Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002 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 1.1 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 GNU Free Documentation License.
printf style output.
scanf style input.
This library is free; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.
Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.
The precise conditions of the license for the GNU MP library are found in the
Lesser General Public License version 2.1 that accompanies the source code,
see COPYING.LIB. Certain demonstration programs are provided under the
terms of the plain General Public License version 2, see COPYING.
GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers. It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types.
Many applications use just a few hundred bits of precision; but some applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum.
The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance).
There is carefully optimized assembly code for these CPUs: ARM, DEC Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2 and Athlon, 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 a mailing list for GMP users. To join it, send a mail to
gmp-request@swox.com with the word subscribe in the message
body (not in the subject line).
For up-to-date information on GMP, please see the GMP web pages at
http://swox.com/gmp/
The latest version of the library is available at
ftp://ftp.gnu.org/gnu/gmp
Many sites around the world mirror ftp.gnu.org, please use a mirror
near you, see http://www.gnu.org/order/ftp.html for a full list.
Everyone should read GMP Basics. If you need to install the library yourself, then read Installing GMP. If you have a system with multiple ABIs, then read ABI and ISA, for the compiler options that must be used on applications.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
GMP has an autoconf/automake/libtool based configuration system. On a
Unix-like system a basic build can be done with
./configure make
Some self-tests can be run with
make check
And you can install (under /usr/local by default) with
make install
If you experience problems, please report them to bug-gmp@gnu.org. See Reporting Bugs, for information on what to include in useful bug reports.
All the usual autoconf configure options are available, run ./configure
--help for a summary. The file INSTALL.autoconf has some generic
installation information too.
configure requires various Unix-like tools. On an MS-DOS system DJGPP
can be used, and on MS Windows Cygwin or MINGW can be used,
http://www.cygnus.com/cygwin http://www.delorie.com/djgpp http://www.mingw.org
The macos directory contains an unsupported port to MacOS 9 on Power
Macintosh, see macos/README. Note that MacOS X "Darwin" should use
the normal Unix-style ./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.
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-4.1/configure
Not all make programs have the necessary features (VPATH) to
support this. In particular, SunOS and Slowaris make have bugs that
make them unable to build in a separate directory. Use GNU make
instead.
--disable-shared, --disable-static
--build=CPU-VENDOR-OS
--build. By default ./configure uses the output from running
./config.guess. On some systems ./config.guess can determine
the exact CPU type, on others it will be necessary to give it explicitly. For
example,
./configure --build=ultrasparc-sun-solaris2.7
In all cases the OS part is important, since it controls how libtool
generates shared libraries. Running ./config.guess is the simplest way
to see what it should be, if you don't know already.
--host=CPU-VENDOR-OS
--build
and the system where the library will run is given by --host. For
example when using a FreeBSD Athlon system to build GNU/Linux m68k binaries,
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
Compiler tools are sought first with the host system type as a prefix. For
example m68k-mac-linux-gnu-ranlib is tried, then plain
ranlib. This makes it possible for a set of cross-compiling tools
to co-exist with native tools. The prefix is the argument to --host,
and this can be an alias, such as m68k-linux. But note that tools
don't have to be setup this way, it's enough to just have a PATH with a
suitable cross-compiling cc etc.
Compiling for a different CPU in the same family as the build system is a form
of cross-compilation, though very possibly this would merely be special
options on a native compiler. In any case ./configure avoids depending
on being able to run code on the build system, which is important when
creating binaries for a newer CPU since they very possibly won't run on the
build system.
In all cases the compiler must be able to produce an executable (of whatever
format) from a standard C main. Although only object files will go to
make up libgmp, ./configure uses linking tests for various
purposes, such as determining what functions are available on the host system.
Currently a warning is given unless an explicit --build is used when
cross-compiling, because it may not be possible to correctly guess the build
system type if the PATH has only a cross-compiling cc.
Note that the --target option is not appropriate for GMP. It's for use
when building compiler tools, with --host being where they will run,
and --target what they'll produce code for. Ordinary programs or
libraries like GMP are only interested in the --host part, being where
they'll run. (Some past versions of GMP used --target incorrectly.)
The following CPUs have specific support. See configure.in for details
of what code and compiler options they select.
alpha,
alphaev5,
alphaev56,
alphapca56,
alphapca57,
alphaev6,
alphaev67,
alphaev68
c90,
j90,
t90,
sv1
hppa1.0,
hppa1.1,
hppa2.0,
hppa2.0n,
hppa2.0w
mips,
mips3,
mips64
m68k,
m68000,
m68010,
m68020,
m68030,
m68040,
m68060,
m68302,
m68360,
m88k,
m88110
power,
power1,
power2,
power2sc
powerpc,
powerpc64,
powerpc401,
powerpc403,
powerpc405,
powerpc505,
powerpc601,
powerpc602,
powerpc603,
powerpc603e,
powerpc604,
powerpc604e,
powerpc620,
powerpc630,
powerpc740,
powerpc7400,
powerpc7450,
powerpc750,
powerpc801,
powerpc821,
powerpc823,
powerpc860,
sparc,
sparcv8,
microsparc,
supersparc,
sparcv9,
ultrasparc,
ultrasparc2,
ultrasparc2i,
ultrasparc3,
sparc64
i386,
i486,
i586,
pentium,
pentiummmx,
pentiumpro,
pentium2,
pentium3,
pentium4,
k6,
k62,
k63,
athlon
a29k,
arm,
clipper,
i960,
ns32k,
pyramid,
sh,
sh2,
vax,
z8k
CPUs not listed will use generic C code.
none. For example,
./configure --host=none-unknown-freebsd3.5
Note that this will run quite slowly, but it should be portable and should at
least make it possible to get something running if all else fails.
ABI
./configure --host=mips64-sgi-irix6 ABI=n32
See ABI and ISA, for the available choices on relevant CPUs, and what
applications need to do.
CC, CFLAGS
gcc normally preferred if it's present. The usual
CC=whatever can be passed to ./configure to choose something
different.
For some systems, default compiler flags are set based on the CPU and
compiler. The usual CFLAGS="-whatever" can be passed to
./configure to use something different or to set good flags for systems
GMP doesn't otherwise know.
The CC and CFLAGS used are printed during ./configure,
and can be found in each generated Makefile. This is the easiest way
to check the defaults when considering changing or adding something.
Note that when CC and CFLAGS are specified on a system
supporting multiple ABIs it's important to give an explicit
ABI=whatever, since GMP can't determine the ABI just from the flags and
won't be able to select the correct assembler code.
If just CC is selected then normal default CFLAGS for that
compiler will be used (if GMP recognises it). For example CC=gcc can
be used to force the use of GCC, with default flags (and default ABI).
CPPFLAGS
-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.
--enable-cxx
--enable-cxx, in which case a
C++ compiler will be required. As a convenience --enable-cxx=detect
can be used to enable C++ support only if a compiler can be found. The C++
support consists of a library libgmpxx.la and header file
gmpxx.h.
A separate libgmpxx.la has been adopted rather than having C++ objects
within libgmp.la in order to ensure dynamic linked C programs aren't
bloated by a dependency on the C++ standard library, and to avoid any chance
that the C++ compiler could be required when linking plain C programs.
libgmpxx.la will use certain internals from libgmp.la and can
only be expected to work with libgmp.la from the same GMP version.
Future changes to the relevant internals will be accompanied by renaming, so a
mismatch will cause unresolved symbols rather than perhaps mysterious
misbehaviour.
In general libgmpxx.la will be usable only with the C++ compiler that
built it, since name mangling and runtime support are usually incompatible
between different compilers.
CXX, CXXFLAGS
CXX and CXXFLAGS in the usual way. The default for
CXX is the first compiler that works from a list of likely candidates,
with g++ normally preferred when available. The default for
CXXFLAGS is to try CFLAGS, CFLAGS without -g, then
for g++ either -g -O2 or -O2, or for other compilers
-g or nothing. Trying CFLAGS this way is convenient when using
gcc and g++ together, since the flags for gcc will
usually suit g++.
It's important that the C and C++ compilers match, meaning their startup and
runtime support routines are compatible and that they generate code in the
same ABI (if there's a choice of ABIs on the system). ./configure
isn't currently able to check these things very well itself, so for that
reason --disable-cxx is the default, to avoid a build failure due to a
compiler mismatch. Perhaps this will change in the future.
Incidentally, it's normally not good enough to set CXX to the same as
CC. Although gcc for instance recognises foo.cc as
C++ code, only g++ will invoke the linker the right way when
building an executable or shared library from object files.
--enable-alloca=<choice>
GMP allocates temporary workspace using one of the following three methods,
which can be selected with for instance
--enable-alloca=malloc-reentrant.
alloca - C library or compiler builtin.
malloc-reentrant - the heap, in a re-entrant fashion.
malloc-notreentrant - the heap, with global variables.
For convenience, the following choices are also available.
--disable-alloca is the same as --enable-alloca=no.
yes - a synonym for alloca.
no - a synonym for malloc-reentrant.
reentrant - alloca if available, otherwise
malloc-reentrant. This is the default.
notreentrant - alloca if available, otherwise
malloc-notreentrant.
alloca is reentrant and fast, and is recommended, but when working with
large numbers it can overflow the available stack space, in which case one of
the two malloc methods will need to be used. Alternately it might be possible
to increase available stack with limit, ulimit or
setrlimit, or under DJGPP with stubedit or
_stklen. Note that depending on the system the only indication of
stack overflow might be a segmentation violation.
malloc-reentrant is, as the name suggests, reentrant and thread safe,
but malloc-notreentrant is faster and should be used if reentrancy is
not required.
The two malloc methods in fact use the memory allocation functions selected by
mp_set_memory_functions, these being malloc and friends by
default. See Custom Allocation.
An additional choice --enable-alloca=debug is available, to help when
debugging memory related problems (see Debugging).
--disable-fft
--enable-mpbsd
libmp) and header file
(mp.h) are built and installed only if --enable-mpbsd is used.
See BSD Compatible Functions.
--enable-mpfr
The optional MPFR functions are built and installed only if
--enable-mpfr is used. These are in a separate library
libmpfr.a and are documented separately too (see Introduction to MPFR).
--enable-assert
--enable-profiling=prof/gprof
prof or gprof.
This adds -p or -pg respectively to CFLAGS, and for some
systems adds corresponding mcount calls to the assembler code.
See Profiling.
MPN_PATH
sparcv8 has
MPN_PATH="sparc32/v8 sparc32 generic"
which means look first for v8 code, then plain sparc32 (which is v7), and
finally fall back on generic C. Knowledgeable users with special requirements
can specify a different path. Normally this is completely unnecessary.
The demos subdirectory has some sample programs using GMP. These
aren't built or installed, but there's a Makefile with rules for them.
For instance,
make pexpr ./pexpr 68^975+10
gmp.texi. The usual automake
targets are available to make PostScript gmp.ps and/or DVI
gmp.dvi.
HTML can be produced with makeinfo --html, see Generating HTML. Or alternately
texi2html, see Texinfo to HTML.
PDF can be produced with texi2dvi --pdf (see PDF) or with pdftex.
Some supplementary notes can be found in the doc subdirectory.
ABI (Application Binary Interface) refers to the calling conventions between functions, meaning what registers are used and what sizes the various C data types are. ISA (Instruction Set Architecture) refers to the instructions and registers a CPU has available.
Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
latter for compatibility with older CPUs in the family. GMP supports some
CPUs like this in both ABIs. In fact within GMP ABI means a
combination of chip ABI, plus how GMP chooses to use it. For example in some
32-bit ABIs, GMP may support a limb as either a 32-bit long or a 64-bit
long long.
By default GMP chooses the best ABI available for a given system, and this
generally gives significantly greater speed. But an ABI can be chosen
explicitly to make GMP compatible with other libraries, or particular
application requirements. For example,
./configure ABI=32
In all cases it's vital that all object code used in a given program is compiled for the same ABI.
Usually a limb is implemented as a long. When a long long limb
is used this is encoded in the generated gmp.h. This is convenient for
applications, but it does mean that gmp.h will vary, and can't be just
copied around. gmp.h remains compiler independent though, since all
compilers for a particular ABI will be expected to use the same limb type.
Currently no attempt is made to follow whatever conventions a system has for
installing library or header files built for a particular ABI. This will
probably only matter when installing multiple builds of GMP, and it might be
as simple as configuring with a special libdir, or it might require
more than that. Note that builds for different ABIs need to done separately,
with a fresh ./configure and make each.
hppa2.0*)
ABI=2.0w
cc. gcc support for this is in progress.
Applications must be compiled with
cc +DD64
ABI=2.0n
long long. This is available on HP-UX 10 or up when using
cc. No gcc support is planned for this. Applications
must be compiled with
cc +DA2.0 +e
ABI=1.0
All three ABIs are available for CPUs hppa2.0w and hppa2.0, but
for CPU hppa2.0n only 2.0n or 1.0 are allowed.
mips*-*-irix[6789])
gcc
is required (2.95 for instance).
ABI=n32
long long. Applications must be compiled with
gcc -mabi=n32 cc -n32
ABI=64
gcc -mabi=64 cc -64
Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have the necessary
support for n32 or 64 and so only gets a 32-bit limb and the MIPS 2 code.
powerpc64*)
ABI=aix64
powerpc64*-*-aix*. Applications must be compiled (and linked) with
gcc -maix64 xlc -q64
ABI=32L
long long in
64-bit registers. Applications must be compiled with
gcc -mpowerpc64
ABI=32
sparcv9 and ultrasparc*)
ABI=64
cc is required. Applications must be compiled with
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9 cc -xarch=v9
ABI=32
gcc -mv8plus cc -xarch=v8plus
gcc 2.8 and earlier only supports -mv8 though.
Don't be confused by the names of these sparc -m and -x options,
they're called arch but they effectively control the ABI.
On Solaris 2.7 with the kernel in 32-bit-mode, a normal native build will
reject ABI=64 because the resulting executables won't run.
ABI=64 can still be built if desired by making it look like a
cross-compile, for example
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
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. The GMP 4 series
will be upwardly binary compatible in each release and will be upwardly binary
compatible with all of the GMP 3 series. Additional function interfaces may
be added in each release, so on systems where libtool versioning is not fully
checked by the loader an auxiliary mechanism may be needed to express that a
dynamic linked application depends on a new enough GMP.
An auxiliary mechanism may also be needed to express that libgmpxx.la
(from --enable-cxx, see 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.
Using DESTDIR or a prefix override with make install and
a shared libgmpxx may run into a libtool relinking problem, see
Known Build Problems.
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 necessitate
i386 for x86s, or plain sparc (meaning V7) for SPARCs.
Users who care about speed will want GMP built for their exact CPU type, to
make 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.
Note that gmp.h is a generated file, and will be architecture and ABI
dependent.
*-*-aix[34]* shared libraries are disabled by default, since
some versions of the native ar fail on the convenience libraries
used. A shared build can be attempted with
./configure --enable-shared --disable-static
Note that the --disable-static is necessary because in a shared build
libtool makes libgmp.a a symlink to libgmp.so, apparently for
the benefit of old versions of ld which only recognise .a,
but unfortunately this is done even if a fully functional ld is
available.
arm*-*-*, 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.
*-*-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. --enable-cxx cannot be used when
building a DLL, since libtool doesn't currently support C++ DLLs. This might
change in the future.
GCC is recommended for compiling GMP, but the resulting DLL can be used with
any compiler. On mingw only the standard Windows libraries will be needed, on
Cygwin the usual cygwin runtime will be required.
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.
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* 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 assembler 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.
sparcv8 or supersparc on relevant systems will give a
significant performance increase over the V7 code.
/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).
i386 selects generic code which will run reasonably well on all x86
chips.
i586, pentium or pentiummmx code is good for the intended
P5 Pentium chips, but quite slow when run on Intel P6 class chips (PPro, P-II,
P-III). i386 is a better choice when making binaries that must run on
both.
pentium4 and an SSE2 capable assembler are important for best results
on Pentium 4. The specific code is for instance roughly a 2x to
3x speedup over the generic i386 code.
Old versions of gas don't support MMX instructions, in particular
version 1.92.3 that comes with FreeBSD 2.2.8 doesn't (and unfortunately
there's no newer assembler for that system).
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.
-march=pentiumpro
mpz/powm.c when
-march=pentiumpro was used, so for relevant CPUs that option is only in
the default CFLAGS for GCC 2.95.4 and up.
You might find more up-to-date information at http://swox.com/gmp/.
bash 2.03 is unable to run the configure
script, it exits silently, having died writing a preamble to
config.log. Use bash 2.04 or higher.
make all was found to run out of memory during the final
libgmp.la link on one system tested, despite having 64Mb available. A
separate make libgmp.la helped, perhaps recursing into the various
subdirectories uses up memory.
DESTDIR and shared libgmpxx
make install DESTDIR=/my/staging/area or the same with a prefix
override to install to a temporary directory is not fully supported by current
versions of libtool when building a shared version of a library which depends
on another being built at the same time, like libgmpxx and
libgmp.
The problem is that libgmpxx is relinked at the install stage to ensure
that if the system puts a hard-coded path to libgmp within
libgmpxx then that path will be correct. Naturally the linker is
directed to look only at the final location, not the staging area, so if
libgmp is not already in that final location then the link will fail.
On systems which don't hard-code library paths, for instance SVR4 style
systems such as GNU/Linux, a workaround is to insert a suitable -L in
the relink_command of libgmpxx.la after building but before
installing. This can be automated with something like
sed '/^relink_command/s:libgmp.la:-L /my/staging/area libgmp.la:' \
<libgmpxx.la >libgmpxx.new
mv libgmpxx.new libgmpxx.la
strip
GNU binutils strip should not be used on the static libraries
libgmp.a and libmp.a, neither directly nor via make
install-strip. It can be used on the shared libraries libgmp.so and
libmp.so though.
Currently (binutils 2.10.0), strip unpacks an archive then operates
on the files, but GMP contains multiple object files of the same name
(eg. three versions of init.o), and they overwrite each other, leaving
only the one that happens to be last.
If stripped static libraries are wanted, the suggested workaround is to build
normally, strip the separate object files, and do another make all to
rebuild. Alternately CFLAGS with -g omitted can always be used
if it's just debugging which is unwanted.
make syntax error
make
is unable to handle the long dependencies list for libgmp.la. The
symptom is a "syntax error" on the following line of the top-level
Makefile.
libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
Either use GNU Make, or as a workaround remove
$(libgmp_la_DEPENDENCIES) from that line (which will make the initial
build work, but if any recompiling is done libgmp.la might not be
rebuilt).
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.)
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.
gmp_randinit_lc_2exp_size. The exact cause is unknown,
--disable-shared is recommended.
libgmp, libtool creates wrapper scripts
like t-mul for programs that would normally be t-mul.exe, in
order to setup the right library paths etc. This works fine, but the absence
of t-mul.exe etc causes make to think they need recompiling
every time, which is an annoyance when re-running a make check.
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.
All declarations needed to use GMP are collected in the include file
gmp.h. It is designed to work with both C and C++ compilers.
#include <gmp.h>
Note however that prototypes for GMP functions with FILE * parameters
are only provided if <stdio.h> is included too.
#include <stdio.h> #include <gmp.h>
Likewise <stdarg.h> (or <varargs.h>) is required for prototypes
with va_list parameters, such as gmp_vprintf. And
<obstack.h> for prototypes with struct obstack parameters, such
as gmp_obstack_printf, when available.
All programs using GMP must link against the libgmp library. On a
typical Unix-like system this can be done with -lgmp, for example
gcc myprogram.c -lgmp
GMP C++ functions are in a separate libgmpxx library. This is built
and installed if C++ support has been enabled (see Build Options). For
example,
g++ mycxxprog.cc -lgmpxx -lgmp
GMP is built using Libtool and an application can use that to link if desired, see Shared library support for GNU
If GMP has been installed to a non-standard location then it may be necessary
to use -I and -L compiler options to point to the right
directories, and some sort of run-time path for a shared library. Consult
your compiler documentation, for instance Introduction.
In this manual, integer usually means a multiple precision integer, as
defined by the GMP library. The C data type for such integers is mpz_t.
Here are some examples of how to declare such integers:
mpz_t sum;
struct foo { mpz_t x, y; };
mpz_t vec[20];
Rational number means a multiple precision fraction. The C data type
for these fractions is mpq_t. For example:
mpq_t quotient;
Floating point number or Float for short, is an arbitrary precision
mantissa with a limited precision exponent. The C data type for such objects
is mpf_t.
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.
There are six classes of functions in the GMP library:
mpz_. The associated type is mpz_t. There are about 150
functions in this class.
mpq_. The associated type is mpq_t. There are about 40
functions in this class, but the integer functions can be used for arithmetic
on the numerator and denominator separately.
mpf_. The associated type is mpf_t. There are about 60
functions is this class.
itom, madd, and
mult. The associated type is MINT.
mpn_. The associated type is array of mp_limb_t. There are
about 30 (hard-to-use) functions in this class.
GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last.
GMP lets you use the same variable for both input and output in one call. For
example, the main function for integer multiplication, mpz_mul, can be
used to square x and put the result back in x with
mpz_mul (x, x, x);
Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details.
A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times.
For efficiency reasons, avoid excessive initializing and clearing. In
general, initialize near the start of a function and clear near the end. For
example,
void
foo (void)
{
mpz_t n;
int i;
mpz_init (n);
for (i = 1; i < 100; i++)
{
mpz_mul (n, ...);
mpz_fdiv_q (n, ...);
...
}
mpz_clear (n);
}
When a GMP variable is used as a function parameter, it's effectively a
call-by-reference, meaning if the function stores a value there it will change
the original in the caller. Parameters which are input-only can be designated
const to provoke a compiler error or warning on attempting to modify
them.
When a function is going to return a GMP result, it should designate a
parameter that it sets, like the library functions do. More than one value
can be returned by having more than one output parameter, again like the
library functions. A return of an mpz_t etc doesn't return the
object, only a pointer, and this is almost certainly not what's wanted.
