/* 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.
+Copyright 1991, 1993-1997, 1999-2016, 2020 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.
+it under the terms of either:
+
+ * 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.
+
+or
+
+ * the GNU General Public License as published by the Free Software
+ Foundation; either version 2 of the License, or (at your option) any
+ later version.
+
+or both in parallel, as here.
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.
+or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 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/. */
+You should have received copies of the GNU General Public License and the
+GNU Lesser General Public License along with the GNU MP Library. If not,
+see https://www.gnu.org/licenses/. */
#ifndef __GMP_H__
#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 __GNU_MP__ 6
-#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
+#include <limits.h>
/* Instantiated by configure. */
#if ! defined (__GMP_WITHIN_CONFIGURE)
#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.
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
#endif
#if __GMP_LIBGMP_DLL
-#if __GMP_WITHIN_GMP
+#ifdef __GMP_WITHIN_GMP
/* compiling to go into a DLL libgmp */
#define __GMP_DECLSPEC __GMP_DECLSPEC_EXPORT
#else
typedef __mpz_struct mpz_t[1];
typedef mp_limb_t * mp_ptr;
-typedef __gmp_const mp_limb_t * mp_srcptr;
+typedef 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
/* Types for function declarations in gmp files. */
/* ??? Should not pollute user name space with these ??? */
-typedef __gmp_const __mpz_struct *mpz_srcptr;
+typedef const __mpz_struct *mpz_srcptr;
typedef __mpz_struct *mpz_ptr;
-typedef __gmp_const __mpf_struct *mpf_srcptr;
+typedef const __mpf_struct *mpf_srcptr;
typedef __mpf_struct *mpf_ptr;
-typedef __gmp_const __mpq_struct *mpq_srcptr;
+typedef 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
+#ifdef __GMP_WITHIN_GMPXX
/* compiling to go into a DLL libgmpxx */
#define __GMP_DECLSPEC_XX __GMP_DECLSPEC_EXPORT
#else
#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,
|| defined (__STDIO__) /* Apple MPW MrC */ \
|| defined (_MSL_STDIO_H) /* Metrowerks */ \
|| defined (_STDIO_H_INCLUDED) /* QNX4 */ \
- || defined (_ISO_STDIO_ISO_H) /* Sun C++ */
+ || defined (_ISO_STDIO_ISO_H) /* Sun C++ */ \
+ || defined (__STDIO_LOADED) /* VMS */ \
+ || defined (__DEFINED_FILE) /* musl */
#define _GMP_H_HAVE_FILE 1
#endif
#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.
+/* The prototypes for gmp_vprintf etc are provided only if va_list is defined,
+ via an application having included <stdarg.h>. Usually va_list is a typedef
+ so can't be tested directly, but C99 specifies that va_start is a macro.
<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. */
+ whole <stdarg.h> anyway. */
#ifdef va_start
#define _GMP_H_HAVE_VA_LIST 1
__GMP_ATTRIBUTE_PURE. */
#if defined (__cplusplus)
+#if __cplusplus >= 201103L
+#define __GMP_NOTHROW noexcept
+#else
#define __GMP_NOTHROW throw ()
+#endif
#else
#define __GMP_NOTHROW
#endif
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)
+#if (defined __GNUC_STDC_INLINE__) || (__GNUC__ == 4 && __GNUC_MINOR__ == 2) \
+ || (defined __GNUC_GNU_INLINE__ && defined __cplusplus)
#define __GMP_EXTERN_INLINE extern __inline__ __attribute__ ((__gnu_inline__))
#else
#define __GMP_EXTERN_INLINE extern __inline__
#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)
#endif
-/* Allow direct user access to numerator and denominator of a mpq_t object. */
+/* Allow direct user access to numerator and denominator of an mpq_t object. */
#define mpq_numref(Q) (&((Q)->_mp_num))
#define mpq_denref(Q) (&((Q)->_mp_den))
#endif
#define mp_set_memory_functions __gmp_set_memory_functions
-__GMP_DECLSPEC void mp_set_memory_functions __GMP_PROTO ((void *(*) (size_t),
+__GMP_DECLSPEC void mp_set_memory_functions (void *(*) (size_t),
void *(*) (void *, size_t, size_t),
- void (*) (void *, size_t))) __GMP_NOTHROW;
+ 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;
+__GMP_DECLSPEC void mp_get_memory_functions (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;
+__GMP_DECLSPEC extern 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;
+__GMP_DECLSPEC extern const char * 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, ...));
+__GMP_DECLSPEC void gmp_randinit (gmp_randstate_t, gmp_randalg_t, ...);
#define gmp_randinit_default __gmp_randinit_default
-__GMP_DECLSPEC void gmp_randinit_default __GMP_PROTO ((gmp_randstate_t));
+__GMP_DECLSPEC void gmp_randinit_default (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));
+__GMP_DECLSPEC void gmp_randinit_lc_2exp (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));
+__GMP_DECLSPEC int gmp_randinit_lc_2exp_size (gmp_randstate_t, mp_bitcnt_t);
#define gmp_randinit_mt __gmp_randinit_mt
-__GMP_DECLSPEC void gmp_randinit_mt __GMP_PROTO ((gmp_randstate_t));
+__GMP_DECLSPEC void gmp_randinit_mt (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 *));
+__GMP_DECLSPEC void gmp_randinit_set (gmp_randstate_t, const __gmp_randstate_struct *);
#define gmp_randseed __gmp_randseed
-__GMP_DECLSPEC void gmp_randseed __GMP_PROTO ((gmp_randstate_t, mpz_srcptr));
+__GMP_DECLSPEC void gmp_randseed (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));
+__GMP_DECLSPEC void gmp_randseed_ui (gmp_randstate_t, unsigned long int);
#define gmp_randclear __gmp_randclear
-__GMP_DECLSPEC void gmp_randclear __GMP_PROTO ((gmp_randstate_t));
+__GMP_DECLSPEC void gmp_randclear (gmp_randstate_t);
#define gmp_urandomb_ui __gmp_urandomb_ui
-__GMP_DECLSPEC unsigned long gmp_urandomb_ui __GMP_PROTO ((gmp_randstate_t, unsigned long));
+__GMP_DECLSPEC unsigned long gmp_urandomb_ui (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));
+__GMP_DECLSPEC unsigned long gmp_urandomm_ui (gmp_randstate_t, unsigned long);
/**************** Formatted output routines. ****************/
#define gmp_asprintf __gmp_asprintf
-__GMP_DECLSPEC int gmp_asprintf __GMP_PROTO ((char **, __gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_asprintf (char **, const char *, ...);
#define gmp_fprintf __gmp_fprintf
#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC int gmp_fprintf __GMP_PROTO ((FILE *, __gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_fprintf (FILE *, 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 *, ...));
+__GMP_DECLSPEC int gmp_obstack_printf (struct obstack *, 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));
+__GMP_DECLSPEC int gmp_obstack_vprintf (struct obstack *, const char *, va_list);
#endif
#define gmp_printf __gmp_printf
-__GMP_DECLSPEC int gmp_printf __GMP_PROTO ((__gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_printf (const char *, ...);
#define gmp_snprintf __gmp_snprintf
-__GMP_DECLSPEC int gmp_snprintf __GMP_PROTO ((char *, size_t, __gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_snprintf (char *, size_t, const char *, ...);
#define gmp_sprintf __gmp_sprintf
-__GMP_DECLSPEC int gmp_sprintf __GMP_PROTO ((char *, __gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_sprintf (char *, 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));
+__GMP_DECLSPEC int gmp_vasprintf (char **, 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));
+__GMP_DECLSPEC int gmp_vfprintf (FILE *, 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));
+__GMP_DECLSPEC int gmp_vprintf (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));
+__GMP_DECLSPEC int gmp_vsnprintf (char *, size_t, 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));
+__GMP_DECLSPEC int gmp_vsprintf (char *, const char *, va_list);
#endif
#define gmp_fscanf __gmp_fscanf
#ifdef _GMP_H_HAVE_FILE
-__GMP_DECLSPEC int gmp_fscanf __GMP_PROTO ((FILE *, __gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_fscanf (FILE *, const char *, ...);
#endif
#define gmp_scanf __gmp_scanf
-__GMP_DECLSPEC int gmp_scanf __GMP_PROTO ((__gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_scanf (const char *, ...);
#define gmp_sscanf __gmp_sscanf
-__GMP_DECLSPEC int gmp_sscanf __GMP_PROTO ((__gmp_const char *, __gmp_const char *, ...));
+__GMP_DECLSPEC int gmp_sscanf (const char *, 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));
+__GMP_DECLSPEC int gmp_vfscanf (FILE *, 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));
+__GMP_DECLSPEC int gmp_vscanf (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));
+__GMP_DECLSPEC int gmp_vsscanf (const char *, const char *, va_list);
#endif
#define _mpz_realloc __gmpz_realloc
#define mpz_realloc __gmpz_realloc
-__GMP_DECLSPEC void *_mpz_realloc __GMP_PROTO ((mpz_ptr, mp_size_t));
+__GMP_DECLSPEC void *_mpz_realloc (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));
+__GMP_DECLSPEC void mpz_abs (mpz_ptr, mpz_srcptr);
#endif
#define mpz_add __gmpz_add
-__GMP_DECLSPEC void mpz_add __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_add (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));
+__GMP_DECLSPEC void mpz_add_ui (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));
+__GMP_DECLSPEC void mpz_addmul (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));
+__GMP_DECLSPEC void mpz_addmul_ui (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));
+__GMP_DECLSPEC void mpz_and (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));
+__GMP_DECLSPEC void mpz_array_init (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));
+__GMP_DECLSPEC void mpz_bin_ui (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));
+__GMP_DECLSPEC void mpz_bin_uiui (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));
+__GMP_DECLSPEC void mpz_cdiv_q (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));
+__GMP_DECLSPEC void mpz_cdiv_q_2exp (mpz_ptr, mpz_srcptr, mp_bitcnt_t);
#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));
+__GMP_DECLSPEC unsigned long int mpz_cdiv_q_ui (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));
+__GMP_DECLSPEC void mpz_cdiv_qr (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));
+__GMP_DECLSPEC unsigned long int mpz_cdiv_qr_ui (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));
+__GMP_DECLSPEC void mpz_cdiv_r (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));
+__GMP_DECLSPEC void mpz_cdiv_r_2exp (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));
+__GMP_DECLSPEC unsigned long int mpz_cdiv_r_ui (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;
+__GMP_DECLSPEC unsigned long int mpz_cdiv_ui (mpz_srcptr, unsigned long int) __GMP_ATTRIBUTE_PURE;
#define mpz_clear __gmpz_clear
-__GMP_DECLSPEC void mpz_clear __GMP_PROTO ((mpz_ptr));
+__GMP_DECLSPEC void mpz_clear (mpz_ptr);
#define mpz_clears __gmpz_clears
-__GMP_DECLSPEC void mpz_clears __GMP_PROTO ((mpz_ptr, ...));
+__GMP_DECLSPEC void mpz_clears (mpz_ptr, ...);
#define mpz_clrbit __gmpz_clrbit
-__GMP_DECLSPEC void mpz_clrbit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
+__GMP_DECLSPEC void mpz_clrbit (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;
+__GMP_DECLSPEC int mpz_cmp (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;
+__GMP_DECLSPEC int mpz_cmp_d (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;
+__GMP_DECLSPEC int _mpz_cmp_si (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;
+__GMP_DECLSPEC int _mpz_cmp_ui (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;
+__GMP_DECLSPEC int mpz_cmpabs (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;
+__GMP_DECLSPEC int mpz_cmpabs_d (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;
+__GMP_DECLSPEC int mpz_cmpabs_ui (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));
+__GMP_DECLSPEC void mpz_com (mpz_ptr, mpz_srcptr);
#define mpz_combit __gmpz_combit
-__GMP_DECLSPEC void mpz_combit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
+__GMP_DECLSPEC void mpz_combit (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;
+__GMP_DECLSPEC int mpz_congruent_p (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;
+__GMP_DECLSPEC int mpz_congruent_2exp_p (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;
+__GMP_DECLSPEC int mpz_congruent_ui_p (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));
+__GMP_DECLSPEC void mpz_divexact (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));
+__GMP_DECLSPEC void mpz_divexact_ui (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;
+__GMP_DECLSPEC int mpz_divisible_p (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;
+__GMP_DECLSPEC int mpz_divisible_ui_p (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;
+__GMP_DECLSPEC int mpz_divisible_2exp_p (mpz_srcptr, mp_bitcnt_t) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
#define mpz_dump __gmpz_dump
-__GMP_DECLSPEC void mpz_dump __GMP_PROTO ((mpz_srcptr));
+__GMP_DECLSPEC void mpz_dump (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));
+__GMP_DECLSPEC void *mpz_export (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));
+__GMP_DECLSPEC void mpz_fac_ui (mpz_ptr, unsigned long int);
+
+#define mpz_2fac_ui __gmpz_2fac_ui
+__GMP_DECLSPEC void mpz_2fac_ui (mpz_ptr, unsigned long int);
+
+#define mpz_mfac_uiui __gmpz_mfac_uiui
+__GMP_DECLSPEC void mpz_mfac_uiui (mpz_ptr, unsigned long int, unsigned long int);
+
+#define mpz_primorial_ui __gmpz_primorial_ui
+__GMP_DECLSPEC void mpz_primorial_ui (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));
+__GMP_DECLSPEC void mpz_fdiv_q (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));
+__GMP_DECLSPEC void mpz_fdiv_q_2exp (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));
+__GMP_DECLSPEC unsigned long int mpz_fdiv_q_ui (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));
+__GMP_DECLSPEC void mpz_fdiv_qr (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));
+__GMP_DECLSPEC unsigned long int mpz_fdiv_qr_ui (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));
+__GMP_DECLSPEC void mpz_fdiv_r (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));
+__GMP_DECLSPEC void mpz_fdiv_r_2exp (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));
+__GMP_DECLSPEC unsigned long int mpz_fdiv_r_ui (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;
+__GMP_DECLSPEC unsigned long int mpz_fdiv_ui (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));
+__GMP_DECLSPEC void mpz_fib_ui (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));
+__GMP_DECLSPEC void mpz_fib2_ui (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;
+__GMP_DECLSPEC int mpz_fits_sint_p (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;
+__GMP_DECLSPEC int mpz_fits_slong_p (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;
+__GMP_DECLSPEC int mpz_fits_sshort_p (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;
+__GMP_DECLSPEC int mpz_fits_uint_p (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;
+__GMP_DECLSPEC int mpz_fits_ulong_p (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;
+__GMP_DECLSPEC int mpz_fits_ushort_p (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));
+__GMP_DECLSPEC void mpz_gcd (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));
+__GMP_DECLSPEC unsigned long int mpz_gcd_ui (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));
+__GMP_DECLSPEC void mpz_gcdext (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;
+__GMP_DECLSPEC double mpz_get_d (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));
+__GMP_DECLSPEC double mpz_get_d_2exp (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;
+__GMP_DECLSPEC /* signed */ long int mpz_get_si (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));
+__GMP_DECLSPEC char *mpz_get_str (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;
+__GMP_DECLSPEC unsigned long int mpz_get_ui (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;
+__GMP_DECLSPEC mp_limb_t mpz_getlimbn (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;
+__GMP_DECLSPEC mp_bitcnt_t mpz_hamdist (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 *));
+__GMP_DECLSPEC void mpz_import (mpz_ptr, size_t, int, size_t, int, size_t, const void *);
#define mpz_init __gmpz_init
-__GMP_DECLSPEC void mpz_init __GMP_PROTO ((mpz_ptr));
+__GMP_DECLSPEC void mpz_init (mpz_ptr) __GMP_NOTHROW;
#define mpz_init2 __gmpz_init2
-__GMP_DECLSPEC void mpz_init2 __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
+__GMP_DECLSPEC void mpz_init2 (mpz_ptr, mp_bitcnt_t);
#define mpz_inits __gmpz_inits
-__GMP_DECLSPEC void mpz_inits __GMP_PROTO ((mpz_ptr, ...));
+__GMP_DECLSPEC void mpz_inits (mpz_ptr, ...) __GMP_NOTHROW;
#define mpz_init_set __gmpz_init_set
-__GMP_DECLSPEC void mpz_init_set __GMP_PROTO ((mpz_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_init_set (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));
+__GMP_DECLSPEC void mpz_init_set_d (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));
+__GMP_DECLSPEC void mpz_init_set_si (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));
+__GMP_DECLSPEC int mpz_init_set_str (mpz_ptr, 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));
+__GMP_DECLSPEC void mpz_init_set_ui (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 *));
+__GMP_DECLSPEC size_t mpz_inp_raw (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));
+__GMP_DECLSPEC size_t mpz_inp_str (mpz_ptr, FILE *, int);
#endif
#define mpz_invert __gmpz_invert
-__GMP_DECLSPEC int mpz_invert __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
+__GMP_DECLSPEC int mpz_invert (mpz_ptr, mpz_srcptr, mpz_srcptr);
#define mpz_ior __gmpz_ior
-__GMP_DECLSPEC void mpz_ior __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_ior (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;
+__GMP_DECLSPEC int mpz_jacobi (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;
+__GMP_DECLSPEC int mpz_kronecker_si (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;
+__GMP_DECLSPEC int mpz_kronecker_ui (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;
+__GMP_DECLSPEC int mpz_si_kronecker (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;
+__GMP_DECLSPEC int mpz_ui_kronecker (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));
+__GMP_DECLSPEC void mpz_lcm (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));
+__GMP_DECLSPEC void mpz_lcm_ui (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));
+__GMP_DECLSPEC void mpz_lucnum_ui (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));
+__GMP_DECLSPEC void mpz_lucnum2_ui (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;
+__GMP_DECLSPEC int mpz_millerrabin (mpz_srcptr, int) __GMP_ATTRIBUTE_PURE;
#define mpz_mod __gmpz_mod
-__GMP_DECLSPEC void mpz_mod __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_mod (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));
+__GMP_DECLSPEC void mpz_mul (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));
+__GMP_DECLSPEC void mpz_mul_2exp (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));
+__GMP_DECLSPEC void mpz_mul_si (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));
+__GMP_DECLSPEC void mpz_mul_ui (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));
+__GMP_DECLSPEC void mpz_neg (mpz_ptr, mpz_srcptr);
#endif
#define mpz_nextprime __gmpz_nextprime
-__GMP_DECLSPEC void mpz_nextprime __GMP_PROTO ((mpz_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_nextprime (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));
+__GMP_DECLSPEC size_t mpz_out_raw (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));
+__GMP_DECLSPEC size_t mpz_out_str (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;
+__GMP_DECLSPEC int mpz_perfect_power_p (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;
+__GMP_DECLSPEC int mpz_perfect_square_p (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;
+__GMP_DECLSPEC mp_bitcnt_t mpz_popcount (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));
+__GMP_DECLSPEC void mpz_pow_ui (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));
+__GMP_DECLSPEC void mpz_powm (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));
+__GMP_DECLSPEC void mpz_powm_sec (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));
+__GMP_DECLSPEC void mpz_powm_ui (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;
+__GMP_DECLSPEC int mpz_probab_prime_p (mpz_srcptr, int) __GMP_ATTRIBUTE_PURE;
#define mpz_random __gmpz_random
-__GMP_DECLSPEC void mpz_random __GMP_PROTO ((mpz_ptr, mp_size_t));
+__GMP_DECLSPEC void mpz_random (mpz_ptr, mp_size_t);
#define mpz_random2 __gmpz_random2
-__GMP_DECLSPEC void mpz_random2 __GMP_PROTO ((mpz_ptr, mp_size_t));
+__GMP_DECLSPEC void mpz_random2 (mpz_ptr, mp_size_t);
#define mpz_realloc2 __gmpz_realloc2
-__GMP_DECLSPEC void mpz_realloc2 __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
+__GMP_DECLSPEC void mpz_realloc2 (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));
+__GMP_DECLSPEC mp_bitcnt_t mpz_remove (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));
+__GMP_DECLSPEC int mpz_root (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));
+__GMP_DECLSPEC void mpz_rootrem (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));
+__GMP_DECLSPEC void mpz_rrandomb (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;
+__GMP_DECLSPEC mp_bitcnt_t mpz_scan0 (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;
+__GMP_DECLSPEC mp_bitcnt_t mpz_scan1 (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));
+__GMP_DECLSPEC void mpz_set (mpz_ptr, mpz_srcptr);
#define mpz_set_d __gmpz_set_d
-__GMP_DECLSPEC void mpz_set_d __GMP_PROTO ((mpz_ptr, double));
+__GMP_DECLSPEC void mpz_set_d (mpz_ptr, double);
#define mpz_set_f __gmpz_set_f
-__GMP_DECLSPEC void mpz_set_f __GMP_PROTO ((mpz_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpz_set_f (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));
+__GMP_DECLSPEC void mpz_set_q (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));
+__GMP_DECLSPEC void mpz_set_si (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));
+__GMP_DECLSPEC int mpz_set_str (mpz_ptr, const char *, int);
#define mpz_set_ui __gmpz_set_ui
-__GMP_DECLSPEC void mpz_set_ui __GMP_PROTO ((mpz_ptr, unsigned long int));
+__GMP_DECLSPEC void mpz_set_ui (mpz_ptr, unsigned long int);
#define mpz_setbit __gmpz_setbit
-__GMP_DECLSPEC void mpz_setbit __GMP_PROTO ((mpz_ptr, mp_bitcnt_t));
+__GMP_DECLSPEC void mpz_setbit (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;
+__GMP_DECLSPEC size_t mpz_size (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;
+__GMP_DECLSPEC size_t mpz_sizeinbase (mpz_srcptr, int) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
#define mpz_sqrt __gmpz_sqrt
-__GMP_DECLSPEC void mpz_sqrt __GMP_PROTO ((mpz_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_sqrt (mpz_ptr, mpz_srcptr);
#define mpz_sqrtrem __gmpz_sqrtrem
-__GMP_DECLSPEC void mpz_sqrtrem __GMP_PROTO ((mpz_ptr, mpz_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_sqrtrem (mpz_ptr, mpz_ptr, mpz_srcptr);
#define mpz_sub __gmpz_sub
-__GMP_DECLSPEC void mpz_sub __GMP_PROTO ((mpz_ptr, mpz_srcptr, mpz_srcptr));
+__GMP_DECLSPEC void mpz_sub (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));
+__GMP_DECLSPEC void mpz_sub_ui (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));
+__GMP_DECLSPEC void mpz_ui_sub (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));
+__GMP_DECLSPEC void mpz_submul (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));
+__GMP_DECLSPEC void mpz_submul_ui (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;
+__GMP_DECLSPEC void mpz_swap (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;
+__GMP_DECLSPEC unsigned long int mpz_tdiv_ui (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));
+__GMP_DECLSPEC void mpz_tdiv_q (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));
+__GMP_DECLSPEC void mpz_tdiv_q_2exp (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));
+__GMP_DECLSPEC unsigned long int mpz_tdiv_q_ui (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));
+__GMP_DECLSPEC void mpz_tdiv_qr (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));
+__GMP_DECLSPEC unsigned long int mpz_tdiv_qr_ui (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));
+__GMP_DECLSPEC void mpz_tdiv_r (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));
+__GMP_DECLSPEC void mpz_tdiv_r_2exp (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));
+__GMP_DECLSPEC unsigned long int mpz_tdiv_r_ui (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;
+__GMP_DECLSPEC int mpz_tstbit (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));
+__GMP_DECLSPEC void mpz_ui_pow_ui (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));
+__GMP_DECLSPEC void mpz_urandomb (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));
+__GMP_DECLSPEC void mpz_urandomm (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));
+__GMP_DECLSPEC void mpz_xor (mpz_ptr, mpz_srcptr, mpz_srcptr);
+
+#define mpz_limbs_read __gmpz_limbs_read
+__GMP_DECLSPEC mp_srcptr mpz_limbs_read (mpz_srcptr);
+
+#define mpz_limbs_write __gmpz_limbs_write
+__GMP_DECLSPEC mp_ptr mpz_limbs_write (mpz_ptr, mp_size_t);
+
+#define mpz_limbs_modify __gmpz_limbs_modify
+__GMP_DECLSPEC mp_ptr mpz_limbs_modify (mpz_ptr, mp_size_t);
+
+#define mpz_limbs_finish __gmpz_limbs_finish
+__GMP_DECLSPEC void mpz_limbs_finish (mpz_ptr, mp_size_t);
+#define mpz_roinit_n __gmpz_roinit_n
+__GMP_DECLSPEC mpz_srcptr mpz_roinit_n (mpz_ptr, mp_srcptr, mp_size_t);
+
+#define MPZ_ROINIT_N(xp, xs) {{0, (xs),(xp) }}
/**************** 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));
+__GMP_DECLSPEC void mpq_abs (mpq_ptr, mpq_srcptr);
#endif
#define mpq_add __gmpq_add
