothing but endorsements of your Modified Version by various
+ parties--for example, statements of peer review or that the text
+ has been approved by an organization as the authoritative
+ definition of a standard.
+
+ You may add a passage of up to five words as a Front-Cover Text,
+ and a passage of up to 25 words as a Back-Cover Text, to the end
+ of the list of Cover Texts in the Modified Version. Only one
+ passage of Front-Cover Text and one of Back-Cover Text may be
+ added by (or through arrangements made by) any one entity. If the
+ Document already includes a cover text for the same cover,
+ previously added by you or by arrangement made by the same entity
+ you are acting on behalf of, you may not add another; but you may
+ replace the old one, on explicit permission from the previous
+ publisher that added the old one.
+
+ The author(s) and publisher(s) of the Document do not by this
+ License give permission to use their names for publicity for or to
+ assert or imply endorsement of any Modified Version.
+
+ 5. COMBINING DOCUMENTS
+
+ You may combine the Document with other documents released under
+ this License, under the terms defined in section 4 above for
+ modified versions, provided that you include in the combination
+ all of the Invariant Sections of all of the original documents,
+ unmodified, and list them all as Invariant Sections of your
+ combined work in its license notice, and that you preserve all
+ their Warranty Disclaimers.
+
+ The combined work need only contain one copy of this License, and
+ multiple identical Invariant Sections may be replaced with a single
+ copy. If there are multiple Invariant Sections with the same name
+ but different contents, make the title of each such section unique
+ by adding at the end of it, in parentheses, the name of the
+ original author or publisher of that section if known, or else a
+ unique number. Make the same adjustment to the section titles in
+ the list of Invariant Sections in the license notice of the
+ combined work.
+
+ In the combination, you must combine any sections Entitled
+ "History" in the various original documents, forming one section
+ Entitled "History"; likewise combine any sections Entitled
+ "Acknowledgements", and any sections Entitled "Dedications". You
+ must delete all sections Entitled "Endorsements."
+
+ 6. COLLECTIONS OF DOCUMENTS
+
+ You may make a collection consisting of the Document and other
+ documents released under this License, and replace the individual
+ copies of this License in the various documents with a single copy
+ that is included in the collection, provided that you follow the
+ rules of this License for verbatim copying of each of the
+ documents in all other respects.
+
+ You may extract a single document from such a collection, and
+ distribute it individually under this License, provided you insert
+ a copy of this License into the extracted document, and follow
+ this License in all other respects regarding verbatim copying of
+ that document.
+
+ 7. AGGREGATION WITH INDEPENDENT WORKS
+
+ A compilation of the Document or its derivatives with other
+ separate and independent documents or works, in or on a volume of
+ a storage or distribution medium, is called an "aggregate" if the
+ copyright resulting from the compilation is not used to limit the
+ legal rights of the compilation's users beyond what the individual
+ works permit. When the Document is included in an aggregate, this
+ License does not apply to the other works in the aggregate which
+ are not themselves derivative works of the Document.
+
+ If the Cover Text requirement of section 3 is applicable to these
+ copies of the Document, then if the Document is less than one half
+ of the entire aggregate, the Document's Cover Texts may be placed
+ on covers that bracket the Document within the aggregate, or the
+ electronic equivalent of covers if the Document is in electronic
+ form. Otherwise they must appear on printed covers that bracket
+ the whole aggregate.
+
+ 8. TRANSLATION
+
+ Translation is considered a kind of modification, so you may
+ distribute translations of the Document under the terms of section
+ 4. Replacing Invariant Sections with translations requires special
+ permission from their copyright holders, but you may include
+ translations of some or all Invariant Sections in addition to the
+ original versions of these Invariant Sections. You may include a
+ 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
+ original versions of those notices and disclaimers. In case of a
+ disagreement between the translation and the original version of
+ this License or a notice or disclaimer, the original version will
+ prevail.
+
+ If a section in the Document is Entitled "Acknowledgements",
+ "Dedications", or "History", the requirement (section 4) to
+ Preserve its Title (section 1) will typically require changing the
+ actual title.
+
+ 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.
+
+ However, if you cease all violation of this License, then your
+ license from a particular copyright holder is reinstated (a)
+ provisionally, unless and until the copyright holder explicitly
+ and finally terminates your license, and (b) permanently, if the
+ copyright holder fails to notify you of the violation by some
+ reasonable means prior to 60 days after the cessation.
+
+ Moreover, your license from a particular copyright holder is
+ reinstated permanently if the copyright holder notifies you of the
+ violation by some reasonable means, this is the first time you have
+ received notice of violation of this License (for any work) from
+ that copyright holder, and you cure the violation prior to 30 days
+ after your receipt of the notice.
+
+ Termination of your rights under this section does not terminate
+ the licenses of parties who have received copies or rights from
+ you under this License. If your rights have been terminated and
+ not permanently reinstated, receipt of a copy of some or all of
+ the same material does not give you any rights to use it.
+
+ 10. FUTURE REVISIONS OF THIS LICENSE
+
+ The Free Software Foundation may publish new, revised versions of
+ the GNU Free Documentation License from time to time. Such new
+ versions will be similar in spirit to the present version, but may
+ differ in detail to address new problems or concerns. See
+ `http://www.gnu.org/copyleft/'.
+
+ Each version of the License is given a distinguishing version
+ number. If the Document specifies that a particular numbered
+ version of this License "or any later version" applies to it, you
+ have the option of following the terms and conditions either of
+ that specified version or of any later version that has been
+ published (not as a draft) by the Free Software Foundation. If
+ the Document does not specify a version number of this License,
+ you may choose any version ever published (not as a draft) by the
+ Free Software Foundation. If the Document specifies that a proxy
+ can decide which future versions of this License can be used, that
+ proxy's public statement of acceptance of a version permanently
+ authorizes you to choose that version for the Document.
+
+ 11. RELICENSING
+
+ "Massive Multiauthor Collaboration Site" (or "MMC Site") means any
+ World Wide Web server that publishes copyrightable works and also
+ provides prominent facilities for anybody to edit those works. A
+ 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)
+
+
projects / xonotic / xonotic.git / commitdiff