Here's an example accepting an mpz_t parameter, doing a calculation,
and storing the result to the indicated parameter.
void
foo (mpz_t result, const mpz_t param, unsigned long n)
{
unsigned long i;
mpz_mul_ui (result, param, n);
for (i = 1; i < n; i++)
mpz_add_ui (result, result, i*7);
}
int
main (void)
{
mpz_t r, n;
mpz_init (r);
mpz_init_set_str (n, "123456", 0);
foo (r, n, 20L);
gmp_printf ("%Zd\n", r);
return 0;
}
foo works even if the mainline passes the same variable for
param and result, just like the library functions. But
sometimes it's tricky to make that work, and an application might not want to
bother supporting that sort of thing.
For interest, the GMP types mpz_t etc are implemented as one-element
arrays of certain structures. This is why declaring a variable creates an
object with the fields GMP needs, but then using it as a parameter passes a
pointer to the object. Note that the actual fields in each mpz_t etc
are for internal use only and should not be accessed directly by code that
expects to be compatible with future GMP releases.
The GMP types like mpz_t are small, containing only a couple of sizes,
and pointers to allocated data. Once a variable is initialized, GMP takes
care of all space allocation. Additional space is allocated whenever a
variable doesn't have enough.
mpz_t and mpq_t variables never reduce their allocated space.
Normally this is the best policy, since it avoids frequent reallocation.
Applications that need to return memory to the heap at some particular point
can use mpz_realloc2, or clear variables no longer needed.
mpf_t variables, in the current implementation, use a fixed amount of
space, determined by the chosen precision and allocated at initialization, so
their size doesn't change.
All memory is allocated using malloc and friends by default, but this
can be changed, see Custom Allocation. Temporary memory on the stack is
also used (via alloca), but this can be changed at build-time if
desired, see Build Options.
GMP is reentrant and thread-safe, with some exceptions:
--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.
mpz_random and the other old random number functions use a global
random state and are hence not reentrant. The newer random number functions
that accept a gmp_randstate_t parameter can be used instead.
mp_set_memory_functions uses global variables to store the selected
memory allocation functions.
mp_set_memory_functions (or malloc and friends by default) are
not reentrant, then GMP will not be reentrant either.
fwrite are not reentrant then the
GMP I/O functions using them will not be reentrant either.
gmp_randstate_t simultaneously,
since this involves an update of that variable.
<ctype.h> macros use per-file static
variables and may not be reentrant, depending whether the compiler optimizes
away fetches from them. The GMP text-based input functions are affected.
| const int mp_bits_per_limb | Global Constant |
| The number of bits per limb. |
| __GNU_MP_VERSION | Macro |
| __GNU_MP_VERSION_MINOR | Macro |
| __GNU_MP_VERSION_PATCHLEVEL | Macro |
| The major and minor GMP version, and patch level, respectively, as integers. For GMP i.j, these numbers will be i, j, and 0, respectively. For GMP i.j.k, these numbers will be i, j, and k, respectively. |
| const char * const gmp_version | Global Constant |
The GMP version number, as a null-terminated string, in the form "i.j" or
"i.j.k". This release is "4.1".
|
This version of GMP is upwardly binary compatible with all 4.x and 3.x versions, and upwardly compatible at the source level with all 2.x versions, with the following exceptions.
mpn_gcd had its source arguments swapped as of GMP 3.0, for consistency
with other mpn functions.
mpf_get_prec counted precision slightly differently in GMP 3.0 and
3.0.1, but in 3.1 reverted to the 2.x style.
There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 4. Please see the GMP 2 manual for details.
The Berkeley MP compatibility library (see BSD Compatible Functions) is
source and binary compatible with the standard libmp.
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.
A language interpreter might want to keep a free list or stack of
initialized variables ready for use. It should be possible to integrate
something like that with a garbage collector too.
mpz_t or mpq_t variable used to hold successively increasing
values will have its memory repeatedly realloced, which could be quite
slow or could fragment memory, depending on the C library. If an application
can estimate the final size then mpz_init2 or mpz_realloc2 can
be called to allocate the necessary space from the beginning
(see Initializing Integers).
It doesn't matter if a size set with mpz_init2 or mpz_realloc2
is too small, since all functions will do a further reallocation if necessary.
Badly overestimating memory required will waste space though.
2exp functions
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
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.
mpz_abs, mpq_abs, mpf_abs, mpz_neg, mpq_neg
and mpf_neg are fast when used for in-place operations like
mpz_abs(x,x), since in the current implementation only a single field
of x needs changing. On suitable compilers (GCC for instance) this is
inlined too.
mpz_add_ui, mpz_sub_ui, mpf_add_ui and mpf_sub_ui
benefit from an in-place operation like mpz_add_ui(x,x,y), since
usually only one or two limbs of x will need to be changed. The same
applies to the full precision mpz_add etc if y is small. If
y is big then cache locality may be helped, but that's all.
mpz_mul is currently the opposite, a separate destination is slightly
better. A call like mpz_mul(x,x,y) will, unless y is only one
limb, make a temporary copy of x before forming the result. Normally
that copying will only be a tiny fraction of the time for the multiply, so
this is not a particularly important consideration.
mpz_set, mpq_set, mpq_set_num, mpf_set, etc, make
no attempt to recognise a copy of something to itself, so a call like
mpz_set(x,x) will be wasteful. Naturally that would never be written
deliberately, but if it might arise from two pointers to the same object then
a test to avoid it might be desirable.
if (x != y) mpz_set (x, y);
Note that it's never worth introducing extra mpz_set calls just to get
in-place operations. If a result should go to a particular variable then just
direct it there and let GMP take care of data movement.
mpz_divisible_ui_p and mpz_congruent_ui_p are the best functions
for testing whether an mpz_t is divisible by an individual small
integer. They use an algorithm which is faster than mpz_tdiv_ui, but
which gives no useful information about the actual remainder, only whether
it's zero (or a particular value).
However when testing divisibility by several small integers, it's best to take a remainder modulo their product, to save multi-precision operations. For instance to test whether a number is divisible by any of 23, 29 or 31 take a remainder modulo 23*29*31 = 20677 and then test that.
The division functions like mpz_tdiv_q_ui which give a quotient as well
as a remainder are generally a little slower than the remainder-only functions
like mpz_tdiv_ui. If the quotient is only rarely wanted then it's
probably best to just take a remainder and then go back and calculate the
quotient if and when it's wanted (mpz_divexact_ui can be used if the
remainder is zero).
mpq functions operate on mpq_t values with no common factors
in the numerator and denominator. Common factors are checked-for and cast out
as necessary. In general, cancelling factors every time is the best approach
since it minimizes the sizes for subsequent operations.
However, applications that know something about the factorization of the values they're working with might be able to avoid some of the GCDs used for canonicalization, or swap them for divisions. For example when multiplying by a prime it's enough to check for factors of it in the denominator instead of doing a full GCD. Or when forming a big product it might be known that very little cancellation will be possible, and so canonicalization can be left to the end.
The mpq_numref and mpq_denref macros give access to the
numerator and denominator to do things outside the scope of the supplied
mpq functions. See Applying Integer Functions.
The canonical form for rationals allows mixed-type mpq_t and integer
additions or subtractions to be done directly with multiples of the
denominator. This will be somewhat faster than mpq_add. For example,
/* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z);
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.
Maybe we can hope octal will one day become the normal base for everyday use, as proposed by King Charles XII of Sweden and later reformers.
--enable-alloca choices in
Build Options, for how to address this.
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
Allocation Debugging, or
man 3 malloc. Other possibilities, in no particular order, include
http://www.inf.ethz.ch/personal/biere/projects/ccmalloc http://quorum.tamu.edu/jon/gnu (debauch) http://dmalloc.com http://www.perens.com/FreeSoftware (electric fence) http://packages.debian.org/fda http://www.gnupdate.org/components/leakbug http://people.redhat.com/~otaylor/memprof http://www.cbmamiga.demon.co.uk/mpatrol
The GMP default allocation routines in memory.c also have a simple
sentinel scheme which can be enabled with #define DEBUG in that file.
This is mainly designed for detecting buffer overruns during GMP development,
but might find other uses.
-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.
.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.
mpz, mpq, mpf and mpfr 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-4.1/configure
This works via VPATH, and might require GNU make.
Alternately it might be possible to change the .c.lo rules
appropriately.
--enable-assert is available to add some consistency
checks to the library (see Build Options). These are likely to be of
limited value to most applications. Assertion failures are just as likely to
indicate memory corruption as a library or compiler bug.
Applications using the low-level mpn functions, however, will benefit
from --enable-assert since it adds checks on the parameters of most
such functions, many of which have subtle restrictions on their usage. Note
however that only the generic C code has checks, not the assembler code, so
CPU none should be used for maximum checking.
--enable-alloca=debug arranges that each block of
temporary memory in GMP is allocated with a separate call to malloc (or
the allocation function set with mp_set_memory_functions).
This can help a malloc debugger detect accesses outside the intended bounds,
or detect memory not released. In a normal build, on the other hand,
temporary memory is allocated in blocks which GMP divides up for its own use,
or may be allocated with a compiler builtin alloca which will go
nowhere near any malloc debugger hooks.
A build of GMP with checking within GMP itself can be made. This will run
very very slowly. Configure with
./configure --host=none-pc-linux-gnu CC=checkergcc
--host=none must be used, since the GMP assembler 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.
Current versions (20020226 snapshot) don't support MMX or SSE, so GMP must be
configured for an x86 without those (eg. plain i386), or with a special
MPN_PATH that excludes those subdirectories (see Build Options).
Running a program under a profiler is a good way to find where it's spending most time and where improvements can be best sought.
Depending on the system, it may be possible to get a flat profile, meaning
simple timer sampling of the program counter, with no special GMP build
options, just a -p when compiling the mainline. This is a good way to
ensure minimum interference with normal operation. The necessary symbol type
and size information exists in most of the GMP assembler code.
The --enable-profiling build option can be used to add suitable
compiler flags, either for prof (-p) or gprof
(-pg), see Build Options. Which of the two is available and what
they do will depend on the system, and possibly on support available in
libc. For some systems appropriate corresponding mcount calls
are added to the assembler code too.
On x86 systems prof gives call counting, so that average time spent
in a function can be determined. gprof, where supported, adds call
graph construction, so for instance calls to mpn_add_n from
mpz_add and from mpz_mul can be differentiated.
On x86 and 68k systems -pg and -fomit-frame-pointer are
incompatible, so the latter is not used when gprof profiling is
selected, which may result in poorer code generation. If prof
profiling is selected instead it should still be possible to use
gprof, but only the gprof -p flat profile and call counts can
be expected to be valid, not the gprof -q call graph.
Autoconf based applications can easily check whether GMP is installed. The
only thing to be noted is that GMP library symbols from version 3 onwards have
prefixes like __gmpz. The following therefore would be a simple test,
AC_CHECK_LIB(gmp, __gmpz_init)
This just uses the default AC_CHECK_LIB actions for found or not found,
but an application that must have GMP would want to generate an error if not
found. For example,
AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR( [GNU MP not found, see http://swox.com/gmp])])
If functions added in some particular version of GMP are required, then one of
those can be used when checking. For example mpz_mul_si was added in
GMP 3.1,
AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR( [GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])
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.
An application that can use either GMP 2 or 3 will need to test for
__gmpz_init (GMP 3 and up) or mpz_init (GMP 2), and it's also
worth checking for libgmp2 since Debian GNU/Linux systems used that
name in the past. For example,
AC_CHECK_LIB(gmp, __gmpz_init, ,
[AC_CHECK_LIB(gmp, mpz_init, ,
[AC_CHECK_LIB(gmp2, mpz_init)])])
In general it's suggested that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions.
Occasionally an application will need or want to know the size of a type at
configuration or preprocessing time, not just with sizeof in the code.
This can be done in the normal way with mp_limb_t etc, but GMP 4.0 or
up is best for this, since prior versions needed certain -D defines on
systems using a long long limb. The following would suit Autoconf 2.50
or up,
AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
The optional mpfr functions are provided in a separate
libmpfr.a, and this might be from GMP with --enable-mpfr or
from MPFR installed separately. Either way libmpfr depends on
libgmp, it doesn't stand alone. Currently only a static
libmpfr.a will be available, not a shared library, since upward binary
compatibility is not guaranteed.
AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR( [Need MPFR either from GNU MP 4 or separate MPFR package. See http://www.mpfr.org or http://swox.com/gmp])
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 Known Build Problems, or perhaps Notes for Particular Systems. You may also want to check http://swox.com/gmp/ for patches for this release.
Please include the following in any report,
where in gdb, or $C in adb).
straces.
gcc, get the version
with gcc -v, otherwise perhaps what `which cc`, or similar.
uname -a.
./config.guess, and from running
./configfsf.guess (might be the same).
configure, then the contents of
config.log.
asm file not assembling, then the contents
of config.m4 and the offending line or lines from the temporary
mpn/tmp-<file>.s.
Please make an effort to produce a self-contained report, with something definite that can be tested or debugged. Vague queries or piecemeal messages are difficult to act on and don't help the development effort.
It is not uncommon that an observed problem is actually due to a bug in the compiler; the GMP code tends to explore interesting corners in compilers.
If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won't do anything about it (except maybe ask you to send a better report).
Send your report to: bug-gmp@gnu.org.
If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.
This chapter describes the GMP functions for performing integer arithmetic.
These functions start with the prefix mpz_.
GMP integers are stored in objects of type mpz_t.
The functions for integer arithmetic assume that all integer objects are
initialized. You do that by calling the function mpz_init. For
example,
{
mpz_t integ;
mpz_init (integ);
...
mpz_add (integ, ...);
...
mpz_sub (integ, ...);
/* Unless the program is about to exit, do ... */
mpz_clear (integ);
}
As you can see, you can store new values any number of times, once an object is initialized.
| void mpz_init (mpz_t integer) | Function |
| Initialize integer, and set its value to 0. |
| void mpz_init2 (mpz_t integer, unsigned long n) | Function |
|
Initialize integer, with space for n bits, and set its value to 0.
n is only the initial space, integer will grow automatically in
the normal way, if necessary, for subsequent values stored. |
| void mpz_clear (mpz_t integer) | Function |
Free the space occupied by integer. Call this function for all
mpz_t variables when you are done with them.
|
| void mpz_realloc2 (mpz_t integer, unsigned long n) | Function |
|
Change the space allocated for integer to n bits. The value in
integer is preserved if it fits, or is set to 0 if not.
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. |
| void mpz_array_init (mpz_t integer_array[], size_t array_size, mp_size_t fixed_num_bits) | Function |
|
This is a special type of initialization. Fixed space of
fixed_num_bits bits is allocated to each of the array_size
integers in integer_array.
The space will not be automatically increased, unlike the normal
For other functions, or if in doubt, the suggestion is to calculate in a
regular
|
| void * _mpz_realloc (mpz_t integer, mp_size_t new_alloc) | Function |
|
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.
|
These functions assign new values to already initialized integers (see Initializing Integers).
| void mpz_set (mpz_t rop, mpz_t op) | Function |
| void mpz_set_ui (mpz_t rop, unsigned long int op) | Function |
| void mpz_set_si (mpz_t rop, signed long int op) | Function |
| void mpz_set_d (mpz_t rop, double op) | Function |
| void mpz_set_q (mpz_t rop, mpq_t op) | Function |
| void mpz_set_f (mpz_t rop, mpf_t op) | Function |
|
Set the value of rop from op.
|
| int mpz_set_str (mpz_t rop, char *str, int base) | Function |
|
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 36. If base is 0, the actual base is determined
from the leading characters: if the first two characters are "0x" or "0X",
hexadecimal is assumed, otherwise if the first character is "0", octal is
assumed, otherwise decimal is assumed.
This function returns 0 if the entire string is a valid number in base base. Otherwise it returns -1. [It turns out that it is not entirely true that this function ignores
white-space. It does ignore it between digits, but not after a minus sign or
within or after "0x". 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. Send your opinion of this change to
bug-gmp@gnu.org. Do you really want it to accept |
| void mpz_swap (mpz_t rop1, mpz_t rop2) | Function |
| Swap the values rop1 and rop2 efficiently. |
For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there. These functions'
names have the form mpz_init_set...
Here is an example of using one:
{
mpz_t pie;
mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
...
mpz_sub (pie, ...);
...
mpz_clear (pie);
}
Once the integer has been initialized by any of the mpz_init_set...
functions, it can be used as the source or destination operand for the ordinary
integer functions. Don't use an initialize-and-set function on a variable
already initialized!
| void mpz_init_set (mpz_t rop, mpz_t op) | Function |
| void mpz_init_set_ui (mpz_t rop, unsigned long int op) | Function |
| void mpz_init_set_si (mpz_t rop, signed long int op) | Function |
| void mpz_init_set_d (mpz_t rop, double op) | Function |
| Initialize rop with limb space and set the initial numeric value from op. |
| int mpz_init_set_str (mpz_t rop, char *str, int base) | Function |
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 |
This section describes functions for converting GMP integers to standard C types. Functions for converting to GMP integers are described in Assigning Integers and I/O of Integers.
| unsigned long int mpz_get_ui (mpz_t op) | Function |
Return the value of op as an unsigned long.
If op is too big to fit an |
| signed long int mpz_get_si (mpz_t op) | Function |
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 |
| double mpz_get_d (mpz_t op) | Function |
Convert op to a double.
|
| double mpz_get_d_2exp (signed long int *exp, mpz_t op) | Function |
| Find d and exp such that d times 2 raised to exp, with 0.5<=abs(d)<1, is a good approximation to op. |
| char * mpz_get_str (char *str, int base, mpz_t op) | Function |
|
Convert op to a string of digits in base base. The base may vary
from 2 to 36.
If str is If str is not A pointer to the result string is returned, being either the allocated block, or the given str. |
| mp_limb_t mpz_getlimbn (mpz_t op, mp_size_t n) | Function |
|
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.
|
| 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 |
| Set rop to op1 + op2. |
| 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 |
| Set rop to op1 - op2. |
| 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 |
| Set rop to op1 times op2. |
| 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 |
| Set rop to rop + op1 times op2. |
| 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 |
| Set rop to rop - op1 times op2. |
| void mpz_mul_2exp (mpz_t rop, mpz_t op1, unsigned long int op2) | Function |
| Set rop to op1 times 2 raised to op2. This operation can also be defined as a left shift by op2 bits. |
| void mpz_neg (mpz_t rop, mpz_t op) | Function |
| Set rop to -op. |
| void mpz_abs (mpz_t rop, mpz_t op) | Function |
| Set rop to the absolute value of op. |
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.
| void mpz_cdiv_q (mpz_t q, mpz_t n, mpz_t d) | Function |
| void mpz_cdiv_r (mpz_t r, mpz_t n, mpz_t d) | Function |
| void mpz_cdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d) | Function |
| unsigned long int mpz_cdiv_q_ui (mpz_t q, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_cdiv_r_ui (mpz_t r, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_cdiv_qr_ui (mpz_t q, mpz_t r, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_cdiv_ui (mpz_t n, unsigned long int d) | Function |
| void mpz_cdiv_q_2exp (mpz_t q, mpz_t n, unsigned long int b) | Function |
| void mpz_cdiv_r_2exp (mpz_t r, mpz_t n, unsigned long int b) | Function |
| void mpz_fdiv_q (mpz_t q, mpz_t n, mpz_t d) | Function |
| void mpz_fdiv_r (mpz_t r, mpz_t n, mpz_t d) | Function |
| void mpz_fdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d) | Function |
| unsigned long int mpz_fdiv_q_ui (mpz_t q, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_fdiv_r_ui (mpz_t r, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_fdiv_qr_ui (mpz_t q, mpz_t r, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_fdiv_ui (mpz_t n, unsigned long int d) | Function |
| void mpz_fdiv_q_2exp (mpz_t q, mpz_t n, unsigned long int b) | Function |
| void mpz_fdiv_r_2exp (mpz_t r, mpz_t n, unsigned long int b) | Function |
| void mpz_tdiv_q (mpz_t q, mpz_t n, mpz_t d) | Function |
| void mpz_tdiv_r (mpz_t r, mpz_t n, mpz_t d) | Function |
| void mpz_tdiv_qr (mpz_t q, mpz_t r, mpz_t n, mpz_t d) | Function |
| unsigned long int mpz_tdiv_q_ui (mpz_t q, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_tdiv_r_ui (mpz_t r, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_tdiv_qr_ui (mpz_t q, mpz_t r, mpz_t n, unsigned long int d) | Function |
| unsigned long int mpz_tdiv_ui (mpz_t n, unsigned long int d) | Function |
| void mpz_tdiv_q_2exp (mpz_t q, mpz_t n, unsigned long int b) | Function |
| void mpz_tdiv_r_2exp (mpz_t r, mpz_t n, unsigned long int b) | Function |
|
Divide n by d, forming a quotient q and/or remainder
r. For the
In all cases q and r will satisfy n=q*d+r, and r will satisfy 0<=abs(r)<abs(d). The For the The |
| void mpz_mod (mpz_t r, mpz_t n, mpz_t d) | Function |
| unsigned long int mpz_mod_ui (mpz_t r, mpz_t n, unsigned long int d) | Function |
Set r to n mod d. The sign of the divisor is
ignored; the result is always non-negative.
|
| 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) | Function |
|
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. |
| 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, unsigned long int b) | Function |
Return non-zero if n is exactly divisible by d, or in the case of
mpz_divisible_2exp_p by 2^b.
|
| 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, unsigned long int b) | Function |
Return non-zero if n is congruent to c modulo d, or in the
case of mpz_congruent_2exp_p modulo 2^b.
|
| 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 |
|
Set rop to (base raised to exp)
modulo mod.