-__GMP_DECLSPEC void mpq_add __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_add (mpq_ptr, mpq_srcptr, mpq_srcptr);
#define mpq_canonicalize __gmpq_canonicalize
-__GMP_DECLSPEC void mpq_canonicalize __GMP_PROTO ((mpq_ptr));
+__GMP_DECLSPEC void mpq_canonicalize (mpq_ptr);
#define mpq_clear __gmpq_clear
-__GMP_DECLSPEC void mpq_clear __GMP_PROTO ((mpq_ptr));
+__GMP_DECLSPEC void mpq_clear (mpq_ptr);
#define mpq_clears __gmpq_clears
-__GMP_DECLSPEC void mpq_clears __GMP_PROTO ((mpq_ptr, ...));
+__GMP_DECLSPEC void mpq_clears (mpq_ptr, ...);
#define mpq_cmp __gmpq_cmp
-__GMP_DECLSPEC int mpq_cmp __GMP_PROTO ((mpq_srcptr, mpq_srcptr)) __GMP_ATTRIBUTE_PURE;
+__GMP_DECLSPEC int mpq_cmp (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;
+__GMP_DECLSPEC int _mpq_cmp_si (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;
+__GMP_DECLSPEC int _mpq_cmp_ui (mpq_srcptr, unsigned long int, unsigned long int) __GMP_ATTRIBUTE_PURE;
+
+#define mpq_cmp_z __gmpq_cmp_z
+__GMP_DECLSPEC int mpq_cmp_z (mpq_srcptr, mpz_srcptr) __GMP_ATTRIBUTE_PURE;
#define mpq_div __gmpq_div
-__GMP_DECLSPEC void mpq_div __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_div (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));
+__GMP_DECLSPEC void mpq_div_2exp (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;
+__GMP_DECLSPEC int mpq_equal (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));
+__GMP_DECLSPEC void mpq_get_num (mpz_ptr, mpq_srcptr);
#define mpq_get_den __gmpq_get_den
-__GMP_DECLSPEC void mpq_get_den __GMP_PROTO ((mpz_ptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_get_den (mpz_ptr, mpq_srcptr);
#define mpq_get_d __gmpq_get_d
-__GMP_DECLSPEC double mpq_get_d __GMP_PROTO ((mpq_srcptr)) __GMP_ATTRIBUTE_PURE;
+__GMP_DECLSPEC double mpq_get_d (mpq_srcptr) __GMP_ATTRIBUTE_PURE;
#define mpq_get_str __gmpq_get_str
-__GMP_DECLSPEC char *mpq_get_str __GMP_PROTO ((char *, int, mpq_srcptr));
+__GMP_DECLSPEC char *mpq_get_str (char *, int, mpq_srcptr);
#define mpq_init __gmpq_init
-__GMP_DECLSPEC void mpq_init __GMP_PROTO ((mpq_ptr));
+__GMP_DECLSPEC void mpq_init (mpq_ptr);
#define mpq_inits __gmpq_inits
-__GMP_DECLSPEC void mpq_inits __GMP_PROTO ((mpq_ptr, ...));
+__GMP_DECLSPEC void mpq_inits (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));
+__GMP_DECLSPEC size_t mpq_inp_str (mpq_ptr, FILE *, int);
#endif
#define mpq_inv __gmpq_inv
-__GMP_DECLSPEC void mpq_inv __GMP_PROTO ((mpq_ptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_inv (mpq_ptr, mpq_srcptr);
#define mpq_mul __gmpq_mul
-__GMP_DECLSPEC void mpq_mul __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_mul (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));
+__GMP_DECLSPEC void mpq_mul_2exp (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));
+__GMP_DECLSPEC void mpq_neg (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));
+__GMP_DECLSPEC size_t mpq_out_str (FILE *, int, mpq_srcptr);
#endif
#define mpq_set __gmpq_set
-__GMP_DECLSPEC void mpq_set __GMP_PROTO ((mpq_ptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_set (mpq_ptr, mpq_srcptr);
#define mpq_set_d __gmpq_set_d
-__GMP_DECLSPEC void mpq_set_d __GMP_PROTO ((mpq_ptr, double));
+__GMP_DECLSPEC void mpq_set_d (mpq_ptr, double);
#define mpq_set_den __gmpq_set_den
-__GMP_DECLSPEC void mpq_set_den __GMP_PROTO ((mpq_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpq_set_den (mpq_ptr, mpz_srcptr);
#define mpq_set_f __gmpq_set_f
-__GMP_DECLSPEC void mpq_set_f __GMP_PROTO ((mpq_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpq_set_f (mpq_ptr, mpf_srcptr);
#define mpq_set_num __gmpq_set_num
-__GMP_DECLSPEC void mpq_set_num __GMP_PROTO ((mpq_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpq_set_num (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));
+__GMP_DECLSPEC void mpq_set_si (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));
+__GMP_DECLSPEC int mpq_set_str (mpq_ptr, 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));
+__GMP_DECLSPEC void mpq_set_ui (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));
+__GMP_DECLSPEC void mpq_set_z (mpq_ptr, mpz_srcptr);
#define mpq_sub __gmpq_sub
-__GMP_DECLSPEC void mpq_sub __GMP_PROTO ((mpq_ptr, mpq_srcptr, mpq_srcptr));
+__GMP_DECLSPEC void mpq_sub (mpq_ptr, mpq_srcptr, mpq_srcptr);
#define mpq_swap __gmpq_swap
-__GMP_DECLSPEC void mpq_swap __GMP_PROTO ((mpq_ptr, mpq_ptr)) __GMP_NOTHROW;
+__GMP_DECLSPEC void mpq_swap (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));
+__GMP_DECLSPEC void mpf_abs (mpf_ptr, mpf_srcptr);
#define mpf_add __gmpf_add
-__GMP_DECLSPEC void mpf_add __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_add (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));
+__GMP_DECLSPEC void mpf_add_ui (mpf_ptr, mpf_srcptr, unsigned long int);
#define mpf_ceil __gmpf_ceil
-__GMP_DECLSPEC void mpf_ceil __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_ceil (mpf_ptr, mpf_srcptr);
#define mpf_clear __gmpf_clear
-__GMP_DECLSPEC void mpf_clear __GMP_PROTO ((mpf_ptr));
+__GMP_DECLSPEC void mpf_clear (mpf_ptr);
#define mpf_clears __gmpf_clears
-__GMP_DECLSPEC void mpf_clears __GMP_PROTO ((mpf_ptr, ...));
+__GMP_DECLSPEC void mpf_clears (mpf_ptr, ...);
#define mpf_cmp __gmpf_cmp
-__GMP_DECLSPEC int mpf_cmp __GMP_PROTO ((mpf_srcptr, mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
+__GMP_DECLSPEC int mpf_cmp (mpf_srcptr, mpf_srcptr) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
+
+#define mpf_cmp_z __gmpf_cmp_z
+__GMP_DECLSPEC int mpf_cmp_z (mpf_srcptr, mpz_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;
+__GMP_DECLSPEC int mpf_cmp_d (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;
+__GMP_DECLSPEC int mpf_cmp_si (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;
+__GMP_DECLSPEC int mpf_cmp_ui (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));
+__GMP_DECLSPEC void mpf_div (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));
+__GMP_DECLSPEC void mpf_div_2exp (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));
+__GMP_DECLSPEC void mpf_div_ui (mpf_ptr, mpf_srcptr, unsigned long int);
#define mpf_dump __gmpf_dump
-__GMP_DECLSPEC void mpf_dump __GMP_PROTO ((mpf_srcptr));
+__GMP_DECLSPEC void mpf_dump (mpf_srcptr);
#define mpf_eq __gmpf_eq
-__GMP_DECLSPEC int mpf_eq __GMP_PROTO ((mpf_srcptr, mpf_srcptr, unsigned long int)) __GMP_ATTRIBUTE_PURE;
+__GMP_DECLSPEC int mpf_eq (mpf_srcptr, mpf_srcptr, mp_bitcnt_t) __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;
+__GMP_DECLSPEC int mpf_fits_sint_p (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;
+__GMP_DECLSPEC int mpf_fits_slong_p (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;
+__GMP_DECLSPEC int mpf_fits_sshort_p (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;
+__GMP_DECLSPEC int mpf_fits_uint_p (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;
+__GMP_DECLSPEC int mpf_fits_ulong_p (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;
+__GMP_DECLSPEC int mpf_fits_ushort_p (mpf_srcptr) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
#define mpf_floor __gmpf_floor
-__GMP_DECLSPEC void mpf_floor __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_floor (mpf_ptr, mpf_srcptr);
#define mpf_get_d __gmpf_get_d
-__GMP_DECLSPEC double mpf_get_d __GMP_PROTO ((mpf_srcptr)) __GMP_ATTRIBUTE_PURE;
+__GMP_DECLSPEC double mpf_get_d (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));
+__GMP_DECLSPEC double mpf_get_d_2exp (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;
+__GMP_DECLSPEC mp_bitcnt_t mpf_get_default_prec (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;
+__GMP_DECLSPEC mp_bitcnt_t mpf_get_prec (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;
+__GMP_DECLSPEC long mpf_get_si (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));
+__GMP_DECLSPEC char *mpf_get_str (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;
+__GMP_DECLSPEC unsigned long mpf_get_ui (mpf_srcptr) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
#define mpf_init __gmpf_init
-__GMP_DECLSPEC void mpf_init __GMP_PROTO ((mpf_ptr));
+__GMP_DECLSPEC void mpf_init (mpf_ptr);
#define mpf_init2 __gmpf_init2
-__GMP_DECLSPEC void mpf_init2 __GMP_PROTO ((mpf_ptr, mp_bitcnt_t));
+__GMP_DECLSPEC void mpf_init2 (mpf_ptr, mp_bitcnt_t);
#define mpf_inits __gmpf_inits
-__GMP_DECLSPEC void mpf_inits __GMP_PROTO ((mpf_ptr, ...));
+__GMP_DECLSPEC void mpf_inits (mpf_ptr, ...);
#define mpf_init_set __gmpf_init_set
-__GMP_DECLSPEC void mpf_init_set __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_init_set (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));
+__GMP_DECLSPEC void mpf_init_set_d (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));
+__GMP_DECLSPEC void mpf_init_set_si (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));
+__GMP_DECLSPEC int mpf_init_set_str (mpf_ptr, 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));
+__GMP_DECLSPEC void mpf_init_set_ui (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));
+__GMP_DECLSPEC size_t mpf_inp_str (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;
+__GMP_DECLSPEC int mpf_integer_p (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));
+__GMP_DECLSPEC void mpf_mul (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));
+__GMP_DECLSPEC void mpf_mul_2exp (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));
+__GMP_DECLSPEC void mpf_mul_ui (mpf_ptr, mpf_srcptr, unsigned long int);
#define mpf_neg __gmpf_neg
-__GMP_DECLSPEC void mpf_neg __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_neg (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));
+__GMP_DECLSPEC size_t mpf_out_str (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));
+__GMP_DECLSPEC void mpf_pow_ui (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));
+__GMP_DECLSPEC void mpf_random2 (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));
+__GMP_DECLSPEC void mpf_reldiff (mpf_ptr, mpf_srcptr, mpf_srcptr);
#define mpf_set __gmpf_set
-__GMP_DECLSPEC void mpf_set __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_set (mpf_ptr, mpf_srcptr);
#define mpf_set_d __gmpf_set_d
-__GMP_DECLSPEC void mpf_set_d __GMP_PROTO ((mpf_ptr, double));
+__GMP_DECLSPEC void mpf_set_d (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;
+__GMP_DECLSPEC void mpf_set_default_prec (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));
+__GMP_DECLSPEC void mpf_set_prec (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;
+__GMP_DECLSPEC void mpf_set_prec_raw (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));
+__GMP_DECLSPEC void mpf_set_q (mpf_ptr, mpq_srcptr);
#define mpf_set_si __gmpf_set_si
-__GMP_DECLSPEC void mpf_set_si __GMP_PROTO ((mpf_ptr, signed long int));
+__GMP_DECLSPEC void mpf_set_si (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));
+__GMP_DECLSPEC int mpf_set_str (mpf_ptr, const char *, int);
#define mpf_set_ui __gmpf_set_ui
-__GMP_DECLSPEC void mpf_set_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
+__GMP_DECLSPEC void mpf_set_ui (mpf_ptr, unsigned long int);
#define mpf_set_z __gmpf_set_z
-__GMP_DECLSPEC void mpf_set_z __GMP_PROTO ((mpf_ptr, mpz_srcptr));
+__GMP_DECLSPEC void mpf_set_z (mpf_ptr, mpz_srcptr);
#define mpf_size __gmpf_size
-__GMP_DECLSPEC size_t mpf_size __GMP_PROTO ((mpf_srcptr)) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
+__GMP_DECLSPEC size_t mpf_size (mpf_srcptr) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
#define mpf_sqrt __gmpf_sqrt
-__GMP_DECLSPEC void mpf_sqrt __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_sqrt (mpf_ptr, mpf_srcptr);
#define mpf_sqrt_ui __gmpf_sqrt_ui
-__GMP_DECLSPEC void mpf_sqrt_ui __GMP_PROTO ((mpf_ptr, unsigned long int));
+__GMP_DECLSPEC void mpf_sqrt_ui (mpf_ptr, unsigned long int);
#define mpf_sub __gmpf_sub
-__GMP_DECLSPEC void mpf_sub __GMP_PROTO ((mpf_ptr, mpf_srcptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_sub (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));
+__GMP_DECLSPEC void mpf_sub_ui (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;
+__GMP_DECLSPEC void mpf_swap (mpf_ptr, mpf_ptr) __GMP_NOTHROW;
#define mpf_trunc __gmpf_trunc
-__GMP_DECLSPEC void mpf_trunc __GMP_PROTO ((mpf_ptr, mpf_srcptr));
+__GMP_DECLSPEC void mpf_trunc (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));
+__GMP_DECLSPEC void mpf_ui_div (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));
+__GMP_DECLSPEC void mpf_ui_sub (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));
+__GMP_DECLSPEC void mpf_urandomb (mpf_t, gmp_randstate_t, mp_bitcnt_t);
/************ Low level positive-integer (i.e. N) routines. ************/
#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));
+__GMP_DECLSPEC mp_limb_t mpn_add (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;
+__GMP_DECLSPEC mp_limb_t mpn_add_1 (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));
+__GMP_DECLSPEC mp_limb_t mpn_add_n (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));
+__GMP_DECLSPEC mp_limb_t mpn_addmul_1 (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;
+__GMP_DECLSPEC int mpn_cmp (mp_srcptr, mp_srcptr, mp_size_t) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
#endif
+#define mpn_zero_p __MPN(zero_p)
+#if __GMP_INLINE_PROTOTYPES || defined (__GMP_FORCE_mpn_zero_p)
+__GMP_DECLSPEC int mpn_zero_p (mp_srcptr, mp_size_t) __GMP_NOTHROW __GMP_ATTRIBUTE_PURE;
+#endif
+
+#define mpn_divexact_1 __MPN(divexact_1)
+__GMP_DECLSPEC void mpn_divexact_1 (mp_ptr, mp_srcptr, mp_size_t, mp_limb_t);
+
#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));
+__GMP_DECLSPEC mp_limb_t mpn_divexact_by3c (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));
+__GMP_DECLSPEC mp_limb_t mpn_divrem (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));
+__GMP_DECLSPEC mp_limb_t mpn_divrem_1 (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));
+__GMP_DECLSPEC mp_limb_t mpn_divrem_2 (mp_ptr, mp_size_t, mp_ptr, mp_size_t, mp_srcptr);
+
+#define mpn_div_qr_1 __MPN(div_qr_1)
+__GMP_DECLSPEC mp_limb_t mpn_div_qr_1 (mp_ptr, mp_limb_t *, mp_srcptr, mp_size_t, mp_limb_t);
+
+#define mpn_div_qr_2 __MPN(div_qr_2)
+__GMP_DECLSPEC mp_limb_t mpn_div_qr_2 (mp_ptr, mp_ptr, mp_srcptr, 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));
+__GMP_DECLSPEC mp_size_t mpn_gcd (mp_ptr, mp_ptr, mp_size_t, mp_ptr, mp_size_t);
+
+#define mpn_gcd_11 __MPN(gcd_11)
+__GMP_DECLSPEC mp_limb_t mpn_gcd_11 (mp_limb_t, mp_limb_t) __GMP_ATTRIBUTE_PURE;
#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;
+__GMP_DECLSPEC mp_limb_t mpn_gcd_1 (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));
+__GMP_DECLSPEC mp_limb_t mpn_gcdext_1 (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));
+__GMP_DECLSPEC mp_size_t mpn_gcdext (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));
+__GMP_DECLSPEC size_t mpn_get_str (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;
+__GMP_DECLSPEC mp_bitcnt_t mpn_hamdist (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));
+__GMP_DECLSPEC mp_limb_t mpn_lshift (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;
+__GMP_DECLSPEC mp_limb_t mpn_mod_1 (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));
+__GMP_DECLSPEC mp_limb_t mpn_mul (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));
+__GMP_DECLSPEC mp_limb_t mpn_mul_1 (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));
+__GMP_DECLSPEC void mpn_mul_n (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));
+__GMP_DECLSPEC void mpn_sqr (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));
+__GMP_DECLSPEC mp_limb_t mpn_neg (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
+__GMP_DECLSPEC void mpn_com (mp_ptr, mp_srcptr, mp_size_t);
#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;
+__GMP_DECLSPEC int mpn_perfect_square_p (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;
+__GMP_DECLSPEC int mpn_perfect_power_p (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;
+__GMP_DECLSPEC mp_bitcnt_t mpn_popcount (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));
+__GMP_DECLSPEC mp_size_t mpn_pow_1 (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;
+__GMP_DECLSPEC mp_limb_t mpn_preinv_mod_1 (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));
+__GMP_DECLSPEC void mpn_random (mp_ptr, mp_size_t);
#define mpn_random2 __MPN(random2)
-__GMP_DECLSPEC void mpn_random2 __GMP_PROTO ((mp_ptr, mp_size_t));
+__GMP_DECLSPEC void mpn_random2 (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));
+__GMP_DECLSPEC mp_limb_t mpn_rshift (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;
+__GMP_DECLSPEC mp_bitcnt_t mpn_scan0 (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;
+__GMP_DECLSPEC mp_bitcnt_t mpn_scan1 (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));
+__GMP_DECLSPEC mp_size_t mpn_set_str (mp_ptr, const unsigned char *, size_t, int);
+
+#define mpn_sizeinbase __MPN(sizeinbase)
+__GMP_DECLSPEC size_t mpn_sizeinbase (mp_srcptr, mp_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));
+__GMP_DECLSPEC mp_size_t mpn_sqrtrem (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));
+__GMP_DECLSPEC mp_limb_t mpn_sub (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;
+__GMP_DECLSPEC mp_limb_t mpn_sub_1 (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));
+__GMP_DECLSPEC mp_limb_t mpn_sub_n (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));
+__GMP_DECLSPEC mp_limb_t mpn_submul_1 (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));
+__GMP_DECLSPEC void mpn_tdiv_qr (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));
+__GMP_DECLSPEC void mpn_and_n (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));
+__GMP_DECLSPEC void mpn_andn_n (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));
+__GMP_DECLSPEC void mpn_nand_n (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));
+__GMP_DECLSPEC void mpn_ior_n (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));
+__GMP_DECLSPEC void mpn_iorn_n (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));
+__GMP_DECLSPEC void mpn_nior_n (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));
+__GMP_DECLSPEC void mpn_xor_n (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));
+__GMP_DECLSPEC void mpn_xnor_n (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));
+__GMP_DECLSPEC void mpn_copyi (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));
+__GMP_DECLSPEC void mpn_copyd (mp_ptr, mp_srcptr, mp_size_t);
#define mpn_zero __MPN(zero)
-__GMP_DECLSPEC void mpn_zero __GMP_PROTO ((mp_ptr, mp_size_t));
+__GMP_DECLSPEC void mpn_zero (mp_ptr, mp_size_t);
+
+#define mpn_cnd_add_n __MPN(cnd_add_n)
+__GMP_DECLSPEC mp_limb_t mpn_cnd_add_n (mp_limb_t, mp_ptr, mp_srcptr, mp_srcptr, mp_size_t);
+#define mpn_cnd_sub_n __MPN(cnd_sub_n)
+__GMP_DECLSPEC mp_limb_t mpn_cnd_sub_n (mp_limb_t, mp_ptr, mp_srcptr, mp_srcptr, mp_size_t);
+
+#define mpn_sec_add_1 __MPN(sec_add_1)
+__GMP_DECLSPEC mp_limb_t mpn_sec_add_1 (mp_ptr, mp_srcptr, mp_size_t, mp_limb_t, mp_ptr);
+#define mpn_sec_add_1_itch __MPN(sec_add_1_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_add_1_itch (mp_size_t) __GMP_ATTRIBUTE_PURE;
+
+#define mpn_sec_sub_1 __MPN(sec_sub_1)
+__GMP_DECLSPEC mp_limb_t mpn_sec_sub_1 (mp_ptr, mp_srcptr, mp_size_t, mp_limb_t, mp_ptr);
+#define mpn_sec_sub_1_itch __MPN(sec_sub_1_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_sub_1_itch (mp_size_t) __GMP_ATTRIBUTE_PURE;
+
+#define mpn_cnd_swap __MPN(cnd_swap)
+__GMP_DECLSPEC void mpn_cnd_swap (mp_limb_t, volatile mp_limb_t *, volatile mp_limb_t *, mp_size_t);
+
+#define mpn_sec_mul __MPN(sec_mul)
+__GMP_DECLSPEC void mpn_sec_mul (mp_ptr, mp_srcptr, mp_size_t, mp_srcptr, mp_size_t, mp_ptr);
+#define mpn_sec_mul_itch __MPN(sec_mul_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_mul_itch (mp_size_t, mp_size_t) __GMP_ATTRIBUTE_PURE;
+
+#define mpn_sec_sqr __MPN(sec_sqr)
+__GMP_DECLSPEC void mpn_sec_sqr (mp_ptr, mp_srcptr, mp_size_t, mp_ptr);
+#define mpn_sec_sqr_itch __MPN(sec_sqr_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_sqr_itch (mp_size_t) __GMP_ATTRIBUTE_PURE;
+
+#define mpn_sec_powm __MPN(sec_powm)
+__GMP_DECLSPEC void mpn_sec_powm (mp_ptr, mp_srcptr, mp_size_t, mp_srcptr, mp_bitcnt_t, mp_srcptr, mp_size_t, mp_ptr);
+#define mpn_sec_powm_itch __MPN(sec_powm_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_powm_itch (mp_size_t, mp_bitcnt_t, mp_size_t) __GMP_ATTRIBUTE_PURE;
+
+#define mpn_sec_tabselect __MPN(sec_tabselect)
+__GMP_DECLSPEC void mpn_sec_tabselect (volatile mp_limb_t *, volatile const mp_limb_t *, mp_size_t, mp_size_t, mp_size_t);
+
+#define mpn_sec_div_qr __MPN(sec_div_qr)
+__GMP_DECLSPEC mp_limb_t mpn_sec_div_qr (mp_ptr, mp_ptr, mp_size_t, mp_srcptr, mp_size_t, mp_ptr);
+#define mpn_sec_div_qr_itch __MPN(sec_div_qr_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_div_qr_itch (mp_size_t, mp_size_t) __GMP_ATTRIBUTE_PURE;
+#define mpn_sec_div_r __MPN(sec_div_r)
+__GMP_DECLSPEC void mpn_sec_div_r (mp_ptr, mp_size_t, mp_srcptr, mp_size_t, mp_ptr);
+#define mpn_sec_div_r_itch __MPN(sec_div_r_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_div_r_itch (mp_size_t, mp_size_t) __GMP_ATTRIBUTE_PURE;
+
+#define mpn_sec_invert __MPN(sec_invert)
+__GMP_DECLSPEC int mpn_sec_invert (mp_ptr, mp_ptr, mp_srcptr, mp_size_t, mp_bitcnt_t, mp_ptr);
+#define mpn_sec_invert_itch __MPN(sec_invert_itch)
+__GMP_DECLSPEC mp_size_t mpn_sec_invert_itch (mp_size_t) __GMP_ATTRIBUTE_PURE;
+
/**************** mpz inlines ****************/
int
mpz_fits_uint_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_UINT_MAX);
+ __GMPZ_FITS_UTYPE_P (__gmp_z, UINT_MAX);
}
#endif
int
mpz_fits_ulong_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_ULONG_MAX);
+ __GMPZ_FITS_UTYPE_P (__gmp_z, ULONG_MAX);
}
#endif
int
mpz_fits_ushort_p (mpz_srcptr __gmp_z) __GMP_NOTHROW
{
- __GMPZ_FITS_UTYPE_P (__gmp_z, __GMP_USHRT_MAX);
+ __GMPZ_FITS_UTYPE_P (__gmp_z, USHRT_MAX);
}
#endif
/* 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". */
+ "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);
mp_bitcnt_t __gmp_result;
__gmp_usize = __gmp_u->_mp_size;
- __gmp_result = (__gmp_usize < 0 ? __GMP_ULONG_MAX : 0);
+ __gmp_result = (__gmp_usize < 0 ? ~ __GMP_CAST (mp_bitcnt_t, 0) : __GMP_CAST (mp_bitcnt_t, 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_mpn_zero_p)
+#if ! defined (__GMP_FORCE_mpn_zero_p)
+__GMP_EXTERN_INLINE
+#endif
+int
+mpn_zero_p (mp_srcptr __gmp_p, mp_size_t __gmp_n) __GMP_NOTHROW
+{
+ /* if (__GMP_LIKELY (__gmp_n > 0)) */
+ do {
+ if (__gmp_p[--__gmp_n] != 0)
+ return 0;
+ } while (__gmp_n != 0);
+ return 1;
+}
+#endif
+
#if defined (__GMP_EXTERN_INLINE) || defined (__GMP_FORCE_mpn_sub)
#if ! defined (__GMP_FORCE_mpn_sub)
__GMP_EXTERN_INLINE
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;
+ while (*__gmp_up == 0) /* Low zero limbs are unchanged by negation. */
+ {
+ *__gmp_rp = 0;
+ if (!--__gmp_n) /* All zero */
+ return 0;
+ ++__gmp_up; ++__gmp_rp;
+ }
+
+ *__gmp_rp = (- *__gmp_up) & GMP_NUMB_MASK;
+
+ if (--__gmp_n) /* Higher limbs get complemented. */
+ mpn_com (++__gmp_rp, ++__gmp_up, __gmp_n);
+
+ return 1;
}
#endif
#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)) \
+#define mpz_cmp_si(Z,SI) \
+ (__builtin_constant_p ((SI) >= 0) && (SI) >= 0 \
+ ? mpz_cmp_ui (Z, __GMP_CAST (unsigned long, 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) \
+#define mpq_cmp_ui(Q,NUI,DUI) \
+ (__builtin_constant_p (NUI) && (NUI) == 0 ? mpq_sgn (Q) \
+ : __builtin_constant_p ((NUI) == (DUI)) && (NUI) == (DUI) \
+ ? mpz_cmp (mpq_numref (Q), mpq_denref (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 CC and CFLAGS which were used to build this version of GMP */
-#define __GMP_CC "gcc -std=gnu99"
-#define __GMP_CFLAGS "-O2 -pedantic -m64 -mtune=k8 -march=k8"
+#define __GMP_CC "gcc"
+#define __GMP_CFLAGS "-O2 -pedantic -fomit-frame-pointer -m64 -mtune=znver2 -march=znver2"
-/* 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
+/* Major version number is the value of __GNU_MP__ too, above. */
+#define __GNU_MP_VERSION 6
+#define __GNU_MP_VERSION_MINOR 2
#define __GNU_MP_VERSION_PATCHLEVEL 1
-#define __GMP_MP_RELEASE (__GNU_MP_VERSION * 10000 + __GNU_MP_VERSION_MINOR * 100 + __GNU_MP_VERSION_PATCHLEVEL)
+#define __GNU_MP_RELEASE (__GNU_MP_VERSION * 10000 + __GNU_MP_VERSION_MINOR * 100 + __GNU_MP_VERSION_PATCHLEVEL)
#define __GMP_H__
#endif /* __GMP_H__ */
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
+This is gmp.info, produced by makeinfo version 6.7 from gmp.texi.
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.1.
- 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::.
+ Copyright 1991, 1993-2016, 2018-2020 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.
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.
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.1.
- 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::.
+ Copyright 1991, 1993-2016, 2018-2020 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:
* 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.
+* 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.
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.
+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.
+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'.
+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.
+
+ More precisely, the GNU MP library is dual licensed, under the
+conditions of the GNU Lesser General Public License version 3 (see
+'COPYING.LESSERv3'), or the GNU General Public License version 2 (see
+'COPYINGv2'). This is the recipient's choice, and the recipient also
+has the additional option of applying later versions of these licenses.