Negative exp is supported if an inverse base^-1 mod
mod exists (see |
| void mpz_pow_ui (mpz_t rop, mpz_t base, unsigned long int exp) | Function |
| void mpz_ui_pow_ui (mpz_t rop, unsigned long int base, unsigned long int exp) | Function |
| Set rop to base raised to exp. The case 0^0 yields 1. |
| int mpz_root (mpz_t rop, mpz_t op, unsigned long int n) | Function |
| 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. |
| void mpz_sqrt (mpz_t rop, mpz_t op) | Function |
| Set rop to the truncated integer part of the square root of op. |
| void mpz_sqrtrem (mpz_t rop1, mpz_t rop2, mpz_t op) | Function |
Set rop1 to the truncated integer part
of the square root of op, like mpz_sqrt. Set rop2 to the
remainder op-rop1*rop1, which will be zero if op is a
perfect square.
If rop1 and rop2 are the same variable, the results are undefined. |
| int mpz_perfect_power_p (mpz_t op) | Function |
|
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. |
| int mpz_perfect_square_p (mpz_t op) | Function |
| 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. |
| int mpz_probab_prime_p (mpz_t n, int reps) | Function |
|
Determine whether n is prime. Return 2 if n is definitely prime,
return 1 if n is probably prime (without being certain), or return 0 if
n is definitely composite.
This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. reps controls how many such tests are done, 5 to 10 is a reasonable number, more will reduce the chances of a composite being returned as "probably prime". Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime. |
| void mpz_nextprime (mpz_t rop, mpz_t op) | Function |
|
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. |
| void mpz_gcd (mpz_t rop, mpz_t op1, mpz_t op2) | Function |
| 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. |
| unsigned long int mpz_gcd_ui (mpz_t rop, mpz_t op1, unsigned long int op2) | Function |
Compute the greatest common divisor of op1 and op2. If
rop is not NULL, store the result there.
If the result is small enough to fit in an |
| void mpz_gcdext (mpz_t g, mpz_t s, mpz_t t, mpz_t a, mpz_t b) | Function |
|
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.
g is always positive, even if one or both of a and b are
negative.
If t is |
| 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) | Function |
| 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. |
| int mpz_invert (mpz_t rop, mpz_t op1, mpz_t op2) | Function |
| 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. |
| int mpz_jacobi (mpz_t a, mpz_t b) | Function |
| Calculate the Jacobi symbol (a/b). This is defined only for b odd. |
| int mpz_legendre (mpz_t a, mpz_t p) | Function |
| 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. |
| 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 |
|
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 For more information see Henri Cohen section 1.4.2 (see References),
or any number theory textbook. See also the example program
|
| unsigned long int mpz_remove (mpz_t rop, mpz_t op, mpz_t f) | Function |
| Remove all occurrences of the factor f from op and store the result in rop. Return the multiplicity of f in op. |
| void mpz_fac_ui (mpz_t rop, unsigned long int op) | Function |
| Set rop to op!, the factorial of op. |
| void mpz_bin_ui (mpz_t rop, mpz_t n, unsigned long int k) | Function |
| void mpz_bin_uiui (mpz_t rop, unsigned long int n, unsigned long int k) | Function |
Compute the binomial coefficient n over
k and store the result in rop. Negative values of n are
supported by mpz_bin_ui, using the identity
bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1 section 1.2.6
part G.
|
| 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) | Function |
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 |
| 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) | Function |
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 The Fibonacci numbers and Lucas numbers are related sequences, so it's never
necessary to call both |
| int mpz_cmp (mpz_t op1, mpz_t op2) | Function |
| int mpz_cmp_d (mpz_t op1, double op2) | Function |
| int mpz_cmp_si (mpz_t op1, signed long int op2) | Macro |
| int mpz_cmp_ui (mpz_t op1, unsigned long int op2) | Macro |
|
Compare op1 and op2. Return a positive value if op1 >
op2, zero if op1 = op2, or a negative value if
op1 < op2.
Note that |
| 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 |
|
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).
Note that |
| int mpz_sgn (mpz_t op) | Macro |
|
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. |
These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0.
| void mpz_and (mpz_t rop, mpz_t op1, mpz_t op2) | Function |
| Set rop to op1 logical-and op2. |
| void mpz_ior (mpz_t rop, mpz_t op1, mpz_t op2) | Function |
| Set rop to op1 inclusive-or op2. |
| void mpz_xor (mpz_t rop, mpz_t op1, mpz_t op2) | Function |
| Set rop to op1 exclusive-or op2. |
| void mpz_com (mpz_t rop, mpz_t op) | Function |
| Set rop to the one's complement of op. |
| unsigned long int mpz_popcount (mpz_t op) | Function |
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 MAX_ULONG, the largest
possible unsigned long.
|
| unsigned long int mpz_hamdist (mpz_t op1, mpz_t op2) | Function |
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 MAX_ULONG, the largest
possible unsigned long.
|
| unsigned long int mpz_scan0 (mpz_t op, unsigned long int starting_bit) | Function |
| unsigned long int mpz_scan1 (mpz_t op, unsigned long int starting_bit) | Function |
|
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 MAX_ULONG is returned. This will happen
in |
| void mpz_setbit (mpz_t rop, unsigned long int bit_index) | Function |
| Set bit bit_index in rop. |
| void mpz_clrbit (mpz_t rop, unsigned long int bit_index) | Function |
| Clear bit bit_index in rop. |
| int mpz_tstbit (mpz_t op, unsigned long int bit_index) | Function |
| Test bit bit_index in op and return 0 or 1 accordingly. |
Functions that perform input from a stdio stream, and functions that output to
a stdio stream. Passing a NULL pointer for a stream argument to any of
these functions will make them read from stdin and write to
stdout, respectively.
When using any of these functions, it is a good idea to include stdio.h
before gmp.h, since that will allow gmp.h to define prototypes
for these functions.
| size_t mpz_out_str (FILE *stream, int base, mpz_t op) | Function |
|
Output op on stdio stream stream, as a string of digits in base
base. The base may vary from 2 to 36.
Return the number of bytes written, or if an error occurred, return 0. |
| size_t mpz_inp_str (mpz_t rop, FILE *stream, int base) | Function |
|
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 36. If base is 0, the actual base is determined from the
leading characters: if the first two characters are `0x' or `0X', hexadecimal
is assumed, otherwise if the first character is `0', octal is assumed,
otherwise decimal is assumed.
Return the number of bytes read, or if an error occurred, return 0. |
| size_t mpz_out_raw (FILE *stream, mpz_t op) | Function |
|
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 Return the number of bytes written, or if an error occurred, return 0. The output of this can not be read by |
| size_t mpz_inp_raw (mpz_t rop, FILE *stream) | Function |
Input from stdio stream stream in the format written by
mpz_out_raw, and put the result in rop. Return the number of
bytes read, or if an error occurred, return 0.
This routine can read the output from |
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 Random Number Functions for more information on how to use and not to use random number functions.
| void mpz_urandomb (mpz_t rop, gmp_randstate_t state, unsigned long int n) | Function |
|
Generate a uniformly distributed random integer in the range 0 to 2^n-1, inclusive.
The variable state must be initialized by calling one of the
|
| void mpz_urandomm (mpz_t rop, gmp_randstate_t state, mpz_t n) | Function |
|
Generate a uniform random integer in the range 0 to n-1,
inclusive.
The variable state must be initialized by calling one of the
|
| void mpz_rrandomb (mpz_t rop, gmp_randstate_t state, unsigned long int n) | Function |
|
Generate a random integer with long strings of zeros and ones in the
binary representation. Useful for testing functions and algorithms,
since this kind of random numbers have proven to be more likely to
trigger corner-case bugs. The random number will be in the range
0 to 2^n-1, inclusive.
The variable state must be initialized by calling one of the
|
| void mpz_random (mpz_t rop, mp_size_t max_size) | Function |
|
Generate a random integer of at most max_size limbs. The generated
random number doesn't satisfy any particular requirements of randomness.
Negative random numbers are generated when max_size is negative.
This function is obsolete. Use |
| void mpz_random2 (mpz_t rop, mp_size_t max_size) | Function |
|
Generate a random integer of at most max_size limbs, with long strings
of zeros and ones in the binary representation. Useful for testing functions
and algorithms, since this kind of random numbers have proven to be more
likely to trigger corner-case bugs. Negative random numbers are generated
when max_size is negative.
This function is obsolete. Use |
mpz_t variables can be converted to and from arbitrary words of binary
data with the following functions.
| void mpz_import (mpz_t rop, size_t count, int order, int size, int endian, size_t nails, const void *op) | Function |
|
Set rop from an array of word data at op.
The parameters specify the format of the data. count many words are read, each size bytes. order can be 1 for most significant word first or -1 for least significant first. Within each word endian can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant nails bits of each word are skipped, this can be 0 to use the full words. There are no data alignment restrictions on op, any address is allowed. Here's an example converting an array of unsigned long a[20]; mpz_t z; mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a); This example assumes the full |
| void *mpz_export (void *rop, size_t *count, int order, int size, int endian, size_t nails, mpz_t op) | Function |
|
Fill rop with word data from op.
The parameters specify the format of the data produced. Each word will be size bytes and order can be 1 for most significant word first or -1 for least significant first. Within each word endian can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant nails bits of each word are unused and set to zero, this can be 0 to produce full words. The number of words produced is written to 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 There are no data alignment restrictions on rop, any address is allowed. The sign of op is ignored, just the absolute value is used. When an application is allocating space itself the required size can be
determined with a calculation like the following. Since numb = 8*size - nail; count = (mpz_sizeinbase (z, 2) + numb-1) / numb; p = malloc (count * size); |
| 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) | Function |
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.
|
| int mpz_odd_p (mpz_t op) | Macro |
| int mpz_even_p (mpz_t op) | Macro |
| Determine whether op is odd or even, respectively. Return non-zero if yes, zero if no. These macros evaluate their argument more than once. |
| size_t mpz_size (mpz_t op) | Function |
| Return the size of op measured in number of limbs. If op is zero, the returned value will be zero. |
| size_t mpz_sizeinbase (mpz_t op, int base) | Function |
|
Return the size of op measured in number of digits in base base.
The base may vary from 2 to 36. The sign of op is ignored, just the
absolute value is used. The result will be exact or 1 too big. If base
is a power of 2, the result will always be exact. If op is zero the
return value is always 1.
This function is useful in order to allocate the right amount of space before
converting op to a string. The right amount of allocation is normally
two more than the value returned by |
This chapter describes the GMP functions for performing arithmetic on rational
numbers. These functions start with the prefix mpq_.
Rational numbers are stored in objects of type mpq_t.
All rational arithmetic functions assume operands have a canonical form, and canonicalize their result. The canonical from means that the denominator and the numerator have no common factors, and that the denominator is positive. Zero has the unique representation 0/1.
Pure assignment functions do not canonicalize the assigned variable. It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable.
| void mpq_canonicalize (mpq_t op) | Function |
| Remove any factors that are common to the numerator and denominator of op, and make the denominator positive. |
| void mpq_init (mpq_t dest_rational) | Function |
Initialize dest_rational 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.
|
| void mpq_clear (mpq_t rational_number) | Function |
Free the space occupied by rational_number. Make sure to call this
function for all mpq_t variables when you are done with them.
|
| void mpq_set (mpq_t rop, mpq_t op) | Function |
| void mpq_set_z (mpq_t rop, mpz_t op) | Function |
| Assign rop from op. |
| void mpq_set_ui (mpq_t rop, unsigned long int op1, unsigned long int op2) | Function |
| void mpq_set_si (mpq_t rop, signed long int op1, unsigned long int op2) | Function |
Set the value of rop to op1/op2. Note that if op1 and
op2 have common factors, rop has to be passed to
mpq_canonicalize before any operations are performed on rop.
|
| int mpq_set_str (mpq_t rop, char *str, int base) | Function |
|
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 (see Rational Number Functions), or if
not then The numerator and optional denominator are parsed the same as in
The return value is 0 if the entire string is a valid number, or -1 if not. |
| void mpq_swap (mpq_t rop1, mpq_t rop2) | Function |
| Swap the values rop1 and rop2 efficiently. |
| double mpq_get_d (mpq_t op) | Function |
Convert op to a double.
|
| void mpq_set_d (mpq_t rop, double op) | Function |
| void mpq_set_f (mpq_t rop, mpf_t op) | Function |
| Set rop to the value of op, without rounding. |
| char * mpq_get_str (char *str, int base, mpq_t op) | Function |
Convert op to a string of digits in base base. The base may vary
from 2 to 36. The string will be of the form num/den, or if the
denominator is 1 then just num.
If str is If str is not mpz_sizeinbase (mpq_numref(op), base) + mpz_sizeinbase (mpq_denref(op), base) + 3 The three extra bytes are for a possible minus sign, possible slash, and the null-terminator. A pointer to the result string is returned, being either the allocated block, or the given str. |
| void mpq_add (mpq_t sum, mpq_t addend1, mpq_t addend2) | Function |
| Set sum to addend1 + addend2. |
| void mpq_sub (mpq_t difference, mpq_t minuend, mpq_t subtrahend) | Function |
| Set difference to minuend - subtrahend. |
| void mpq_mul (mpq_t product, mpq_t multiplier, mpq_t multiplicand) | Function |
| Set product to multiplier times multiplicand. |
| void mpq_mul_2exp (mpq_t rop, mpq_t op1, unsigned long int op2) | Function |
| Set rop to op1 times 2 raised to op2. |
| void mpq_div (mpq_t quotient, mpq_t dividend, mpq_t divisor) | Function |
| Set quotient to dividend/divisor. |
| void mpq_div_2exp (mpq_t rop, mpq_t op1, unsigned long int op2) | Function |
| Set rop to op1 divided by 2 raised to op2. |
| void mpq_neg (mpq_t negated_operand, mpq_t operand) | Function |
| Set negated_operand to -operand. |
| void mpq_abs (mpq_t rop, mpq_t op) | Function |
| Set rop to the absolute value of op. |
| void mpq_inv (mpq_t inverted_number, mpq_t number) | Function |
| Set inverted_number to 1/number. If the new denominator is zero, this routine will divide by zero. |
| int mpq_cmp (mpq_t op1, mpq_t op2) | Function |
|
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, |
| 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 |
|
Compare op1 and num2/den2. Return a positive value if
op1 > num2/den2, zero if op1 =
num2/den2, and a negative value if op1 <
num2/den2.
num2 and den2 are allowed to have common factors. These functions are implemented as a macros and evaluate their arguments multiple times. |
| int mpq_sgn (mpq_t op) | Macro |
|
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. |
| int mpq_equal (mpq_t op1, mpq_t op2) | Function |
Return non-zero if op1 and op2 are equal, zero if they are
non-equal. Although mpq_cmp can be used for the same purpose, this
function is much faster.
|
The set of mpq functions is quite small. In particular, there are few
functions for either input or output. The following functions give direct
access to the numerator and denominator of an mpq_t.
Note that if an assignment to the numerator and/or denominator could take an
mpq_t out of the canonical form described at the start of this chapter
(see Rational Number Functions) then mpq_canonicalize must be
called before any other mpq functions are applied to that mpq_t.
| mpz_t mpq_numref (mpq_t op) | Macro |
| mpz_t mpq_denref (mpq_t op) | Macro |
Return a reference to the numerator and denominator of op, respectively.
The mpz functions can be used on the result of these macros.
|
| 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 |
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.
|
When using any of these functions, it's a good idea to include stdio.h
before gmp.h, since that will allow gmp.h to define prototypes
for these functions.
Passing a NULL pointer for a stream argument to any of these
functions will make them read from stdin and write to stdout,
respectively.
| size_t mpq_out_str (FILE *stream, int base, mpq_t op) | Function |
Output op on stdio stream stream, as a string of digits in base
base. The base may vary from 2 to 36. Output is in the form
num/den or if the denominator is 1 then just num.
Return the number of bytes written, or if an error occurred, return 0. |
| size_t mpq_inp_str (mpq_t rop, FILE *stream, int base) | Function |
|
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 The base can be between 2 and 36, or can be 0 in which case the leading
characters of the string determine the base, |
GMP floating point numbers are stored in objects of type mpf_t and
functions operating on them have an mpf_ prefix.
The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on most
systems. In the current implementation the exponent is a count of limbs, so
for example on a 32-bit system this means a range of roughly
2^-68719476768 to 2^68719476736, or on a 64-bit system
this will be greater. Note however mpf_get_str can only return an
exponent which fits an mp_exp_t and currently mpf_set_str
doesn't accept exponents bigger than a long.
Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high.
All calculations are performed to the precision of the destination variable. Each function is defined to calculate with "infinite precision" followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition.
The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details.
mpf functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the exponent or
results will be unpredictable. This might change in a future release.
Note that the mpf functions are not intended as a smooth
extension to IEEE P754 arithmetic. In particular results obtained on one
computer often differ from the results on a computer with a different word
size.
| void mpf_set_default_prec (unsigned long int prec) | Function |
Set the default precision to be at least prec bits. All
subsequent calls to mpf_init will use this precision, but previously
initialized variables are unaffected.
|
| unsigned long int mpf_get_default_prec (void) | Function |
| Return the default default precision actually used. |
An mpf_t object must be initialized before storing the first value in
it. The functions mpf_init and mpf_init2 are used for that
purpose.
| void mpf_init (mpf_t x) | Function |
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.
|
| void mpf_init2 (mpf_t x, unsigned long int prec) | Function |
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.
|
| void mpf_clear (mpf_t x) | Function |
Free the space occupied by x. Make sure to call this function for all
mpf_t variables when you are done with them.
|
Here is an example on how to initialize floating-point variables:
{
mpf_t x, y;
mpf_init (x); /* use default precision */
mpf_init2 (y, 256); /* precision at least 256 bits */
...
/* Unless the program is about to exit, do ... */
mpf_clear (x);
mpf_clear (y);
}
The following three functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers.
| unsigned long int mpf_get_prec (mpf_t op) | Function |
| Return the current precision of op, in bits. |
| void mpf_set_prec (mpf_t rop, unsigned long int prec) | Function |
|
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 |
| void mpf_set_prec_raw (mpf_t rop, unsigned long int prec) | Function |
|
Set the precision of rop to be at least prec bits,
without changing the memory allocated.
prec must be no more than the allocated precision for rop, that
being the precision when rop was initialized, or in the most recent
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
|
These functions assign new values to already initialized floats (see Initializing Floats).
| void mpf_set (mpf_t rop, mpf_t op) | Function |
| void mpf_set_ui (mpf_t rop, unsigned long int op) | Function |
| void mpf_set_si (mpf_t rop, signed long int op) | Function |
| void mpf_set_d (mpf_t rop, double op) | Function |
| void mpf_set_z (mpf_t rop, mpz_t op) | Function |
| void mpf_set_q (mpf_t rop, mpq_t op) | Function |
| Set the value of rop from op. |
| int mpf_set_str (mpf_t rop, char *str, int base) | Function |
Set the value of rop from the string in str. The string is of the
form M@N or, if the base is 10 or less, alternatively MeN.
M is the mantissa and N is the exponent. The mantissa is always
in the specified base. The exponent is either in the specified base or, if
base is negative, in decimal. The decimal point expected is taken from
the current locale, on systems providing localeconv.
The argument base may be in the ranges 2 to 36, or -36 to -2. Negative values are used to specify that the exponent is in decimal. Unlike the corresponding 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 This function returns 0 if the entire string is a valid number in base base. Otherwise it returns -1. |
| void mpf_swap (mpf_t rop1, mpf_t rop2) | Function |
| Swap rop1 and rop2 efficiently. Both the values and the precisions of the two variables are swapped. |
For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there. These functions'
names have the form mpf_init_set...
Once the float has been initialized by any of the mpf_init_set...
functions, it can be used as the source or destination operand for the ordinary
float functions. Don't use an initialize-and-set function on a variable
already initialized!
| void mpf_init_set (mpf_t rop, mpf_t op) | Function |
| void mpf_init_set_ui (mpf_t rop, unsigned long int op) | Function |
| void mpf_init_set_si (mpf_t rop, signed long int op) | Function |
| void mpf_init_set_d (mpf_t rop, double op) | Function |
|
Initialize rop and set its value from op.
The precision of rop will be taken from the active default precision, as
set by |
| int mpf_init_set_str (mpf_t rop, char *str, int base) | Function |
Initialize rop and set its value from the string in str. See
mpf_set_str above for details on the assignment operation.
Note that rop is initialized even if an error occurs. (I.e., you have to
call The precision of rop will be taken from the active default precision, as
set by |
| double mpf_get_d (mpf_t op) | Function |
Convert op to a double.
|
| double mpf_get_d_2exp (signed long int exp, mpf_t op) | Function |
Find d and exp such that d times 2
raised to exp, with 0.5<=abs(d)<1, is a good
approximation to op. This is similar to the standard C function
frexp.
|
| long mpf_get_si (mpf_t op) | Function |
| unsigned long mpf_get_ui (mpf_t op) | Function |
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 |
| char * mpf_get_str (char *str, mp_exp_t *expptr, int base, size_t n_digits, mpf_t op) | Function |
|
Convert op to a string of digits in base base. base can be
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.
If str is If str is not The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit. The applicable exponent is written through
the expptr pointer. For example, the number 3.1416 would be returned as
string When op is zero, an empty string is produced and the exponent returned is 0. A pointer to the result string is returned, being either the allocated block or the given str. |
| 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 |
| Set rop to op1 + op2. |
| 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 |
| Set rop to op1 - op2. |
| 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 |
| Set rop to op1 times op2. |
Division is undefined if the divisor is zero, and passing a zero divisor to the divide functions will make these functions intentionally divide by zero. This lets the user handle arithmetic exceptions in these functions in the same manner as other arithmetic exceptions.
| 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 |
| Set rop to op1/op2. |
| void mpf_sqrt (mpf_t rop, mpf_t op) | Function |
| void mpf_sqrt_ui (mpf_t rop, unsigned long int op) | Function |
| Set rop to the square root of op. |
| void mpf_pow_ui (mpf_t rop, mpf_t op1, unsigned long int op2) | Function |
| Set rop to op1 raised to the power op2. |
| void mpf_neg (mpf_t rop, mpf_t op) | Function |
| Set rop to -op. |
| void mpf_abs (mpf_t rop, mpf_t op) | Function |
| Set rop to the absolute value of op. |
| void mpf_mul_2exp (mpf_t rop, mpf_t op1, unsigned long int op2) | Function |
| Set rop to op1 times 2 raised to op2. |
| void mpf_div_2exp (mpf_t rop, mpf_t op1, unsigned long int op2) | Function |
| Set rop to op1 divided by 2 raised to op2. |
| 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) | Function |
| Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. |
| int mpf_eq (mpf_t op1, mpf_t op2, unsigned long int op3) | Function |
|
Return non-zero if the first op3 bits of op1 and op2 are
equal, zero otherwise. I.e., test of op1 and op2 are approximately
equal.
Caution: Currently only whole limbs are compared, and only in an exact fashion. In the future values like 1000 and 0111 may be considered the same to 3 bits (on the basis that their difference is that small). |
| void mpf_reldiff (mpf_t rop, mpf_t op1, mpf_t op2) | Function |
| Compute the relative difference between op1 and op2 and store the result in rop. This is abs(op1-op2)/op1. |
| int mpf_sgn (mpf_t op) | Macro |
|
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. |
Functions that perform input from a stdio stream, and functions that output to
a stdio stream. Passing a NULL pointer for a stream argument to
any of these functions will make them read from stdin and write to
stdout, respectively.
When using any of these functions, it is a good idea to include stdio.h
before gmp.h, since that will allow gmp.h to define prototypes
for these functions.
| size_t mpf_out_str (FILE *stream, int base, size_t n_digits, mpf_t op) | Function |
|
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 Up to n_digits will be printed from the mantissa, except that no more digits than are accurately representable by op will be printed. n_digits can be 0 to select that accurate maximum. |
| size_t mpf_inp_str (mpf_t rop, FILE *stream, int base) | Function |
Read a string in base base from stream, and put the read float in
rop. The string is of the form M@N or, if the base is 10 or
less, alternatively MeN. M is the mantissa and N is the
exponent. The mantissa is always in the specified base. The exponent is
either in the specified base or, if base is negative, in decimal. The
decimal point expected is taken from the current locale, on systems providing
localeconv.
The argument base may be in the ranges 2 to 36, or -36 to -2. Negative values are used to specify that the exponent is in decimal. Unlike the corresponding Return the number of bytes read, or if an error occurred, return 0. |
| 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) | Function |
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.
|
| int mpf_integer_p (mpf_t op) | Function |
| Return non-zero if op is an integer. |
| 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 |
| Return non-zero if op would fit in the respective C data type, when truncated to an integer. |
| void mpf_urandomb (mpf_t rop, gmp_randstate_t state, unsigned long int nbits) | Function |
|
Generate a uniformly distributed random float in rop, such that 0
<= rop < 1, with nbits significant bits in the mantissa.
The variable state must be initialized by calling one of the
|
| void mpf_random2 (mpf_t rop, mp_size_t max_size, mp_exp_t exp) | Function |
| 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. This function is 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 chapter describes low-level GMP functions, used to implement the high-level GMP functions, but also intended for time-critical user code.
These functions start with the prefix mpn_.
The mpn functions are designed to be as fast as possible, not
to provide a coherent calling interface. The different functions have somewhat
similar interfaces, but there are variations that make them hard to use. These
functions do as little as possible apart from the real multiple precision
computation, so that no time is spent on things that not all callers need.
A source operand is specified by a pointer to the least significant limb and a limb count. A destination operand is specified by just a pointer. It is the responsibility of the caller to ensure that the destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform computations on subranges of an argument, and store the result into a subrange of a destination.
A common requirement for all functions is that each source area needs at least one limb. No size argument may be zero. Unless otherwise stated, in-place operations are allowed where source and destination are the same, but not where they only partly overlap.
The mpn functions are the base for the implementation of the
mpz_, mpf_, and mpq_ functions.
This example adds the number beginning at s1p and the number beginning at
s2p and writes the sum at destp. All areas have n limbs.
cy = mpn_add_n (destp, s1p, s2p, n)
In the notation used here, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {s1p, s1n}.
| mp_limb_t mpn_add_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n) | Function |
|
Add {s1p, n} and {s2p, n}, and write the n
least significant limbs of the result to rp. Return carry, either 0 or
1.
This is the lowest-level function for addition. It is the preferred function
for addition, since it is written in assembly for most CPUs. For addition of
a variable to itself (i.e., s1p equals s2p, use |
| mp_limb_t mpn_add_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n, mp_limb_t s2limb) | Function |
| Add {s1p, n} and s2limb, and write the n least significant limbs of the result to rp. Return carry, either 0 or 1. |
| 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) | Function |
|
Add {s1p, s1n} and {s2p, s2n}, and write the
s1n least significant limbs of the result to rp. Return carry,
either 0 or 1.
This function requires that s1n is greater than or equal to s2n. |
| mp_limb_t mpn_sub_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n) | Function |
|
Subtract {s2p, n} from {s1p, n}, and write the
n least significant limbs of the result to rp. Return borrow,
either 0 or 1.
This is the lowest-level function for subtraction. It is the preferred function for subtraction, since it is written in assembly for most CPUs. |
| mp_limb_t mpn_sub_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n, mp_limb_t s2limb) | Function |
| Subtract s2limb from {s1p, n}, and write the n least significant limbs of the result to rp. Return borrow, either 0 or 1. |
| mp_limb_t mpn_sub (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t s1n, const mp_limb_t *s2p, mp_size_t s2n) | Function |
|
Subtract {s2p, s2n} from {s1p, s1n}, and write the
s1n least significant limbs of the result to rp. Return borrow,
either 0 or 1.
This function requires that s1n is greater than or equal to s2n. |
| void mpn_mul_n (mp_limb_t *rp, const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n) | Function |
|
Multiply {s1p, n} and {s2p, n}, and write the
2*n-limb result to rp.
The destination has to have space for 2*n limbs, even if the product's most significant limb is zero. |
| mp_limb_t mpn_mul_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n, mp_limb_t s2limb) | Function |
|
Multiply {s1p, n} by s2limb, and write the n least
significant limbs of the product to rp. Return the most significant
limb of the product. {s1p, n} and {rp, n} are
allowed to overlap provided rp <= s1p.
This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs. Don't call this function if s2limb is a power of 2; use |
| mp_limb_t mpn_addmul_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n, mp_limb_t s2limb) | Function |
|
Multiply {s1p, n} and s2limb, and add the n least
significant limbs of the product to {rp, n} and write the result
to rp. Return the most significant limb of the product, plus carry-out
from the addition.
This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs. |
| mp_limb_t mpn_submul_1 (mp_limb_t *rp, const mp_limb_t *s1p, mp_size_t n, mp_limb_t s2limb) | Function |
|
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, minus
borrow-out from the subtraction.
This is a low-level function that is a building block for general multiplication and division as well as other operations in GMP. It is written in assembly for most CPUs. |
| 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) | Function |
|
Multiply {s1p, s1n} and {s2p, s2n}, and write the
result to rp. Return the most significant limb of the result.
The destination has to have space for s1n + s2n limbs, even if the result might be one limb smaller. This function requires that s1n is greater than or equal to s2n. The destination must be distinct from both input operands. |
| void mpn_tdiv_qr (mp_limb_t *qp, mp_limb_t *rp, mp_size_t qxn, const mp_limb_t *np, mp_size_t nn, const mp_limb_t *dp, mp_size_t dn) | Function |
|
Divide {np, nn} by {dp, dn} and put the quotient
at {qp, nn-dn+1} and the remainder at {rp,
dn}. The quotient is rounded towards 0.
No overlap is permitted between arguments. nn must be greater than or equal to dn. The most significant limb of dp must be non-zero. The qxn operand must be zero. |
| 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) | Function |
[This function is obsolete. Please call mpn_tdiv_qr instead for best
performance.]
Divide {rs2p, rs2n} by {s3p, s3n}, and write the quotient at r1p, with the exception of the most significant limb, which is returned. The remainder replaces the dividend at rs2p; it will be s3n limbs long (i.e., as many limbs as the divisor). In addition to an integer quotient, qxn fraction limbs are developed, and stored after the integral limbs. For most usages, qxn will be zero. It is required that rs2n is greater than or equal to s3n. It is required that the most significant bit of the divisor is set. If the quotient is not needed, pass rs2p + s3n as r1p. Aside from that special case, no overlap between arguments is permitted. Return the most significant limb of the quotient, either 0 or 1. The area at r1p needs to be rs2n - s3n + qxn limbs large. |
| mp_limb_t mpn_divrem_1 (mp_limb_t *r1p, mp_size_t qxn, mp_limb_t *s2p, mp_size_t s2n, mp_limb_t s3limb) | Function |
| mp_limb_t mpn_divmod_1 (mp_limb_t *r1p, mp_limb_t *s2p, mp_size_t s2n, mp_limb_t s3limb) | Macro |
|
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.
The areas at r1p and s2p have to be identical or completely separate, not partially overlapping. |
| 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) | Function |
[This function is obsolete. Please call mpn_tdiv_qr instead for best
performance.]
|
| mp_limb_t mpn_divexact_by3 (mp_limb_t *rp, mp_limb_t *sp, mp_size_t n) | Macro |
| mp_limb_t mpn_divexact_by3c (mp_limb_t *rp, mp_limb_t *sp, mp_size_t n, mp_limb_t carry) | Function |
|
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.
These routines use a multiply-by-inverse and will be faster than
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^mp_bits_per_limb. 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_limb_t mpn_mod_1 (mp_limb_t *s1p, mp_size_t s1n, mp_limb_t s2limb) | Function |
| Divide {s1p, s1n} by s2limb, and return the remainder. s1n can be zero. |
| mp_limb_t mpn_bdivmod (mp_limb_t *rp, mp_limb_t *s1p, mp_size_t s1n, const mp_limb_t *s2p, mp_size_t s2n, unsigned long int d) | Function |
This function puts the low
floor(d/mp_bits_per_limb) limbs of q =
{s1p, s1n}/{s2p, s2n} mod 2^d at
rp, and returns the high d mod mp_bits_per_limb bits of
q.
{s1p, s1n} - q * {s2p, s2n} mod 2^(s1n* This function requires that s1n * This interface is preliminary. It might change incompatibly in future revisions. |
| mp_limb_t mpn_lshift (mp_limb_t *rp, const mp_limb_t *sp, mp_size_t n, unsigned int count) | Function |
|
Shift {sp, n} left by count bits, and write the result to
{rp, n}. The bits shifted out at the left are returned in the
least significant count bits of the return value (the rest of the return
value is zero).
count must be in the range 1 to This function is written in assembly for most CPUs. |
| mp_limb_t mpn_rshift (mp_limb_t *rp, const mp_limb_t *sp, mp_size_t n, unsigned int count) | Function |
|
Shift {sp, n} right by count bits, and write the result to
{rp, n}. The bits shifted out at the right are returned in the
most significant count bits of the return value (the rest of the return
value is zero).
count must be in the range 1 to This function is written in assembly for most CPUs. |
| int mpn_cmp (const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n) | Function |
| 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. |
| mp_size_t mpn_gcd (mp_limb_t *rp, mp_limb_t *s1p, mp_size_t s1n, mp_limb_t *s2p, mp_size_t s2n) | Function |
|
Set {rp, retval} to the greatest common divisor of {s1p,
s1n} and {s2p, s2n}. The result can be up to s2n
limbs, the return value is the actual number produced. Both source operands
are destroyed.
{s1p, s1n} must have at least as many bits as {s2p, s2n}. {s2p, s2n} must be odd. Both operands must have non-zero most significant limbs. No overlap is permitted between {s1p, s1n} and {s2p, s2n}. |
| mp_limb_t mpn_gcd_1 (const mp_limb_t *s1p, mp_size_t s1n, mp_limb_t s2limb) | Function |
| Return the greatest common divisor of {s1p, s1n} and s2limb. Both operands must be non-zero. |
| mp_size_t mpn_gcdext (mp_limb_t *r1p, mp_limb_t *r2p, mp_size_t *r2n, mp_limb_t *s1p, mp_size_t s1n, mp_limb_t *s2p, mp_size_t s2n) | Function |
|
Calculate the greatest common divisor of {s1p, s1n} and
{s2p, s2n}. Store the gcd at {r1p, retval} and
the first cofactor at {r2p, *r2n}, with *r2n negative if
the cofactor is negative. r1p and r2p should each have room for
s1n+1 limbs, but the return value and value stored through
r2n indicate the actual number produced.
{s1p, s1n} >= {s2p, s2n} is required, and both must be non-zero. The regions {s1p, s1n+1} and {s2p, s2n+1} are destroyed (i.e. the operands plus an extra limb past the end of each). The cofactor r1 will satisfy r2*s1 + k*s2 = r1. The second cofactor k is not calculated but can easily be obtained from (r1 - r2*s1) / s2. |
| mp_size_t mpn_sqrtrem (mp_limb_t *r1p, mp_limb_t *r2p, const mp_limb_t *sp, mp_size_t n) | Function |
|
Compute the square root of {sp, n} and put the result at
{r1p, ceil(n/2)} and the remainder at {r2p,
retval}. r2p needs space for n limbs, but the return value
indicates how many are produced.
The most significant limb of {sp, n} must be non-zero. The areas {r1p, ceil(n/2)} and {sp, n} must be completely separate. The areas {r2p, n} and {sp, n} must be either identical or completely separate. If the remainder is not wanted then r2p can be A return value of zero indicates a perfect square. See also
|
| mp_size_t mpn_get_str (unsigned char *str, int base, mp_limb_t *s1p, mp_size_t s1n) | Function |
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.
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. |
| mp_size_t mpn_set_str (mp_limb_t *rp, const char *str, size_t strsize, int base) | Function |
|
Convert bytes {str,strsize} in the given base to limbs at
rp.
str[0] is the most significant byte and str[strsize-1] is the least significant. Each byte should be a value in the range 0 to base-1, not an ASCII character. base can vary from 2 to 256. The return value is the number of limbs written to rp. If the most significant input byte is non-zero then the high limb at rp will be non-zero, and only that exact number of limbs will be required there. If the most significant input byte is zero then there may be high zero limbs written to rp and included in the return value. strsize must be at least 1, and no overlap is permitted between {str,strsize} and the result at rp. |
| unsigned long int mpn_scan0 (const mp_limb_t *s1p, unsigned long int bit) | Function |
|
Scan s1p from bit position bit for the next clear bit.
It is required that there be a clear bit within the area at s1p at or beyond bit position bit, so that the function has something to return. |
| unsigned long int mpn_scan1 (const mp_limb_t *s1p, unsigned long int bit) | Function |
|
Scan s1p from bit position bit for the next set bit.
It is required that there be a set bit within the area at s1p at or beyond bit position bit, so that the function has something to return. |
| void mpn_random (mp_limb_t *r1p, mp_size_t r1n) | Function |
| void mpn_random2 (mp_limb_t *r1p, mp_size_t r1n) | Function |
Generate a random number of length r1n and store it at r1p. The
most significant limb is always non-zero. mpn_random generates
uniformly distributed limb data, mpn_random2 generates long strings of
zeros and ones in the binary representation.
|
| unsigned long int mpn_popcount (const mp_limb_t *s1p, mp_size_t n) | Function |
| Count the number of set bits in {s1p, n}. |
| unsigned long int mpn_hamdist (const mp_limb_t *s1p, const mp_limb_t *s2p, mp_size_t n) | Function |
| Compute the hamming distance between {s1p, n} and {s2p, n}. |
| int mpn_perfect_square_p (const mp_limb_t *s1p, mp_size_t n) | Function |
| Return non-zero iff {s1p, n} is a perfect square. |
Everything in this section is highly experimental and may disappear or be subject to incompatible changes in a future version of GMP.
Nails are an experimental feature whereby a few bits are left unused at the
top of each mp_limb_t. This can significantly improve carry handling
on some processors.
All the mpn functions accepting limb data will expect the nail bits to
be zero on entry, and will return data with the nails similarly all zero.
This applies both to limb vectors and to single limb arguments.
Nails can be enabled by configuring with --enable-nails. By default
the number of bits will be chosen according to what suits the host processor,
but a particular number can be selected with --enable-nails=N.
At the mpn level, a nail build is neither source nor binary compatible with a non-nail build, strictly speaking. But programs acting on limbs only through the mpn functions are likely to work equally well with either build, and judicious use of the definitions below should make any program compatible with either build, at the source level.
For the higher level routines, meaning mpz etc, a nail build should be
fully source and binary compatible with a non-nail build.
| GMP_NAIL_BITS | Macro |
| GMP_NUMB_BITS | Macro |
| GMP_LIMB_BITS | Macro |
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 |
| GMP_NAIL_MASK | Macro |
| GMP_NUMB_MASK | Macro |
Bit masks for the nail and number parts of a limb. GMP_NAIL_MASK is 0
when nails are not in use.
|
| GMP_NUMB_MAX | Macro |
The maximum value that can be stored in the number part of a limb. This is
the same as GMP_NUMB_MASK, but can be used for clarity when doing
comparisons rather than bit-wise operations.
|
The term "nails" comes from finger or toe nails, which are at the ends of a limb (arm or leg). "numb" is short for number, but is also how the developers felt after trying for a long time to come up with sensible names for these things.
In the future (the distant future most likely) a non-zero nail might be permitted, giving non-unique representations for numbers in a limb vector. This would help vector processors since carries would only ever need to propagate one or two limbs.
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 Integer Random Numbers, and 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.
| void gmp_randinit_default (gmp_randstate_t state) | Function |
| Initialize state with a default algorithm. This will be a compromise between speed and randomness, and is recommended for applications with no special requirements. |
| void gmp_randinit_lc_2exp (gmp_randstate_t state, mpz_t a, unsigned long c, unsigned long m2exp) | Function |
|
Initialize state with a linear congruential algorithm X = (a*X + c) mod 2^m2exp.
The low bits of X in this algorithm are not very random. The least significant bit will have a period no more than 2, and the second bit no more than 4, etc. For this reason only the high half of each X is actually used. When a random number of more than m2exp/2 bits is to be generated, multiple iterations of the recurrence are used and the results concatenated. |
| int gmp_randinit_lc_2exp_size (gmp_randstate_t state, unsigned long size) | Function |
Initialize state for a linear congruential algorithm as per
gmp_randinit_lc_2exp. a, c and m2exp are selected
from a table, chosen so that size bits (or more) of each X will
be used, ie. m2exp >= size/2.
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. |
| void gmp_randinit (gmp_randstate_t state, gmp_randalg_t alg, ...) | Function |
|
This function is obsolete.
Initialize state with an algorithm selected by alg. The only
choice is
|
| void gmp_randclear (gmp_randstate_t state) | Function |
| Free all memory occupied by state. |
| void gmp_randseed (gmp_randstate_t state, mpz_t seed) | Function |
| void gmp_randseed_ui (gmp_randstate_t state, unsigned long int seed) | Function |
|
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 |
gmp_printf and friends accept format strings similar to the standard C
printf (see 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, and N for an mp_limb_t array.
Z, Q and N behave like integers. Q will print a
/ and a denominator, if needed. F behaves like a float. For
example,
mpz_t z;
gmp_printf ("%s is an mpz %Zd\n", "here", z);
mpq_t q;
gmp_printf ("a hex rational: %#40Qx\n", q);
mpf_t f;
int n;
gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
const mp_limb_t *ptr;
mp_size_t size;
gmp_printf ("limb array %Nx\n", ptr, size);
For N the limbs are expected least significant first, as per the
mpn functions (see Low-level Functions). A negative size can be
given to print the value as a negative.
All the standard C printf types behave the same as the C library
printf, and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are simply
handed to printf and only the GMP extensions handled directly.
The flags accepted are as follows. GLIBC style ' is only for the
standard C types (not the GMP types), and only if the C library supports it.
0pad with zeros (rather than spaces) #show the base with 0x,0Xor0+always show a sign (space) show a space or a -sign'group digits, GLIBC style (not GMP types)
The optional width and precision can be given as a number within the format
string, or as a * to take an extra parameter of type int, the
same as the standard printf.
The standard types accepted are as follows. h and l are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the output.
hshorthhcharjintmax_toruintmax_tllongorwchar_tllsame as LLlong longorlong doubleqquad_toru_quad_ttptrdiff_tzsize_t
The GMP types are
Fmpf_t, float conversionsQmpq_t, integer conversionsNmp_limb_tarray, integer conversionsZmpz_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.
aAhex floats, GLIBC style ccharacter ddecimal integer eEscientific format float ffixed point float isame as dgGfixed or scientific float mstrerrorstring, GLIBC stylenstore characters written so far ooctal integer ppointer sstring uunsigned integer xXhex 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.
n can be used with any type, even the GMP types.
Other types or conversions that might be accepted by the C library
printf cannot be used through gmp_printf, this includes for
instance extensions registered with GLIBC register_printf_function.
Also currently there's no support for POSIX $ style numbered arguments
(perhaps this will be added in the future).
The precision field has it's usual meaning for integer Z and float
F types, but is currently undefined for Q and should not be used
with that.
mpf_t conversions only ever generate as many digits as can be
accurately represented by the operand, the same as mpf_get_str does.
Zeros will be used if necessary to pad to the requested precision. This
happens even for an f conversion of an mpf_t which is an
integer, for instance 2^1024 in an mpf_t of 128 bits
precision will only produce about 20 digits, then pad with zeros to the
decimal point. An empty precision field like %.Fe or %.Ff can
be used to specifically request just the significant digits.
The decimal point character (or string) is taken from the current locale
settings on systems which provide localeconv (see Locales and Internationalization). The C
library will normally do the same for standard float output.
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 Variadic Functions, or man 3
va_start.
It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions.