+(The reason for this dual licensing is to make it possible to use the
+library with programs which are licensed under GPL version 2, but which
+for historical or other reasons do not allow use under later versions of
+the GPL).
+
+ Programs which are not part of the library itself, such as
+demonstration programs and the GMP testsuite, are licensed under the
+terms of the GNU General Public License version 3 (see 'COPYINGv3'), or
+any later version.
\1f
File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top
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.
+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
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.
+ There is assembly code for these CPUs: ARM Cortex-A9, Cortex-A15, and
+generic ARM, DEC Alpha 21064, 21164, and 21264, AMD K8 and K10 (sold
+under many brands, e.g. Athlon64, Phenom, Opteron) Bulldozer, and
+Bobcat, Intel Pentium, Pentium Pro/II/III, Pentium 4, Core2, Nehalem,
+Sandy bridge, Haswell, generic x86, Intel IA-64, Motorola/IBM PowerPC 32
+and 64 such as POWER970, POWER5, POWER6, and POWER7, MIPS 32-bit and
+64-bit, SPARC 32-bit ad 64-bit with special support for all UltraSPARC
+models. There is also assembly code for many obsolete CPUs.
For up-to-date information on GMP, please see the GMP web pages at
- `http://gmplib.org/'
+ <https://gmplib.org/>
The latest version of the library is available at
- `ftp://ftp.gnu.org/gnu/gmp/'
+ <https://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.
+ Many sites around the world mirror 'ftp.gnu.org', please use a mirror
+near you, see <https://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/'.
+ <https://gmplib.org/mailman/listinfo/>.
- The proper place for bug reports is <gmp-bugs@gmplib.org>. See
-*Note Reporting Bugs:: for information about reporting bugs.
+ 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
+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
make check
-And you can install (under `/usr/local' by default) with
+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
+<gmp-bugs@gmplib.org>. See *note Reporting Bugs::, for information on
what to include in useful bug reports.
* Menu:
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
+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
+ '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',
+ 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,
+ 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
+ /my/sources/gmp-6.2.1/configure
- Not all `make' programs have the necessary features (`VPATH') to
- support this. In particular, SunOS and Slowaris `make' have bugs
+ 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'
+ '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' is
+ architecture-dependent since it encodes 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'
+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
+ '--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
+ 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'
+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
+ '--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.
+ 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.
+ './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.
+ 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
+ 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'.
+ 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.
+ 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.)
+ '--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 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
+ The following CPUs have specific support. See 'configure.ac' for
details of what code and compiler options they select.
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64,
viac3, viac32
- * Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
- z8k
+ * Other: arm, sh, sh2, vax,
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
+ desired, the generic C code can be selected with the configure
+ '--disable-assembly'.
- 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.
+ 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
+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.)
+ 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'
+'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
./configure --host=mips64-sgi-irix6 ABI=n32
- See *Note ABI and ISA::, for the available choices on relevant
+ See *note ABI and ISA::, for the available choices on relevant
CPUs, and what applications need to do.
-`CC', `CFLAGS'
+'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
+ 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
+ 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
+ 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
+ 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
+ '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).
+ 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
+'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.
+ Preprocessing is done separately in some configure tests.
-`CC_FOR_BUILD'
+'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
+ 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.
+ 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
+ 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
+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
+ '--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
+ '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
+ 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
+ '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
+ 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'
+'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++'.
+ 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
+ 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
+ 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>'
+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.
+ '--enable-alloca=malloc-reentrant'.
- * `malloc-notreentrant' - the heap, with global variables.
+ * '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'.
+ '--disable-alloca' is the same as 'no'.
- * `yes' - a synonym for `alloca'.
+ * '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'.
- * `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
+ '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
+ '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
+ 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
+ An additional choice '--enable-alloca=debug' is available, to help
when debugging memory related problems (*note Debugging::).
-FFT Multiplication, `--disable-fft'
+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.
+ higher degree 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'
+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'
+Execution Profiling, '--enable-profiling=prof/gprof/instrument'
Enable profiling support, in one of various styles, *note
Profiling::.
-`MPN_PATH'
+'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
+ 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"
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.
+ 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
+ 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.
+ 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'
+ 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
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'.
+ 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
+ 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.
+ 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
+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.
+to done separately, with a fresh './configure' and 'make' each.
-AMD64 (`x86_64')
+AMD64 ('x86_64')
On AMD64 systems supporting both 32-bit and 64-bit modes for
applications, the following ABI choices are available.
- `ABI=64'
+ '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
+ Applications will usually not need special compiler flags, but
+ for reference the option is
gcc -m64
- `ABI=32'
+ '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
+ 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
+ (In GCC 2.95 and earlier there's no '-m32' option, it's the
only mode.)
+ 'ABI=x32'
+ The x32 ABI uses 64-bit limbs but 32-bit pointers. Like the
+ 64-bit ABI, it makes full use of the chip's arithmetic
+ capabilities. This ABI is not supported by all operating
+ systems.
-HPPA 2.0 (`hppa2.0*', `hppa64')
+ gcc -mx32
- `ABI=2.0w'
+
+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'
+ '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.
+ 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
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
+ 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'
+ '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
+ 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
+ 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'.
+ corresponding 'ABI'.
-IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
+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
+ '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
+ '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
+ On other IA-64 systems, GNU/Linux for instance, 'ABI=64' is the
only choice.
-MIPS under IRIX 6 (`mips*-*-irix[6789]')
+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'
+ '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.
+ 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
+ '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'
+ 'ABI=64'
The 64-bit ABI is 64-bit pointers and integers. Applications
must be compiled with
and the MIPS 2 code.
-PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970', `power4', `power5')
-
- `ABI=aix64'
+PowerPC 64 ('powerpc64', 'powerpc620', 'powerpc630', 'powerpc970', 'power4', 'power5')
+ 'ABI=mode64'
The AIX 64 ABI uses 64-bit limbs and pointers and is the
- default on PowerPC 64 `*-*-aix*' systems. Applications must
+ 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
+ On 64-bit GNU/Linux, BSD, and Mac OS X/Darwin systems, the
+ applications must be compiled with
gcc -m64
- `ABI=mode32'
- The `mode32' ABI uses a 64-bit `long long' limb but with the
+ '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.
+ conventions. This is the default for systems where the true
+ 64-bit ABI is unavailable. No special compiler options are
+ typically needed for applications. This ABI is not available
+ under AIX.
- `ABI=32'
+ '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.
-
+ GMP's speed is greatest for the 'mode64' ABI, the 'mode32' ABI is
+ 2nd best. In 'ABI=32' only the 32-bit ISA is used and this doesn't
+ make full use of a 64-bit chip.
-Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
- `ABI=64'
+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
+ higher, or Sun 'cc' is required. On GNU/Linux, depending on
+ the default 'gcc' mode, applications must be compiled with
gcc -m64
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
+ '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'
+ 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
+ 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.
+ 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
+ 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
+ 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
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
+ 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
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.
+'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.
+'--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,
+ 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.
+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').
+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
+'/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
+ 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'
+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'.
+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
================================
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
+ 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.
+ 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
+ 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.
+ 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
+ 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'.
+ default but must be selected by defining '__USE_STD_IOSTREAM'.
Configure with for instance
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
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.
+ 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.
+
+FreeBSD 7.x, 8.x, 9.0, 9.1, 9.2
+ 'm4' in these releases of FreeBSD has an eval function which
+ ignores its 2nd and 3rd arguments, which 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 FreeBSD 9.3 and 10.0, so either upgrade or use GNU m4. Note
+ that the FreeBSD package system installs GNU m4 under the name
+ 'gm4', which GMP cannot guess.
+
+FreeBSD 7.x, 8.x, 9.x
+ GMP releases starting with 6.0 do not support 'ABI=32' on
+ FreeBSD/amd64 prior to release 10.0 of the system. The cause is a
+ broken 'limits.h', which GMP no longer works around.
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/'
+ <https://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
+ 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
+ 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.
+ 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'.
+ 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
+ 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
+ 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'.
+ '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'.
+
+NetBSD 5.x
+ 'm4' in these releases of NetBSD has an eval function which ignores
+ its 2nd and 3rd arguments, which 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
+ NetBSD 6, so either upgrade or use GNU m4. Note that the NetBSD
+ package system installs GNU m4 under the name 'gm4', which GMP
+ cannot guess.
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'.
+ '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
+ 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'.
+ suggestion is to use the generic C code ('--disable-assembly'),
+ 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 '--disable-assembly'.
Sparc CPU Types
- `sparcv8' or `supersparc' on relevant systems will give a
+ 'sparcv8' or 'supersparc' on relevant systems will give a
significant performance increase over the V7 code selected by plain
- `sparc'.
+ '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
+ "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),
+ '-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.
+ purposes. In these cases the only suggestion currently is to build
+ GMP with '--disable-assembly' 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
+ '/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
+ '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
+ (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.
+ 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
+ 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
+ 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
2.5 Known Build Problems
========================
-You might find more up-to-date information at `http://gmplib.org/'.
+You might find more up-to-date information at <https://gmplib.org/>.
Compiler link options
The version of libtool currently in use rather aggressively strips
./configure CC=gcc-with-my-options
-DJGPP (`*-*-msdosdjgpp*')
- The DJGPP port of `bash' 2.03 is unable to run the `configure'
+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.
+ '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
+ '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.
+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.
+ 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
+'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'.
+ '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'
+ '$(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*')
+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
+ MacOS X using the native 'cc' (which is a modified GCC), not a
plain GCC. A static-only build should work though
- (`--disable-shared').
+ ('--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.)
+ 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
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.
+ 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.
+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
============================
For optimal performance, build GMP for the exact CPU type of the target
-computer, see *Note Build Options::.
+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,
+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.
+ 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.
+ To use the results, put the output in the file indicated in the
+'Parameters for ...' header. Then recompile from scratch.
- The `tuneup' program takes one useful parameter, `-f NNN', which
+ 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.
+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.*
+*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:
=========================
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.
+'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.
+ 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.
+ Likewise '<stdarg.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
+ 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
+ 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.
+ 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
+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
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'. Here are some examples of how to declare such integers:
mpz_t sum;
mpz_t vec[20];
"Rational number" means a multiple precision fraction. The C data
-type for these fractions is `mpq_t'. For example:
+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:
+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.
+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'.
+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.
+'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 .
+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:
+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.
+ 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
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
+ '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
+ 'mpq_'. The associated type is 'mpq_t'. There are about 35
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
+ 'mpf_'. The associated type is 'mpf_t'. There are about 70
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::)
+ 4. 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 60 (hard-to-use) functions in this
+ class. (*note Low-level 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
+ 5. Miscellaneous functions. Functions for setting up custom
allocation and functions for generating random numbers. (*note
Custom Allocation::, and *note Random Number Functions::)
========================
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.
+This notation is by analogy with the assignment operator.
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', 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
+ 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
mpz_clear (n);
}
+ GMP types like 'mpz_t' are implemented as one-element arrays of
+certain structures. Declaring a variable creates an object with the
+fields GMP needs, but variables are normally manipulated by using the
+pointer to the object. For both behavior and efficiency reasons, it is
+discouraged to make copies of the GMP object itself (either directly or
+via aggregate objects containing such GMP objects). If copies are done,
+all of them must be used read-only; using a copy as the output of some
+function will invalidate all the other copies. 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: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics
=========================
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.
+call-by-reference, meaning that when 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.
+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
+ Here's an example accepting an 'mpz_t' parameter, doing a
calculation, and storing the result to the indicated parameter.
void
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.
+ Our function 'foo' works even if its caller 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.
+ Since GMP types are implemented as one-element arrays, using a GMP
+variable as a parameter passes a pointer to the object. Hence the
+call-by-reference.
\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,
+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.
+ '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.
+point can use 'mpz_realloc2', or clear variables no longer needed.
- `mpf_t' variables, in the current implementation, use a fixed amount
+ '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::.
+ 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
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),
+ * 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.
+ * '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
+ * '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
+ functions that accept a 'gmp_randstate_t' parameter can be used
instead.
- * `gmp_randinit' (obsolete) returns an error indication through a
+ * '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'
+ advised to use 'gmp_randinit_default' or 'gmp_randinit_lc_2exp'
instead.
- * `mp_set_memory_functions' uses global variables to store the
+ * '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
+ '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
+ * 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.
+ simultaneously, but it's not safe for one to read while 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
-- 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.
+ "i.j.k". This release is "6.2.1". Note that the format "i.j" was
+ used, before version 4.3.0, when k was zero.
-- Macro: __GMP_CC
-- Macro: __GMP_CFLAGS
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
+This version of GMP is upwardly binary compatible with all 5.x, 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.
+ * '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
+ * '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.
+ * 'mpn_bdivmod', documented as preliminary in GMP 4, has been
+ removed.
- The Berkeley MP compatibility library (*note BSD Compatible
-Functions::) is source and binary compatible with the standard `libmp'.
+ 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 5.
+Please see the GMP 2 manual for details.
\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.
+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
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
+ * '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'
+ '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
+ * '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'.
+ * 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.
+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
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.
+ 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
+ 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.
+ '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
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
+ 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
+ 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.
+ 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.
+'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
+'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.
+ 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
+ '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
+ '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
+ '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
+ 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
+ '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).
+ 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
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
+ 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).
+ 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
+ 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
cancellation will be possible, and so canonicalization can be left
to the end.
- The `mpq_numref' and `mpq_denref' macros give access to the
+ 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::.
+ supplied 'mpq' functions. *Note Applying Integer Functions::.
- The canonical form for rationals allows mixed-type `mpq_t' and
+ 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_add'. For example,
/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
mpz_submul (mpq_numref(q), mpq_denref(q), z);
Number Sequences
- Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
+ 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.
+ wanted it's probably best to get a starting point and iterate from
+ there.
Text Input/Output
Hexadecimal or octal are suggested for input or output in text
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
+ '--enable-alloca' choices in *note Build Options::, for how to
address this.
- In new enough versions of GCC, `-fstack-check' may be able to
+ 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.
+ 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
+ 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
+ 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/'
+ <http://cs.ecs.baylor.edu/~donahoo/tools/ccmalloc/>
+ <http://dmalloc.com/>
+ <https://wiki.gnome.org/Apps/MemProf>
- 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.
+ 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
+ '-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.
+ 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.
+ 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
+ /my/source/dir/gmp-6.2.1/configure
- This works via `VPATH', and might require GNU `make'. Alternately
- it might be possible to change the `.c.lo' rules appropriately.
+ 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::).
+ 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.
+ 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
+ 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.
+ code has checks, not the assembly code, so '--disable-assembly'
+ should be used for maximum checking.
Temporary Memory Checking
- The build option `--enable-alloca=debug' arranges that each block
+ 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').
+ '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.
+ 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
+ To summarize the above, a GMP build for maximum debuggability would
+ be
./configure --disable-shared --enable-assert \
- --enable-alloca=debug --host=none CFLAGS=-g
+ --enable-alloca=debug --disable-assembly CFLAGS=-g
- For C++, add `--enable-cxx CXXFLAGS=-g'.
+ For C++, add '--enable-cxx CXXFLAGS=-g'.
Checker
- The GCC checker (`http://savannah.nongnu.org/projects/checker/')
+ The GCC checker (<https://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
+ ./configure --disable-assembly 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.
+ '--disable-assembly' 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').
+ Valgrind (<http://valgrind.org/>) is a memory checker for x86, ARM,
+ MIPS, PowerPC, and S/390. 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.
+
+ Valgrind does not always support every possible instruction, in
+ particular ones recently added to an ISA. Valgrind might therefore
+ be incompatible with a recent GMP or even a less recent GMP which
+ is compiled using a recent GCC.
+
+ GMP's assembly code sometimes promotes a read of the limbs to some
+ larger size, for efficiency. GMP will do this even at the start
+ and end of a multilimb operand, using naturally aligned operations
+ on the larger type. This may lead to benign reads outside of
+ allocated areas, triggering complaints from Valgrind. Valgrind's
+ option '--partial-loads-ok=yes' should help.
Other Problems
Any suspected bug in GMP itself should be isolated to make sure
- it's not an application problem, see *Note Reporting Bugs::.
+ it's not an application problem, see *note Reporting Bugs::.
\1f
File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics
spending most time and where improvements can be best sought. The
profiling choices for a GMP build are as follows.
-`--disable-profiling'
+'--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.
+ 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.
+ 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'.
+'--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
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.
+ '-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
+ 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'.
+'--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.
+ 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.
+ '-pg' and therefore not be included in the call counts.
- On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
+ 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.
+ 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
+'--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
This instrumenting is not normally a standard system feature and
will require support from an external library, such as
- `http://sourceforge.net/projects/fnccheck/'
+ <https://sourceforge.net/projects/fnccheck/>
- This should be included in `LIBS' during the GMP configure so that
+ This should be included in 'LIBS' during the GMP configure so that
test programs will link. For example,
./configure --enable-profiling=instrument LIBS=-lfc
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
+ 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
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,
+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
+ 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/])])
+ [AC_MSG_ERROR([GNU MP not found, see https://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'
+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/])])
+ [GNU MP not found, or not 3.1 or up, see https://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
+ 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
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,
+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>])
3.15 Emacs
==========
-<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
+<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',
+following in your '.emacs',
(eval-after-load "info-look"
'(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
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.
+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.
+ 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 <https://gmplib.org/> for patches for this
+release.
Please include the following in any report,
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').
+ it's informative ('where' in 'gdb', or '$C' in 'adb').
+
+ * Please do not send core dumps, executables or 'strace's.
- * Please do not send core dumps, executables or `strace's.
+ * The 'configure' options you used when building GMP, if any.
- * The configuration options you used when building GMP, if any.
+ * The output from 'configure', as printed to stdout, with any options
+ used.
- * The name of the compiler and its version. For `gcc', get the
- version with `gcc -v', otherwise perhaps `what `which cc`', or
+ * 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 'uname -a'.
- * The output from running `./config.guess', and from running
- `./configfsf.guess' (might be the same).
+ * 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 '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'.
+ * 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
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).
+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>.
*******************
This chapter describes the GMP functions for performing integer
-arithmetic. These functions start with the prefix `mpz_'.
+arithmetic. These functions start with the prefix 'mpz_'.
- GMP integers are stored in objects of type `mpz_t'.
+ GMP integers are stored in objects of type 'mpz_t'.
* Menu:
============================
The functions for integer arithmetic assume that all integer objects are
-initialized. You do that by calling the function `mpz_init'. For
+initialized. You do that by calling the function 'mpz_init'. For
example,
{
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
+ 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
+ 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
+ While N defines 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
+ 'mpz_init2' makes it possible to avoid such reallocations if a
maximum size is known in advance.
+ In preparation for an operation, GMP often allocates one limb more
+ than ultimately needed. To make sure GMP will not perform
+ reallocation for X, you need to add the number of bits in
+ 'mp_limb_t' to N.
+
-- Function: void mpz_clear (mpz_t X)
- Free the space occupied by X. Call this function for all `mpz_t'
+ 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'
+ Free the space occupied by a NULL-terminated list of 'mpz_t'
variables.
-- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N)
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
+ 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.
5.2 Assignment Functions
========================
-These functions assign new values to already initialized integers
-(*note Initializing Integers::).
+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 (mpz_t ROP, const 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)
+ -- Function: void mpz_set_q (mpz_t ROP, const mpq_t OP)
+ -- Function: void mpz_set_f (mpz_t ROP, const 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
+ '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)
+ -- Function: int mpz_set_str (mpz_t ROP, const 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.
+ 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.
+ 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.
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...'
+These functions' names have the form 'mpz_init_set...'
Here is an example of using one:
mpz_clear (pie);
}
-Once the integer has been initialized by any of the `mpz_init_set...'
+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!
+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 (mpz_t ROP, const 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
+ -- Function: int mpz_init_set_str (mpz_t ROP, const 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.)
+ 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::.
+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'.
+ -- Function: unsigned long int mpz_get_ui (const 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
+ 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.
+ -- Function: signed long int mpz_get_si (const 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'.
+ 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
+ -- Function: double mpz_get_d (const mpz_t OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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
+ -- Function: double mpz_get_d_2exp (signed long int *EXP, const mpz_t
+ OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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'.
+ 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.).
+ This is similar to the standard C 'frexp' function (*note
+ (libc)Normalization Functions::).
- -- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
+ -- Function: char * mpz_get_str (char *STR, int BASE, const 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.
+ 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
+ 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
+ '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'.
+ 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.
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)
+ -- Function: void mpz_add (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_add_ui (mpz_t ROP, const 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)
+ -- Function: void mpz_sub (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_sub_ui (mpz_t ROP, const mpz_t OP1, unsigned long
+ int OP2)
+ -- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, const
+ 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)
+ -- Function: void mpz_mul (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_mul_si (mpz_t ROP, const mpz_t OP1, long int OP2)
+ -- Function: void mpz_mul_ui (mpz_t ROP, const 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)
+ -- Function: void mpz_addmul (mpz_t ROP, const mpz_t OP1, const mpz_t
+ OP2)
+ -- Function: void mpz_addmul_ui (mpz_t ROP, const 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)
+ -- Function: void mpz_submul (mpz_t ROP, const mpz_t OP1, const mpz_t
+ OP2)
+ -- Function: void mpz_submul_ui (mpz_t ROP, const 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)
+ -- Function: void mpz_mul_2exp (mpz_t ROP, const 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)
+ -- Function: void mpz_neg (mpz_t ROP, const mpz_t OP)
Set ROP to -OP.
- -- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
+ -- Function: void mpz_abs (mpz_t ROP, const mpz_t OP)
Set ROP to the absolute value of OP.
\1f
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 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.
+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,
+ -- Function: void mpz_cdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_cdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
+ mpz_t D)
+
+ -- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
- -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
+ -- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, const 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,
+ const mpz_t N, unsigned long int D)
+ -- Function: unsigned long int mpz_cdiv_ui (const 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,
+ -- Function: void mpz_cdiv_q_2exp (mpz_t Q, const mpz_t N,
+ mp_bitcnt_t B)
+ -- Function: void mpz_cdiv_r_2exp (mpz_t R, const mpz_t N,
+ mp_bitcnt_t B)
+
+ -- Function: void mpz_fdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_fdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
+ mpz_t D)
+
+ -- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
- -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
+ -- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, const 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,
+ const mpz_t N, unsigned long int D)
+ -- Function: unsigned long int mpz_fdiv_ui (const 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,
+ -- Function: void mpz_fdiv_q_2exp (mpz_t Q, const mpz_t N,
+ mp_bitcnt_t B)
+ -- Function: void mpz_fdiv_r_2exp (mpz_t R, const mpz_t N,
+ mp_bitcnt_t B)
+
+ -- Function: void mpz_tdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_tdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
+ mpz_t D)
+
+ -- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
- -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
+ -- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, const 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,
+ const mpz_t N, unsigned long int D)
+ -- Function: unsigned long int mpz_tdiv_ui (const 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)
+
+ -- Function: void mpz_tdiv_q_2exp (mpz_t Q, const mpz_t N,
+ mp_bitcnt_t B)
+ -- Function: void mpz_tdiv_r_2exp (mpz_t R, const 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
+ '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".
+ * '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".
+ * '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".
+ * '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,
+ 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
+ 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 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
+ 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.
+ '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,
+ -- Function: void mpz_mod (mpz_t R, const mpz_t N, const mpz_t D)
+ -- Function: unsigned long int mpz_mod_ui (mpz_t R, const 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.
+ 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
+ '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.
+ -- Function: void mpz_divexact (mpz_t Q, const mpz_t N, const mpz_t D)
+ -- Function: void mpz_divexact_ui (mpz_t Q, const 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)
+ -- Function: int mpz_divisible_p (const mpz_t N, const mpz_t D)
+ -- Function: int mpz_divisible_ui_p (const mpz_t N, unsigned long int
+ D)
+ -- Function: int mpz_divisible_2exp_p (const 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.
+ '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)
+ -- Function: int mpz_congruent_p (const mpz_t N, const mpz_t C, const
+ mpz_t D)
+ -- Function: int mpz_congruent_ui_p (const mpz_t N, unsigned long int
+ C, unsigned long int D)
+ -- Function: int mpz_congruent_2exp_p (const mpz_t N, const 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.
+ '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
+ 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.
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)
+ -- Function: void mpz_powm (mpz_t ROP, const mpz_t BASE, const mpz_t
+ EXP, const mpz_t MOD)
+ -- Function: void mpz_powm_ui (mpz_t ROP, const mpz_t BASE, unsigned
+ long int EXP, const 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
+ Negative EXP is supported if the 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)
+ -- Function: void mpz_powm_sec (mpz_t ROP, const mpz_t BASE, const
+ mpz_t EXP, const mpz_t MOD)
Set ROP to (BASE raised to EXP) modulo MOD.
It is required that EXP > 0 and that MOD is odd.
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_pow_ui (mpz_t ROP, const 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.
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: int mpz_root (mpz_t ROP, const 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,
+ -- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, const mpz_t U,
unsigned long int N)
- Set ROOT to the truncated integer part of the Nth root of U. Set
+ 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_sqrt (mpz_t ROP, const 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)
+ -- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, const 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
+ 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)
+ -- Function: int mpz_perfect_power_p (const 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.
+ 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
+ -- Function: int mpz_perfect_square_p (const 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
5.9 Number Theoretic Functions
==============================
- -- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
+ -- Function: int mpz_probab_prime_p (const 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.
+ 0 if N is definitely non-prime.
- 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".
+ This function performs some trial divisions, a Baillie-PSW probable
+ prime test, then REPS-24 Miller-Rabin probabilistic primality
+ tests. A higher REPS value will reduce the chances of a non-prime
+ being identified as "probably prime". A composite number will be
+ identified as a prime with an asymptotic probability of less than
+ 4^(-REPS). Reasonable values of REPS are between 15 and 50.
- 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.
+ GMP versions up to and including 6.1.2 did not use the Baillie-PSW
+ primality test. In those older versions of GMP, this function
+ performed REPS Miller-Rabin tests.
- -- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
+ -- Function: void mpz_nextprime (mpz_t ROP, const 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)
+ -- Function: void mpz_gcd (mpz_t ROP, const mpz_t OP1, const 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.
+ Except if both inputs are zero; then this function defines gcd(0,0)
+ = 0.
- -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
+ -- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, const 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.
+ 'NULL', store the result there.
- If the result is small enough to fit in an `unsigned long int', it
+ 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)
+ -- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, const mpz_t A,
+ const 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).
+ 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
+ (or zero if both inputs are zero). The values in S and T are
+ chosen such that normally, abs(S) < abs(B) / (2 G) and abs(T) <
+ abs(A) / (2 G), and these relations define S and T uniquely. There
+ are a few exceptional cases:
- If T is `NULL' then that value is not computed.
+ If abs(A) = abs(B), then S = 0, T = sgn(B).
- -- 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.
+ Otherwise, S = sgn(A) if B = 0 or abs(B) = 2 G, and T = sgn(B) if A
+ = 0 or abs(A) = 2 G.
+
+ In all cases, S = 0 if and only if G = abs(B), i.e., if B divides A
+ or A = B = 0.
+
+ If T or G is 'NULL' then that value is not computed.
+
+ -- Function: void mpz_lcm (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_lcm_ui (mpz_t ROP, const 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)
+ -- Function: int mpz_invert (mpz_t ROP, const mpz_t OP1, const 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.
+ satisfy 0 <= ROP < abs(OP2) (with ROP = 0 possible only when
+ abs(OP2) = 1, i.e., in the somewhat degenerate zero ring). If an
+ inverse doesn't exist the return value is zero and ROP is
+ undefined. The behaviour of this function is undefined when OP2 is
+ zero.
- -- Function: int mpz_jacobi (mpz_t A, mpz_t B)
+ -- Function: int mpz_jacobi (const mpz_t A, const 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
+ -- Function: int mpz_legendre (const mpz_t A, const 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)
+ -- Function: int mpz_kronecker (const mpz_t A, const mpz_t B)
+ -- Function: int mpz_kronecker_si (const mpz_t A, long B)
+ -- Function: int mpz_kronecker_ui (const mpz_t A, unsigned long B)
+ -- Function: int mpz_si_kronecker (long A, const mpz_t B)
+ -- Function: int mpz_ui_kronecker (unsigned long A, const 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.
+ 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'.
+ 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
+ -- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, const mpz_t OP, const
+ 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_fac_ui (mpz_t ROP, unsigned long int N)
+ -- Function: void mpz_2fac_ui (mpz_t ROP, unsigned long int N)
+ -- Function: void mpz_mfac_uiui (mpz_t ROP, unsigned long int N,
+ unsigned long int M)
+ Set ROP to the factorial of N: 'mpz_fac_ui' computes the plain
+ factorial N!, 'mpz_2fac_ui' computes the double-factorial N!!, and
+ 'mpz_mfac_uiui' the M-multi-factorial N!^(M).
+
+ -- Function: void mpz_primorial_ui (mpz_t ROP, unsigned long int N)
+ Set ROP to the primorial of N, i.e. the product of all positive
+ prime numbers <=N.
- -- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
+ -- Function: void mpz_bin_ui (mpz_t ROP, const 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
+ 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].
+ '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
+ 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].
+ -- 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]
+ 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
+ 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
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.
+ -- Function: int mpz_cmp (const mpz_t OP1, const mpz_t OP2)
+ -- Function: int mpz_cmp_d (const mpz_t OP1, double OP2)
+ -- Macro: int mpz_cmp_si (const mpz_t OP1, signed long int OP2)
+ -- Macro: int mpz_cmp_ui (const 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
+ '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)
+ -- Function: int mpz_cmpabs (const mpz_t OP1, const mpz_t OP2)
+ -- Function: int mpz_cmpabs_d (const mpz_t OP1, double OP2)
+ -- Function: int mpz_cmpabs_ui (const 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
+ 'mpz_cmpabs_d' can be called with an infinity, but results are
undefined for a NaN.
- -- Macro: int mpz_sgn (mpz_t OP)
+ -- Macro: int mpz_sgn (const 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.
+ 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
(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)
+ -- Function: void mpz_and (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise-and OP2.
- -- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
+ -- Function: void mpz_ior (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise inclusive-or OP2.
- -- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
+ -- Function: void mpz_xor (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise exclusive-or OP2.
- -- Function: void mpz_com (mpz_t ROP, mpz_t OP)
+ -- Function: void mpz_com (mpz_t ROP, const 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)
+ -- Function: mp_bitcnt_t mpz_popcount (const 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 (const mpz_t OP1, const 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 (const mpz_t OP, mp_bitcnt_t
+ STARTING_BIT)
+ -- Function: mp_bitcnt_t mpz_scan1 (const 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
+ 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)
-- 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)
+ -- Function: int mpz_tstbit (const mpz_t OP, mp_bitcnt_t BIT_INDEX)
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.
\1f
===============================
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.
+output to a stdio stream, of 'mpz' numbers. 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
+'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)
+ See also *note Formatted Output:: and *note Formatted Input::.
+
+ -- Function: size_t mpz_out_str (FILE *STREAM, int BASE, const 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.
+ 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.
+ 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.
+ 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.
+ 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)
+ -- Function: size_t mpz_out_raw (FILE *STREAM, const 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'.
+ The output can be read with 'mpz_inp_raw'.
- Return the number of bytes written, or if an error occurred,
- return 0.
+ 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,
+ 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
+ '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
+ 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.
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.
+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)
2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
+ 'gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
- -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
+ -- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE, const
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::)
+ 'gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
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.
+ more likely to trigger corner-case bugs. The random number will be
+ in the range 2^(N-1) to 2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
+ '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.
+ randomness. Negative random numbers are generated when MAX_SIZE is
+ negative.
- This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
+ 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.
+ 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.
+ 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
+'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,
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.
+ 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'.
+ 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
+ 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
+ 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'.
+ '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)
+ size_t SIZE, int ENDIAN, size_t NAILS, const mpz_t OP)
Fill ROP with word data from OP.
The parameters specify the format of the data produced. Each word
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.
+ 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.
+ 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
+ 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
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).
+ '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;
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)
+ -- Function: int mpz_fits_ulong_p (const mpz_t OP)
+ -- Function: int mpz_fits_slong_p (const mpz_t OP)
+ -- Function: int mpz_fits_uint_p (const mpz_t OP)
+ -- Function: int mpz_fits_sint_p (const mpz_t OP)
+ -- Function: int mpz_fits_ushort_p (const mpz_t OP)
+ -- Function: int mpz_fits_sshort_p (const 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 (const mpz_t OP)
+ -- Macro: int mpz_even_p (const 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 (const 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.
+ 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
+ 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
+ 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.)
-- 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'.
+ *This is an obsolete function. Do not use it.*
-- 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.
+ '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)
+ -- Function: mp_limb_t mpz_getlimbn (const 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'.
+ '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)
+ -- Function: size_t mpz_size (const mpz_t OP)
Return the size of OP measured in number of limbs. If OP is zero,
the returned value will be zero.
+ -- Function: const mp_limb_t * mpz_limbs_read (const mpz_t X)
+ Return a pointer to the limb array representing the absolute value
+ of X. The size of the array is 'mpz_size(X)'. Intended for read
+ access only.
+
+ -- Function: mp_limb_t * mpz_limbs_write (mpz_t X, mp_size_t N)
+ -- Function: mp_limb_t * mpz_limbs_modify (mpz_t X, mp_size_t N)
+ Return a pointer to the limb array, intended for write access. The
+ array is reallocated as needed, to make room for N limbs. Requires
+ N > 0. The 'mpz_limbs_modify' function returns an array that holds
+ the old absolute value of X, while 'mpz_limbs_write' may destroy
+ the old value and return an array with unspecified contents.
+
+ -- Function: void mpz_limbs_finish (mpz_t X, mp_size_t S)
+ Updates the internal size field of X. Used after writing to the
+ limb array pointer returned by 'mpz_limbs_write' or
+ 'mpz_limbs_modify' is completed. The array should contain abs(S)
+ valid limbs, representing the new absolute value for X, and the
+ sign of X is taken from the sign of S. This function never
+ reallocates X, so the limb pointer remains valid.
+
+ void foo (mpz_t x)
+ {
+ mp_size_t n, i;
+ mp_limb_t *xp;
+
+ n = mpz_size (x);
+ xp = mpz_limbs_modify (x, 2*n);
+ for (i = 0; i < n; i++)
+ xp[n+i] = xp[n-1-i];
+ mpz_limbs_finish (x, mpz_sgn (x) < 0 ? - 2*n : 2*n);
+ }
+
+ -- Function: mpz_srcptr mpz_roinit_n (mpz_t X, const mp_limb_t *XP,
+ mp_size_t XS)
+ Special initialization of X, using the given limb array and size.
+ X should be treated as read-only: it can be passed safely as input
+ to any mpz function, but not as an output. The array XP must point
+ to at least a readable limb, its size is abs(XS), and the sign of X
+ is the sign of XS. For convenience, the function returns X, but
+ cast to a const pointer type.
+
+ void foo (mpz_t x)
+ {
+ static const mp_limb_t y[3] = { 0x1, 0x2, 0x3 };
+ mpz_t tmp;
+ mpz_add (x, x, mpz_roinit_n (tmp, y, 3));
+ }
+
+ -- Macro: mpz_t MPZ_ROINIT_N (mp_limb_t *XP, mp_size_t XS)
+ This macro expands to an initializer which can be assigned to an
+ mpz_t variable. The limb array XP must point to at least a
+ readable limb, moreover, unlike the 'mpz_roinit_n' function, the
+ array must be normalized: if XS is non-zero, then 'XP[abs(XS)-1]'
+ must be non-zero. Intended primarily for constant values. Using
+ it for non-constant values requires a C compiler supporting C99.
+
+ void foo (mpz_t x)
+ {
+ static const mp_limb_t ya[3] = { 0x1, 0x2, 0x3 };
+ static const mpz_t y = MPZ_ROINIT_N ((mp_limb_t *) ya, 3);
+
+ mpz_add (x, x, y);
+ }
+
\1f
File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
***************************
This chapter describes the GMP functions for performing arithmetic on
-rational numbers. These functions start with the prefix `mpq_'.
+rational numbers. These functions start with the prefix 'mpq_'.
- Rational numbers are stored in objects of type `mpq_t'.
+ 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
+form, and canonicalize their result. The canonical form means that the
denominator and the numerator have no common factors, and that the
denominator is positive. Zero has the unique representation 0/1.
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.
+ Remove any factors that are common to the numerator and denominator
+ of OP, and make the denominator positive.
* Menu:
===========================================
-- 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.
+ 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
+ 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.
+ 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'
+ 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)
+ -- Function: void mpq_set (mpq_t ROP, const mpq_t OP)
+ -- Function: void mpq_set_z (mpq_t ROP, const mpz_t OP)
Assign ROP from OP.
-- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
-- 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
+ 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)
+ -- Function: int mpq_set_str (mpq_t ROP, const 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 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.
+ '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.
+ 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.
6.2 Conversion Functions
========================
- -- Function: double mpq_get_d (mpq_t OP)
- Convert OP to a `double', truncating if necessary (ie. rounding
+ -- Function: double mpq_get_d (const mpq_t OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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.
+ 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)
+ -- Function: void mpq_set_f (mpq_t ROP, const 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'.
+ -- Function: char * mpq_get_str (char *STR, int BASE, const mpq_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. The string will be of the
+ form 'num/den', or if the denominator is 1 then just 'num'.
+
+ 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
+ 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
+ '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
+ 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)
6.3 Arithmetic Functions
========================
- -- Function: void mpq_add (mpq_t SUM, mpq_t ADDEND1, mpq_t ADDEND2)
+ -- Function: void mpq_add (mpq_t SUM, const mpq_t ADDEND1, const mpq_t
+ ADDEND2)
Set SUM to ADDEND1 + ADDEND2.
- -- Function: void mpq_sub (mpq_t DIFFERENCE, mpq_t MINUEND, mpq_t
- SUBTRAHEND)
+ -- Function: void mpq_sub (mpq_t DIFFERENCE, const mpq_t MINUEND, const
+ mpq_t SUBTRAHEND)
Set DIFFERENCE to MINUEND - SUBTRAHEND.
- -- Function: void mpq_mul (mpq_t PRODUCT, mpq_t MULTIPLIER, mpq_t
- MULTIPLICAND)
+ -- Function: void mpq_mul (mpq_t PRODUCT, const mpq_t MULTIPLIER, const
+ mpq_t MULTIPLICAND)
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
- -- Function: void mpq_mul_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
+ -- Function: void mpq_mul_2exp (mpq_t ROP, const 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)
+ -- Function: void mpq_div (mpq_t QUOTIENT, const mpq_t DIVIDEND, const
+ mpq_t DIVISOR)
Set QUOTIENT to DIVIDEND/DIVISOR.
- -- Function: void mpq_div_2exp (mpq_t ROP, mpq_t OP1, mp_bitcnt_t OP2)
+ -- Function: void mpq_div_2exp (mpq_t ROP, const 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)
+ -- Function: void mpq_neg (mpq_t NEGATED_OPERAND, const mpq_t OPERAND)
Set NEGATED_OPERAND to -OPERAND.
- -- Function: void mpq_abs (mpq_t ROP, mpq_t OP)
+ -- Function: void mpq_abs (mpq_t ROP, const mpq_t OP)
Set ROP to the absolute value of OP.
- -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, mpq_t NUMBER)
+ -- Function: void mpq_inv (mpq_t INVERTED_NUMBER, const mpq_t NUMBER)
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
this routine will divide by zero.
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.
+ -- Function: int mpq_cmp (const mpq_t OP1, const mpq_t OP2)
+ -- Function: int mpq_cmp_z (const mpq_t OP1, const mpz_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'.
+ 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)
+ -- Macro: int mpq_cmp_ui (const mpq_t OP1, unsigned long int NUM2,
+ unsigned long int DEN2)
+ -- Macro: int mpq_cmp_si (const 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.
These functions are implemented as a macros and evaluate their
arguments multiple times.
- -- Macro: int mpq_sgn (mpq_t OP)
+ -- Macro: int mpq_sgn (const 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.
+ argument multiple times.
- -- Function: int mpq_equal (mpq_t OP1, mpq_t OP2)
+ -- Function: int mpq_equal (const mpq_t OP1, const 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,
+ non-equal. Although 'mpq_cmp' can be used for the same purpose,
this function is much faster.
\1f
6.5 Applying Integer Functions to Rationals
===========================================
-The set of `mpq' functions is quite small. In particular, there are few
+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'.
+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'.
+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)
+ -- Macro: mpz_t mpq_numref (const mpq_t OP)
+ -- Macro: mpz_t mpq_denref (const mpq_t OP)
Return a reference to the numerator and denominator of OP,
- respectively. The `mpz' functions can be used on the result of
+ 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)
+ -- Function: void mpq_get_num (mpz_t NUMERATOR, const mpq_t RATIONAL)
+ -- Function: void mpq_get_den (mpz_t DENOMINATOR, const mpq_t RATIONAL)
+ -- Function: void mpq_set_num (mpq_t RATIONAL, const mpz_t NUMERATOR)
+ -- Function: void mpq_set_den (mpq_t RATIONAL, const 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.
+ 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.
+Functions that perform input from a stdio stream, and functions that
+output to a stdio stream, of 'mpq' numbers. 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.
- Passing a `NULL' pointer for a STREAM argument to any of these
-functions will make them read from `stdin' and write to `stdout',
-respectively.
+ See also *note Formatted Output:: and *note Formatted Input::.
- -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, mpq_t OP)
+ -- Function: size_t mpq_out_str (FILE *STREAM, int BASE, const 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'.
+ BASE. The base argument may vary from 2 to 62 or from -2 to -36.
+ Output is in the form 'num/den' or if the denominator is 1 then
+ just 'num'.
+
+ 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.
+ 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.
+ 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 62, or can be 0 in which case the
+ leading characters of the string determine the base, '0x' or '0X'
+ for hexadecimal, '0b' and '0B' for binary, '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.
+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 mantissa of each float has a user-selectable precision, in
+practice only limited by available memory. Each variable has its own
+precision, and that can be increased or decreased at any time. This
+selectable precision is a minimum value, GMP rounds it up to a whole
+limb.
- The exponent of each float is a fixed precision, one machine word on
+ The accuracy of a calculation is determined by the priorly set
+precision of the destination variable and the numeric values of the
+input variables. Input variables' set precisions do not affect
+calculations (except indirectly as their values might have been affected
+when they were assigned).
+
+ The exponent of each float has 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
+2^-68719476768 to 2^68719476736, or on a 64-bit system this will be much
+greater. Note however that '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 track of 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 variable's
+selected precision is high. This is a performance optimization; it does
+not affect the numeric results.
+
+ Internally, GMP sometimes calculates with higher precision than that
+of the destination variable in order to limit errors. Final results are
+always truncated to the destination variable's precision.
+
+ The mantissa is stored in binary. 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
+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.)
+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.
+ The '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.
- Note that the `mpf' functions are _not_ intended as a smooth
+ 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.
+ New projects should consider using the GMP extension library MPFR
+(<http://mpfr.org>) instead. MPFR provides well-defined precision and
+accurate rounding, and thereby naturally extends IEEE P754.
+
* Menu:
* Initializing Floats::
-- 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
+ 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
+ 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
+ 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'.
+ '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.
+ 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
+ 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'.
+ 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.
+ 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'
+ Free the space occupied by a NULL-terminated list of 'mpf_t'
variables.
Here is an example on how to initialize floating-point variables:
the computation precision closely match the actual accurate part of the
numbers.
- -- Function: mp_bitcnt_t mpf_get_prec (mpf_t OP)
+ -- Function: mp_bitcnt_t mpf_get_prec (const 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
+ 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)
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'.
+ '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
+ 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_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
+ '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.
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 (mpf_t ROP, const 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)
+ -- Function: void mpf_set_z (mpf_t ROP, const mpz_t OP)
+ -- Function: void mpf_set_q (mpf_t ROP, const mpq_t OP)
Set the value of ROP from OP.
- -- Function: int mpf_set_str (mpf_t ROP, char *STR, int BASE)
+ -- Function: int mpf_set_str (mpf_t ROP, const 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'.
+ 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
represent the usual 10..35 while lower-case letter represent
36..61.
- Unlike the corresponding `mpz' function, the base will not be
+ 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.
+ 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?]
+ 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.
+ 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
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...'
+These functions' names have the form 'mpf_init_set...'
- Once the float has been initialized by any of the `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!
+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 (mpf_t ROP, const 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'.
+ precision, as set by 'mpf_set_default_prec'.
- -- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
+ -- Function: int mpf_init_set_str (mpf_t ROP, const 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.
+ '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.)
+ 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'.
+ 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
+ -- Function: double mpf_get_d (const mpf_t OP)
+ Convert OP to a 'double', truncating if necessary (i.e. rounding
towards zero).
- If the exponent in OP is too big or too small to fit a `double'
+ 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
+ -- Function: double mpf_get_d_2exp (signed long int *EXP, const mpf_t
+ OP)
+ Convert OP to a 'double', truncating if necessary (i.e. 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'.
+ 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.).
+ This is similar to the standard C 'frexp' function (*note
+ (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
+ -- Function: long mpf_get_si (const mpf_t OP)
+ -- Function: unsigned long mpf_get_ui (const 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
+ 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)
+ -- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int BASE,
+ size_t N_DIGITS, const 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.
+ 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.
+ 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
+ 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
+ '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
+ 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.
+ 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
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)
+ -- Function: void mpf_add (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_add_ui (mpf_t ROP, const 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)
+ -- Function: void mpf_sub (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, const
+ mpf_t OP2)
+ -- Function: void mpf_sub_ui (mpf_t ROP, const 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)
+ -- Function: void mpf_mul (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_mul_ui (mpf_t ROP, const 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
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)
+ -- Function: void mpf_div (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
+ -- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, const
+ mpf_t OP2)
+ -- Function: void mpf_div_ui (mpf_t ROP, const 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 (mpf_t ROP, const 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)
+ -- Function: void mpf_pow_ui (mpf_t ROP, const 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)
+ -- Function: void mpf_neg (mpf_t ROP, const mpf_t OP)
Set ROP to -OP.
- -- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
+ -- Function: void mpf_abs (mpf_t ROP, const 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)
+ -- Function: void mpf_mul_2exp (mpf_t ROP, const 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)
+ -- Function: void mpf_div_2exp (mpf_t ROP, const mpf_t OP1, mp_bitcnt_t
+ OP2)
Set ROP to OP1 divided by 2 raised to OP2.
\1f
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.
+ -- Function: int mpf_cmp (const mpf_t OP1, const mpf_t OP2)
+ -- Function: int mpf_cmp_z (const mpf_t OP1, const mpz_t OP2)
+ -- Function: int mpf_cmp_d (const mpf_t OP1, double OP2)
+ -- Function: int mpf_cmp_ui (const mpf_t OP1, unsigned long int OP2)
+ -- Function: int mpf_cmp_si (const 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
+ '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.
+ -- Function: int mpf_eq (const mpf_t OP1, const mpf_t OP2, mp_bitcnt_t
+ op3)
+ *This function is mathematically ill-defined and should not be
+ used.*
- 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.
+ Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
+ zero otherwise. Note that numbers like e.g., 256 (binary
+ 100000000) and 255 (binary 11111111) will never be equal by this
+ function's measure, and furthermore that 0 will only be equal to
+ itself.
- -- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
+ -- Function: void mpf_reldiff (mpf_t ROP, const mpf_t OP1, const 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)
+ -- Macro: int mpf_sgn (const 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.
+ This function is actually implemented as a macro. It evaluates its
+ argument multiple times.
\1f
File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point 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.
+output to a stdio stream, of 'mpf' numbers. 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
+'stdio.h' before 'gmp.h', since that will allow 'gmp.h' to define
prototypes for these functions.
+ See also *note Formatted Output:: and *note Formatted Input::.
+
-- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
- N_DIGITS, mpf_t OP)
+ N_DIGITS, const 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'.
+ 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.
+ 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
-- 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
+ 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'.
+ 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
+ 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.
+ 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.
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: void mpf_ceil (mpf_t ROP, const mpf_t OP)
+ -- Function: void mpf_floor (mpf_t ROP, const mpf_t OP)
+ -- Function: void mpf_trunc (mpf_t ROP, const 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)
+ -- Function: int mpf_integer_p (const 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)
+ -- Function: int mpf_fits_ulong_p (const mpf_t OP)
+ -- Function: int mpf_fits_slong_p (const mpf_t OP)
+ -- Function: int mpf_fits_uint_p (const mpf_t OP)
+ -- Function: int mpf_fits_sint_p (const mpf_t OP)
+ -- Function: int mpf_fits_ushort_p (const mpf_t OP)
+ -- Function: int mpf_fits_sshort_p (const 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.
+ <= ROP < 1, with NBITS significant bits in the mantissa or less if
+ the precision of ROP is smaller.
The variable STATE must be initialized by calling one of the
- `gmp_randinit' functions (*Note Random State Initialization::)
+ '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
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.
+ 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
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_'.
+ These functions start with the prefix 'mpn_'.
- The `mpn' functions are designed to be as fast as possible, *not* to
+ 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
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.
+ 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.
+ 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
+ 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
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.
+ 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.
+ 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)
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.
+ 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)
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,
+ -- Function: mp_limb_t 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.
+ This is equivalent to calling 'mpn_sub_n' with a N-limb zero
+ minuend and passing {SP, N} as subtrahend. Return borrow, either 0
+ or 1.
- -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P,
- const mp_limb_t *S2P, mp_size_t N)
+ -- 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.
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'.
+ 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)
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
+ The destination has to have space for 2N limbs, even if the
result's most significant limb is zero. No overlap is permitted
between the destination and the source.
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.
+ '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.
+ 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. {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
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.
+ from the subtraction. {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 and division as well as other operations in GMP.
- It is written in assembly for most CPUs.
+ 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,
-- 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
+ [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).
+ 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,
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.
+ 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.
+ 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.
+ '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
+ [This function is obsolete. Please call 'mpn_tdiv_qr' instead for
best performance.]
+ -- Function: void mpn_divexact_1 (mp_limb_t * RP, const mp_limb_t * SP,
+ mp_size_t N, mp_limb_t D)
+ Divide {SP, N} by D, expecting it to divide exactly, and writing
+ the result to {RP, N}. If D doesn't divide exactly, the value
+ written to {RP, N} is undefined. The areas at RP and SP have to be
+ identical or completely separate, not partially overlapping.
+
-- 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)
+ -- 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
+ '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
+ 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.
+ '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).
+ 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,
+ -- Function: mp_limb_t mpn_mod_1 (const 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_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
+ 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).
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)
+ -- Function: int mpn_zero_p (const mp_limb_t *SP, mp_size_t N)
+ Test {SP, N} and return 1 if the operand is zero, 0 otherwise.
+
+ -- 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}.
+ It is required that XN >= YN > 0, the most significant limb of {YP,
+ YN} must be non-zero, and at least one of the two operands must be
+ odd. 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)
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}.
+ mp_size_t *SN, mp_limb_t *UP, mp_size_t UN, mp_limb_t *VP,
+ mp_size_t VN)
+ Let U be defined by {UP, UN} and let V be defined by {VP, VN}.
- 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
+ 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.
+ division will be exact). It is required that UN >= VN > 0, and the
+ most significant limb of {VP, VN} must be non-zero.
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.
+ negative; when this happens *SN will be negative. The area at GP
+ should have room for VN limbs and the area at SP should have room
+ for VN+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).
+ Both source operands are destroyed.
- Compatibility note: GMP 4.3.0 and 4.3.1 defined S less strictly.
+ Compatibility notes: GMP 4.3.0 and 4.3.1 defined S less strictly.
Earlier as well as later GMP releases define S as described here.
+ GMP releases before GMP 4.3.0 required additional space for both
+ input and output areas. More precisely, the areas {UP, UN+1} and
+ {VP, VN+1} were destroyed (i.e. the operands plus an extra limb
+ past the end of each), and the areas pointed to by GP and SP should
+ each have room for UN+1 limbs.
-- 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.
+ 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
+ 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'.
+ 'mpn_perfect_square_p'.
+
+ -- Function: size_t mpn_sizeinbase (const mp_limb_t *XP, mp_size_t N,
+ int BASE)
+ Return the size of {XP,N} measured in number of digits in the given
+ BASE. BASE can vary from 2 to 62. Requires N > 0 and XP[N-1] > 0.
+ The result will be either exact or 1 too big. If BASE is a power
+ of 2, the result is always exact.
-- 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.
+ 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 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.
*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.
+ STR[0] is the most significant input byte and STR[STRSIZE-1] is the
+ least significant input byte. 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.
+ The converted value is {RP,RN} where RN is the return value. If
+ the most significant input byte STR[0] is non-zero, then RP[RN-1]
+ will be non-zero, else RP[RN-1] and some number of subsequent limbs
+ may be zero.
- If the most significant input byte is zero then there may be high
- zero limbs written to RP and included in the return value.
+ The area at RP has to have space for the largest possible number
+ with STRSIZE digits in the chosen base, plus one extra limb.
- STRSIZE must be at least 1, and no overlap is permitted between
- {STR,STRSIZE} and the result at RP.
+ The input must have at least one byte, 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)
-- 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
+ 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'
+ '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
-- 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.
+ Return non-zero iff {S1P, N} is a perfect square. The most
+ significant limb of the input {S1P, N} must be non-zero.
- -- 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_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)
+ -- 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)
+ -- 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_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}.
+ 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)
Zero {RP, N}.
-8.1 Nails
+8.1 Low-level functions for cryptography
+========================================
+
+The functions prefixed with 'mpn_sec_' and 'mpn_cnd_' are designed to
+perform the exact same low-level operations 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. These functions are intended for
+cryptographic purposes, where resilience to side-channel attacks is
+desired.