The file based functions gmp_printf and gmp_fprintf will return
-1 to indicate a write error. All the functions can return -1
if the C library printf variant in use returns -1, but this
shouldn't normally occur.
| int gmp_printf (const char *fmt, ...) | Function |
| int gmp_vprintf (const char *fmt, va_list ap) | Function |
Print to the standard output stdout. Return the number of characters
written, or -1 if an error occurred.
|
| int gmp_fprintf (FILE *fp, const char *fmt, ...) | Function |
| int gmp_vfprintf (FILE *fp, const char *fmt, va_list ap) | Function |
| Print to the stream fp. Return the number of characters written, or -1 if an error occurred. |
| int gmp_sprintf (char *buf, const char *fmt, ...) | Function |
| int gmp_vsprintf (char *buf, const char *fmt, va_list ap) | Function |
|
Form a null-terminated string in buf. Return the number of characters
written, excluding the terminating null.
No overlap is permitted between the space at buf and the string fmt. These functions are not recommended, since there's no protection against exceeding the space available at buf. |
| 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) | Function |
|
Form a null-terminated string in buf. No more than size bytes
will be written. To get the full output, size must be enough for the
string and null-terminator.
The return value is the total number of characters which ought to have been produced, excluding the terminating null. If retval >= size then the actual output has been truncated to the first size-1 characters, and a null appended. No overlap is permitted between the region {buf,size} and the fmt string. Notice the return value is in ISO C99 |
| int gmp_asprintf (char **pp, const char *fmt, ...) | Function |
| int gmp_vasprintf (char *pp, const char *fmt, va_list ap) | Function |
|
Form a null-terminated string in a block of memory obtained from the current
memory allocation function (see Custom Allocation). The block will be the
size of the string and null-terminator. Put the address of the block in
*pp. Return the number of characters produced, excluding the
null-terminator.
Unlike the C library |
| int gmp_obstack_printf (struct obstack *ob, const char *fmt, ...) | Function |
| int gmp_obstack_vprintf (struct obstack *ob, const char *fmt, va_list ap) | Function |
Append to the current obstack object, in the same style as
obstack_printf. Return the number of characters written. A
null-terminator is not written.
fmt cannot be within the current obstack object, since the 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 Obstacks. |
The following functions are provided in libgmpxx, which is built if C++
support is enabled (see Build Options). Prototypes are available from
<gmp.h>.
| ostream& operator<< (ostream& stream, mpz_t op) | Function |
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 |
| ostream& operator<< (ostream& stream, mpq_t op) | Function |
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 In hex or octal, op is printed as a signed value, the same as for
decimal. If |
| ostream& operator<< (ostream& stream, mpf_t op) | Function |
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 current
locale, on systems providing localeconv.
Hex and octal are supported, unlike the standard
|
These operators mean that GMP types can be printed in the usual C++ way, for
example,
mpz_t z; int n; ... cout << "iteration " << n << " value " << z << "\n";
But note that ostream output (and istream input, see C++ Formatted Input) is the only overloading available and using for instance
+ with an mpz_t will have unpredictable results.
gmp_scanf and friends accept format strings similar to the standard C
scanf (see Formatted Input). A format specification is of the form
% [flags] [width] [type] conv
GMP adds types Z, Q and F for mpz_t, mpq_t
and mpf_t respectively. Z and Q behave like integers.
Q will read a / and a denominator, if present. F behaves
like a float.
GMP variables don't require an & when passed to gmp_scanf, since
they're already "call-by-reference". For example,
/* to read say "a(5) = 1234" */
int n;
mpz_t z;
gmp_scanf ("a(%d) = %Zd\n", &n, z);
mpq_t q1, q2;
gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
/* to read say "topleft (1.55,-2.66)" */
mpf_t x, y;
char buf[32];
gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
All the standard C scanf types behave the same as in the C library
scanf, and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are simply
handed to scanf and only the GMP extensions handled directly.
The flags accepted are as follows. a and ' will depend on
support from the C library, and ' cannot be used with GMP types.
*read but don't store aallocate a buffer (string conversions) 'group digits, GLIBC style (not GMP types)
The standard types accepted are as follows. h and l are
portable, the rest will depend on the compiler (or include files) for the type
and the C library for the input.
hshorthhcharjintmax_toruintmax_tllongorwchar_tllsame as LLlong longorlong doubleqquad_toru_quad_ttptrdiff_tzsize_t
The GMP types are
Fmpf_t, float conversionsQmpq_t, integer conversionsZmpz_t, integer conversions
The conversions accepted are as follows. p and [ will depend on
support from the C library, the rest are standard.
ccharacter or characters ddecimal integer eEfgGfloat iinteger with base indicator ncharacters written so far ooctal integer ppointer sstring of non-whitespace characters udecimal integer xXhex 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 e or E
for the exponent in scientific format.
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 (see strtoul, Parsing of Integers). For GMP types there are no overflows, and
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 (see Rational Number Functions).
Qi will read a base specification separately for the numerator and
denominator. For example 0x10/11 would be 16/11, whereas
0x10/0x11 would be 16/17.
n can be used with any of the types above, even the GMP types.
* to suppress assignment is allowed, though the field would then do
nothing at all.
Other conversions or types that might be accepted by the C library
scanf cannot be used through gmp_scanf.
Whitespace is read and discarded before a field, except for c and
[ conversions.
For float conversions, the decimal point character (or string) expected is
taken from the current locale settings on systems which provide
localeconv (see Locales and Internationalization). The C library will normally do the same for
standard float input.
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 Variadic Functions, or man 3
va_start.
It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions.
No overlap is permitted between the fmt string and any of the results produced.
| int gmp_scanf (const char *fmt, ...) | Function |
| int gmp_vscanf (const char *fmt, va_list ap) | Function |
Read from the standard input stdin.
|
| int gmp_fscanf (FILE *fp, const char *fmt, ...) | Function |
| int gmp_vfscanf (FILE *fp, const char *fmt, va_list ap) | Function |
| Read from the stream fp. |
| int gmp_sscanf (const char *s, const char *fmt, ...) | Function |
| int gmp_vsscanf (const char *s, const char *fmt, va_list ap) | Function |
| Read from a null-terminated string s. |
The return value from each of these functions is the same as the standard C99
scanf, namely the number of fields successfully parsed and stored.
%n fields and fields read but suppressed by * don't count
towards the return value.
If end of file or file error, or end of string, is reached when a match is
required, and when no previous non-suppressed fields have matched, then the
return value is EOF instead of 0. A match is required for a literal character
in the format string or a field other than %n. Whitespace in the
format string is only an optional match and won't induce an EOF in this
fashion. Leading whitespace read and discarded for a field doesn't count as a
match.
The following functions are provided in libgmpxx, which is built only
if C++ support is enabled (see Build Options). Prototypes are available
from <gmp.h>.
| istream& operator>> (istream& stream, mpz_t rop) | Function |
Read rop from stream, using its ios formatting settings.
|
| istream& operator>> (istream& stream, mpq_t rop) | Function |
Read rop from stream, using its ios formatting settings.
An integer like |
| istream& operator>> (istream& stream, mpf_t rop) | Function |
Read rop from stream, using its ios formatting settings.
Hex or octal floats are not supported, but might be in the future. |
These operators mean that GMP types can be read in the usual C++ way, for
example,
mpz_t z; ... cin >> z;
But note that istream input (and ostream output, see C++ Formatted Output) is the only overloading available and using for instance
+ with an mpz_t will have unpredictable results.
This chapter describes the C++ class based interface to GMP.
All GMP C language types and functions can be used in C++ programs, since
gmp.h has extern "C" qualifiers, but the class interface offers
overloaded functions and operators which may be more convenient.
Due to the implementation of this interface, a reasonably recent C++ compiler is required, one supporting namespaces, partial specialization of templates and member templates. For GCC this means version 2.91 or later.
Everything described in this chapter is to be considered preliminary and might be subject to incompatible changes if some unforeseen difficulty reveals itself.
All the C++ classes and functions are available with
#include <gmpxx.h>
The classes defined are
| mpz_class | Class |
| mpq_class | Class |
| mpf_class | Class |
The standard operators and various standard functions are overloaded to allow
arithmetic with these classes. For example,
int
main (void)
{
mpz_class a, b, c;
a = 1234;
b = "-5678";
c = a+b;
cout << "sum is " << c << "\n";
cout << "absolute value is " << abs(c) << "\n";
return 0;
}
An important feature of the implementation is that an expression like
a=b+c results in a single call to the corresponding mpz_add,
without using a temporary for the b+c part. Expressions which by their
nature imply intermediate values, like a=b*c+d*e, still use temporaries
though.
The classes can be freely intermixed in expressions, as can the classes and
the standard types long, unsigned long and double.
Smaller types like int or float can also be intermixed, since
C++ will promote them.
Note that bool is not accepted directly, but must be explicitly cast to
an int first. This is because C++ will automatically convert any
pointer to a bool, so if GMP accepted bool it would make all
sorts of invalid class and pointer combinations compile but almost certainly
not do anything sensible.
Conversions back from the classes to standard C++ types aren't done
automatically, instead member functions like get_si are provided (see
the following sections for details).
Also there are no automatic conversions from the classes to the corresponding GMP C types, instead a reference to the underlying C object can be obtained with the following functions,
| mpz_t mpz_class::get_mpz_t () | Function |
| mpq_t mpq_class::get_mpq_t () | Function |
| mpf_t mpf_class::get_mpf_t () | Function |
These can be used to call a C function which doesn't have a C++ class
interface. For example to set a to the GCD of b and c,
mpz_class a, b, c; ... mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
In the other direction, a class can be initialized from the corresponding GMP
C type, or assigned to if an explicit constructor is used. In both cases this
makes a copy of the value, it doesn't create any sort of association. For
example,
mpz_t z; // ... init and calculate z ... mpz_class x(z); mpz_class y; y = mpz_class (z);
There are no namespace setups in gmpxx.h, all types and functions are
simply put into the global namespace. This is what gmp.h has done in
the past, and continues to do for compatibility. The extras provided by
gmpxx.h follow GMP naming conventions and are unlikely to clash with
anything.
| void mpz_class::mpz_class (type n) | Function |
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 (see Assigning Integers).
|
| void mpz_class::mpz_class (mpz_t z) | Function |
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.
|
| void mpz_class::mpz_class (const char *s) | Function |
| void mpz_class::mpz_class (const char *s, int base) | Function |
| void mpz_class::mpz_class (const string& s) | Function |
| void mpz_class::mpz_class (const string& s, int base) | Function |
Construct an mpz_class converted from a string using
mpz_set_str, (see Assigning Integers). If the base is not
given then 0 is used.
|
| mpz_class operator/ (mpz_class a, mpz_class d) | Function |
| mpz_class operator% (mpz_class a, mpz_class d) | Function |
Divisions involving mpz_class round towards zero, as per the
mpz_tdiv_q and mpz_tdiv_r functions (see Integer Division).
This corresponds to the rounding used for plain int calculations on
most machines.
The mpz_class q, a, d; ... mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t()); |
| mpz_class abs (mpz_class op1) | Function |
| int cmp (mpz_class op1, type op2) | Function |
| int cmp (type op1, mpz_class op2) | Function |
| double mpz_class::get_d (void) | Function |
| long mpz_class::get_si (void) | Function |
| unsigned long mpz_class::get_ui (void) | 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 |
| int sgn (mpz_class op) | Function |
| mpz_class sqrt (mpz_class op) | Function |
|
These functions provide a C++ class interface to the corresponding GMP C
routines.
|
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.
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.
| void mpq_class::mpq_class (type op) | Function |
| void mpq_class::mpq_class (integer num, integer den) | Function |
Construct an mpq_class. The initial value can be a single value of any
type, or a pair of integers (mpz_class or standard C++ integer types)
representing a fraction, except that long long and long double
are not supported. For example,
mpq_class q (99); mpq_class q (1.75); mpq_class q (1, 3); |
| void mpq_class::mpq_class (mpq_t q) | Function |
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.
|
| void mpq_class::mpq_class (const char *s) | Function |
| void mpq_class::mpq_class (const char *s, int base) | Function |
| void mpq_class::mpq_class (const string& s) | Function |
| void mpq_class::mpq_class (const string& s, int base) | Function |
Construct an mpq_class converted from a string using
mpq_set_str, (see Initializing Rationals). If the base is
not given then 0 is used.
|
| void mpq_class::canonicalize () | Function |
Put an mpq_class into canonical form, as per Rational Number Functions. All arithmetic operators require their operands in canonical
form, and will return results in canonical form.
|
| 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 |
| int sgn (mpq_class op) | Function |
|
These functions provide a C++ class interface to the corresponding GMP C
routines.
|
| mpz_class& mpq_class::get_num () | Function |
| mpz_class& mpq_class::get_den () | Function |
Get a reference to an mpz_class which is the numerator or denominator
of an mpq_class. This can be used both for read and write access. If
the object returned is modified, it modifies the original mpq_class.
If direct manipulation might produce a non-canonical value, then
|
| mpz_t mpq_class::get_num_mpz_t () | Function |
| mpz_t mpq_class::get_den_mpz_t () | Function |
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
|
| istream& operator>> (istream& stream, mpq_class& rop); | Function |
Read rop from stream, using its ios formatting settings,
the same as mpq_t operator>> (see C++ Formatted Input).
If the rop read might not be in canonical form then
|
When an expression requires the use of temporary intermediate mpf_class
values, like f=g*h+x*y, those temporaries will have the same precision
as the destination f. Explicit constructors can be used if this
doesn't suit.
| mpf_class::mpf_class (type op) | Function |
| mpf_class::mpf_class (type op, unsigned long prec) | Function |
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 mpf_class f(1.5); // default precision mpf_class f(1.5, 500); // 500 bits (at least) mpf_class f(x); // precision of x mpf_class f(abs(x)); // precision of x mpf_class f(-g, 1000); // 1000 bits (at least) mpf_class f(x+y); // greater of precisions of x and y |
| mpf_class abs (mpf_class op) | Function |
| mpf_class ceil (mpf_class op) | Function |
| int cmp (mpf_class op1, type op2) | Function |
| int cmp (type op1, mpf_class op2) | Function |
| 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 |
| unsigned long mpf_class::get_ui (void) | 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 |
| int sgn (mpf_class op) | Function |
| mpf_class sqrt (mpf_class op) | Function |
| mpf_class trunc (mpf_class op) | Function |
|
These functions provide a C++ class interface to the corresponding GMP C
routines.
The accuracy provided by |
| unsigned long int mpf_class::get_prec () | Function |
| void mpf_class::set_prec (unsigned long prec) | Function |
| void mpf_class::set_prec_raw (unsigned long prec) | Function |
Get or set the current precision of an mpf_class.
The restrictions described for |
The C++ class interface to MPFR is provided if MPFR is enabled (see Build Options). This interface must be regarded as preliminary and possibly
subject to incompatible changes in the future, since MPFR itself is
preliminary. All definitions can be obtained with
#include <mpfrxx.h>
This defines
| mpfr_class | Class |
which behaves similarly to mpf_class (see C++ Interface Floats).
| gmp_randclass | Class |
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::gmp_randclass (void (*randinit) (gmp_randstate_t, ...), ...) | Function |
Construct a gmp_randclass, using a call to the given randinit
function (see Random State Initialization). The arguments expected are
the same as randinit, but with mpz_class instead of mpz_t.
For example,
gmp_randclass r1 (gmp_randinit_default); gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32); gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
|
| gmp_randclass::gmp_randclass (gmp_randalg_t alg, ...) | Function |
Construct a gmp_randclass using the same parameters as
gmp_randinit (see Random State Initialization). This function is
obsolete and the above randinit style should be preferred.
|
| void gmp_randclass::seed (unsigned long int s) | Function |
| void gmp_randclass::seed (mpz_class s) | Function |
| Seed a random number generator. See see Random Number Functions, for how to choose a good seed. |
| mpz_class gmp_randclass::get_z_bits (unsigned long bits) | Function |
| mpz_class gmp_randclass::get_z_bits (mpz_class bits) | Function |
| Generate a random integer with a specified number of bits. |
| mpz_class gmp_randclass::get_z_range (mpz_class n) | Function |
| Generate a random integer in the range 0 to n-1 inclusive. |
| mpf_class gmp_randclass::get_f () | Function |
| mpf_class gmp_randclass::get_f (unsigned long prec) | Function |
Generate a random float f in the range 0 <= f < 1. f
will be to prec bits precision, or if prec is not given then to
the precision of the destination. For example,
gmp_randclass r; ... mpf_class f (0, 512); // 512 bits precision f = r.get_f(); // random number, 512 bits |
mpq_class and Templated Reading
mpq_class
requires a canonicalize call if inputs read with operator>>
might be non-canonical. This can lead to incorrect results.
operator>> behaves as it does for reasons of efficiency. A
canonicalize can be quite time consuming on large operands, and is best
avoided if it's not necessary.
But this potential difficulty reduces the usefulness of mpq_class.
Perhaps a mechanism to tell operator>> what to do will be adopted in
the future, maybe a preprocessor define, a global flag, or an ios flag
pressed into service. Or maybe, at the risk of inconsistency, the
mpq_class operator>> could canonicalize and leave mpq_t
operator>> not doing so, for use on those occasions when that's
acceptable. Send feedback or alternate ideas to bug-gmp@gnu.org.
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.
T
intended to be some numeric type,
template <class T> T fun (const T &, const T &);
When used with, say, plain mpz_class variables, it works fine: T
is resolved as mpz_class.
mpz_class f(1), g(2); fun (f, g); // Good
But when one of the arguments is an expression, it doesn't work.
mpz_class f(1), g(2), h(3); fun (f, g+h); // Bad
This is because g+h ends up being a certain expression template type
internal to gmpxx.h, which the C++ template resolution rules are unable
to automatically convert to mpz_class. The workaround is simply to add
an explicit cast.
mpz_class f(1), g(2), h(3); fun (f, mpz_class(g+h)); // Good
Similarly, within fun it may be necessary to cast an expression to type
T when calling a templated fun2.
template <class T>
void fun (T f, T g)
{
fun2 (f, f+g); // Bad
}
template <class T>
void fun (T f, T g)
{
fun2 (f, T(f+g)); // Good
}
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 (see 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.
| MINT * itom (signed short int initial_value) | Function |
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.
|
| MINT * xtom (char *initial_value) | Function |
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.
|
| void move (MINT *src, MINT *dest) | Function |
| Set dest to src by copying. Both variables must be previously initialized. |
| void madd (MINT *src_1, MINT *src_2, MINT *destination) | Function |
| Add src_1 and src_2 and put the sum in destination. |
| void msub (MINT *src_1, MINT *src_2, MINT *destination) | Function |
| Subtract src_2 from src_1 and put the difference in destination. |
| void mult (MINT *src_1, MINT *src_2, MINT *destination) | Function |
| Multiply src_1 and src_2 and put the product in destination. |
| 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) | Function |
|
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. |
| void msqrt (MINT *op, MINT *root, MINT *remainder) | Function |
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. |
| void pow (MINT *base, MINT *exp, MINT *mod, MINT *dest) | Function |
| Set dest to (base raised to exp) modulo mod. |
| void rpow (MINT *base, signed short int exp, MINT *dest) | Function |
| Set dest to base raised to exp. |
| void gcd (MINT *op1, MINT *op2, MINT *res) | Function |
| Set res to the greatest common divisor of op1 and op2. |
| int mcmp (MINT *op1, MINT *op2) | Function |
| Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2. |
| void min (MINT *dest) | Function |
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.
|
| void mout (MINT *src) | Function |
Output src to stdout, as a decimal string. Also output a newline.
|
| char * mtox (MINT *op) | Function |
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.
|
| void mfree (MINT *op) | Function |
De-allocate, the space used by op. This function should only be
passed a value returned by itom or xtom.
|
By default GMP uses malloc, realloc and free for memory
allocation, and if they fail GMP prints a message to the standard error output
and terminates the program.
Alternate functions can be specified to allocate memory in a different way or to have a different error action on running out of memory.
This feature is available in the Berkeley compatibility library (see BSD Compatible Functions) as well as the main GMP library.
| 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)) |
Function |
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 Be sure to call |
The functions supplied should fit the following declarations:
| void * allocate_function (size_t alloc_size) | Function |
| Return a pointer to newly allocated space with at least alloc_size bytes. |
| void * reallocate_function (void *ptr, size_t old_size, size_t new_size) | 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. ptr is never |
| void deallocate_function (void *ptr, size_t size) | Function |
|
De-allocate the space pointed to by ptr.
ptr is never |
A byte here means the unit used by the sizeof operator.
The old_size parameters to reallocate_function and
deallocate_function are passed for convenience, but of course can be
ignored if not needed. The default functions using malloc and friends
for instance don't use them.
No error return is allowed from any of these functions, if they return then
they must have performed the specified operation. In particular note that
allocate_function or reallocate_function mustn't return
NULL.
Getting a different fatal error action is a good use for custom allocation
functions, for example giving a graphical dialog rather than the default print
to stderr. How much is possible when genuinely out of memory is
another question though.
There's currently no defined way for the allocation functions to recover from
an error such as out of memory, they must terminate program execution. A
longjmp or throwing a C++ exception will have undefined results. This
may change in the future.
GMP may use allocated blocks to hold pointers to other allocated blocks. This will limit the assumptions a conservative garbage collection scheme can make.
Since the default GMP allocation uses malloc and friends, those
functions will be linked in even if the first thing a program does is an
mp_set_memory_functions. It's necessary to change the GMP sources if
this is a problem.
The following packages and projects offer access to GMP from languages other
than C, though perhaps with varying levels of functionality and efficiency.
mpeval.
demos/perl in the GMP sources.
This chapter is an introduction to some of the algorithms used for various GMP operations. The code is likely to be hard to understand without knowing something about the algorithms.
Some GMP internals are mentioned, but applications that expect to be compatible with future GMP releases should take care to use only the documented functions.
NxN limb multiplications and squares are done using one of four algorithms, as the size N increases.
Algorithm Threshold Basecase (none) Karatsuba MUL_KARATSUBA_THRESHOLDToom-3 MUL_TOOM3_THRESHOLDFFT MUL_FFT_THRESHOLD
Similarly for squaring, with the SQR thresholds. Note though that the
FFT is only used if GMP is configured with --enable-fft, see Build Options.
NxM multiplications of operands with different sizes above
MUL_KARATSUBA_THRESHOLD are currently done by splitting into MxM
pieces. The Karatsuba and Toom-3 routines then operate only on equal size
operands. This is not very efficient, and is slated for improvement in the
future.
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
(see References), and the mpn/generic/mul_basecase.c code.
Assembler implementations of mpn_mul_basecase are essentially the same
as the generic C code, but have all the usual assembler tricks and
obscurities introduced for speed.
A square can be done in roughly half the time of a multiply, by using the fact
that the cross products above and below the diagonal are the same. A triangle
of products below the diagonal is formed, doubled (left shift by one bit), and
then the products on the diagonal added. This can be seen in
mpn/generic/sqr_basecase.c. Again the assembler implementations take
essentially the same approach.
u0 u1 u2 u3 u4 +---+---+---+---+---+ u0 | d | | | | | +---+---+---+---+---+ u1 | | d | | | | +---+---+---+---+---+ u2 | | | d | | | +---+---+---+---+---+ u3 | | | | d | | +---+---+---+---+---+ u4 | | | | | d | +---+---+---+---+---+
In practice squaring isn't a full 2x faster than multiplying, it's
usually around 1.5x. Less than 1.5x probably indicates
mpn_sqr_basecase wants improving on that CPU.
On some CPUs mpn_mul_basecase can be faster than the generic C
mpn_sqr_basecase. SQR_BASECASE_THRESHOLD is the size at which
to use mpn_sqr_basecase, this will be zero if that routine should be
used always.
The Karatsuba multiplication algorithm is described in Knuth section 4.3.3 part A, and various other textbooks. A brief description is given here.
The inputs x and y are treated as each split into two parts of
equal length (or the most significant part one limb shorter if N is odd).
high low +----------+----------+ | x1 | x0 | +----------+----------+ +----------+----------+ | y1 | y0 | +----------+----------+
Let b be the power of 2 where the split occurs, ie. if x0 is
k limbs (y0 the same) then
b=2^(k*mp_bits_per_limb).
With that x=x1*b+x0 and y=y1*b+y0, and the
following holds,
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
This formula means doing only three multiplies of (N/2)x(N/2) limbs,
whereas a basecase multiply of NxN limbs is equivalent to four
multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent
the positions where the three products must be added.
high low
+--------+--------+ +--------+--------+
| x1*y1 | | x0*y0 |
+--------+--------+ +--------+--------+
+--------+--------+
add | x1*y1 |
+--------+--------+
+--------+--------+
add | x0*y0 |
+--------+--------+
+--------+--------+
sub | (x1-x0)*(y1-y0) |
+--------+--------+
The term (x1-x0)*(y1-y0) is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb additions, rather than 6*k, but in GMP extra function call overheads outweigh the saving.
Squaring is similar to multiplying, but with x=y the formula reduces to
an equivalent with three squares,
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
The final result is accumulated from those three squares the same way as for the three multiplies above. The middle term (x1-x0)^2 is now always positive.
A similar formula for both multiplying and squaring can be constructed with a middle term (x1+x0)*(y1+y0). But those sums can exceed k limbs, leading to more carry handling and additions than the form above.
Karatsuba multiplication is asymptotically an O(N^1.585) algorithm, the exponent being log(3)/log(2), representing 3 multiplies each 1/2 the size of the inputs. This is a big improvement over the basecase multiply at O(N^2) and the advantage soon overcomes the extra additions Karatsuba performs.
MUL_KARATSUBA_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. Clearly per-crossproduct speedups in the basecase code reduce a
and decrease the threshold, but linear style speedups reducing b will
actually increase the threshold. 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.
The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom-Cook and FFT algorithms. A description of Toom-Cook 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 others).
high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+
These parts are treated as the coefficients of two polynomials
X(t) = x2*t^2 + x1*t + x0 Y(t) = y2*t^2 + y1*t + y0
Again let b equal the power of 2 which is the size of the x0, x1, y0 and y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb). With this x=X(b) and y=Y(b).
Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients
are
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
The w[i] are going to be determined, and when they are they'll give
the final result using w=W(b), since
x*y=X(b)*Y(b)=W(b). The coefficients will be roughly
b^2 each, and the final W(b) will be an addition like,
high low
+-------+-------+
| w4 |
+-------+-------+
+--------+-------+
| w3 |
+--------+-------+
+--------+-------+
| w2 |
+--------+-------+
+--------+-------+
| w1 |
+--------+-------+
+-------+-------+
| w0 |
+-------+-------+
The w[i] coefficients could be formed by a simple set of cross products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but this would need all nine x[i]*y[j] for i,j=0,1,2, and would be equivalent merely to a basecase multiply. Instead the following approach is used.
X(t) and Y(t) are evaluated and multiplied at 5 points, giving values of W(t) at those points. The points used can be chosen in various ways, but in GMP the following are used
Point Value t=0 x0*y0, which gives w0 immediately t=2 (4*x2+2*x1+x0)*(4*y2+2*y1+y0) t=1 (x2+x1+x0)*(y2+y1+y0) t=1/2 (x2+2*x1+4*x0)*(y2+2*y1+4*y0) t=inf x2*y2, which gives w4 immediately
At t=1/2 the value calculated is actually 16*X(1/2)*Y(1/2), giving a value for 16*W(1/2), and this is always an integer. 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).
Now 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 16*W(1/2) = w4 + 2*w3 + 4*w2 + 8*w1 + 16*w0 W(1) = w4 + w3 + w2 + w1 + w0 W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0 W(inf) = w4
This is a set of five equations in five unknowns, and some elementary linear algebra quickly isolates each w[i], by subtracting multiples of one equation from another.
In the code the set of five values W(0),...,W(inf)
will represent those certain linear combinations. By adding or subtracting
one from another as necessary, values which are each w[i] alone are
arrived at. This involves only a few subtractions of small multiples (some of
which are powers of 2), and so is fast. A couple of divisions remain by
powers of 2 and one division by 3 (or by 6 rather), and that last uses the
special mpn_divexact_by3 (see Exact Division).
In the code the values w4, w2 and w0 are formed in the
destination with pointers E, C and A, and w3 and
w1 in temporary space D and B are added to them. There
are extra limbs tD, tC and tB at the high end of
w3, w2 and w1 which are handled separately. The final
addition then is as follows.
high low
+-------+-------+-------+-------+-------+-------+
| E | C | A |
+-------+-------+-------+-------+-------+-------+
+------+-------++------+-------+
| D || B |
+------+-------++------+-------+
-- -- --
|tD| |tC| |tB|
-- -- --
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 can be chosen to make the linear equations come out with a convenient set of steps for isolating the w[i].
In mpn/generic/mul_n.c the interpolate3 routine performs the
interpolation. The open-coded one-pass version may be a bit hard to
understand, the steps performed can be better seen in the USE_MORE_MPN
version.
Squaring follows the same procedure as multiplication, but there's only one
X(t) and it's evaluated at 5 points, and those values squared to give
values of W(t). The interpolation is then identical, and in fact the
same interpolate3 subroutine is used for both squaring and multiplying.
Toom-3 is asymptotically O(N^1.465), the exponent being log(5)/log(3), representing 5 recursive multiplies of 1/3 the original size. This is an improvement over Karatsuba at O(N^1.585), though Toom-Cook does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size.
Near the crossover between Toom-3 and Karatsuba there's generally a range of
sizes where the difference between the two is small.
MUL_TOOM3_THRESHOLD is a somewhat arbitrary point in that range and
successive runs of the tune program can give different values due to small
variations in measuring. A graph of time versus size for the two shows the
effect, see tune/README.
At the fairly small sizes where the Toom-3 thresholds occur it's worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there's a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted.
The formula given above for the Karatsuba algorithm has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening.
An alternate view of Toom-3 can be found in Zuras (see References), using a vector to represent the x and y splits and a matrix multiplication for the evaluation and interpolation stages. The matrix inverses are not meant to be actually used, and they have elements with values much greater than in fact arise in the interpolation steps. The diagram shown for the 3-way is attractive, but again doesn't have to be implemented that way and for example with a bit of rearrangement just one division by 6 can be done.
At large to very large sizes a Fermat style FFT multiplication is used, following Schönhage and Strassen (see References). Descriptions of FFTs in various forms can be found in many textbooks, for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief description of the form used in GMP is given here.
The multiplication done is x*y mod 2^N+1, for a given N. A full product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x and y with high zero limbs. The modular product is the native form for the algorithm, so padding to get a full product is unavoidable.
The algorithm follows a split, evaluate, pointwise multiply, interpolate and
combine similar to that described above for Karatsuba and Toom-3. A k
parameter controls the split, with an FFT-k splitting into 2^k
pieces of M=N/2^k bits each. N must be a multiple of
(2^k)*mp_bits_per_limb so
the split falls on limb boundaries, avoiding bit shifts in the split and
combine stages.
The evaluations, pointwise multiplications, and interpolation, are all done
modulo 2^N'+1 where N' is 2M+k+3 rounded up to a
multiple of 2^k and of mp_bits_per_limb. The results of
interpolation will be the following negacyclic convolution of the input
pieces, and the choice of N' ensures these sums aren't truncated.
---
\ b
w[n] = / (-1) * x[i] * y[j]
---
i+j==b*2^k+n
b=0,1
The points used for the evaluation are g^i for i=0 to 2^k-1 where g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces necessary cancellations at the interpolation stage, and it's also a power of 2 so the fast fourier transforms used for the evaluation and interpolation do only shifts, adds and negations.
The pointwise multiplications are done modulo 2^N'+1 and either recurse into a further FFT or use a plain multiplication (Toom-3, Karatsuba or basecase), whichever is optimal at the size N'. The interpolation is an inverse fast fourier transform. The resulting set of sums of x[i]*y[j] are added at appropriate offsets to give the final result.
Squaring is the same, but x is the only input so it's one transform at the evaluate stage and the pointwise multiplies are squares. The interpolation is the same.
For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm, the exponent representing 2^k recursed
modular multiplies each 1/2^(k-1) the size of the original.
Each successive k is an asymptotic improvement, but overheads mean each
is only faster at bigger and bigger sizes. In the code, MUL_FFT_TABLE
and SQR_FFT_TABLE are the thresholds where each k is used. Each
new k effectively swaps some multiplying for some shifts, adds and
overheads.
A mod 2^N+1 product can be formed with a normal
NxN->2N bit multiply plus a subtraction, so an FFT
and Toom-3 etc can be compared directly. A k=4 FFT at
O(N^1.333) can be expected to be the first faster than Toom-3 at
O(N^1.465). In practice this is what's found, with
MUL_FFT_MODF_THRESHOLD and SQR_FFT_MODF_THRESHOLD being between
300 and 1000 limbs, depending on the CPU. So far it's been found that only
very large FFTs recurse into pointwise multiplies above these sizes.
When an FFT is to give a full product, the change of N to 2N
doesn't alter the theoretical complexity for a given k, but for the
purposes of considering where an FFT might be first used it can be assumed
that the FFT is recursing into a normal multiply and that on that basis it's
doing 2^k recursed multiplies each 1/2^(k-2) the size of
the inputs, making it O(N^(k/(k-2))). This would mean
k=7 at O(N^1.4) would be the first FFT faster than Toom-3.
In practice MUL_FFT_THRESHOLD and SQR_FFT_THRESHOLD have been
found to be in the k=8 range, somewhere between 3000 and 10000 limbs.
The way N is split into 2^k pieces and then 2M+k+3 is
rounded up to a multiple of 2^k and mp_bits_per_limb means that
when 2^k>=mp_bits_per_limb the effective N is a
multiple of 2^(2k-1) bits. The +k+3 means some values of
N just under such a multiple will be rounded to the next. The
complexity calculations above assume that a favourable size is used, meaning
one which isn't padded through rounding, and it's also assumed that the extra
+k+3 bits are negligible at typical FFT sizes.
The practical effect of the 2^(2k-1) constraint is to introduce a
step-effect into measured speeds. For example k=8 will round N
up to a multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb
groups of sizes for which mpn_mul_n runs at the same speed. Or for
k=9 groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In
practice it's been found each k is used at quite small multiples of its
size constraint and so the step effect is quite noticeable in a time versus
size graph.
The threshold determinations currently measure at the mid-points of size
steps, but this is sub-optimal since at the start of a new step it can happen
that it's better to go back to the previous k for a while. Something
more sophisticated for MUL_FFT_TABLE and SQR_FFT_TABLE will be
needed.
The 3-way Toom-Cook algorithm described above (see Toom-Cook 3-Way Multiplication) generalizes to split into an arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C. This is not currently used, though it's possible a Toom-4 might fit in between Toom-3 and the FFTs. The notes here are merely for interest.
In general a split into r+1 pieces is made, and evaluations and pointwise multiplications done at 2*r+1 points. A 4-way split does 7 pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise multiplications count towards big-O complexity, but the time spent in the evaluate and interpolate stages grows with r and has a significant practical impact, with the asymptotic advantage of each r realized only at bigger and bigger sizes. The overheads grow as O(N*r), whereas in an r=2^k FFT they grow only as O(N*log(r)).
Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4 uses -r,...,0,...,r and the latter saves some small multiplies in the evaluate stage (or rather trades them for additions), and has a further saving of nearly half the interpolate steps. The idea is to separate odd and even final coefficients and then perform algorithm C steps C7 and C8 on them separately. The divisors at step C7 become j^2 and the multipliers at C8 become 2*t*j-j^2.
Splitting odd and even parts through positive and negative points can be thought of as using -1 as a square root of unity. If a 4th root of unity was available then a further split and speedup would be possible, but no such root exists for plain integers. Going to complex integers with i=sqrt(-1) doesn't help, essentially because in cartesian form it takes three real multiplies to do a complex multiply. The existence of 2^k'th roots of unity in a suitable ring or field lets the fast fourier transform keep splitting and get to O(N*log(r)).
Floating point FFTs use complex numbers approximating Nth roots of unity. Some processors have special support for such FFTs. But these are not used in GMP since it's very difficult to guarantee an exact result (to some number of bits). An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm, but is of course of vital importance to GMP.
Nx1 division is implemented using repeated 2x1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU.
The multiply by inverse follows section 8 of "Division by Invariant Integers
using Multiplication" by Granlund and Montgomery (see References) and is
implemented as udiv_qrnnd_preinv in gmp-impl.h. The idea is to
have a fixed-point approximation to 1/d (see invert_limb) and
then multiply by the high limb (plus one bit) of the dividend to get a
quotient q. With d normalized (high bit set), q is no
more than 1 too small. Subtracting q*d from the dividend gives a
remainder, and reveals whether q or q-1 is correct.
The result is a division done with two multiplications and four or five arithmetic operations. On CPUs with low latency multipliers this can be much faster than a hardware divide, though the cost of calculating the inverse at the start may mean it's only better on inputs bigger than say 4 or 5 limbs.
When a divisor must be normalized, either for the generic C
__udiv_qrnnd_c or the multiply by inverse, the division performed is
actually a*2^k by d*2^k where a is the dividend and
k is the power necessary to have the high bit of d*2^k set.
The bit shifts for the dividend are usually accomplished "on the fly"
meaning by extracting the appropriate bits at each step. Done this way the
quotient limbs come out aligned ready to store. When only the remainder is
wanted, an alternative is to take the dividend limbs unshifted and calculate
r = a mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can help on CPUs with poor bit shifts or
few registers.
The multiply by inverse can be done two limbs at a time. The calculation is basically the same, but the inverse is two limbs and the divisor treated as if padded with a low zero limb. This means more work, since the inverse will need a 2x2 multiply, but the four 1x1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2x1 calculating q*d. The net effect is to process two limbs with roughly the same two multiplies worth of latency that one limb at a time gives. This extends to 3 or 4 limbs at a time, though the extra work to apply the inverse will almost certainly soon reach the limits of multiplier throughput.
A similar approach in reverse can be taken to process just half a limb at a time if the divisor is only a half limb. In this case the 1x1 multiply for the inverse effectively becomes two (1/2)x1 for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it's not pipelined.
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 Single Limb Division) whichever is faster. Such a quotient is sometimes one too big, requiring an addback of the divisor, but that happens rarely.
With Q=N-M being the number of quotient limbs, this is an O(Q*M) algorithm and will run at a speed similar to a basecase QxM multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs.
For divisors larger than DIV_DC_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 (see References).
The algorithm consists essentially of recognising that a 2NxN division can be done with the basecase division algorithm (see Basecase Division), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)x(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (see Multiplication Algorithms). The "digits" of the quotient are formed by recursive Nx(N/2) divisions.
If the (N/2)x(N/2) multiplies are done with a basecase multiplication
then the work is about the same as a basecase division, but with more function
call overheads and with some subtractions separated from the multiplies.
These overheads mean that it's only when N/2 is above
MUL_KARATSUBA_THRESHOLD that divide and conquer is of use.
DIV_DC_THRESHOLD is based on the divisor size N, so it will be somewhere
above twice MUL_KARATSUBA_THRESHOLD, but how much above depends on the
CPU. An optimized mpn_mul_basecase can lower DIV_DC_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.
Newton's method used for division is asymptotically O(M(N)) and should therefore be superior to divide and conquer, but it's believed this would only be for large to very large N.
A so-called exact division is when the dividend is known to be an exact multiple of the divisor. Jebelean's exact division algorithm uses this knowledge to make some significant optimizations (see References).
The idea can be illustrated in decimal for example with 368154 divided by 543. Because the low digit of the dividend is 4, the low digit of the quotient must be 8. This is arrived at from 4*7 mod 10, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from the dividend leaving 363810. Notice the low digit has become zero.
The procedure is repeated at the second digit, with the next quotient digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving 325800. And finally at the third digit with quotient digit 6 (8*7 mod 10), subtracting 6*543=3258 leaving 0. So the quotient is 678.
Notice however that the multiplies and subtractions don't need to extend past the low three digits of the dividend, since that's enough to determine the three quotient digits. For the last quotient digit no subtraction is needed at all. On a 2NxN division like this one, only about half the work of a normal basecase division is necessary.
For an NxM exact division producing Q=N-M quotient limbs, the saving over a normal basecase division is in two parts. Firstly, each of the Q quotient limbs needs only one multiply, not a 2x1 divide and multiply. Secondly, the crossproducts are reduced when Q>M to Q*M-M*(M+1)/2, or when Q<=M to Q*(Q-1)/2. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number.
The modular inverse used is calculated efficiently by modlimb_invert in
gmp-impl.h. This does four multiplies for a 32-bit limb, or six for a
64-bit limb. tune/modlinv.c has some alternate implementations that
might suit processors better at bit twiddling than multiplying.
The sub-quadratic exact division described by Jebelean in "Exact Division with Karatsuba Complexity" is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (see Divide and Conquer Division), but operating from low to high. A further possibility not currently implemented is "Bidirectional Exact Integer Division" by Krandick and Jebelean which forms quotient limbs from both the high and low ends of the dividend, and can halve once more the number of crossproducts needed in a 2NxN division.
A special case exact division by 3 exists in mpn_divexact_by3,
supporting Toom-3 multiplication and mpq canonicalizations. It forms
quotient digits with a multiply by the modular inverse of 3 (which is
0xAA..AAB) and uses two comparisons to determine a borrow for the next
limb. The multiplications don't need to be on the dependent chain, as long as
the effect of the borrows is applied. Only a few optimized assembler
implementations currently exist.
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, then 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_bdivmod, mpn_divexact_by3, mpn_modexact_1_odd and the
redc function in mpz_powm 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_bdivmod in mpn_gcd, and the use of
mpn_modexact_1_odd by mpn_gcd_1 and mpz_kronecker_ui etc
(see 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 (see 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. Other such
factors include 5, 17 and 257, but no particular use has been found for this.
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.
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 operands a and b using gcd(a,b) = gcd(min(a,b),abs(a-b)), and also that if a and b are first made odd then abs(a-b) is even and factors of two can be discarded.
Variants like letting a-b become negative and doing a different next step are of interest only as far as they suit particular CPUs, since on small operands it's machine dependent factors that determine performance.
The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using a mod b but this has so far been found to be slower 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.
For sizes above GCD_ACCEL_THRESHOLD, GMP uses the Accelerated GCD
algorithm described independently by Weber and Jebelean (the latter as the
"Generalized Binary" algorithm), see References. This algorithm is
still O(N^2), but is much faster than the binary algorithm since it
does fewer multi-precision operations. It consists of alternating the
k-ary reduction by Sorenson, and a "dmod" exact remainder reduction.
For operands u and v the k-ary reduction replaces
u with n*v-d*u where n and d are single limb
values chosen to give two trailing zero limbs on that value, which can be
stripped. n and d are calculated using an algorithm similar to
half of a two limb GCD (see find_a in mpn/generic/gcd.c).
When u and v differ in size by more than a certain number of bits, a dmod is performed to zero out bits at the low end of the larger. It consists of an exact remainder style division applied to an appropriate number of bits (see Exact Division, and see Exact Remainder). This is faster than a k-ary reduction but useful only when the operands differ in size. There's a dmod after each k-ary reduction, and if the dmod leaves the operands still differing in size then it's repeated.
The k-ary reduction step can introduce spurious factors into the GCD calculated, and these are eliminated at the end by taking GCDs with the original inputs gcd(u,gcd(v,g)) using the binary algorithm. Since g is almost always small this takes very little time.
At small sizes the algorithm needs a good implementation of find_a. At
larger sizes it's dominated by mpn_addmul_1 applying n and
d.
The extended GCD calculates gcd(a,b) and also cofactors x and
y satisfying a*x+b*y=gcd(a,b). Lehmer's
multi-step improvement of the extended Euclidean algorithm is used. See Knuth
section 4.5.2 algorithm L, and mpn/generic/gcdext.c. This is an
O(N^2) algorithm.
The multipliers at each step are found using single limb calculations for
sizes up to GCDEXT_THRESHOLD, or double limb calculations above that.
The single limb code is faster but doesn't produce full-limb multipliers,
hence not making full use of the mpn_addmul_1 calls.