+
+ These functions are less efficient than their "leaky" counterparts;
+their performance for operands of the sizes typically used for
+cryptographic applications is between 15% and 100% worse. For larger
+operands, these functions might be inadequate, since they rely on
+asymptotically elementary algorithms.
+
+ These functions do not make any explicit allocations. Those of these
+functions that need scratch space accept a scratch space operand. This
+convention allows callers to keep sensitive data in designated memory
+areas. Note however that compilers may choose to spill scalar values
+used within these functions to their stack frame and that such scalars
+may contain sensitive data.
+
+ In addition to these specially crafted functions, the following 'mpn'
+functions are naturally side-channel resistant: 'mpn_add_n',
+'mpn_sub_n', 'mpn_lshift', 'mpn_rshift', 'mpn_zero', 'mpn_copyi',
+'mpn_copyd', 'mpn_com', and the logical function ('mpn_and_n', etc).
+
+ There are some exceptions from the side-channel resilience: (1) Some
+assembly implementations of 'mpn_lshift' identify shift-by-one as a
+special case. This is a problem iff the shift count is a function of
+sensitive data. (2) Alpha ev6 and Pentium4 using 64-bit limbs have
+leaky 'mpn_add_n' and 'mpn_sub_n'. (3) Alpha ev6 has a leaky
+'mpn_mul_1' which also makes 'mpn_sec_mul' on those systems unsafe.
+
+ -- Function: mp_limb_t mpn_cnd_add_n (mp_limb_t CND, mp_limb_t *RP,
+ const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N)
+ -- Function: mp_limb_t mpn_cnd_sub_n (mp_limb_t CND, mp_limb_t *RP,
+ const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N)
+ These functions do conditional addition and subtraction. If CND is
+ non-zero, they produce the same result as a regular 'mpn_add_n' or
+ 'mpn_sub_n', and if CND is zero, they copy {S1P,N} to the result
+ area and return zero. The functions are designed to have timing
+ and memory access patterns depending only on size and location of
+ the data areas, but independent of the condition CND. Like for
+ 'mpn_add_n' and 'mpn_sub_n', on most machines, the timing will also
+ be independent of the actual limb values.
+
+ -- Function: mp_limb_t mpn_sec_add_1 (mp_limb_t *RP, const mp_limb_t
+ *AP, mp_size_t N, mp_limb_t B, mp_limb_t *TP)
+ -- Function: mp_limb_t mpn_sec_sub_1 (mp_limb_t *RP, const mp_limb_t
+ *AP, mp_size_t N, mp_limb_t B, mp_limb_t *TP)
+ Set R to A + B or A - B, respectively, where R = {RP,N}, A =
+ {AP,N}, and B is a single limb. Returns carry.
+
+ These functions take O(N) time, unlike the leaky functions
+ 'mpn_add_1' which are O(1) on average. They require scratch space
+ of 'mpn_sec_add_1_itch(N)' and 'mpn_sec_sub_1_itch(N)' limbs,
+ respectively, to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to be at most N limbs, and increase
+ monotonously in the operand size.
+
+ -- Function: void mpn_cnd_swap (mp_limb_t CND, volatile mp_limb_t *AP,
+ volatile mp_limb_t *BP, mp_size_t N)
+ If CND is non-zero, swaps the contents of the areas {AP,N} and
+ {BP,N}. Otherwise, the areas are left unmodified. Implemented
+ using logical operations on the limbs, with the same memory
+ accesses independent of the value of CND.
+
+ -- Function: void mpn_sec_mul (mp_limb_t *RP, const mp_limb_t *AP,
+ mp_size_t AN, const mp_limb_t *BP, mp_size_t BN, mp_limb_t
+ *TP)
+ -- Function: mp_size_t mpn_sec_mul_itch (mp_size_t AN, mp_size_t BN)
+ Set R to A * B, where A = {AP,AN}, B = {BP,BN}, and R = {RP,AN+BN}.
+
+ It is required that AN >= BN > 0.
+
+ No overlapping between R and the input operands is allowed. For A
+ = B, use 'mpn_sec_sqr' for optimal performance.
+
+ This function requires scratch space of 'mpn_sec_mul_itch(AN, BN)'
+ limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ sizes.
+
+ -- Function: void mpn_sec_sqr (mp_limb_t *RP, const mp_limb_t *AP,
+ mp_size_t AN, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_sqr_itch (mp_size_t AN)
+ Set R to A^2, where A = {AP,AN}, and R = {RP,2AN}.
+
+ It is required that AN > 0.
+
+ No overlapping between R and the input operands is allowed.
+
+ This function requires scratch space of 'mpn_sec_sqr_itch(AN)'
+ limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ size.
+
+ -- Function: void mpn_sec_powm (mp_limb_t *RP, const mp_limb_t *BP,
+ mp_size_t BN, const mp_limb_t *EP, mp_bitcnt_t ENB, const
+ mp_limb_t *MP, mp_size_t N, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_powm_itch (mp_size_t BN, mp_bitcnt_t
+ ENB, size_t N)
+ Set R to (B raised to E) modulo M, where R = {RP,N}, M = {MP,N},
+ and E = {EP,ceil(ENB / 'GMP\_NUMB\_BITS')}.
+
+ It is required that B > 0, that M > 0 is odd, and that E < 2^ENB,
+ with ENB > 0.
+
+ No overlapping between R and the input operands is allowed.
+
+ This function requires scratch space of 'mpn_sec_powm_itch(BN, ENB,
+ N)' limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ sizes.
+
+ -- Function: void mpn_sec_tabselect (mp_limb_t *RP, const mp_limb_t
+ *TAB, mp_size_t N, mp_size_t NENTS, mp_size_t WHICH)
+ Select entry WHICH from table TAB, which has NENTS entries, each N
+ limbs. Store the selected entry at RP.
+
+ This function reads the entire table to avoid side-channel
+ information leaks.
+
+ -- Function: mp_limb_t mpn_sec_div_qr (mp_limb_t *QP, mp_limb_t *NP,
+ mp_size_t NN, const mp_limb_t *DP, mp_size_t DN, mp_limb_t
+ *TP)
+ -- Function: mp_size_t mpn_sec_div_qr_itch (mp_size_t NN, mp_size_t DN)
+
+ Set Q to the truncated quotient N / D and R to N modulo D, where N
+ = {NP,NN}, D = {DP,DN}, Q's most significant limb is the function
+ return value and the remaining limbs are {QP,NN-DN}, and R =
+ {NP,DN}.
+
+ It is required that NN >= DN >= 1, and that DP[DN-1] != 0. This
+ does not imply that N >= D since N might be zero-padded.
+
+ Note the overlapping between N and R. No other operand overlapping
+ is allowed. The entire space occupied by N is overwritten.
+
+ This function requires scratch space of 'mpn_sec_div_qr_itch(NN,
+ DN)' limbs to be passed in the TP parameter.
+
+ -- Function: void mpn_sec_div_r (mp_limb_t *NP, mp_size_t NN, const
+ mp_limb_t *DP, mp_size_t DN, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_div_r_itch (mp_size_t NN, mp_size_t DN)
+
+ Set R to N modulo D, where N = {NP,NN}, D = {DP,DN}, and R =
+ {NP,DN}.
+
+ It is required that NN >= DN >= 1, and that DP[DN-1] != 0. This
+ does not imply that N >= D since N might be zero-padded.
+
+ Note the overlapping between N and R. No other operand overlapping
+ is allowed. The entire space occupied by N is overwritten.
+
+ This function requires scratch space of 'mpn_sec_div_r_itch(NN,
+ DN)' limbs to be passed in the TP parameter.
+
+ -- Function: int mpn_sec_invert (mp_limb_t *RP, mp_limb_t *AP, const
+ mp_limb_t *MP, mp_size_t N, mp_bitcnt_t NBCNT, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_invert_itch (mp_size_t N)
+ Set R to the inverse of A modulo M, where R = {RP,N}, A = {AP,N},
+ and M = {MP,N}. *This function's interface is preliminary.*
+
+ If an inverse exists, return 1, otherwise return 0 and leave R
+ undefined. In either case, the input A is destroyed.
+
+ It is required that M is odd, and that NBCNT >= ceil(\log(A+1)) +
+ ceil(\log(M+1)). A safe choice is NBCNT = 2 * N * GMP_NUMB_BITS,
+ but a smaller value might improve performance if M or A are known
+ to have leading zero bits.
+
+ This function requires scratch space of 'mpn_sec_invert_itch(N)'
+ limbs to be passed in the TP parameter.
+
+
+8.2 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
+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.
+ 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
+ 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'.
+'--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
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.
+ 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
+ '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.
+ 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
+ '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
+ 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
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.
+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.
+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
+*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.
+'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:
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'.
+ '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.
+ -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, const
+ 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
-- 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.
+ '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, i.e.
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.
+ 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)
*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
+ 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_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
+ '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).
+ 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.
9.2 Random State Seeding
========================
- -- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
+ -- Function: void gmp_randseed (gmp_randstate_t STATE, const 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.
+ 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
-- 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
+ Return a uniformly distributed random number of N bits, i.e. 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'.
+ 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.
+ 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
10.1 Format Strings
===================
-`gmp_printf' and friends accept format strings similar to the standard C
-`printf' (*note Formatted Output: (libc)Formatted Output.). A format
+'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,
+ 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);
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
+ 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
+ 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.
+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.
+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'
+ # 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)
+ (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'.
+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
+ 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.
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.
+ 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
f fixed point float
i same as d
g G fixed or scientific float
- m `strerror' string, GLIBC style
+ m 'strerror' string, GLIBC style
n store characters written so far
o octal integer
p pointer
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'.
+ '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
+ '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.
+ '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
+'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
+ The precision field has its 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.
+ '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
+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.
+empty precision field like '%.Fe' or '%.Ff' can be used to specifically
+request just the significant digits. Without any dot and thus no
+precision field, a precision value of 6 will be used. Note that these
+rules mean that '%Ff', '%.Ff', and '%.0Ff' will all be different.
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.
+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
+ The format string is only interpreted as plain 'char's, multibyte
characters are not recognised. Perhaps this will change in the future.
\1f
==============
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'.
+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
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
+ 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
+ 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, ...)
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_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.
+ 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 >=
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.
+ 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.
+ 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
+ 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.
*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.
+ 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.
+ *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
+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>'.
+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.
+ -- Function: ostream& operator<< (ostream& STREAM, const 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.
+ 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.
+ -- Function: ostream& operator<< (ostream& STREAM, const 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'.
+ 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
+ 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.
+ -- Function: ostream& operator<< (ostream& STREAM, const 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.
+ 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'.
+ 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'.
+ '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.
...
cout << "iteration " << n << " value " << z << "\n";
- But note that `ostream' output (and `istream' input, *note C++
+ 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::.
+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
11.1 Formatted Input Strings
============================
-`gmp_scanf' and friends accept format strings similar to the standard C
-`scanf' (*note Formatted Input: (libc)Formatted Input.). A format
+'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 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
+ 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" */
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
+ 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.
+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.
+ 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
+ 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.
Q mpq_t, integer conversions
Z mpz_t, integer conversions
- The conversions accepted are as follows. `p' and `[' will depend on
+ 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
+ e E f g float
+ G
i integer with base indicator
n characters read so far
o octal 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
+ '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
+ 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
+ '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.
+ '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::).
+ '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'
+ '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
+ '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'.
+'scanf' cannot be used through 'gmp_scanf'.
- Whitespace is read and discarded before a field, except for `c' and
-`[' conversions.
+ 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:
+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
+ The format string is only interpreted as plain 'char's, multibyte
characters are not recognised. Perhaps this will change in the future.
\1f
==============================
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'.
+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
-- Function: int gmp_scanf (const char *FMT, ...)
-- Function: int gmp_vscanf (const char *FMT, va_list AP)
- Read from the standard input `stdin'.
+ 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 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
+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
+ 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
+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.
+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.
+ 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
+ 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'.
+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>'.
+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.
+ 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
+ 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.
+ 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.
+ 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.
+ 'operator>>' does.
- Note that digit grouping specified by the `istream' locale is
+ 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,
+ 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++
+ 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::.
+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
+File: gmp.info, Node: C++ Class Interface, Next: Custom Allocation, 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
+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.
+of templates and member templates.
*Everything described in this chapter is to be considered preliminary
and might be subject to incompatible changes if some unforeseen
#include <gmpxx.h>
- Programs should be linked with the `libgmpxx' and `libgmp'
-libraries. For example,
+ Programs should be linked with the 'libgmpxx' and 'libgmp' libraries.
+For example,
g++ mycxxprog.cc -lgmpxx -lgmp
}
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.
+'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++
+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
+ 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
+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
-- 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',
+interface. For example to set 'a' to the GCD of 'b' and 'c',
mpz_class a, b, c;
...
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.
+ 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
+ -- Function: 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, although conversions from 'mpq_class' and 'mpf_class'
+ are 'explicit'. 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: explicit mpz_class::mpz_class (const 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::).
+ -- Function: explicit mpz_class::mpz_class (const char *S, int BASE =
+ 0)
+ -- Function: explicit 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='.
+ 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 (const char *STR)
+ With C++11 compilers, integers can be constructed with the syntax
+ '123_mpz' which is equivalent to 'mpz_class("123")'.
-- 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.
+ 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
+ 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: mpz_class abs (mpz_class OP)
-- 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)
+
+ -- Function: mpz_class gcd (mpz_class OP1, mpz_class OP2)
+ -- Function: mpz_class lcm (mpz_class OP1, mpz_class OP2)
+ -- Function: mpz_class mpz_class::factorial (type OP)
+ -- Function: mpz_class factorial (mpz_class OP)
+ -- Function: mpz_class mpz_class::primorial (type OP)
+ -- Function: mpz_class primorial (mpz_class OP)
+ -- Function: mpz_class mpz_class::fibonacci (type OP)
+ -- Function: mpz_class fibonacci (mpz_class OP)
+
+ -- Function: void mpz_class::swap (mpz_class& OP)
+ -- Function: void swap (mpz_class& OP1, mpz_class& OP2)
These functions provide a C++ class interface to the corresponding
- GMP C routines.
+ GMP C routines. Calling 'factorial' or 'primorial' on a negative
+ number is undefined.
- `cmp' can be used with any of the classes or the standard C++
- types, except `long long' and `long double'.
+ '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
+ 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'.
+ 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.
+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,
+ -- Function: mpq_class::mpq_class (type OP)
+ -- Function: mpq_class::mpq_class (integer NUM, integer DEN)
+ Construct an 'mpq_class'. The initial value can be a single value
+ of any type (conversion from 'mpf_class' is 'explicit'), 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: explicit mpq_class::mpq_class (const 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: explicit mpq_class::mpq_class (const char *S, int BASE =
+ 0)
+ -- Function: explicit 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::).
- -- 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='.
- If the string is not a valid rational, an `std::invalid_argument'
- exception is thrown. The same applies to `operator='.
+ -- Function: mpq_class operator"" _mpq (const char *STR)
+ With C++11 compilers, integral rationals can be constructed with
+ the syntax '123_mpq' which is equivalent to 'mpq_class(123_mpz)'.
+ Other rationals can be built as '-1_mpq/2' or '0xb_mpq/123456_mpz'.
-- Function: void mpq_class::canonicalize ()
- Put an `mpq_class' into canonical form, as per *Note Rational
+ 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)
+
+ -- Function: void mpq_class::swap (mpq_class& OP)
+ -- Function: void swap (mpq_class& OP1, mpq_class& OP2)
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'.
+ '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
+ 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'.
+ original 'mpq_class'.
If direct manipulation might produce a non-canonical value, then
- `mpq_class::canonicalize' must be called before further operations.
+ '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'.
+ 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.
+ '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::).
+ 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.
+ '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
=========================
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
+'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++
+ -- Function: mpf_class::mpf_class (type OP)
+ -- Function: mpf_class::mpf_class (type OP, mp_bitcnt_t 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.
+ 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(-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
+ -- Function: explicit mpf_class::mpf_class (const mpf_t F)
+ -- Function: mpf_class::mpf_class (const mpf_t F, mp_bitcnt_t PREC)
+ Construct an 'mpf_class' from an 'mpf_t'. The value in F is copied
+ into the new 'mpf_class', there won't be any permanent association
+ between it and F.
+
+ If PREC is given, the initial precision is that value, in bits. If
+ PREC is not given, then the initial precision is that of F.
+
+ -- Function: explicit mpf_class::mpf_class (const char *S)
+ -- Function: mpf_class::mpf_class (const char *S, mp_bitcnt_t PREC, int
+ BASE = 0)
+ -- Function: explicit mpf_class::mpf_class (const string& S)
+ -- Function: mpf_class::mpf_class (const string& S, mp_bitcnt_t 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.
+ '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='.
+ If the string is not a valid float, an 'std::invalid_argument'
+ exception is thrown. The same applies to 'operator='.
+
+ -- Function: mpf_class operator"" _mpf (const char *STR)
+ With C++11 compilers, floats can be constructed with the syntax
+ '1.23e-1_mpf' which is equivalent to 'mpf_class("1.23e-1")'.
-- Function: mpf_class& mpf_class::operator= (type OP)
- Convert and store the given OP value to an `mpf_class' object. The
+ 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
+ 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
+ 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
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
+ '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 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: void mpf_class::swap (mpf_class& OP)
+ -- Function: void swap (mpf_class& OP1, mpf_class& OP2)
-- 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'.
+ '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.
+ 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'.
+ 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.
+ 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
-- 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'.
+ 'gmp_randclass' to hold an algorithm selection and current state,
+ as per 'gmp_randstate_t'.
- -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
+ -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
(gmp_randstate_t, ...), ...)
- Construct a `gmp_randclass', using a call to the given RANDINIT
+ 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,
+ 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.
+ '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: 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.
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 (mp_bitcnt_t BITS)
-- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
Generate a random integer with a specified number of bits.
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)
+ -- Function: mpf_class gmp_randclass::get_f (mp_bitcnt_t 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,
12.6 C++ Interface Limitations
==============================
-`mpq_class' and Templated Reading
+'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
+ '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
+ '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
+ '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
+ 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>.
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.
+ 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,
+ 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'.
+ 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
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.
+ 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'.
+ 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, 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.
+C++11
+ C++11 provides several new ways in which types can be inferred:
+ 'auto', 'decltype', etc. While they can be very convenient, they
+ don't mix well with expression templates. In this example, the
+ addition is performed twice, as if we had defined 'sum' as a macro.
- -- Function: void gcd (MINT *OP1, MINT *OP2, MINT *RES)
- Set RES to the greatest common divisor of OP1 and OP2.
+ mpz_class z = 33;
+ auto sum = z + z;
+ mpz_class prod = sum * sum;
- -- 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.
+ This other example may crash, though some compilers might make it
+ look like it is working, because the expression 'z+z' goes out of
+ scope before it is evaluated.
- -- 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.
+ mpz_class z = 33;
+ auto sum = z + z + z;
+ mpz_class prod = sum * 2;
- -- 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'.*
+ It is thus strongly recommended to avoid 'auto' anywhere a GMP C++
+ expression may appear.
\1f
-File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: BSD Compatible Functions, Up: Top
+File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top
-14 Custom Allocation
+13 Custom Allocation
********************
-By default GMP uses `malloc', `realloc' and `free' for memory
+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.
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.
+ 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
+ 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
+ *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.*
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.
+ 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.
+ 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
+ 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.
+ 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.
+ The REALLOCATE_FUNCTION parameter OLD_SIZE and the FREE_FUNCTION
+parameter SIZE are passed for convenience, but of course they can be
+ignored if not needed by an implementation. 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'.
+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
+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.
+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
+ 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
+'mp_set_memory_functions'. It's necessary to change the GMP sources if
this is a problem.
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',
+ 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,
\1f
File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
-15 Language Bindings
+14 Language Bindings
********************
The following packages and projects offer access to GMP from languages
* GMP C++ class interface, *note C++ Class Interface::
Straightforward interface, expression templates to eliminate
temporaries.
-
- * ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
+ * ALP <https://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/'
+ * CLN <https://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/'
+ * Linbox <http://www.linalg.org/>
Sparse vectors and matrices.
-
- * NTL `http://www.shoup.net/ntl/'
+ * NTL <http://www.shoup.net/ntl/>
A C++ number theory library.
-Fortran
- * Omni F77 `http://phase.hpcc.jp/Omni/home.html'
- Arbitrary precision floats.
+Eiffel
+ * Eiffelroom <http://www.eiffelroom.org/node/442>
Haskell
- * Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
+ * Glasgow Haskell Compiler <https://www.haskell.org/ghc/>
Java
- * Kaffe `http://www.kaffe.org/'
-
- * Kissme `http://kissme.sourceforge.net/'
+ * Kaffe <https://github.com/kaffe/kaffe>
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'
+ * GNU Common Lisp <https://www.gnu.org/software/gcl/gcl.html>
+ * Librep <http://librep.sourceforge.net/>
+ * XEmacs (21.5.18 beta and up) <https://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/'
+ * MLton compiler <http://mlton.org/>
Objective Caml
- * MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en'
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
+ * MLGMP <https://opam.ocaml.org/packages/mlgmp/>
+ * Numerix <http://pauillac.inria.fr/~quercia/>
Optionally using GMP.
Oz
- * Mozart `http://www.mozart-oz.org/'
+ * Mozart <https://mozart.github.io/>
Pascal
- * GNU Pascal Compiler `http://www.gnu-pascal.de/'
+ * GNU Pascal Compiler <http://www.gnu-pascal.de/>
GMP unit.
-
- * Numerix `http://pauillac.inria.fr/~quercia/'
+ * Numerix <http://pauillac.inria.fr/~quercia/>
For Free Pascal, optionally using GMP.
Perl
- * GMP module, see `demos/perl' in the GMP sources (*note
+ * 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/'
+ * Math::GMP <https://www.cpan.org/>
+ Compatible with Math::BigInt, but not as many functions as the
+ GMP module above.
+ * Math::BigInt::GMP <https://www.cpan.org/>
Plug Math::GMP into normal Math::BigInt operations.
Pike
- * mpz module in the standard distribution,
- `http://pike.ida.liu.se/'
+ * pikempz module in the standard distribution,
+ <https://pike.lysator.liu.se/>
Prolog
- * SWI Prolog `http://www.swi-prolog.org/'
+ * SWI Prolog <http://www.swi-prolog.org/>
Arbitrary precision floats.
Python
- * mpz module in the standard distribution,
- `http://www.python.org/'
+ * GMPY <https://code.google.com/p/gmpy/>
- * GMPY `http://gmpy.sourceforge.net/'
+Ruby
+ * <https://rubygems.org/gems/gmp>
Scheme
- * GNU Guile (upcoming 1.8)
- `http://www.gnu.org/software/guile/guile.html'
-
- * RScheme `http://www.rscheme.org/'
-
- * STklos `http://www.stklos.org/'
+ * GNU Guile <https://www.gnu.org/software/guile/guile.html>
+ * RScheme <https://www.rscheme.org/>
+ * STklos <http://www.stklos.net/>
Smalltalk
- * GNU Smalltalk
- `http://www.smalltalk.org/versions/GNUSmalltalk.html'
+ * GNU Smalltalk <http://smalltalk.gnu.org/>
Other
- * Axiom `http://savannah.nongnu.org/projects/axiom'
+ * Axiom <https://savannah.nongnu.org/projects/axiom>
Computer algebra using GCL.
-
- * DrGenius `http://drgenius.seul.org/'
+ * DrGenius <http://drgenius.seul.org/>
Geometry system and mathematical programming language.
-
- * GiNaC `http://www.ginac.de/'
+ * GiNaC <httsp://www.ginac.de/>
C++ computer algebra using CLN.
-
- * GOO `http://www.googoogaga.org/'
+ * GOO <https://www.eecs.berkeley.edu/~jrb/goo/>
Dynamic object oriented language.
-
- * Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
+ * Maxima <https://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/'
+ * Regina <http://regina.sourceforge.net/>
Topological calculator.
-
- * Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
+ * Yacas <http://yacas.sourceforge.net>
Yet another computer algebra system.
-
\1f
File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
-16 Algorithms
+15 Algorithms
*************
This chapter is an introduction to some of the algorithms used for
\1f
File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
-16.1 Multiplication
+15.1 Multiplication
===================
-NxN limb multiplications and squares are done using one of five
+NxN limb multiplications and squares are done using one of seven
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'
+ Karatsuba 'MUL_TOOM22_THRESHOLD'
+ Toom-3 'MUL_TOOM33_THRESHOLD'
+ Toom-4 'MUL_TOOM44_THRESHOLD'
+ Toom-6.5 'MUL_TOOM6H_THRESHOLD'
+ Toom-8.5 'MUL_TOOM8H_THRESHOLD'
+ FFT 'MUL_FFT_THRESHOLD'
- Similarly for squaring, with the `SQR' thresholds.
+ 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
+'MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired
algorithms or directly with FFT, depending on operand size (*note
Unbalanced Multiplication::).
* Karatsuba Multiplication::
* Toom 3-Way Multiplication::
* Toom 4-Way Multiplication::
+* Higher degree Toom'n'half::
* 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
+15.1.1 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.
+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
+ 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
+ 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
+This can be seen in 'mpn/generic/sqr_basecase.c'. Again the assembly
implementations take essentially the same approach.
u0 u1 u2 u3 u4
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.
+'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
+ 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
+15.1.2 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.
+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).
| y1 | y0 |
+----------+----------+
- Let b be the power of 2 where the split occurs, ie. if x0 is k limbs
+ Let b be the power of 2 where the split occurs, i.e. 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,
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
+ 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.
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
+ 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
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
+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'.
+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
+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
+15.1.3 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.
+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).
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).
+ Let b equal the power of 2 which is the size of the x0, x1, y0 and y1
+pieces, i.e. 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
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,
+ 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=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).
+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
+ 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(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::).
+ 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].
+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
+and in fact the same 'toom_interpolate_5pts' subroutine is used for both
squaring and multiplying.
Toom-3 is asymptotically O(N^1.465), the exponent being
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
+'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'.
+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.
+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
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.
+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
+File: gmp.info, Node: Toom 4-Way Multiplication, Next: Higher degree Toom'n'half, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
-16.1.4 Toom 4-Way Multiplication
+15.1.4 Toom 4-Way Multiplication
--------------------------------
Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
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,
+ 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
size each.
\1f
-File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
+File: gmp.info, Node: Higher degree Toom'n'half, Next: FFT Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
+
+15.1.5 Higher degree Toom'n'half
+--------------------------------
+
+The Toom algorithms described above (*note Toom 3-Way Multiplication::,
+*note Toom 4-Way Multiplication::) generalizes to split into an
+arbitrary number of pieces. In general a split of two equally long
+operands into r pieces leads to evaluations and pointwise
+multiplications done at 2*r-1 points. To fully exploit symmetries it
+would be better to have a multiple of 4 points, that's why for higher
+degree Toom'n'half is used.