When a CPU has a data-dependent multiplier, meaning one which is faster on
operands with fewer bits, the extra work in the double-limb calculation might
only save some looping overheads, leading to a large GCDEXT_THRESHOLD.
Currently the single limb calculation doesn't optimize for the small quotients
that often occur, and this can lead to unusually low values of
GCDEXT_THRESHOLD, depending on the CPU.
An analysis of double-limb calculations can be found in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean (see References). The code in GMP was developed independently.
It should be noted that when a double limb calculation is used, it's used for the whole of that GCD, it doesn't fall back to single limb part way through. This is because as the algorithm proceeds, the inputs a and b are reduced, but the cofactors x and y grow, so the multipliers at each step are applied to a roughly constant total number of limbs.
mpz_jacobi and mpz_kronecker are currently implemented with a
simple binary algorithm similar to that described for the GCDs (see 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.
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.
Modular powering is implemented using a 2^k-ary sliding window algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 (see References). k is chosen according to the size of the exponent. Larger exponents use larger values of k, the choice being made to minimize the average number of multiplications that must supplement the squaring.
The modular multiplies and squares use either a simple division or the REDC
method by Montgomery (see References). REDC is a little faster,
essentially saving N single limb divisions in a fashion similar to an exact
remainder (see Exact Remainder). The current REDC has some limitations.
It's only O(N^2) so above POWM_THRESHOLD division becomes faster
and is used. It doesn't attempt to detect small bases, but rather always uses
a REDC form, which is usually a full size operand. And lastly it's only
applied to odd moduli.
Square roots are taken using the "Karatsuba Square Root" algorithm by Paul Zimmermann (see References). This 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.
In the Karatsuba multiplication range this is an O(1.5*M(N/2)) algorithm, where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
The algorithm does all its calculations in integers and the resulting
mpn_sqrtrem is used for both mpz_sqrt and mpf_sqrt.
The extended precision given by mpf_sqrt_ui is obtained by
padding with zero limbs.
Integer Nth roots are taken using Newton's method with the following
iteration, where A is the input and n is the root to be taken.
1 A
a[i+1] = - * ( --------- + (n-1)*a[i] )
n a[i]^(n-1)
The initial approximation a[1] is generated bitwise by successively powering a trial root with or without new 1 bits, aiming to be just above the true root. The iteration converges quadratically when started from a good approximation. When n is large more initial bits are needed to get good convergence. The current implementation is not particularly well optimized.
mpz_perfect_square_p is able to quickly exclude most non-squares by
checking whether the input is a quadratic residue modulo some small integers.
The first test is modulo 256 which means simply examining the least
significant byte. Only 44 different values occur as the low byte of a square,
so 82.8% of non-squares can be immediately excluded. Similar tests modulo
primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits
then primes up to 53 are used, excluding 99.99%. A single Nx1
remainder using PP from gmp-impl.h quickly gives all these
remainders.
A square root must still be taken for any value that passes the residue tests, to verify it's really a square and not one of the 0.086% (or 0.000156% for 64 bits) non-squares that get through. See Square Root Algorithm.
Detecting perfect powers is required by some factorization algorithms.
Currently mpz_perfect_power_p is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(See Nth Root Algorithm.)
If a prime divisor p with multiplicity e can be found, then only roots which are divisors of e need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked.
Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation.
Conversions from binary to a power-of-2 radix use a simple and fast O(N) bit extraction algorithm.
Conversions from binary to other radices use one of two algorithms. Sizes
below GET_STR_PRECOMPUTE_THRESHOLD use a basic O(N^2) method.
Repeated divisions by b^n are made, where b is the radix and
n is the biggest power that fits in a limb. But instead of simply
using the remainder r from such divisions, an extra divide step is done
to give a fractional limb representing r/b^n. The digits of r
can then be extracted using multiplications by b rather than divisions.
Special case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.
Above GET_STR_PRECOMPUTE_THRESHOLD a sub-quadratic algorithm is used.
For an input t, powers b^(n*2^i) of the radix are
calculated, until a power between t and sqrt(t) is
reached. t is then divided by that largest power, giving a quotient
which is the digits above that power, and a remainder which is those below.
These two parts are in turn divided by the second highest power, and so on
recursively. When a piece has been divided down to less than
GET_STR_DC_THRESHOLD limbs, the basecase algorithm described above is
used.
The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (see Divide and Conquer Division), and big divisions tend to have less overheads than lots of separate single limb divisions anyway. But in any case the cost of calculating the powers b^(n*2^i) must first be overcome.
GET_STR_PRECOMPUTE_THRESHOLD and GET_STR_DC_THRESHOLD represent
the same basic thing, the point where it becomes worth doing a big division to
cut the input in half. GET_STR_PRECOMPUTE_THRESHOLD includes the cost
of calculating the radix power required, whereas GET_STR_DC_THRESHOLD
assumes that's already available, which is the case when recursing.
Since the base case produces digits from least to most significant but they
want to be stored from most to least, it's necessary to calculate in advance
how many digits there will be, or at least be sure not to underestimate that.
For GMP the number of input bits is multiplied by chars_per_bit_exactly
from mp_bases, rounding up. The result is either correct or one too
big.
Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100... and 99... is only in the last few bits and the work to identify 99... might well be almost as much as a full conversion.
mpf_get_str doesn't currently use the algorithm described here, it
multiplies or divides by a power of b to move the radix point to the
just above the highest non-zero digit (or at worst one above that location),
then multiplies by b^n to bring out digits. This is O(N^2) and
is certainly not optimal.
The r/b^n scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn't give a noticable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power.
Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, ie. the highest power would be an b^(n*k) approximately equal to sqrt(t), not restricted to a 2^i factor. That ought to smooth out a graph of times against sizes, but may or may not be a net speedup.
Conversions from a power-of-2 radix into binary use a simple and fast O(N) bitwise concatenation algorithm.
Conversions from other radices use one of two algorithms. Sizes below
SET_STR_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 (see References). Some
special case code is provided for decimal, giving the compiler a chance to
optimize multiplications by 10.
Above SET_STR_THRESHOLD a sub-quadratic algorithm is used. First
groups of n digits are converted into limbs. Then adjacent limbs are
combined into limb pairs with x*b^n+y, where x and y
are the limbs. Adjacent limb pairs are combined into quads similarly with
x*b^(2n)+y. This continues until a single block remains, that
being the result.
The advantage of this method is that the multiplications for each x are
big blocks, allowing Karatsuba and higher algorithms to be used. But the cost
of calculating the powers b^(n*2^i) must be overcome.
SET_STR_THRESHOLD usually ends up quite big, around 5000 digits, and on
some processors much bigger still.
SET_STR_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_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 10, or more).
Factorials n! are calculated by a simple product from 1 to n, but arranged into certain sub-products.
First as many factors as fit in a limb are accumulated, then two of those multiplied to give a 2-limb product. When two 2-limb products are ready they're multiplied to a 4-limb product, and when two 4-limbs are ready they're multiplied to an 8-limb product, etc. A stack of outstanding products is built up, with two of the same size multiplied together when ready.
Arranging for multiplications to have operands the same (or nearly the same) size means the Karatsuba and higher multiplication algorithms can be used. And even on sizes below the Karatsuba threshold an NxN multiply will give a basecase multiply more to work on.
An obvious improvement not currently implemented would be to strip factors of 2 from the products and apply them at the end with a bit shift. Another possibility would be to determine the prime factorization of the result (which can be done easily), and use a powering method, at each stage squaring then multiplying in those primes with a 1 in their exponent at that point. The advantage would be some multiplies turned into squares.
Binomial coefficients C(n,k) are calculated
by first arranging k <= n/2 using C(n,k) = C(n,n-k) if necessary, and then
evaluating the following product simply from i=2 to i=k.
k (n-k+i)
C(n,k) = (n-k+1) * prod -------
i=2 i
It's easy to show that each denominator i will divide the product so far, so the exact division algorithm is used (see Exact Division).
The numerators n-k+i and denominators i are first accumulated
into as many fit a limb, to save multi-precision operations, though for
mpz_bin_ui this applies only to the divisors, since n is an
mpz_t and n-k+i in general won't fit in a limb at all.
An obvious improvement would be to strip factors of 2 from each multiplier and divisor and count them separately, to be applied with a bit shift at the end. Factors of 3 and perhaps 5 could even be handled similarly. Another possibility, if n is not too big, would be to determine the prime factorization of the result based on the factorials involved, and power up those primes appropriately. This would help most when k is near n/2.
The Fibonacci functions mpz_fib_ui and mpz_fib2_ui are designed
for calculating isolated F[n] or F[n],F[n-1]
values efficiently.
For small n, a table of single limb values in __gmp_fib_table is
used. On a 32-bit limb this goes up to F[47], or on a 64-bit limb
up to F[93]. For convenience the table starts at F[-1].
Beyond the table, values are generated with a binary powering algorithm,
calculating a pair F[n] and F[n-1] working from high to
low across the bits of n. The formulas used are
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1]
At each step, k is the high b bits of n. If the next bit of n is 0 then F[2k],F[2k-1] is used, or if it's a 1 then F[2k+1],F[2k] is used, and the process repeated until all bits of n are incorporated. Notice these formulas require just two squares per bit of n.
It'd be possible to handle the first few n above the single limb table with simple additions, using the defining Fibonacci recurrence F[k+1]=F[k]+F[k-1], but this is not done since it usually turns out to be faster for only about 10 or 20 values of n, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table.
Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm.
mpz_fib2_ui returns both F[n] and F[n-1], but
mpz_fib_ui is only interested in F[n]. In this case the last
step of the algorithm can become one multiply instead of two squares. One of
the following two formulas is used, according as n is odd or even.
F[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k
F[2k+1] here is the same as above, just rearranged to be a
multiply. For interest, the 2*(-1)^k term both here and above
can be applied just to the low limb of the calculation, without a carry or
borrow into further limbs, which saves some code size. See comments with
mpz_fib_ui and the internal mpn_fib2_ui for how this is done.
mpz_lucnum2_ui derives a pair of Lucas numbers from a pair of Fibonacci
numbers with the following simple formulas.
L[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1]
mpz_lucnum_ui is only interested in L[n], and some work can be
saved. Trailing zero bits on n can be handled with a single square
each.
L[2k] = L[k]^2 - 2*(-1)^k
And the lowest 1 bit can be handled with one multiply of a pair of Fibonacci
numbers, similar to what mpz_fib_ui does.
L[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^k
The assembler subroutines in GMP are the most significant source of speed at small to moderate sizes. At larger sizes algorithm selection becomes more important, but of course speedups in low level routines will still speed up everything proportionally.
Carry handling and widening multiplies that are important for GMP can't be
easily expressed in C. GCC asm blocks help a lot and are provided in
longlong.h, but hand coding low level routines invariably offers a
speedup over generic C by a factor of anything from 2 to 10.
The various mpn subdirectories contain machine-dependent code, written
in C or assembler. The mpn/generic subdirectory contains default code,
used when there's no machine-specific version of a particular file.
Each mpn subdirectory is for an ISA family. Generally 32-bit and
64-bit variants in a family cannot share code and will have separate
directories. Within a family further subdirectories may exist for CPU
variants.
mpn_addmul_1 and mpn_submul_1 are the most important routines
for overall GMP performance. All multiplications and divisions come down to
repeated calls to these. mpn_add_n, mpn_sub_n,
mpn_lshift and mpn_rshift are next most important.
On some CPUs assembler versions of the internal functions
mpn_mul_basecase and mpn_sqr_basecase give significant speedups,
mainly through avoiding function call overheads. They can also potentially
make better use of a wide superscalar processor.
The restrictions on overlaps between sources and destinations
(see Low-level Functions) are designed to facilitate a variety of
implementations. For example, knowing mpn_add_n won't have partly
overlapping sources and destination means reading can be done far ahead of
writing on superscalar processors, and loops can be vectorized on a vector
processor, depending on the carry handling.
The problem that presents most challenges in GMP is propagating carries from
one limb to the next. In functions like mpn_addmul_1 and
mpn_add_n, carries are the only dependencies between limb operations.
On processors with carry flags, a straightforward CISC style adc is
generally best. AMD K6 mpn_addmul_1 however is an example of an
unusual set of circumstances where a branch works out better.
On RISC processors generally an add and compare for overflow is used. This
sort of thing can be seen in mpn/generic/aors_n.c. Some carry
propagation schemes require 4 instructions, meaning at least 4 cycles per
limb, but other schemes may use just 1 or 2. On wide superscalar processors
performance may be completely determined by the number of dependent
instructions between carry-in and carry-out for each limb.
On vector processors good use can be made of the fact that a carry bit only
very rarely propagates more than one limb. When adding a single bit to a
limb, there's only a carry out if that limb was 0xFF...FF which on
random data will be only 1 in 2^mp_bits_per_limb. mpn/cray/add_n.c is an example of this, it adds
all limbs in parallel, adds one set of carry bits in parallel and then only
rarely needs to fall through to a loop propagating further carries.
On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code
for the RISC style idioms that are necessary to handle carry bits in
C. Often conditional jumps are generated where adc or sbb forms
would be better. And so unfortunately almost any loop involving carry bits
needs to be coded in assembler for best results.
GMP aims to perform well both on operands that fit entirely in L1 cache and those which don't.
Basic routines like mpn_add_n or mpn_lshift are often used on
large operands, so L2 and main memory performance is important for them.
mpn_mul_1 and mpn_addmul_1 are mostly used for multiply and
square basecases, so L1 performance matters most for them, unless assembler
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 which processing the contents of the previous one. Clearly this is only possible if the chip has a lock-up free cache or some sort of prefetch instruction. Most current chips have both these features.
Prefetching sources combines well with loop unrolling, since a prefetch can be initiated once per unrolled loop (or more than once if the loop covers more than one cache line).
On CPUs without write-allocate caches, prefetching destinations will ensure
individual stores don't go further down the cache hierarchy, limiting
bandwidth. Of course for calculations which are slow anyway, like
mpn_divrem_1, write-throughs might be fine.
The distance ahead to prefetch will be determined by memory latency versus throughput. The aim of course is to have data arriving continuously, at peak throughput. Some CPUs have limits on the number of fetches or prefetches in progress.
If a special prefetch instruction doesn't exist then a plain load can be used, but in that case care must be taken not to attempt to read past the end of an operand, since that might produce a segmentation violation.
Some CPUs or systems have hardware that detects sequential memory accesses and initiates suitable cache movements automatically, making life easy.
Floating point arithmetic is used in GMP for multiplications on CPUs with poor
integer multipliers. It's mostly useful for mpn_mul_1,
mpn_addmul_1 and mpn_submul_1 on 64-bit machines, and
mpn_mul_basecase on both 32-bit and 64-bit machines.
With IEEE 53-bit double precision floats, integer multiplications producing up to 53 bits will give exact results. Breaking a 64x64 multiplication into eight 16x32->48 bit pieces is convenient. With some care though six 21x32->53 bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit.
For the mpn_mul_1 family of functions on a 64-bit machine, the
invariant single limb is split at the start, into 3 or 4 pieces. Inside the
loop, the bignum operand is split into 32-bit pieces. Fast conversion of
these unsigned 32-bit pieces to floating point is highly machine-dependent.
In some cases, reading the data into the integer unit, zero-extending to
64-bits, then transferring to the floating point unit back via memory is the
only option.
Converting partial products back to 64-bit limbs is usually best done as a
signed conversion. Since all values are smaller than 2^53, signed
and unsigned are the same, but most processors lack unsigned conversions.
Here is a diagram showing 16x32 bit products for an mpn_mul_1 or
mpn_addmul_1 with a 64-bit limb. The single limb operand V is split
into four 16-bit parts. The multi-limb operand U is split in the loop into
two 32-bit parts.
+---+---+---+---+
|v48|v32|v16|v00| V operand
+---+---+---+---+
+-------+---+---+
x | u32 | u00 | U operand (one limb)
+---------------+
---------------------------------
+-----------+
| u00 x v00 | p00 48-bit products
+-----------+
+-----------+
| u00 x v16 | p16
+-----------+
+-----------+
| u00 x v32 | p32
+-----------+
+-----------+
| u00 x v48 | p48
+-----------+
+-----------+
| u32 x v00 | r32
+-----------+
+-----------+
| u32 x v16 | r48
+-----------+
+-----------+
| u32 x v32 | r64
+-----------+
+-----------+
| u32 x v48 | r80
+-----------+
p32 and r32 can be summed using floating-point addition, and likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from the previous iteration.
For each loop then, four 49-bit quantities are transfered to the integer unit,
aligned as follows,
|-----64bits----|-----64bits----|
+------------+
| p00 + r64' | i00
+------------+
+------------+
| p16 + r80' | i16
+------------+
+------------+
| p32 + r32 | i32
+------------+
+------------+
| p48 + r48 | i48
+------------+
The challenge then is to sum these efficiently and add in a carry limb, generating a low 64-bit result limb and a high 33-bit carry limb (i48 extends 33 bits into the high half).
The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There's generally not much support for propagating the sort of carries that arise in GMP.
SIMD multiplications of say four 16x16 bit multiplies only do as much work as one 32x32 from GMP's point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile.
On the 80x86 chips, MMX has so far found a use in mpn_rshift and
mpn_lshift since it allows 64-bit operations, and is used in a special
case for 16-bit multipliers in the P55 mpn_mul_1. 3DNow and SSE
haven't found a use so far.
Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop taking a checksum of an array of limbs might have a load and an add, but the load wouldn't be for that add, rather for the one next time around the loop. Each load then is effectively scheduled back in the previous iteration, allowing latency to be hidden.
Naturally this is wanted only when doing things like loads or multiplies that take a few cycles to complete, and only where a CPU has multiple functional units so that other work can be done while waiting.
A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling.
Within the loop some moves between registers may be necessary to have the right values in the right places for each iteration. Loop unrolling can help this, with each unrolled block able to use different registers for different values, even if some shuffling is still needed just before going back to the top of the loop.
Loop unrolling consists of replicating code so that several limbs are
processed in each loop. At a minimum this reduces loop overheads by a
corresponding factor, but it can also allow better register usage, for example
alternately using one register combination and then another. Judicious use of
m4 macros can help avoid lots of duplication in the source code.
Unrolling is commonly done to a power of 2 multiple so the number of unrolled loops and the number of remaining limbs can be calculated with a shift and mask. But other multiples can be used too, just by subtracting each n limbs processed from a counter and waiting for less than n remaining (or offsetting the counter by n so it goes negative when there's less than n remaining).
The limbs not a multiple of the unrolling can be handled in various ways, for example
switch statement, providing separate code for each possible excess,
for example an 8-limb unrolling would have separate code for 0 remaining, 1
remaining, etc, up to 7 remaining. This might take a lot of code, but may be
the best way to optimize all cases in combination with a deep pipelined loop.
One way to write the setups and finishups for a pipelined unrolled loop is simply to duplicate the loop at the start and the end, then delete instructions at the start which have no valid antecedents, and delete instructions at the end whose results are unwanted. Sizes not a multiple of the unrolling can then be handled as desired.
This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.
mpz_t variables represent integers using sign and magnitude, in space
dynamically allocated and reallocated. The fields are as follows.
_mp_size
_mp_size set to zero, in which case
the _mp_d data is unused.
_mp_d
mpn functions, so _mp_d[0] is the
least significant limb and _mp_d[ABS(_mp_size)-1] is the most
significant. Whenever _mp_size is non-zero, the most significant limb
is non-zero.
Currently there's always at least one limb allocated, so for instance
mpz_set_ui never needs to reallocate, and mpz_get_ui can fetch
_mp_d[0] unconditionally (though its value is then only wanted if
_mp_size is non-zero).
_mp_alloc
_mp_alloc is the number of limbs currently allocated at _mp_d,
and naturally _mp_alloc >= ABS(_mp_size). When an mpz routine
is about to (or might be about to) increase _mp_size, it checks
_mp_alloc to see whether there's enough space, and reallocates if not.
MPZ_REALLOC is generally used for this.
The various bitwise logical functions like mpz_and behave as if
negative values were twos complement. But sign and magnitude is always used
internally, and necessary adjustments are made during the calculations.
Sometimes this isn't pretty, but sign and magnitude are best for other
routines.
Some internal temporary variables are setup with MPZ_TMP_INIT and these
have _mp_d space obtained from TMP_ALLOC rather than the memory
allocation functions. Care is taken to ensure that these are big enough that
no reallocation is necessary (since it would have unpredictable consequences).
mpq_t variables represent rationals using an mpz_t numerator and
denominator (see Integer Internals).
The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1.
It's believed that casting out common factors at each stage of a calculation
is best in general. A GCD is an O(N^2) operation so it's better to do
a few small ones immediately than to delay and have to do a big one later.
Knowing the numerator and denominator have no common factors can be used for
example in mpq_mul to make only two cross GCDs necessary, not four.
This general approach to common factors is badly sub-optimal in the presence
of simple factorizations or little prospect for cancellation, but GMP has no
way to know when this will occur. As per 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.
Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this.
mpf_t floats have a variable precision mantissa and a single machine
word signed exponent. The mantissa is represented using sign and magnitude.
most least
significant significant
limb limb
_mp_d
|---- _mp_exp ---> |
_____ _____ _____ _____ _____
|_____|_____|_____|_____|_____|
. <------------ radix point
<-------- _mp_size --------->
The fields are as follows.
_mp_size
_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 limbs of result (the most significant being non-zero).
_mp_d
mpn functions, so
_mp_d[0] is the least significant limb and
_mp_d[ABS(_mp_size)-1] the most significant.
The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb.
_mp_prec+1 limbs are allocated to _mp_d, the extra limb being
for convenience (see below). There are no reallocations during a calculation,
only in a change of precision with mpf_set_prec.
_mp_exp
Naturally the exponent can be any value, it doesn't have to fall within the
limbs as the diagram shows, it can be a long way above or a long way below.
Limbs other than those included in the {_mp_d,_mp_size} data
are treated as zero.
The following various points should be noted.
_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.