+
+ Toom'n'half means that the existence of one more piece is considered
+for a single operand. It can be virtual, i.e. zero, or real, when the
+two operand are not exactly balanced. By choosing an even r, Toom-r+1/2
+requires 2r points, a multiple of four.
+
+ The quadruplets of points include 0, inf, +1, -1 and +-2^i, +-2^-i .
+Each of them giving shortcuts for the evaluation phase and for some
+steps in the interpolation phase. Further tricks are used to reduce the
+memory footprint of the whole multiplication algorithm to a memory
+buffer equal in size to the result of the product.
-16.1.5 FFT Multiplication
+ Current GMP uses both Toom-6'n'half and Toom-8'n'half.
+
+\1f
+File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Higher degree Toom'n'half, Up: Multiplication Algorithms
+
+15.1.6 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.
+following Schönhage 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 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
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.
+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
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
+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.
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.
+ 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
+'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.
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
+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 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
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
+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'
+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
+15.1.7 Other Multiplication
---------------------------
The Toom algorithms described above (*note Toom 3-Way Multiplication::,
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)).
+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
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)).
+ 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
\1f
File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
-16.1.7 Unbalanced Multiplication
+15.1.8 Unbalanced Multiplication
--------------------------------
Multiplication of operands with different sizes, both below
-`MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication
+'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.
+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
+15.2 Division Algorithms
========================
* Menu:
\1f
File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
-16.2.1 Single Limb Division
+15.2.1 Single Limb Division
---------------------------
Nx1 division is implemented using repeated 2x1 divisions from high to
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.
+integers" by Möller 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.
+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
+'__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
+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.
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
+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
+15.2.2 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.
+'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
\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
+15.2.3 Divide and Conquer Division
----------------------------------
-For divisors larger than `DC_DIV_QR_THRESHOLD', division is done by
+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::).
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.
+'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
+15.2.4 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.
+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
+ 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
+15.2.5 Exact Division
---------------------
A so-called exact division is when the dividend is known to be an exact
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.
+ 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
+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
+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
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
+ 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
+'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.
-
-This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.8 from
-../../gmp/doc/gmp.texi.
+This is gmp.info, produced by makeinfo version 6.7 from gmp.texi.
- This manual describes how to install and use the GNU multiple
-precision arithmetic library, version 5.0.1.
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.1.
- 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::.
+ Copyright 1991, 1993-2016, 2018-2020 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: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
+
+15.2.6 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.
+
+ The functions 'mpn_divexact_by3', 'mpn_modexact_1_odd' and the
+internal '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
+
+15.2.7 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
+
+15.3 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
+
+15.3.1 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
+
+15.3.2 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 computed 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
+
+15.3.3 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 Schönhage's algorithm and subquadratic integer
+GCD computation" by Möller (*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
+sub-quadratic 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
+
+15.3.4 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
+
+15.3.5 Jacobi Symbol
+--------------------
+
+Jacobi symbol (A/B)
+
+ Initially if either operand fits in a single limb, a 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.
+
+ For inputs larger than 'GCD_DC_THRESHOLD', 'mpz_jacobi',
+'mpz_legendre' and 'mpz_kronecker' are computed via the HGCD (Half GCD)
+function, as a generalization to Lehmer's algorithm.
+
+ Most GCD algorithms reduce a and b by repeatatily computing the
+quotient q = floor(a/b) and iteratively replacing
+
+ a, b = b, a - q * b
+
+ Different algorithms use different methods for calculating q, but the
+core algorithm is the same if we use *note Lehmer's Algorithm:: or *note
+HGCD: Subquadratic GCD.
+
+ At each step it is possible to compute if the reduction inverts the
+Jacobi symbol based on the two least significant bits of A and B. For
+more details see "Efficient computation of the Jacobi symbol" by Möller
+(*note References::).
+
+ A small set of bits is thus used to track state
+ * current sign of result (1 bit)
+
+ * two least significant bits of A and B (4 bits)
+
+ * a pointer to which input is currently the denominator (1 bit)
+
+ In all the routines sign changes for the result are accumulated using
+fast bit twiddling which avoids conditional jumps.
+
+ The final result is calculated after verifying the inputs are coprime
+(GCD = 1) by raising (-1)^e
+
+ Much of the HGCD code is shared directly with the HGCD
+implementations, such as the 2x2 matrix calculation, *Note Lehmer's
+Algorithm:: basecase and 'GCD_DC_THRESHOLD'.
+
+ The asymptotic running time is O(M(N)*log(N)), where M(N) is the time
+for multiplying two N-limb numbers.
+
\1f
File: gmp.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
-16.4 Powering Algorithms
+15.4 Powering Algorithms
========================
* Menu:
\1f
File: gmp.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
-16.4.1 Normal Powering
+15.4.1 Normal Powering
----------------------
-Normal `mpz' or `mpf' powering uses a simple binary algorithm,
+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.
+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
+15.4.2 Modular Powering
-----------------------
Modular powering is implemented using a 2^k-ary sliding window
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 modular multiplies and squarings 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
+15.5 Root Extraction Algorithms
===============================
* Menu:
\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
+15.5.1 Square Root
------------------
Square roots are taken using the "Karatsuba Square Root" algorithm by
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
+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
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.
+ 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.
+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
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.
+'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
+15.5.2 Nth Root
---------------
Integer Nth roots are taken using Newton's method with the following
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.
+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
+15.5.3 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
+ '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%.
+ 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').
+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
+ 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.
+ 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
+of non-squares that get through (i.e. a pseudo-square to all the tested
bases).
- Clearly more residue tests could be done, `mpz_perfect_square_p' only
+ 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
\1f
File: gmp.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
-16.5.4 Perfect Power
+15.5.4 Perfect Power
--------------------
Detecting perfect powers is required by some factorization algorithms.
-Currently `mpz_perfect_power_p' is implemented using repeated Nth root
+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::.)
\1f
File: gmp.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
-16.6 Radix Conversion
+15.6 Radix Conversion
=====================
Radix conversions are less important than other algorithms. A program
\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
+15.6.1 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 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.
+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
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
+ 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.
+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.
+ 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
+ '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'
+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
+'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.
+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.
+identify 99... might well be almost as much as a full conversion.
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
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)
+remainder produced, i.e. 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
+15.6.2 Radix to Binary
----------------------
*This section needs to be rewritten, it currently describes the
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
+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.
+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
+ 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.
+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
+'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
+ '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).
+ 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
+15.7 Other Algorithms
=====================
* Menu:
\1f
File: gmp.info, Node: Prime Testing Algorithm, Next: Factorial Algorithm, Prev: Other Algorithms, Up: Other Algorithms
-16.7.1 Prime Testing
+15.7.1 Prime Testing
--------------------
-The primality testing in `mpz_probab_prime_p' (*note Number Theoretic
+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::).
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.
+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
\1f
File: gmp.info, Node: Factorial Algorithm, Next: Binomial Coefficients Algorithm, Prev: Prime Testing Algorithm, Up: Other Algorithms
-16.7.2 Factorial
+15.7.2 Factorial
----------------
-Factorials are calculated by a combination of removal of twos,
-powering, and binary splitting. The procedure can be best illustrated
+Factorials are calculated by a combination of two algorithms. An idea
+is shared among them: to compute the odd part of the factorial; a final
+step takes account of the power of 2 term, by shifting.
+
+ For small n, the odd factor of n! is computed with the simple
+observation that it is equal to the product of all positive odd numbers
+smaller than n times the odd factor of [n/2]!, where [x] is the integer
+part of x, and so on recursively. 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
+ 23! = (23.21.19.17.15.13.11.9.7.5.3)(11.9.7.5.3)(5.3)2^{19}
+
+ Current code collects all the factors in a single list, with a loop
+and no recursion, and compute the product, with no special care for
+repeated chunks.
-has factors of two removed,
+ When n is larger, computation pass trough prime sieving. An helper
+function is used, as suggested by Peter Luschny:
- 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
+ n
+ -----
+ n! | | L(p,n)
+ msf(n) = -------------- = | | p
+ [n/2]!^2.2^k p=3
-and the resulting terms collected up according to their multiplicity,
+ Where p ranges on odd prime numbers. The exponent k is chosen to
+obtain an odd integer number: k is the number of 1 bits in the binary
+representation of [n/2]. The function L(p,n) can be defined as zero
+when p is composite, and, for any prime p, it is computed with:
- 23! = 2^19.(3.5)^3.(7.9.11)^2.(13.15.17.19.21.23)
+ ---
+ \ n
+ L(p,n) = / [---] mod 2 <= log (n) .
+ --- p^i p
+ i>0
- 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.
+ With this helper function, we are able to compute the odd part of n!
+using the recursion implied by n!=[n/2]!^2*msf(n)*2^k. The recursion
+stops using the small-n algorithm on some [n/2^i].
+
+ Both the above algorithms use binary splitting to compute the product
+of many small factors. At first as many products as possible are
+accumulated in a single register, generating a list of factors that fit
+in a machine word. This list is then split into halves, and the product
+is computed recursively.
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.
+used. And even below the Karatsuba threshold a big block of work can be
+more efficient for the basecase algorithm.
\1f
File: gmp.info, Node: Binomial Coefficients Algorithm, Next: Fibonacci Numbers Algorithm, Prev: Factorial Algorithm, Up: Other Algorithms
-16.7.3 Binomial Coefficients
+15.7.3 Binomial Coefficients
----------------------------
Binomial coefficients C(n,k) are calculated by first arranging k <= n/2
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'
+ 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
+15.7.4 Fibonacci Numbers
------------------------
-The Fibonacci functions `mpz_fib_ui' and `mpz_fib2_ui' are designed for
+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].
+ 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
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.
+ 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
+ '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+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
+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.
+'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
+15.7.5 Lucas Numbers
--------------------
-`mpz_lucnum2_ui' derives a pair of Lucas numbers from a pair of
+'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
+ '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.
+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
+15.7.6 Random Numbers
---------------------
-For the `urandomb' functions, random numbers are generated simply by
+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
+ 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.
+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
+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
+parameters A and C, an 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
+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.
+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
+15.8 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.
+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
+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:
\1f
File: gmp.info, Node: Assembly Code Organisation, Next: Assembly Basics, Prev: Assembly Coding, Up: Assembly Coding
-16.8.1 Code Organisation
+15.8.1 Code Organisation
------------------------
-The various `mpn' subdirectories contain machine-dependent code, written
-in C or assembly. The `mpn/generic' subdirectory contains default code,
+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
+ 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
+ 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.
+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
+15.8.2 Assembly Basics
----------------------
-`mpn_addmul_1' and `mpn_submul_1' are the most important routines for
+'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.
+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,
+'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'.
+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
+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
+15.8.3 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.
+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
+ 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 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.
+'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
+15.8.4 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
+ 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
+'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
+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.
+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
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.
+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
+ 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
\1f
File: gmp.info, Node: Assembly Functional Units, Next: Assembly Floating Point, Prev: Assembly Cache Handling, Up: Assembly Coding
-16.8.5 Functional Units
+15.8.5 Functional Units
-----------------------
When choosing an approach for an assembly loop, consideration is given
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.
+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.
+ 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
\1f
File: gmp.info, Node: Assembly Floating Point, Next: Assembly SIMD Instructions, Prev: Assembly Functional Units, Up: Assembly Coding
-16.8.6 Floating Point
+15.8.6 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 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.
+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
+ 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.
+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
+ 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
+ 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.
| 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.
+ 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,
\1f
File: gmp.info, Node: Assembly SIMD Instructions, Next: Assembly Software Pipelining, Prev: Assembly Floating Point, Up: Assembly Coding
-16.8.7 SIMD Instructions
+15.8.7 SIMD Instructions
------------------------
The single-instruction multiple-data support in current microprocessors
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.
+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'.
+ 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
+15.8.8 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.
+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
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::).
+ 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
+15.8.9 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
+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).
+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.
+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.
+ 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
+ * 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
\1f
File: gmp.info, Node: Assembly Writing Guide, Prev: Assembly Loop Unrolling, Up: Assembly Coding
-16.8.10 Writing Guide
+15.8.10 Writing Guide
---------------------
This is a guide to writing software pipelined loops for processing limb
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.
+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
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.
+ 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.
+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
+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.
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
+ 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
+16 Internals
************
*This chapter is provided only for informational purposes and the
\1f
File: gmp.info, Node: Integer Internals, Next: Rational Internals, Prev: Internals, Up: Internals
-17.1 Integer Internals
+16.1 Integer Internals
======================
-`mpz_t' variables represent integers using sign and magnitude, in space
+'mpz_t' variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated. The fields are as follows.
-`_mp_size'
+'_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.
+ negative integer. Zero is represented by '_mp_size' set to zero,
+ in which case the '_mp_d' data is undefined.
-`_mp_d'
+'_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
+ 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 readable limb, so for
+ instance 'mpz_get_ui' can fetch '_mp_d[0]' unconditionally (though
+ its value is undefined if '_mp_size' is zero).
+
+'_mp_alloc'
+ '_mp_alloc' is the number of limbs currently allocated at '_mp_d',
+ and normally '_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.
+
+ 'mpz_t' variables initialised with the 'mpz_roinit_n' function or
+ the 'MPZ_ROINIT_N' macro have '_mp_alloc = 0' but can have a
+ non-zero '_mp_size'. They can only be used as read-only constants.
+ See *note Integer Special Functions:: for details.
+
+ 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
+ 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.
+ '_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
+16.2 Rational Internals
=======================
-`mpq_t' variables represent rationals using an `mpz_t' numerator and
+'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
+ 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.
+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
+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
+16.3 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
+ 'mpf_t' floats have a variable precision mantissa and a single
machine word signed exponent. The mantissa is represented using sign
and magnitude.
<-------- _mp_size --------->
+
The fields are as follows.
-`_mp_size'
+'_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 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'
+'_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
+ aim is to produce '_mp_prec' limbs of result (the most significant
being non-zero).
-`_mp_d'
+'_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.
+ 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
+ '_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'.
+ calculation, only in a change of precision with 'mpf_set_prec'.
-`_mp_exp'
+'_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
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.
+ '{_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
+ 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.
+ 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 '_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
+ 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'.
+ '_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'
+ 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.
+ 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
+ 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
+ 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
+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'.
+ '_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
+ 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
+ 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
+ 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
+ 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
+ '__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.
+ '_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'.
+ '__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.
+ 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
+ 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
+16.4 Raw Output Internals
=========================
-`mpz_out_raw' uses the following format.
+'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.
+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
+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'.
+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
+16.5 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
+'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
+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
struct __gmp_binary_plus
{
- static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
+ static void eval(mpf_t f, const mpf_t g, const mpf_t h)
+ {
+ mpf_add(f, g, h);
+ }
};
And an "additive expression" object,
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
}
- The seemingly redundant `__gmp_expr<__gmp_binary_expr<...>>' is used
+ 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'.
+type. In fact even 'mpf_class' etc are 'typedef' specializations of
+'__gmp_expr'.
- Next we define assignment of `__gmp_expr' to `mpf_class'.
+ Next we define assignment of '__gmp_expr' to 'mpf_class'.
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
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').
+ 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.
+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:
(expr1, expr2);
}
- And the corresponding specializations of `__gmp_expr::eval':
+ And the corresponding specializations of '__gmp_expr::eval':
template <class T, class U, class Op>
void __gmp_expr
}
The expression is thus recursively evaluated to any level of
-complexity and all subexpressions are evaluated to the precision of `f'.
+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
+Torbjörn 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:
+Torbjörn. 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
+ Gunnar Sjödin and Hans Riesel helped with mathematical problems in
early versions of the library.
Richard Stallman helped with the interface design and revised the
the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
-`mpz_probab_prime_p'.
+'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
+Schönhage-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).
+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 Torbjörn).
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.
+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.
+ 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'.
+ 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).
+ 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
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
+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,
+ 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.
+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.
+ Gerardo Ballabio wrote the 'gmpxx.h' C++ class interface and the C++
+'istream' input routines.
- Jason Moxham rewrote `mpz_fac_ui'.
+ 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.
+ Niels Möller wrote the sub-quadratic GCD, extended GCD and jacobi
+code, the quadratic Hensel division code, and (with Torbjörn) 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, and he is the main author of the
+mini-gmp package used for gmp bootstrapping.
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.
+ 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, mpn_mullo_n, and
+mpn_sqrlo. Marco also wrote the functions mpn_invert and
+mpn_invertappr, and improved the speed of integer root extraction. He
+is the author of mini-mpq, an additional layer to mini-gmp; of most of
+the combinatorial functions and the BPSW primality testing
+implementation, for both the main library and the mini-gmp package.
- David Harvey suggested the internal function `mpn_bdiv_dbm1',
+ 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'.
+'mpn_mul_basecase'. He wrote the internal middle product functions
+'mpn_mulmid_basecase', 'mpn_toom42_mulmid', 'mpn_mulmid_n' and related
+helper routines.
+
+ Martin Boij wrote 'mpn_perfect_power_p'.
+
+ Marc Glisse improved 'gmpxx.h': use fewer temporaries (faster),
+specializations of 'numeric_limits' and 'common_type', C++11 features
+(move constructors, explicit bool conversion, UDL), make the conversion
+from 'mpq_class' to 'mpz_class' explicit, optimize operations where one
+argument is a small compile-time constant, replace some heap allocations
+by stack allocations. He also fixed the eofbit handling of C++ streams,
+and removed one division from 'mpq/aors.c'.
+
+ David S Miller wrote assembly code for SPARC T3 and T4.
- Martin Boij wrote `mpn_perfect_power_p'.
+ Mark Sofroniou cleaned up the types of mul_fft.c, letting it work for
+huge operands.
+
+ Ulrich Weigand ported GMP to the powerpc64le ABI.
(This list is chronological, not ordered after significance. If you
have contributed to GMP but are not listed above, please tell
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
+ The development of GMP 2, 3, and 4.0 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.
+ The development of GMP 4.3, 5.0, and 5.1 was supported in part by the
+Swedish Foundation for Strategic Research.
+
+ 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
* Richard Crandall and Carl Pomerance, "Prime Numbers: A
Computational Perspective", 2nd edition, Springer-Verlag, 2005.
- `http://math.dartmouth.edu/~carlp/'
+ <https://www.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/'
+ <https://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'
+ <https://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/'
+ <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/'
+ * Richard M. Stallman and the GCC Developer Community, "Using the GNU
+ Compiler Collection", Free Software Foundation, 2008, available
+ online <https://gcc.gnu.org/onlinedocs/>, and in the GCC package
+ <https://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'
+ 225-252. Also available online as INRIA Research Report 4475, June
+ 2002, <https://hal.inria.fr/docs/00/07/21/13/PDF/RR-4475.pdf>
* 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'
+ <https://www.mpi-inf.mpg.de/~ziegler/TechRep.ps.gz>
- * Torbjo"rn Granlund and Peter L. Montgomery, "Division by Invariant
+ * Torbjörn 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).
+ <https://gmplib.org/~tege/divcnst-pldi94.pdf>.
- * Niels Mo"ller and Torbjo"rn Granlund, "Improved division by
- invariant integers", to appear.
+ * Niels Möller and Torbjörn Granlund, "Improved division by invariant
+ integers", IEEE Transactions on Computers, 11 June 2010.
+ <https://gmplib.org/~tege/division-paper.pdf>
- * Torbjo"rn Granlund and Niels Mo"ller, "Division of integers large
- and small", to appear.
+ * Torbjörn Granlund and Niels Möller, "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'
+ <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'
+ <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'
+ <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'
+ <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'
+ * 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'
+ <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)
+ <http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/ARTICLES/mt.pdf>
* R. Moenck and A. Borodin, "Fast Modular Transforms via Division",
Proceedings of the 13th Annual IEEE Symposium on Switching and
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.
+ * Niels Möller, "On Schönhage's algorithm and subquadratic integer
+ GCD computation", in Mathematics of Computation, volume 77, January
+ 2008, pp. 589-607,
+ <https://www.ams.org/journals/mcom/2008-77-261/S0025-5718-07-02017-0/home.html>
* 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
+ * Arnold Schönhage and Volker Strassen, "Schnelle Multiplikation
grosser Zahlen", Computing 7, 1971, pp. 281-292.
* Kenneth Weber, "The accelerated integer GCD algorithm", ACM
1995, pp. 111-122.
* Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report
- 3805, November 1999, `http://www.inria.fr/rrrt/rr-3805.html'
+ 3805, November 1999,
+ <https://hal.inria.fr/inria-00072854/PDF/RR-3805.pdf>
* Paul Zimmermann, "A Proof of GMP Fast Division and Square Root
Implementations",
- `http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz'
+ <https://homepages.loria.fr/PZimmermann/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.
IEEE Transactions on Computers, volume 43, number 8, August 1994,
pp. 899-908.
+ * Niels Möller, "Efficient computation of the Jacobi symbol",
+ <https://arxiv.org/abs/1907.07795>
+
\1f
File: gmp.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
Version 1.3, 3 November 2008
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- and finally terminates your license, and (b) permanently, if the
+ 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.
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.
+ 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
+ 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/'.
+ <https://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
+ 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
+ 11. RELICENSING
"Massive Multiauthor Collaboration Site" (or "MMC Site") means any
World Wide Web server that publishes copyrightable works and also
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
====================================================
Free Documentation License''.
If you have Invariant Sections, Front-Cover Texts and Back-Cover
-Texts, replace the "with...Texts." line with this:
+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
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.
+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
* #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)
+* --build: Build Options. (line 51)
+* --disable-fft: Build Options. (line 307)
+* --disable-shared: Build Options. (line 44)
+* --disable-static: Build Options. (line 44)
+* --enable-alloca: Build Options. (line 273)
+* --enable-assert: Build Options. (line 313)
+* --enable-cxx: Build Options. (line 225)
+* --enable-fat: Build Options. (line 160)
+* --enable-profiling: Build Options. (line 317)
* --enable-profiling <1>: Profiling. (line 6)
-* --enable-profiling: Build Options. (line 331)
* --exec-prefix: Build Options. (line 32)
-* --host: Build Options. (line 66)
+* --host: Build Options. (line 65)
* --prefix: Build Options. (line 32)
* -finstrument-functions: Profiling. (line 66)
* 2exp functions: Efficiency. (line 43)
* 68000: Notes for Particular Systems.
- (line 80)
+ (line 94)
* 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)
+ (line 150)
+* ABI: Build Options. (line 167)
+* ABI <1>: ABI and ISA. (line 6)
+* About this manual: Introduction to GMP. (line 57)
* AC_CHECK_LIB: Autoconf. (line 11)
-* AIX <1>: ABI and ISA. (line 184)
-* AIX <2>: Notes for Particular Systems.
+* AIX: ABI and ISA. (line 174)
+* AIX <1>: Notes for Particular Systems.
(line 7)
-* AIX: ABI and ISA. (line 169)
* Algorithms: Algorithms. (line 6)
-* alloca: Build Options. (line 278)
+* alloca: Build Options. (line 273)
* Allocation of memory: Custom Allocation. (line 6)
* AMD64: ABI and ISA. (line 44)
-* Anonymous FTP of latest version: Introduction to GMP. (line 38)
+* Anonymous FTP of latest version: Introduction to GMP. (line 37)
* 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)
+* Arithmetic functions: Integer Arithmetic. (line 6)
+* Arithmetic functions <1>: Rational Arithmetic. (line 6)
+* Arithmetic functions <2>: Float Arithmetic. (line 6)
* ARM: Notes for Particular Systems.
(line 20)
* Assembly cache handling: Assembly Cache Handling.
(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.
+* Assertion checking: Build Options. (line 313)
+* Assertion checking <1>: Debugging. (line 74)
+* Assignment functions: Assigning Integers. (line 6)
+* Assignment functions <1>: Simultaneous Integer Init & Assign.
(line 6)
-* Assignment functions <3>: Simultaneous Integer Init & Assign.
+* Assignment functions <2>: Initializing Rationals.
(line 6)
+* Assignment functions <3>: Assigning Floats. (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)
+ (line 128)
* 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)
+ (line 39)
+* Bit shift left: Integer Arithmetic. (line 38)
+* Bit shift right: Integer Division. (line 74)
* 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.
* Build options: Build Options. (line 6)
* Build problems known: Known Build Problems.
(line 6)
-* Build system: Build Options. (line 52)
+* Build system: Build Options. (line 51)
* Building GMP: Installing GMP. (line 6)
* Bus error: Debugging. (line 7)
-* C compiler: Build Options. (line 182)
-* C++ compiler: Build Options. (line 254)
+* C compiler: Build Options. (line 178)
+* C++ compiler: Build Options. (line 249)
* 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)
+* C++ support: Build Options. (line 225)
+* CC: Build Options. (line 178)
+* CC_FOR_BUILD: Build Options. (line 212)
+* CFLAGS: Build Options. (line 178)
+* Checker: Debugging. (line 110)
+* checkergcc: Debugging. (line 117)
* 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)
+* Comparison functions: Integer Comparisons. (line 6)
+* Comparison functions <1>: Comparing Rationals. (line 6)
+* Comparison functions <2>: 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)
+* Congruence algorithm: Exact Remainder. (line 30)
+* Congruence functions: Integer Division. (line 150)
* Constants: Useful Macros and Constants.
(line 6)
* Contributors: Contributors. (line 6)
(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.
+* Conversion functions: Converting Integers. (line 6)
+* Conversion functions <1>: Rational Conversions.
(line 6)
+* Conversion functions <2>: Converting Floats. (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)
+* CPPFLAGS: Build Options. (line 204)
+* CPU types: Introduction to GMP. (line 24)
+* CPU types <1>: Build Options. (line 107)
+* Cross compiling: Build Options. (line 65)
+* Cryptography functions, low-level: Low-level Functions. (line 507)
* Custom allocation: Custom Allocation. (line 6)
-* CXX: Build Options. (line 254)
-* CXXFLAGS: Build Options. (line 254)
+* CXX: Build Options. (line 249)
+* CXXFLAGS: Build Options. (line 249)
* Cygwin: Notes for Particular Systems.
- (line 43)
+ (line 57)
* Darwin: Known Build Problems.
(line 51)
* Debugging: Debugging. (line 6)
(line 6)
* Digits in an integer: Miscellaneous Integer Functions.
(line 23)
-* Divisibility algorithm: Exact Remainder. (line 29)
-* Divisibility functions: Integer Division. (line 124)
+* Divisibility algorithm: Exact Remainder. (line 30)
+* Divisibility functions: Integer Division. (line 136)
+* Divisibility functions <1>: Integer Division. (line 150)
* 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.
+* Division functions: Integer Division. (line 6)
+* Division functions <1>: Rational Arithmetic. (line 24)
+* Division functions <2>: Float Arithmetic. (line 33)
+* DJGPP: Notes for Particular Systems.
+ (line 57)
+* DJGPP <1>: Known Build Problems.
(line 18)
* DLLs: Notes for Particular Systems.
- (line 56)
-* DocBook: Build Options. (line 354)
-* Documentation formats: Build Options. (line 347)
+ (line 70)
+* DocBook: Build Options. (line 340)
+* Documentation formats: Build Options. (line 333)
* Documentation license: GNU Free Documentation License.
(line 6)
-* DVI: Build Options. (line 350)
+* DVI: Build Options. (line 336)
* Efficiency: Efficiency. (line 6)
* Emacs: Emacs. (line 6)
-* Exact division functions: Integer Division. (line 102)
+* Exact division functions: Integer Division. (line 125)
* Exact remainder: Exact Remainder. (line 6)
* Example programs: Demonstration Programs.
(line 6)
* Exec prefix: Build Options. (line 32)
+* Execution profiling: Build Options. (line 317)
* Execution profiling <1>: Profiling. (line 6)
-* Execution profiling: Build Options. (line 331)
-* Exponentiation functions <1>: Integer Exponentiation.
+* Exponentiation functions: Integer Exponentiation.
(line 6)
-* Exponentiation functions: Float Arithmetic. (line 41)
+* Exponentiation functions <1>: Float Arithmetic. (line 41)
* Export: Integer Import and Export.
(line 45)
* Expression parsing demo: Demonstration Programs.
- (line 18)
+ (line 15)
+* Expression parsing demo <1>: Demonstration Programs.
+ (line 17)
+* Expression parsing demo <2>: Demonstration Programs.
+ (line 19)
* Extended GCD: Number Theoretic Functions.
- (line 45)
+ (line 47)
* Factor removal functions: Number Theoretic Functions.
- (line 90)
+ (line 108)
* Factorial algorithm: Factorial Algorithm. (line 6)
* Factorial functions: Number Theoretic Functions.
- (line 95)
+ (line 116)
* Factorization demo: Demonstration Programs.
- (line 25)
+ (line 22)
* Fast Fourier Transform: FFT Multiplication. (line 6)
-* Fat binary: Build Options. (line 164)
+* Fat binary: Build Options. (line 160)
+* FFT multiplication: Build Options. (line 307)
* 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)
+ (line 136)
* Float arithmetic functions: Float Arithmetic. (line 6)
+* Float assignment functions: Assigning Floats. (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: Initializing Floats. (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 27)
* Float rounding functions: Miscellaneous Float Functions.
(line 9)
-* Float sign tests: Float Comparison. (line 33)
+* Float sign tests: Float Comparison. (line 34)
* Floating point mode: Notes for Particular Systems.
(line 34)
* Floating-point functions: Floating-point Functions.
* 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)
+* FreeBSD: Notes for Particular Systems.
+ (line 43)
+* FreeBSD <1>: Notes for Particular Systems.
+ (line 52)
+* frexp: Converting Integers. (line 43)
+* frexp <1>: Converting Floats. (line 24)
+* FTP of latest version: Introduction to GMP. (line 37)
* Function classes: Function Classes. (line 6)
* FunctionCheck: Profiling. (line 77)
-* GCC Checker: Debugging. (line 115)
+* GCC Checker: Debugging. (line 110)
* GCD algorithms: Greatest Common Divisor Algorithms.
(line 6)
* GCD extended: Number Theoretic Functions.
- (line 45)
+ (line 47)
* GCD functions: Number Theoretic Functions.
(line 30)
-* GDB: Debugging. (line 58)
-* Generic C: Build Options. (line 153)
+* GDB: Debugging. (line 53)
+* Generic C: Build Options. (line 151)
* GMP Perl module: Demonstration Programs.
- (line 35)
+ (line 28)
* 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 Debugger: Debugging. (line 53)
* GNU Free Documentation License: GNU Free Documentation License.
(line 6)
* GNU strip: Known Build Problems.
(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)
+* Heap problems: Debugging. (line 23)
+* Home page: Introduction to GMP. (line 33)
+* Host system: Build Options. (line 65)
+* HP-UX: ABI and ISA. (line 76)
+* HP-UX <1>: ABI and ISA. (line 114)
+* HPPA: ABI and ISA. (line 76)
+* I/O functions: I/O of Integers. (line 6)
+* I/O functions <1>: I/O of Rationals. (line 6)
+* I/O functions <2>: I/O of Floats. (line 6)
* i386: Notes for Particular Systems.
- (line 126)
-* IA-64: ABI and ISA. (line 107)
+ (line 150)
+* IA-64: ABI and ISA. (line 114)
* 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.
+* Initialization functions: Initializing Integers.
(line 6)
-* Initialization functions <2>: Initializing Rationals.
+* Initialization functions <1>: Simultaneous Integer Init & Assign.
(line 6)
-* Initialization functions <3>: Random State Initialization.
+* Initialization functions <2>: Initializing Rationals.
(line 6)
+* Initialization functions <3>: Initializing Floats. (line 6)
* Initialization functions <4>: Simultaneous Float Init & Assign.
(line 6)
-* Initialization functions <5>: Simultaneous Integer Init & Assign.
+* Initialization functions <5>: Random State Initialization.
(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.
+* Input functions: I/O of Integers. (line 6)
+* Input functions <1>: I/O of Rationals. (line 6)
+* Input functions <2>: I/O of Floats. (line 6)
+* Input functions <3>: Formatted Input Functions.
(line 6)
* Install prefix: Build Options. (line 32)
* Installing GMP: Installing GMP. (line 6)
* Integer: Nomenclature and Types.
(line 6)
* Integer arithmetic functions: Integer Arithmetic. (line 6)
+* Integer assignment functions: Assigning Integers. (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 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 initialization functions <1>: Simultaneous Integer Init & Assign.
+ (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.
* Integer special functions: Integer Special Functions.
(line 6)
* Interix: Notes for Particular Systems.
- (line 51)
+ (line 65)
* Internals: Internals. (line 6)
* Introduction: Introduction to GMP. (line 6)
* Inverse modulo functions: Number Theoretic Functions.
- (line 60)
+ (line 74)
+* IRIX: ABI and ISA. (line 139)
* 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)
+ (line 83)
* Karatsuba multiplication: Karatsuba Multiplication.
(line 6)
* Karatsuba square root algorithm: Square Root Algorithm.
(line 6)
* Kronecker symbol functions: Number Theoretic Functions.
- (line 78)
+ (line 95)
* Language bindings: Language Bindings. (line 6)
-* Latest version of GMP: Introduction to GMP. (line 38)
+* Latest version of GMP: Introduction to GMP. (line 37)
* LCM functions: Number Theoretic Functions.
- (line 55)
+ (line 68)
* Least common multiple functions: Number Theoretic Functions.
- (line 55)
+ (line 68)
* Legendre symbol functions: Number Theoretic Functions.
- (line 69)
+ (line 86)
* libgmp: Headers and Libraries.
(line 22)
* libgmpxx: Headers and Libraries.
* Linear congruential algorithm: Random Number Algorithms.
(line 25)
* Linear congruential random numbers: Random State Initialization.
+ (line 18)
+* Linear congruential random numbers <1>: 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)
+* Low-level functions for cryptography: Low-level Functions. (line 507)
* Lucas number algorithm: Lucas Numbers Algorithm.
(line 6)
* Lucas number functions: Number Theoretic Functions.
- (line 119)
+ (line 147)
* MacOS X: Known Build Problems.
(line 51)
-* Mailing lists: Introduction to GMP. (line 45)
-* Malloc debugger: Debugging. (line 30)
-* Malloc problems: Debugging. (line 24)
+* Mailing lists: Introduction to GMP. (line 44)
+* Malloc debugger: Debugging. (line 29)
+* Malloc problems: Debugging. (line 23)
* Memory allocation: Custom Allocation. (line 6)
* Memory management: Memory Management. (line 6)
* Mersenne twister algorithm: Random Number Algorithms.
* Mersenne twister random numbers: Random State Initialization.
(line 13)
* MINGW: Notes for Particular Systems.
- (line 43)
-* MIPS: ABI and ISA. (line 132)
+ (line 57)
+* MIPS: ABI and ISA. (line 139)
* Miscellaneous float functions: Miscellaneous Float Functions.
(line 6)
* Miscellaneous integer functions: Miscellaneous Integer Functions.
(line 6)
* MMX: Notes for Particular Systems.
- (line 132)
+ (line 156)
* Modular inverse functions: Number Theoretic Functions.
- (line 60)
+ (line 74)
* Most significant bit: Miscellaneous Integer Functions.
(line 34)
-* mp.h: BSD Compatible Functions.
- (line 21)
-* MPN_PATH: Build Options. (line 335)
+* MPN_PATH: Build Options. (line 321)
* MS Windows: Notes for Particular Systems.
- (line 56)
+ (line 57)
+* MS Windows <1>: Notes for Particular Systems.
+ (line 70)
* MS-DOS: Notes for Particular Systems.
- (line 43)
+ (line 57)
* Multi-threading: Reentrancy. (line 6)
* Multiplication algorithms: Multiplication Algorithms.
(line 6)
-* Nails: Low-level Functions. (line 478)
-* Native compilation: Build Options. (line 52)
+* Nails: Low-level Functions. (line 686)
+* Native compilation: Build Options. (line 51)
+* NetBSD: Notes for Particular Systems.
+ (line 100)
* NeXT: Known Build Problems.
(line 57)
* Next prime function: Number Theoretic Functions.
(line 6)
* Non-Unix systems: Build Options. (line 11)
* Nth root algorithm: Nth Root Algorithm. (line 6)
-* Number sequences: Efficiency. (line 147)
+* Number sequences: Efficiency. (line 145)
* Number theoretic functions: Number Theoretic Functions.
(line 6)
* Numerator and denominator: Applying Integer Functions.
(line 6)
* obstack output: Formatted Output Functions.
- (line 81)
+ (line 79)
* OpenBSD: Notes for Particular Systems.
- (line 86)
+ (line 109)
* 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: I/O of Integers. (line 6)
+* Output functions <1>: I/O of Rationals. (line 6)
+* Output functions <2>: I/O of Floats. (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)
+ (line 15)
+* Parsing expressions demo <1>: Demonstration Programs.
+ (line 17)
+* Parsing expressions demo <2>: Demonstration Programs.
+ (line 19)
* Particular systems: Notes for Particular Systems.
(line 6)
* Past GMP versions: Compatibility with older versions.
(line 6)
-* PDF: Build Options. (line 350)
+* PDF: Build Options. (line 336)
* Perfect power algorithm: Perfect Power Algorithm.
(line 6)
-* Perfect power functions: Integer Roots. (line 27)
+* Perfect power functions: Integer Roots. (line 28)
* Perfect square algorithm: Perfect Square Algorithm.
(line 6)
-* Perfect square functions: Integer Roots. (line 36)
+* Perfect square functions: Integer Roots. (line 37)
* perl: Demonstration Programs.
- (line 35)
+ (line 28)
* Perl module: Demonstration Programs.
- (line 35)
-* Postscript: Build Options. (line 350)
+ (line 28)
+* Postscript: Build Options. (line 336)
+* Power/PowerPC: Notes for Particular Systems.
+ (line 115)
* 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)
+* Powering functions <1>: Float Arithmetic. (line 41)
+* PowerPC: ABI and ISA. (line 173)
* Precision of floats: Floating-point Functions.
(line 6)
* Precision of hardware floating point: Notes for Particular Systems.
(line 6)
* Prime testing functions: Number Theoretic Functions.
(line 7)
+* Primorial functions: Number Theoretic Functions.
+ (line 121)
* printf formatted output: Formatted Output. (line 6)
* Probable prime testing functions: Number Theoretic Functions.
(line 7)
(line 6)
* Random number algorithms: Random Number Algorithms.
(line 6)
-* Random number functions <1>: Integer Random Numbers.
+* Random number functions: Integer Random Numbers.
(line 6)
-* Random number functions <2>: Miscellaneous Float Functions.
+* Random number functions <1>: Miscellaneous Float Functions.
(line 27)
-* Random number functions: Random Number Functions.
+* Random number functions <2>: Random Number Functions.
(line 6)
* Random number seeding: Random State Seeding.
(line 6)
(line 6)
* Random state: Nomenclature and Types.
(line 46)
-* Rational arithmetic: Efficiency. (line 113)
+* Rational arithmetic: Efficiency. (line 111)
* Rational arithmetic functions: Rational Arithmetic. (line 6)
* Rational assignment functions: Initializing Rationals.
(line 6)
(line 6)
* Rational numerator and denominator: Applying Integer Functions.
(line 6)
-* Rational sign tests: Comparing Rationals. (line 27)
+* Rational sign tests: Comparing Rationals. (line 28)
* Raw output internals: Raw Output Internals.
(line 6)
* Reallocations: Efficiency. (line 30)
* Reentrancy: Reentrancy. (line 6)
-* References: References. (line 6)
+* References: References. (line 5)
* Remove factor functions: Number Theoretic Functions.
- (line 90)
+ (line 108)
* 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)
+* Root extraction functions <1>: Float Arithmetic. (line 37)
+* Root testing functions: Integer Roots. (line 28)
+* Root testing functions <1>: Integer Roots. (line 37)
* Rounding functions: Miscellaneous Float Functions.
(line 9)
* Sample programs: Demonstration Programs.
(line 6)
* Scan bit functions: Integer Logic and Bit Fiddling.
- (line 38)
+ (line 39)
* scanf formatted input: Formatted Input. (line 6)
* SCO: Known Build Problems.
(line 38)
* Sequent Symmetry: Known Build Problems.
(line 68)
* Services for Unix: Notes for Particular Systems.
- (line 51)
+ (line 65)
* 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)
+* Sign tests: Integer Comparisons. (line 28)
+* Sign tests <1>: Comparing Rationals. (line 28)
+* Sign tests <2>: Float Comparison. (line 34)
* 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)
+* Solaris: ABI and ISA. (line 204)
+* Solaris <1>: Known Build Problems.
+ (line 72)
+* Solaris <2>: Known Build Problems.
+ (line 77)
* Sparc: Notes for Particular Systems.
- (line 108)
-* Sparc V9: ABI and ISA. (line 201)
+ (line 127)
+* Sparc <1>: Notes for Particular Systems.
+ (line 132)
+* Sparc V9: ABI and ISA. (line 204)
* 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)
+ (line 156)
+* Stack backtrace: Debugging. (line 45)
+* Stack overflow: Build Options. (line 273)
* Stack overflow <1>: Debugging. (line 7)
-* Stack overflow: Build Options. (line 278)
* Static linking: Efficiency. (line 14)
* stdarg.h: Headers and Libraries.
(line 17)
(line 11)
* Stripped libraries: Known Build Problems.
(line 28)
-* Sun: ABI and ISA. (line 201)
+* Sun: ABI and ISA. (line 204)
* SunOS: Notes for Particular Systems.
- (line 120)
+ (line 144)
* Systems: Notes for Particular Systems.
(line 6)
-* Temporary memory: Build Options. (line 278)
-* Texinfo: Build Options. (line 347)
-* Text input/output: Efficiency. (line 153)
+* Temporary memory: Build Options. (line 273)
+* Texinfo: Build Options. (line 333)
+* Text input/output: Efficiency. (line 151)
* Thread safety: Reentrancy. (line 6)
-* Toom multiplication <1>: Other Multiplication.
+* Toom multiplication: Toom 3-Way Multiplication.
(line 6)
-* Toom multiplication <2>: Toom 4-Way Multiplication.
+* Toom multiplication <1>: Toom 4-Way Multiplication.
(line 6)
-* Toom multiplication: Toom 3-Way Multiplication.
+* Toom multiplication <2>: Higher degree Toom'n'half.
+ (line 6)
+* Toom multiplication <3>: Other Multiplication.
(line 6)
* Types: Nomenclature and Types.
(line 6)
(line 6)
* User-defined precision: Floating-point Functions.
(line 6)
-* Valgrind: Debugging. (line 130)
+* Valgrind: Debugging. (line 125)
* Variable conventions: Variable Conventions.
(line 6)
* Version number: Useful Macros and Constants.
(line 12)
-* Web page: Introduction to GMP. (line 34)
+* Web page: Introduction to GMP. (line 33)
* Windows: Notes for Particular Systems.
- (line 56)
+ (line 57)
+* Windows <1>: Notes for Particular Systems.
+ (line 70)
* x86: Notes for Particular Systems.
- (line 126)
+ (line 150)
* x87: Notes for Particular Systems.
(line 34)
-* XML: Build Options. (line 354)
+* XML: Build Options. (line 340)
\1f
File: gmp.info, Node: Function Index, Prev: Concept Index, Up: Top
\0\b[index\0\b]
* Menu:
+* _mpz_realloc: Integer Special Functions.
+ (line 13)
* __GMP_CC: Useful Macros and Constants.
- (line 23)
+ (line 22)
* __GMP_CFLAGS: Useful Macros and Constants.
- (line 24)
+ (line 23)
* __GNU_MP_VERSION: Useful Macros and Constants.
- (line 10)
+ (line 9)
* __GNU_MP_VERSION_MINOR: Useful Macros and Constants.
- (line 11)
+ (line 10)
* __GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants.
- (line 12)
-* _mpz_realloc: Integer Special Functions.
- (line 51)
+ (line 11)
+* abs: C++ Interface Integers.
+ (line 46)
* abs <1>: C++ Interface Rationals.
- (line 43)
-* abs <2>: C++ Interface Integers.
- (line 42)
-* abs: C++ Interface Floats.
- (line 70)
+ (line 47)
+* abs <2>: C++ Interface Floats.
+ (line 82)
* ceil: C++ Interface Floats.
- (line 71)
-* cmp <1>: C++ Interface Floats.
- (line 72)
+ (line 83)
+* cmp: C++ Interface Integers.
+ (line 47)
+* cmp <1>: C++ Interface Integers.
+ (line 48)
* cmp <2>: C++ Interface Rationals.
- (line 44)
-* cmp <3>: C++ Interface Integers.
- (line 44)
-* cmp: C++ Interface Rationals.
- (line 45)
+ (line 48)
+* cmp <3>: C++ Interface Rationals.
+ (line 49)
+* cmp <4>: C++ Interface Floats.
+ (line 84)
+* cmp <5>: C++ Interface Floats.
+ (line 85)
+* factorial: C++ Interface Integers.
+ (line 71)
+* fibonacci: C++ Interface Integers.
+ (line 75)
* floor: C++ Interface Floats.
- (line 80)
-* gcd: BSD Compatible Functions.
- (line 82)
+ (line 95)
+* gcd: C++ Interface Integers.
+ (line 68)
* gmp_asprintf: Formatted Output Functions.
- (line 65)
+ (line 63)
* gmp_errno: Random State Initialization.
- (line 55)
+ (line 56)
* GMP_ERROR_INVALID_ARGUMENT: Random State Initialization.
- (line 55)
+ (line 56)
* GMP_ERROR_UNSUPPORTED_ARGUMENT: Random State Initialization.
- (line 55)
+ (line 56)
* gmp_fprintf: Formatted Output Functions.
- (line 29)
+ (line 28)
* 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)
+ (line 24)
+* GMP_LIMB_BITS: Low-level Functions. (line 714)
+* GMP_NAIL_BITS: Low-level Functions. (line 712)
+* GMP_NAIL_MASK: Low-level Functions. (line 722)
+* GMP_NUMB_BITS: Low-level Functions. (line 713)
+* GMP_NUMB_MASK: Low-level Functions. (line 723)
+* GMP_NUMB_MAX: Low-level Functions. (line 731)
* gmp_obstack_printf: Formatted Output Functions.
- (line 79)
+ (line 75)
* gmp_obstack_vprintf: Formatted Output Functions.
- (line 81)
+ (line 77)
* 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)
+ (line 23)
* gmp_randclass: C++ Interface Random Numbers.
- (line 7)
+ (line 6)
* gmp_randclass::get_f: C++ Interface Random Numbers.
+ (line 44)
+* gmp_randclass::get_f <1>: C++ Interface Random Numbers.
(line 45)
* gmp_randclass::get_z_bits: C++ Interface Random Numbers.
- (line 39)
+ (line 37)
+* gmp_randclass::get_z_bits <1>: C++ Interface Random Numbers.
+ (line 38)
* gmp_randclass::get_z_range: C++ Interface Random Numbers.
- (line 42)
+ (line 41)
* gmp_randclass::gmp_randclass: C++ Interface Random Numbers.
- (line 13)
+ (line 11)
+* gmp_randclass::gmp_randclass <1>: C++ Interface Random Numbers.
+ (line 26)
* gmp_randclass::seed: C++ Interface Random Numbers.
+ (line 32)
+* gmp_randclass::seed <1>: C++ Interface Random Numbers.
(line 33)
* gmp_randclear: Random State Initialization.
(line 62)
* gmp_randinit: Random State Initialization.
- (line 47)
+ (line 45)
* gmp_randinit_default: Random State Initialization.
- (line 7)
+ (line 6)
* gmp_randinit_lc_2exp: Random State Initialization.
- (line 18)
+ (line 16)
* gmp_randinit_lc_2exp_size: Random State Initialization.
- (line 32)
+ (line 30)
* gmp_randinit_mt: Random State Initialization.
- (line 13)
+ (line 12)
* gmp_randinit_set: Random State Initialization.
- (line 43)
+ (line 41)
* gmp_randseed: Random State Seeding.
- (line 7)
+ (line 6)
* gmp_randseed_ui: Random State Seeding.
- (line 9)
+ (line 8)
* gmp_randstate_t: Nomenclature and Types.
(line 46)
+* GMP_RAND_ALG_DEFAULT: Random State Initialization.
+ (line 50)
+* GMP_RAND_ALG_LC: Random State Initialization.
+ (line 50)
* gmp_scanf: Formatted Input Functions.
- (line 21)
+ (line 20)
* gmp_snprintf: Formatted Output Functions.
- (line 46)
+ (line 44)
* gmp_sprintf: Formatted Output Functions.
- (line 34)
+ (line 33)
* gmp_sscanf: Formatted Input Functions.
- (line 29)
+ (line 28)
* gmp_urandomb_ui: Random State Miscellaneous.
- (line 8)
+ (line 6)
* gmp_urandomm_ui: Random State Miscellaneous.
- (line 14)
+ (line 12)
* gmp_vasprintf: Formatted Output Functions.
- (line 66)
+ (line 64)
* gmp_version: Useful Macros and Constants.
(line 18)
* gmp_vfprintf: Formatted Output Functions.
- (line 30)
+ (line 29)
* gmp_vfscanf: Formatted Input Functions.
- (line 26)
-* gmp_vprintf: Formatted Output Functions.
(line 25)
+* gmp_vprintf: Formatted Output Functions.
+ (line 24)
* gmp_vscanf: Formatted Input Functions.
- (line 22)
+ (line 21)
* gmp_vsnprintf: Formatted Output Functions.
- (line 48)
+ (line 46)
* gmp_vsprintf: Formatted Output Functions.
- (line 35)
+ (line 34)
* 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)
+* hypot: C++ Interface Floats.
+ (line 96)
+* lcm: C++ Interface Integers.
+ (line 69)
+* mpf_abs: Float Arithmetic. (line 46)
+* mpf_add: Float Arithmetic. (line 6)
+* mpf_add_ui: Float Arithmetic. (line 7)
* mpf_ceil: Miscellaneous Float Functions.
- (line 7)
+ (line 6)
* mpf_class: C++ Interface General.
- (line 20)
+ (line 19)
* mpf_class::fits_sint_p: C++ Interface Floats.
- (line 74)
+ (line 87)
* mpf_class::fits_slong_p: C++ Interface Floats.
- (line 75)
+ (line 88)
* mpf_class::fits_sshort_p: C++ Interface Floats.
- (line 76)
+ (line 89)
* mpf_class::fits_uint_p: C++ Interface Floats.
- (line 77)
+ (line 91)
* mpf_class::fits_ulong_p: C++ Interface Floats.
- (line 78)
+ (line 92)
* mpf_class::fits_ushort_p: C++ Interface Floats.
- (line 79)
+ (line 93)
* mpf_class::get_d: C++ Interface Floats.
- (line 82)
+ (line 98)
* mpf_class::get_mpf_t: C++ Interface General.
- (line 66)
+ (line 65)
* mpf_class::get_prec: C++ Interface Floats.
- (line 100)
+ (line 120)
* mpf_class::get_si: C++ Interface Floats.
- (line 83)
+ (line 99)
* mpf_class::get_str: C++ Interface Floats.
- (line 85)
+ (line 100)
* mpf_class::get_ui: C++ Interface Floats.
- (line 86)
+ (line 102)
* mpf_class::mpf_class: C++ Interface Floats.
- (line 38)
+ (line 11)
+* mpf_class::mpf_class <1>: C++ Interface Floats.
+ (line 12)
+* mpf_class::mpf_class <2>: C++ Interface Floats.
+ (line 32)
+* mpf_class::mpf_class <3>: C++ Interface Floats.
+ (line 33)
+* mpf_class::mpf_class <4>: C++ Interface Floats.
+ (line 41)
+* mpf_class::mpf_class <5>: C++ Interface Floats.
+ (line 42)
+* mpf_class::mpf_class <6>: C++ Interface Floats.
+ (line 44)
+* mpf_class::mpf_class <7>: C++ Interface Floats.
+ (line 45)
* mpf_class::operator=: C++ Interface Floats.
- (line 47)
+ (line 59)
* mpf_class::set_prec: C++ Interface Floats.
- (line 101)
+ (line 121)
* mpf_class::set_prec_raw: C++ Interface Floats.
- (line 102)
+ (line 122)
* 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)
+ (line 104)
+* mpf_class::set_str <1>: C++ Interface Floats.
+ (line 105)
+* mpf_class::swap: C++ Interface Floats.
+ (line 109)
+* mpf_clear: Initializing Floats. (line 36)
+* mpf_clears: Initializing Floats. (line 40)
+* mpf_cmp: Float Comparison. (line 6)
* 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_cmp_z: Float Comparison. (line 7)
+* mpf_div: Float Arithmetic. (line 28)
* mpf_div_2exp: Float Arithmetic. (line 53)
-* mpf_div_ui: Float Arithmetic. (line 33)
+* mpf_div_ui: Float Arithmetic. (line 31)
* mpf_eq: Float Comparison. (line 17)
* mpf_fits_sint_p: Miscellaneous Float Functions.
- (line 20)
+ (line 19)
* mpf_fits_slong_p: Miscellaneous Float Functions.
- (line 18)
+ (line 17)
* mpf_fits_sshort_p: Miscellaneous Float Functions.
- (line 22)
+ (line 21)
* mpf_fits_uint_p: Miscellaneous Float Functions.
- (line 19)
+ (line 18)
* mpf_fits_ulong_p: Miscellaneous Float Functions.
- (line 17)
+ (line 16)
* mpf_fits_ushort_p: Miscellaneous Float Functions.
- (line 21)
+ (line 20)
* 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)
+ (line 7)
+* mpf_get_d: Converting Floats. (line 6)
+* mpf_get_default_prec: Initializing Floats. (line 11)
+* mpf_get_d_2exp: Converting Floats. (line 15)
+* mpf_get_prec: Initializing Floats. (line 61)
* mpf_get_si: Converting Floats. (line 27)
-* mpf_get_str: Converting Floats. (line 37)
+* mpf_get_str: Converting Floats. (line 36)
* mpf_get_ui: Converting Floats. (line 28)
-* mpf_init: Initializing Floats. (line 19)
-* mpf_init2: Initializing Floats. (line 26)
+* mpf_init: Initializing Floats. (line 18)
+* mpf_init2: Initializing Floats. (line 25)
+* mpf_inits: Initializing Floats. (line 30)
* mpf_init_set: Simultaneous Float Init & Assign.
- (line 16)
+ (line 15)
* mpf_init_set_d: Simultaneous Float Init & Assign.
- (line 19)
-* mpf_init_set_si: Simultaneous Float Init & Assign.
(line 18)
+* mpf_init_set_si: Simultaneous Float Init & Assign.
+ (line 17)
* mpf_init_set_str: Simultaneous Float Init & Assign.
- (line 25)
+ (line 24)
* 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)
+ (line 16)
+* mpf_inp_str: I/O of Floats. (line 38)
* 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)
+ (line 13)
+* mpf_mul: Float Arithmetic. (line 18)
+* mpf_mul_2exp: Float Arithmetic. (line 49)
+* mpf_mul_ui: Float Arithmetic. (line 19)
+* mpf_neg: Float Arithmetic. (line 43)
* mpf_out_str: I/O of Floats. (line 17)
-* mpf_pow_ui: Float Arithmetic. (line 41)
+* mpf_pow_ui: Float Arithmetic. (line 39)
* 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)
+ (line 35)
+* mpf_reldiff: Float Comparison. (line 28)
+* mpf_set: Assigning Floats. (line 9)
+* mpf_set_d: Assigning Floats. (line 12)
+* mpf_set_default_prec: Initializing Floats. (line 6)
+* mpf_set_prec: Initializing Floats. (line 64)
+* mpf_set_prec_raw: Initializing Floats. (line 71)
+* mpf_set_q: Assigning Floats. (line 14)
+* mpf_set_si: Assigning Floats. (line 11)
+* mpf_set_str: Assigning Floats. (line 17)
+* mpf_set_ui: Assigning Floats. (line 10)
+* mpf_set_z: Assigning Floats. (line 13)
* 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_sqrt: Float Arithmetic. (line 35)
+* mpf_sqrt_ui: Float Arithmetic. (line 36)
+* mpf_sub: Float Arithmetic. (line 11)
+* mpf_sub_ui: Float Arithmetic. (line 14)
+* mpf_swap: Assigning Floats. (line 50)
* 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)
+ (line 8)
+* mpf_ui_div: Float Arithmetic. (line 29)
+* mpf_ui_sub: Float Arithmetic. (line 12)
* 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)
+ (line 25)
+* mpn_add: Low-level Functions. (line 67)
* 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_add_1: Low-level Functions. (line 62)
+* mpn_add_n: Low-level Functions. (line 52)
+* mpn_andn_n: Low-level Functions. (line 462)
+* mpn_and_n: Low-level Functions. (line 447)
+* mpn_cmp: Low-level Functions. (line 293)
+* mpn_cnd_add_n: Low-level Functions. (line 540)
+* mpn_cnd_sub_n: Low-level Functions. (line 542)
+* mpn_cnd_swap: Low-level Functions. (line 567)
+* mpn_com: Low-level Functions. (line 487)
+* mpn_copyd: Low-level Functions. (line 496)
+* mpn_copyi: Low-level Functions. (line 492)
+* mpn_divexact_1: Low-level Functions. (line 231)
+* mpn_divexact_by3: Low-level Functions. (line 238)
+* mpn_divexact_by3c: Low-level Functions. (line 240)
+* mpn_divmod: Low-level Functions. (line 226)
+* mpn_divmod_1: Low-level Functions. (line 210)
+* mpn_divrem: Low-level Functions. (line 183)
+* mpn_divrem_1: Low-level Functions. (line 208)
+* mpn_gcd: Low-level Functions. (line 301)
+* mpn_gcdext: Low-level Functions. (line 316)
+* mpn_gcd_1: Low-level Functions. (line 311)
+* mpn_get_str: Low-level Functions. (line 371)
+* mpn_hamdist: Low-level Functions. (line 436)
+* mpn_iorn_n: Low-level Functions. (line 467)
+* mpn_ior_n: Low-level Functions. (line 452)
+* mpn_lshift: Low-level Functions. (line 269)
+* mpn_mod_1: Low-level Functions. (line 264)
* 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_nand_n: Low-level Functions. (line 472)
+* mpn_neg: Low-level Functions. (line 96)
+* mpn_nior_n: Low-level Functions. (line 477)
+* mpn_perfect_square_p: Low-level Functions. (line 442)
+* mpn_popcount: Low-level Functions. (line 432)
+* mpn_random: Low-level Functions. (line 422)
+* mpn_random2: Low-level Functions. (line 423)
+* mpn_rshift: Low-level Functions. (line 281)
+* mpn_scan0: Low-level Functions. (line 406)
+* mpn_scan1: Low-level Functions. (line 414)
+* mpn_sec_add_1: Low-level Functions. (line 553)
+* mpn_sec_div_qr: Low-level Functions. (line 630)
+* mpn_sec_div_qr_itch: Low-level Functions. (line 633)
+* mpn_sec_div_r: Low-level Functions. (line 649)
+* mpn_sec_div_r_itch: Low-level Functions. (line 651)
+* mpn_sec_invert: Low-level Functions. (line 665)
+* mpn_sec_invert_itch: Low-level Functions. (line 667)
+* mpn_sec_mul: Low-level Functions. (line 574)
+* mpn_sec_mul_itch: Low-level Functions. (line 577)
+* mpn_sec_powm: Low-level Functions. (line 604)
+* mpn_sec_powm_itch: Low-level Functions. (line 607)
+* mpn_sec_sqr: Low-level Functions. (line 590)
+* mpn_sec_sqr_itch: Low-level Functions. (line 592)
+* mpn_sec_sub_1: Low-level Functions. (line 555)
+* mpn_sec_tabselect: Low-level Functions. (line 622)
+* mpn_set_str: Low-level Functions. (line 386)
+* mpn_sizeinbase: Low-level Functions. (line 364)
* 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)
+* mpn_sqrtrem: Low-level Functions. (line 346)
+* mpn_sub: Low-level Functions. (line 88)
+* mpn_submul_1: Low-level Functions. (line 160)
+* mpn_sub_1: Low-level Functions. (line 83)
+* mpn_sub_n: Low-level Functions. (line 74)
+* mpn_tdiv_qr: Low-level Functions. (line 172)
+* mpn_xnor_n: Low-level Functions. (line 482)
+* mpn_xor_n: Low-level Functions. (line 457)
+* mpn_zero: Low-level Functions. (line 500)
+* mpn_zero_p: Low-level Functions. (line 298)
+* mpq_abs: Rational Arithmetic. (line 33)
+* mpq_add: Rational Arithmetic. (line 6)
* mpq_canonicalize: Rational Number Functions.
- (line 22)
+ (line 21)
* mpq_class: C++ Interface General.
- (line 19)
+ (line 18)
* mpq_class::canonicalize: C++ Interface Rationals.
- (line 37)
+ (line 41)
* mpq_class::get_d: C++ Interface Rationals.
- (line 46)
+ (line 51)
* mpq_class::get_den: C++ Interface Rationals.
- (line 58)
+ (line 67)
* mpq_class::get_den_mpz_t: C++ Interface Rationals.
- (line 68)
+ (line 77)
* mpq_class::get_mpq_t: C++ Interface General.
- (line 65)
+ (line 64)
* mpq_class::get_num: C++ Interface Rationals.
- (line 57)
+ (line 66)
* mpq_class::get_num_mpz_t: C++ Interface Rationals.
- (line 67)
+ (line 76)
* mpq_class::get_str: C++ Interface Rationals.
- (line 47)
+ (line 52)
* mpq_class::mpq_class: C++ Interface Rationals.
- (line 22)
+ (line 9)
+* mpq_class::mpq_class <1>: C++ Interface Rationals.
+ (line 10)
+* mpq_class::mpq_class <2>: C++ Interface Rationals.
+ (line 21)
+* mpq_class::mpq_class <3>: C++ Interface Rationals.
+ (line 26)
+* mpq_class::mpq_class <4>: C++ Interface Rationals.
+ (line 28)
* mpq_class::set_str: C++ Interface Rationals.
- (line 49)
+ (line 54)
+* mpq_class::set_str <1>: C++ Interface Rationals.
+ (line 55)
+* mpq_class::swap: C++ Interface Rationals.
+ (line 58)
* mpq_clear: Initializing Rationals.
- (line 16)
+ (line 15)
* 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)
+ (line 19)
+* mpq_cmp: Comparing Rationals. (line 6)
+* mpq_cmp_si: Comparing Rationals. (line 16)
+* mpq_cmp_ui: Comparing Rationals. (line 14)
+* mpq_cmp_z: Comparing Rationals. (line 7)
* mpq_denref: Applying Integer Functions.
- (line 18)
+ (line 16)
* mpq_div: Rational Arithmetic. (line 22)
-* mpq_div_2exp: Rational Arithmetic. (line 25)
+* mpq_div_2exp: Rational Arithmetic. (line 26)
* mpq_equal: Comparing Rationals. (line 33)
* mpq_get_d: Rational Conversions.
- (line 7)
+ (line 6)
* mpq_get_den: Applying Integer Functions.
- (line 24)
+ (line 22)
* mpq_get_num: Applying Integer Functions.
- (line 23)
+ (line 21)
* mpq_get_str: Rational Conversions.
- (line 22)
+ (line 21)
* mpq_init: Initializing Rationals.
- (line 7)
+ (line 6)
* 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)
+ (line 11)
+* mpq_inp_str: I/O of Rationals. (line 32)
+* mpq_inv: Rational Arithmetic. (line 36)
+* mpq_mul: Rational Arithmetic. (line 14)
* mpq_mul_2exp: Rational Arithmetic. (line 18)
-* mpq_neg: Rational Arithmetic. (line 28)
+* mpq_neg: Rational Arithmetic. (line 30)
* mpq_numref: Applying Integer Functions.
- (line 17)
-* mpq_out_str: I/O of Rationals. (line 15)
+ (line 15)
+* mpq_out_str: I/O of Rationals. (line 17)
* mpq_set: Initializing Rationals.
- (line 24)
+ (line 23)
* mpq_set_d: Rational Conversions.
- (line 17)
+ (line 16)
* mpq_set_den: Applying Integer Functions.
- (line 26)
+ (line 24)
* mpq_set_f: Rational Conversions.
- (line 18)
+ (line 17)
* mpq_set_num: Applying Integer Functions.
- (line 25)
+ (line 23)
* mpq_set_si: Initializing Rationals.
- (line 31)
+ (line 29)
* mpq_set_str: Initializing Rationals.
- (line 36)
+ (line 35)
* mpq_set_ui: Initializing Rationals.
- (line 29)
+ (line 27)
* mpq_set_z: Initializing Rationals.
- (line 25)
+ (line 24)
* mpq_sgn: Comparing Rationals. (line 27)
-* mpq_sub: Rational Arithmetic. (line 11)
+* mpq_sub: Rational Arithmetic. (line 10)
* mpq_swap: Initializing Rationals.
- (line 56)
+ (line 54)
* 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_2fac_ui: Number Theoretic Functions.
+ (line 113)
+* mpz_abs: Integer Arithmetic. (line 44)
+* mpz_add: Integer Arithmetic. (line 6)
+* mpz_addmul: Integer Arithmetic. (line 24)
+* mpz_addmul_ui: Integer Arithmetic. (line 26)
+* mpz_add_ui: Integer Arithmetic. (line 7)
* mpz_and: Integer Logic and Bit Fiddling.
- (line 11)
+ (line 10)
* mpz_array_init: Integer Special Functions.
- (line 11)
+ (line 9)
* mpz_bin_ui: Number Theoretic Functions.
- (line 98)
+ (line 124)
* 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)
+ (line 126)
+* mpz_cdiv_q: Integer Division. (line 12)
+* mpz_cdiv_qr: Integer Division. (line 14)
* 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_q_2exp: Integer Division. (line 26)
+* mpz_cdiv_q_ui: Integer Division. (line 17)
+* mpz_cdiv_r: Integer Division. (line 13)
+* mpz_cdiv_r_2exp: Integer Division. (line 29)
* mpz_cdiv_r_ui: Integer Division. (line 19)
* mpz_cdiv_ui: Integer Division. (line 23)
* mpz_class: C++ Interface General.
- (line 18)
+ (line 17)
+* mpz_class::factorial: C++ Interface Integers.
+ (line 70)
+* mpz_class::fibonacci: C++ Interface Integers.
+ (line 74)
* mpz_class::fits_sint_p: C++ Interface Integers.
- (line 45)
+ (line 50)
* mpz_class::fits_slong_p: C++ Interface Integers.
- (line 46)
+ (line 51)
* mpz_class::fits_sshort_p: C++ Interface Integers.
- (line 47)
+ (line 52)
* mpz_class::fits_uint_p: C++ Interface Integers.
- (line 48)
+ (line 54)
* mpz_class::fits_ulong_p: C++ Interface Integers.
- (line 49)
+ (line 55)
* mpz_class::fits_ushort_p: C++ Interface Integers.
- (line 50)
+ (line 56)
* mpz_class::get_d: C++ Interface Integers.
- (line 51)
+ (line 58)
* mpz_class::get_mpz_t: C++ Interface General.
- (line 64)
+ (line 63)
* mpz_class::get_si: C++ Interface Integers.
- (line 52)
+ (line 59)
* mpz_class::get_str: C++ Interface Integers.
- (line 53)
+ (line 60)
* mpz_class::get_ui: C++ Interface Integers.
- (line 54)
+ (line 61)
* mpz_class::mpz_class: C++ Interface Integers.
- (line 7)
+ (line 6)
+* mpz_class::mpz_class <1>: C++ Interface Integers.
+ (line 14)
+* mpz_class::mpz_class <2>: C++ Interface Integers.
+ (line 19)
+* mpz_class::mpz_class <3>: C++ Interface Integers.
+ (line 21)
+* mpz_class::primorial: C++ Interface Integers.
+ (line 72)
* mpz_class::set_str: C++ Interface Integers.
- (line 56)
+ (line 63)
+* mpz_class::set_str <1>: C++ Interface Integers.
+ (line 64)
+* mpz_class::swap: C++ Interface Integers.
+ (line 77)
* mpz_clear: Initializing Integers.
- (line 44)
-* mpz_clears: Initializing Integers.
(line 48)
+* mpz_clears: Initializing Integers.
+ (line 52)
* 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_cmp: Integer Comparisons. (line 6)
+* mpz_cmpabs: Integer Comparisons. (line 17)
+* mpz_cmpabs_d: Integer Comparisons. (line 18)
+* mpz_cmpabs_ui: Integer Comparisons. (line 19)
+* mpz_cmp_d: Integer Comparisons. (line 7)
+* mpz_cmp_si: Integer Comparisons. (line 8)
+* mpz_cmp_ui: Integer Comparisons. (line 9)
* mpz_com: Integer Logic and Bit Fiddling.
- (line 20)
+ (line 19)
* 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_congruent_2exp_p: Integer Division. (line 148)
+* mpz_congruent_p: Integer Division. (line 144)
+* mpz_congruent_ui_p: Integer Division. (line 146)
+* mpz_divexact: Integer Division. (line 122)
+* mpz_divexact_ui: Integer Division. (line 123)
+* mpz_divisible_2exp_p: Integer Division. (line 135)
+* mpz_divisible_p: Integer Division. (line 132)
+* mpz_divisible_ui_p: Integer Division. (line 133)
* mpz_even_p: Miscellaneous Integer Functions.
- (line 18)
+ (line 17)
* mpz_export: Integer Import and Export.
- (line 45)
+ (line 43)
* 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)
+ (line 112)
+* mpz_fdiv_q: Integer Division. (line 33)
+* mpz_fdiv_qr: Integer Division. (line 35)
+* mpz_fdiv_qr_ui: Integer Division. (line 42)
+* mpz_fdiv_q_2exp: Integer Division. (line 47)
+* mpz_fdiv_q_ui: Integer Division. (line 38)
+* mpz_fdiv_r: Integer Division. (line 34)
+* mpz_fdiv_r_2exp: Integer Division. (line 50)
+* mpz_fdiv_r_ui: Integer Division. (line 40)
+* mpz_fdiv_ui: Integer Division. (line 44)
* mpz_fib2_ui: Number Theoretic Functions.
- (line 108)
+ (line 134)
* mpz_fib_ui: Number Theoretic Functions.
- (line 106)
+ (line 133)
* mpz_fits_sint_p: Miscellaneous Integer Functions.
- (line 10)
+ (line 9)
* mpz_fits_slong_p: Miscellaneous Integer Functions.
- (line 8)
+ (line 7)
* mpz_fits_sshort_p: Miscellaneous Integer Functions.
- (line 12)
+ (line 11)
* mpz_fits_uint_p: Miscellaneous Integer Functions.
- (line 9)
+ (line 8)
* mpz_fits_ulong_p: Miscellaneous Integer Functions.
- (line 7)
+ (line 6)
* mpz_fits_ushort_p: Miscellaneous Integer Functions.
- (line 11)
+ (line 10)
* mpz_gcd: Number Theoretic Functions.
- (line 30)
-* mpz_gcd_ui: Number Theoretic Functions.
- (line 35)
+ (line 29)
* 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_gcd_ui: Number Theoretic Functions.
+ (line 35)
* mpz_getlimbn: Integer Special Functions.
- (line 60)
+ (line 22)
+* mpz_get_d: Converting Integers. (line 26)
+* mpz_get_d_2exp: Converting Integers. (line 34)
+* mpz_get_si: Converting Integers. (line 17)
+* mpz_get_str: Converting Integers. (line 46)
+* mpz_get_ui: Converting Integers. (line 10)
* mpz_hamdist: Integer Logic and Bit Fiddling.
- (line 29)
+ (line 28)
* mpz_import: Integer Import and Export.
- (line 11)
+ (line 9)
* mpz_init: Initializing Integers.
- (line 26)
+ (line 25)
* mpz_init2: Initializing Integers.
- (line 33)
+ (line 32)
+* mpz_inits: Initializing Integers.
+ (line 28)
* mpz_init_set: Simultaneous Integer Init & Assign.
- (line 27)
+ (line 26)
* mpz_init_set_d: Simultaneous Integer Init & Assign.
- (line 30)
-* mpz_init_set_si: Simultaneous Integer Init & Assign.
(line 29)
+* mpz_init_set_si: Simultaneous Integer Init & Assign.
+ (line 28)
* mpz_init_set_str: Simultaneous Integer Init & Assign.
- (line 34)
+ (line 33)
* 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)
+ (line 27)
+* mpz_inp_raw: I/O of Integers. (line 61)
+* mpz_inp_str: I/O of Integers. (line 30)
* mpz_invert: Number Theoretic Functions.
- (line 60)
+ (line 72)
* mpz_ior: Integer Logic and Bit Fiddling.
- (line 14)
+ (line 13)
* mpz_jacobi: Number Theoretic Functions.
- (line 66)
+ (line 82)
* mpz_kronecker: Number Theoretic Functions.
- (line 74)
+ (line 90)
* mpz_kronecker_si: Number Theoretic Functions.
- (line 75)
+ (line 91)
* mpz_kronecker_ui: Number Theoretic Functions.
- (line 76)
+ (line 92)
* mpz_lcm: Number Theoretic Functions.
- (line 54)
+ (line 65)
* mpz_lcm_ui: Number Theoretic Functions.
- (line 55)
+ (line 66)
* mpz_legendre: Number Theoretic Functions.
- (line 69)
+ (line 85)
+* mpz_limbs_finish: Integer Special Functions.
+ (line 47)
+* mpz_limbs_modify: Integer Special Functions.
+ (line 40)
+* mpz_limbs_read: Integer Special Functions.
+ (line 34)
+* mpz_limbs_write: Integer Special Functions.
+ (line 39)
* mpz_lucnum2_ui: Number Theoretic Functions.
- (line 119)
+ (line 145)
* 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)
+ (line 144)
+* mpz_mfac_uiui: Number Theoretic Functions.
+ (line 114)
+* mpz_mod: Integer Division. (line 112)
+* mpz_mod_ui: Integer Division. (line 113)
+* mpz_mul: Integer Arithmetic. (line 18)
+* mpz_mul_2exp: Integer Arithmetic. (line 36)
+* mpz_mul_si: Integer Arithmetic. (line 19)
+* mpz_mul_ui: Integer Arithmetic. (line 20)
+* mpz_neg: Integer Arithmetic. (line 41)
* mpz_nextprime: Number Theoretic Functions.
- (line 23)
+ (line 22)
* 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)
+ (line 16)
+* mpz_out_raw: I/O of Integers. (line 45)
+* mpz_out_str: I/O of Integers. (line 17)
* 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)
+ (line 22)
* mpz_powm: Integer Exponentiation.
- (line 8)
+ (line 6)
* mpz_powm_sec: Integer Exponentiation.
- (line 18)
+ (line 16)
* mpz_powm_ui: Integer Exponentiation.
- (line 10)
+ (line 8)
+* mpz_pow_ui: Integer Exponentiation.
+ (line 29)
+* mpz_primorial_ui: Number Theoretic Functions.
+ (line 120)
* mpz_probab_prime_p: Number Theoretic Functions.
- (line 7)
+ (line 6)
* mpz_random: Integer Random Numbers.
- (line 42)
+ (line 41)
* mpz_random2: Integer Random Numbers.
- (line 51)
+ (line 50)
* mpz_realloc2: Initializing Integers.
- (line 52)
+ (line 56)
* mpz_remove: Number Theoretic Functions.
- (line 90)
-* mpz_root: Integer Roots. (line 7)
-* mpz_rootrem: Integer Roots. (line 13)
+ (line 106)
+* mpz_roinit_n: Integer Special Functions.
+ (line 67)
+* MPZ_ROINIT_N: Integer Special Functions.
+ (line 83)
+* mpz_root: Integer Roots. (line 6)
+* mpz_rootrem: Integer Roots. (line 12)
* mpz_rrandomb: Integer Random Numbers.
- (line 31)
+ (line 29)
* mpz_scan0: Integer Logic and Bit Fiddling.
- (line 37)
+ (line 35)
* 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)
+ (line 37)
+* mpz_set: Assigning Integers. (line 9)
* mpz_setbit: Integer Logic and Bit Fiddling.
(line 51)
-* mpz_sgn: Integer Comparisons. (line 28)
-* mpz_si_kronecker: Number Theoretic Functions.
- (line 77)
+* mpz_set_d: Assigning Integers. (line 12)
+* mpz_set_f: Assigning Integers. (line 14)
+* mpz_set_q: Assigning Integers. (line 13)
+* mpz_set_si: Assigning Integers. (line 11)
+* mpz_set_str: Assigning Integers. (line 20)
+* mpz_set_ui: Assigning Integers. (line 10)
+* mpz_sgn: Integer Comparisons. (line 27)
* mpz_size: Integer Special Functions.
- (line 68)
+ (line 30)
* mpz_sizeinbase: Miscellaneous Integer Functions.
- (line 23)
+ (line 22)
+* mpz_si_kronecker: Number Theoretic Functions.
+ (line 93)
* 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_sub: Integer Arithmetic. (line 11)
* mpz_submul: Integer Arithmetic. (line 30)
* mpz_submul_ui: Integer Arithmetic. (line 32)
-* mpz_swap: Assigning Integers. (line 37)
+* mpz_sub_ui: Integer Arithmetic. (line 12)
+* mpz_swap: Assigning Integers. (line 36)
* 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_tdiv_q: Integer Division. (line 54)
+* mpz_tdiv_qr: Integer Division. (line 56)
+* mpz_tdiv_qr_ui: Integer Division. (line 63)
+* mpz_tdiv_q_2exp: Integer Division. (line 68)
+* mpz_tdiv_q_ui: Integer Division. (line 59)
+* mpz_tdiv_r: Integer Division. (line 55)
+* mpz_tdiv_r_2exp: Integer Division. (line 71)
+* mpz_tdiv_r_ui: Integer Division. (line 61)
+* mpz_tdiv_ui: Integer Division. (line 65)
* mpz_tstbit: Integer Logic and Bit Fiddling.
(line 60)
* mpz_ui_kronecker: Number Theoretic Functions.
- (line 78)
+ (line 94)
* mpz_ui_pow_ui: Integer Exponentiation.
- (line 33)
-* mpz_ui_sub: Integer Arithmetic. (line 16)
+ (line 31)
+* mpz_ui_sub: Integer Arithmetic. (line 14)
* mpz_urandomb: Integer Random Numbers.
- (line 14)
+ (line 12)
* mpz_urandomm: Integer Random Numbers.
- (line 23)
+ (line 21)
* 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)
+ (line 16)
+* 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 86)
+* mp_limb_t: Nomenclature and Types.
+ (line 31)
+* mp_set_memory_functions: Custom Allocation. (line 14)
+* mp_size_t: Nomenclature and Types.
+ (line 37)
+* operator"": C++ Interface Integers.
+ (line 29)
+* operator"" <1>: C++ Interface Rationals.
+ (line 36)
+* operator"" <2>: C++ Interface Floats.
+ (line 55)
* operator%: C++ Interface Integers.
- (line 30)
+ (line 34)
* operator/: C++ Interface Integers.
- (line 29)
+ (line 33)
* 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)
+ (line 10)
+* operator<< <1>: C++ Formatted Output.
+ (line 19)
+* operator<< <2>: C++ Formatted Output.
+ (line 32)
+* operator>>: C++ Formatted Input. (line 10)
+* operator>> <1>: C++ Formatted Input. (line 13)
+* operator>> <2>: C++ Formatted Input. (line 24)
+* operator>> <3>: C++ Interface Rationals.
+ (line 86)
+* primorial: C++ Interface Integers.
+ (line 73)
+* sgn: C++ Interface Integers.
+ (line 65)
* 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)
+ (line 56)
+* sgn <2>: C++ Interface Floats.
+ (line 106)
+* sqrt: C++ Interface Integers.
+ (line 66)
+* sqrt <1>: C++ Interface Floats.
+ (line 107)
+* swap: C++ Interface Integers.
+ (line 78)
+* swap <1>: C++ Interface Rationals.
+ (line 59)
+* swap <2>: C++ Interface Floats.
+ (line 110)
* trunc: C++ Interface Floats.
- (line 91)
-* xtom: BSD Compatible Functions.
- (line 34)
-
+ (line 111)
projects / xonotic / xonotic.git / commitdiff