_mp_size count of limbs in use can be less than _mp_prec if
the value can be represented in less. This means low precision values or
small integers stored in a high precision mpf_t can still be operated
on efficiently.
_mp_size can also be greater than _mp_prec. Firstly a value is
allowed to use all of the _mp_prec+1 limbs available at _mp_d,
and secondly when mpf_set_prec_raw lowers _mp_prec it leaves
_mp_size unchanged and so the size can be arbitrarily bigger than
_mp_prec.
_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.
mpf_add and mpf_mul. When differing exponents are
encountered all that's needed is to adjust pointers to line up the relevant
limbs.
Of course mpf_mul_2exp and mpf_div_2exp will require bit shifts,
but the choice is between an exponent in limbs which requires shifts there, or
one in bits which requires them almost everywhere else.
_mp_prec+1 Limbs
_mp_d (_mp_prec+1 rather than just
_mp_prec) helps when an mpf routine might get a carry from its
operation. mpf_add for instance will do an mpn_add of
_mp_prec limbs. If there's no carry then that's the result, but if
there is a carry then it's stored in the extra limb of space and
_mp_size becomes _mp_prec+1.
Whenever _mp_prec+1 limbs are held in a variable, the low limb is not
needed for the intended precision, only the _mp_prec high limbs. But
zeroing it out or moving the rest down is unnecessary. Subsequent routines
reading the value will simply take the high limbs they need, and this will be
_mp_prec if their target has that same precision. This is no more than
a pointer adjustment, and must be checked anyway since the destination
precision can be different from the sources.
Copy functions like mpf_set will retain a full _mp_prec+1 limbs
if available. This ensures that a variable which has _mp_size equal to
_mp_prec+1 will get its full exact value copied. Strictly speaking
this is unnecessary since only _mp_prec limbs are needed for the
application's requested precision, but it's considered that an mpf_set
from one variable into another of the same precision ought to produce an exact
copy.
__GMPF_BITS_TO_PREC converts an application requested precision to an
_mp_prec. The value in bits is rounded up to a whole limb then an
extra limb is added since the most significant limb of _mp_d is only
non-zero and therefore might contain only one bit.
__GMPF_PREC_TO_BITS does the reverse conversion, and removes the extra
limb from _mp_prec before converting to bits. The net effect of
reading back with mpf_get_prec is simply the precision rounded up to a
multiple of mp_bits_per_limb.
Note that the extra limb added here for the high only being non-zero is in
addition to the extra limb allocated to _mp_d. For example with a
32-bit limb, an application request for 250 bits will be rounded up to 8
limbs, then an extra added for the high being only non-zero, giving an
_mp_prec of 9. _mp_d then gets 10 limbs allocated. Reading
back with mpf_get_prec will take _mp_prec subtract 1 limb and
multiply by 32, giving 256 bits.
Strictly speaking, the fact the high limb has at least one bit means that a
float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but
for the purposes of mpf_t it's considered simply to be 64 bits, a nice
multiple of the limb size.
mpz_out_raw uses the following format.
+------+------------------------+ | size | data bytes | +------+------------------------+
The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first.
The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size.
In GMP 1, leading zero bytes were written to pad the data bytes to a multiple
of the limb size. mpz_inp_raw will still accept this, for
compatibility.
The use of "big endian" for both the size and data fields is deliberate, it
makes the data easy to read in a hex dump of a file. Unfortunately it also
means that the limb data must be reversed when reading or writing, so neither
a big endian nor little endian system can just read and write _mp_d.
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 and
mpfr_class the scheme also ensures the precision of the final
destination is used for any temporaries within a statement like
f=w*x+y*z. These are important features which a naive implementation
cannot provide.
A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code.
To perform an operation, say, addition, we first define a "function object"
evaluating it,
struct __gmp_binary_plus
{
static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); }
};
And an "additive expression" object,
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >
operator+(const mpf_class &f, const mpf_class &g)
{
return __gmp_expr
<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g);
}
The seemingly redundant __gmp_expr<__gmp_binary_expr<...>> is used to
encapsulate any possible kind of expression into a single template type. In
fact even mpf_class etc are typedef specializations of
__gmp_expr.
Next we define assignment of __gmp_expr to mpf_class.
template <class T>
mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr)
{
expr.eval(this->get_mpf_t(), this->precision());
return *this;
}
template <class Op>
void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval
(mpf_t f, unsigned long int precision)
{
Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t());
}
where expr.val1 and expr.val2 are references to the expression's
operands (here expr is the __gmp_binary_expr stored within the
__gmp_expr).
This way, the expression is actually evaluated only at the time of assignment,
when the required precision (that of f) is known. Furthermore the
target mpf_t is now available, thus we can call mpf_add directly
with f as the output argument.
Compound expressions are handled by defining operators taking subexpressions
as their arguments, like this:
template <class T, class U>
__gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2)
{
return __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> >
(expr1, expr2);
}
And the corresponding specializations of __gmp_expr::eval:
template <class T, class U, class Op>
void __gmp_expr
<__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval
(mpf_t f, unsigned long int precision)
{
// declare two temporaries
mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision);
Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t());
}
The expression is thus recursively evaluated to any level of complexity and
all subexpressions are evaluated to the precision of f.
Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it. Several other individuals and organizations have contributed to GMP in various ways. Here is a list in chronological order:
Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library.
Richard Stallman contributed to the interface design and revised the first version of this manual.
Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
mpz_probab_prime_p.
Paul Zimmermann of Inria sparked the development of GMP 2, with his comparisons between bignum packages.
Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul)
contributed mpz_gcd, mpz_divexact, mpn_gcd, and
mpn_bdivmod, partially supported by CNPq (Brazil) grant 301314194-2.
Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases.
Joachim Hollman was involved in the design of the mpf interface, and in
the mpz design revisions for version 2.
Bennet Yee contributed the initial versions of mpz_jacobi and
mpz_legendre.
Andreas Schwab contributed the files mpn/m68k/lshift.S and
mpn/m68k/rshift.S (now in .asm form).
The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving).
GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the IDA Center for Computing Sciences, USA.
Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count.
Robert Harley also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3. He also contributed the ARM assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms.
Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square root. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3.
Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions.
Kent Boortz made the Macintosh port.
Kevin Ryde worked on a 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.
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.
GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin Ryde. Torbjorn's work was partially funded by the IDA Center for Computing Sciences, USA.
(This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell tege@swox.com about the omission!)
Thanks goes to Hans Thorsen for donating an SGI system for the GMP test system environment.
Copyright © 2000 Free Software Foundation, Inc. 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.
The purpose of this License is to make a manual, textbook, or other written document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.
This License is a kind of "copyleft", which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.
We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.
This License applies to any manual or other work that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. The "Document", below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as "you".
A "Modified Version" of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.
A "Secondary Section" is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (For example, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.
The "Invariant Sections" are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License.
The "Cover Texts" are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License.
A "Transparent" copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, whose contents can be viewed and edited directly and straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup has been designed to thwart or discourage subsequent modification by readers is not Transparent. A copy that is not "Transparent" is called "Opaque".
Examples of suitable formats for Transparent copies include plain ASCII without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML designed for human modification. Opaque formats include PostScript, PDF, proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML produced by some word processors for output purposes only.
The "Title Page" means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, "Title Page" means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.
You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.
You may also lend copies, under the same conditions stated above, and you may publicly display copies.
If you publish printed copies of the Document numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.
If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.
If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a publicly-accessible computer-network location containing a complete Transparent copy of the Document, free of added material, which the general network-using public has access to download anonymously at no charge using public-standard network protocols. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.
It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.
You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:
If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.
You may add a section entitled "Endorsements", provided it contains nothing but endorsements of your Modified Version by various parties--for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.
You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.
The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.
You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice.
The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.
In the combination, you must combine any sections entitled "History" in the various original documents, forming one section entitled "History"; likewise combine any sections entitled "Acknowledgments", and any sections entitled "Dedications". You must delete all sections entitled "Endorsements."
You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.
You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.
A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, does not as a whole count as a Modified Version of the Document, provided no compilation copyright is claimed for the compilation. Such a compilation is called an "aggregate", and this License does not apply to the other self-contained works thus compiled with the Document, on account of their being thus compiled, if they are not themselves derivative works of the Document.
If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one quarter of the entire aggregate, the Document's Cover Texts may be placed on covers that surround only the Document within the aggregate. Otherwise they must appear on covers around the whole aggregate.
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License provided that you also include the original English version of this License. In case of a disagreement between the translation and the original English version of this License, the original English version will prevail.
You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.
Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.
To use this License in a document you have written, include a copy of
the License in the document and put the following copyright and
license notices just after the title page:
Copyright (C) year your name. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with the Invariant Sections being list their titles, with the Front-Cover Texts being list, and with the Back-Cover Texts being list. A copy of the license is included in the section entitled ``GNU Free Documentation License''.
If you have no Invariant Sections, write "with no Invariant Sections" instead of saying which ones are invariant. If you have no Front-Cover Texts, write "no Front-Cover Texts" instead of "Front-Cover Texts being list"; likewise for Back-Cover Texts.
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.
alloca: Build Options
istream input: C++ Formatted Input
ostream output: C++ Formatted Output
DESTDIR: Known Build Problems
gmp.h: Headers and Libraries
istream input: C++ Formatted Input
mp.h: BSD Compatible Functions
ostream output: C++ Formatted Output
printf formatted output: Formatted Output
scanf formatted input: Formatted Input
*mpz_export: Integer Import and Export
__GNU_MP_VERSION: Useful Macros and Constants
__GNU_MP_VERSION_MINOR: Useful Macros and Constants
__GNU_MP_VERSION_PATCHLEVEL: Useful Macros and Constants
_mpz_realloc: Initializing Integers
abs: C++ Interface Floats, C++ Interface Rationals, C++ Interface Integers
allocate_function: Custom Allocation
ceil: C++ Interface Floats
cmp: C++ Interface Floats, C++ Interface Rationals, C++ Interface Integers
deallocate_function: Custom Allocation
floor: C++ Interface Floats
gcd: BSD Compatible Functions
gmp_asprintf: Formatted Output Functions
gmp_fprintf: Formatted Output Functions
gmp_fscanf: Formatted Input Functions
GMP_LIMB_BITS: Low-level Functions
GMP_NAIL_BITS: Low-level Functions
GMP_NAIL_MASK: Low-level Functions
GMP_NUMB_BITS: Low-level Functions
GMP_NUMB_MASK: Low-level Functions
GMP_NUMB_MAX: Low-level Functions
gmp_obstack_printf: Formatted Output Functions
gmp_obstack_vprintf: Formatted Output Functions
gmp_printf: Formatted Output Functions
gmp_randclass: C++ Interface Random Numbers
gmp_randclass::get_f: C++ Interface Random Numbers
gmp_randclass::get_z_bits: C++ Interface Random Numbers
gmp_randclass::get_z_range: C++ Interface Random Numbers
gmp_randclass::gmp_randclass: C++ Interface Random Numbers
gmp_randclass::seed: C++ Interface Random Numbers
gmp_randclear: Random State Initialization
gmp_randinit: Random State Initialization
gmp_randinit_default: Random State Initialization
gmp_randinit_lc_2exp: Random State Initialization
gmp_randinit_lc_2exp_size: Random State Initialization
gmp_randseed: Random State Seeding
gmp_randseed_ui: Random State Seeding
gmp_scanf: Formatted Input Functions
gmp_snprintf: Formatted Output Functions
gmp_sprintf: Formatted Output Functions
gmp_sscanf: Formatted Input Functions
gmp_vasprintf: Formatted Output Functions
gmp_version: Useful Macros and Constants
gmp_vfprintf: Formatted Output Functions
gmp_vfscanf: Formatted Input Functions
gmp_vprintf: Formatted Output Functions
gmp_vscanf: Formatted Input Functions
gmp_vsnprintf: Formatted Output Functions
gmp_vsprintf: Formatted Output Functions
gmp_vsscanf: Formatted Input Functions
hypot: C++ Interface Floats
itom: BSD Compatible Functions
madd: BSD Compatible Functions
mcmp: BSD Compatible Functions
mdiv: BSD Compatible Functions
mfree: BSD Compatible Functions
min: BSD Compatible Functions
mout: BSD Compatible Functions
move: BSD Compatible Functions
mp_bits_per_limb: Useful Macros and Constants
mp_limb_tmp_set_memory_functions: Custom Allocation
mpf_abs: Float Arithmetic
mpf_add: Float Arithmetic
mpf_add_ui: Float Arithmetic
mpf_ceil: Miscellaneous Float Functions
mpf_class: C++ Interface General
mpf_class::fits_sint_p: C++ Interface Floats
mpf_class::fits_slong_p: C++ Interface Floats
mpf_class::fits_sshort_p: C++ Interface Floats
mpf_class::fits_uint_p: C++ Interface Floats
mpf_class::fits_ulong_p: C++ Interface Floats
mpf_class::fits_ushort_p: C++ Interface Floats
mpf_class::get_d: C++ Interface Floats
mpf_class::get_mpf_t: C++ Interface General
mpf_class::get_prec: C++ Interface Floats
mpf_class::get_si: C++ Interface Floats
mpf_class::get_ui: C++ Interface Floats
mpf_class::mpf_class: C++ Interface Floats
mpf_class::set_prec: C++ Interface Floats
mpf_class::set_prec_raw: C++ Interface Floats
mpf_clear: Initializing Floats
mpf_cmp: Float Comparison
mpf_cmp_d: Float Comparison
mpf_cmp_si: Float Comparison
mpf_cmp_ui: Float Comparison
mpf_div: Float Arithmetic
mpf_div_2exp: Float Arithmetic
mpf_div_ui: Float Arithmetic
mpf_eq: Float Comparison
mpf_fits_sint_p: Miscellaneous Float Functions
mpf_fits_slong_p: Miscellaneous Float Functions
mpf_fits_sshort_p: Miscellaneous Float Functions
mpf_fits_uint_p: Miscellaneous Float Functions
mpf_fits_ulong_p: Miscellaneous Float Functions
mpf_fits_ushort_p: Miscellaneous Float Functions
mpf_floor: Miscellaneous Float Functions
mpf_get_d: Converting Floats
mpf_get_d_2exp: Converting Floats
mpf_get_default_prec: Initializing Floats
mpf_get_prec: Initializing Floats
mpf_get_si: Converting Floats
mpf_get_str: Converting Floats
mpf_get_ui: Converting Floats
mpf_init: Initializing Floats
mpf_init2: Initializing Floats
mpf_init_set: Simultaneous Float Init & Assign
mpf_init_set_d: Simultaneous Float Init & Assign
mpf_init_set_si: Simultaneous Float Init & Assign
mpf_init_set_str: Simultaneous Float Init & Assign
mpf_init_set_ui: Simultaneous Float Init & Assign
mpf_inp_str: I/O of Floats
mpf_integer_p: Miscellaneous Float Functions
mpf_mul: Float Arithmetic
mpf_mul_2exp: Float Arithmetic
mpf_mul_ui: Float Arithmetic
mpf_neg: Float Arithmetic
mpf_out_str: I/O of Floats
mpf_pow_ui: Float Arithmetic
mpf_random2: Miscellaneous Float Functions
mpf_reldiff: Float Comparison
mpf_set: Assigning Floats
mpf_set_d: Assigning Floats
mpf_set_default_prec: Initializing Floats
mpf_set_prec: Initializing Floats
mpf_set_prec_raw: Initializing Floats
mpf_set_q: Assigning Floats
mpf_set_si: Assigning Floats
mpf_set_str: Assigning Floats
mpf_set_ui: Assigning Floats
mpf_set_z: Assigning Floats
mpf_sgn: Float Comparison
mpf_sqrt: Float Arithmetic
mpf_sqrt_ui: Float Arithmetic
mpf_sub: Float Arithmetic
mpf_sub_ui: Float Arithmetic
mpf_swap: Assigning Floats
mpf_tmpf_trunc: Miscellaneous Float Functions
mpf_ui_div: Float Arithmetic
mpf_ui_sub: Float Arithmetic
mpf_urandomb: Miscellaneous Float Functions
mpfr_class: C++ Interface MPFR
mpn_add: Low-level Functions
mpn_add_1: Low-level Functions
mpn_add_n: Low-level Functions
mpn_addmul_1: Low-level Functions
mpn_bdivmod: Low-level Functions
mpn_cmp: Low-level Functions
mpn_divexact_by3: Low-level Functions
mpn_divexact_by3c: Low-level Functions
mpn_divmod: Low-level Functions
mpn_divmod_1: Low-level Functions
mpn_divrem: Low-level Functions
mpn_divrem_1: Low-level Functions
mpn_gcd: Low-level Functions
mpn_gcd_1: Low-level Functions
mpn_gcdext: Low-level Functions
mpn_get_str: Low-level Functions
mpn_hamdist: Low-level Functions
mpn_lshift: Low-level Functions
mpn_mod_1: Low-level Functions
mpn_mul: Low-level Functions
mpn_mul_1: Low-level Functions
mpn_mul_n: Low-level Functions
mpn_perfect_square_p: Low-level Functions
mpn_popcount: Low-level Functions
mpn_random: Low-level Functions
mpn_random2: Low-level Functions
mpn_rshift: Low-level Functions
mpn_scan0: Low-level Functions
mpn_scan1: Low-level Functions
mpn_set_str: Low-level Functions
mpn_sqrtrem: Low-level Functions
mpn_sub: Low-level Functions
mpn_sub_1: Low-level Functions
mpn_sub_n: Low-level Functions
mpn_submul_1: Low-level Functions
mpn_tdiv_qr: Low-level Functions
mpq_abs: Rational Arithmetic
mpq_add: Rational Arithmetic
mpq_canonicalize: Rational Number Functions
mpq_class: C++ Interface General
mpq_class::canonicalize: C++ Interface Rationals
mpq_class::get_d: C++ Interface Rationals
mpq_class::get_den: C++ Interface Rationals
mpq_class::get_den_mpz_t: C++ Interface Rationals
mpq_class::get_mpq_t: C++ Interface General
mpq_class::get_num: C++ Interface Rationals
mpq_class::get_num_mpz_t: C++ Interface Rationals
mpq_class::mpq_class: C++ Interface Rationals
mpq_clear: Initializing Rationals
mpq_cmp: Comparing Rationals
mpq_cmp_si: Comparing Rationals
mpq_cmp_ui: Comparing Rationals
mpq_denref: Applying Integer Functions
mpq_div: Rational Arithmetic
mpq_div_2exp: Rational Arithmetic
mpq_equal: Comparing Rationals
mpq_get_d: Rational Conversions
mpq_get_den: Applying Integer Functions
mpq_get_num: Applying Integer Functions
mpq_get_str: Rational Conversions
mpq_init: Initializing Rationals
mpq_inp_str: I/O of Rationals
mpq_inv: Rational Arithmetic
mpq_mul: Rational Arithmetic
mpq_mul_2exp: Rational Arithmetic
mpq_neg: Rational Arithmetic
mpq_numref: Applying Integer Functions
mpq_out_str: I/O of Rationals
mpq_set: Initializing Rationals
mpq_set_d: Rational Conversions
mpq_set_den: Applying Integer Functions
mpq_set_f: Rational Conversions
mpq_set_num: Applying Integer Functions
mpq_set_si: Initializing Rationals
mpq_set_str: Initializing Rationals
mpq_set_ui: Initializing Rationals
mpq_set_z: Initializing Rationals
mpq_sgn: Comparing Rationals
mpq_sub: Rational Arithmetic
mpq_swap: Initializing Rationals
mpq_tmpz_abs: Integer Arithmetic
mpz_add: Integer Arithmetic
mpz_add_ui: Integer Arithmetic
mpz_addmul: Integer Arithmetic
mpz_addmul_ui: Integer Arithmetic
mpz_and: Integer Logic and Bit Fiddling
mpz_array_init: Initializing Integers
mpz_bin_ui: Number Theoretic Functions
mpz_bin_uiui: Number Theoretic Functions
mpz_cdiv_q: Integer Division
mpz_cdiv_q_2exp: Integer Division
mpz_cdiv_q_ui: Integer Division
mpz_cdiv_qr: Integer Division
mpz_cdiv_qr_ui: Integer Division
mpz_cdiv_r: Integer Division
mpz_cdiv_r_2exp: Integer Division
mpz_cdiv_r_ui: Integer Division
mpz_cdiv_ui: Integer Division
mpz_class: C++ Interface General
mpz_class::fits_sint_p: C++ Interface Integers
mpz_class::fits_slong_p: C++ Interface Integers
mpz_class::fits_sshort_p: C++ Interface Integers
mpz_class::fits_uint_p: C++ Interface Integers
mpz_class::fits_ulong_p: C++ Interface Integers
mpz_class::fits_ushort_p: C++ Interface Integers
mpz_class::get_d: C++ Interface Integers
mpz_class::get_mpz_t: C++ Interface General
mpz_class::get_si: C++ Interface Integers
mpz_class::get_ui: C++ Interface Integers
mpz_class::mpz_class: C++ Interface Integers
mpz_clear: Initializing Integers
mpz_clrbit: Integer Logic and Bit Fiddling
mpz_cmp: Integer Comparisons
mpz_cmp_d: Integer Comparisons
mpz_cmp_si: Integer Comparisons
mpz_cmp_ui: Integer Comparisons
mpz_cmpabs: Integer Comparisons
mpz_cmpabs_d: Integer Comparisons
mpz_cmpabs_ui: Integer Comparisons
mpz_com: Integer Logic and Bit Fiddling
mpz_congruent_2exp_p: Integer Division
mpz_congruent_p: Integer Division
mpz_congruent_ui_p: Integer Division
mpz_divexact: Integer Division
mpz_divexact_ui: Integer Division
mpz_divisible_2exp_p: Integer Division
mpz_divisible_p: Integer Division
mpz_divisible_ui_p: Integer Division
mpz_even_p: Miscellaneous Integer Functions
mpz_fac_ui: Number Theoretic Functions
mpz_fdiv_q: Integer Division
mpz_fdiv_q_2exp: Integer Division
mpz_fdiv_q_ui: Integer Division
mpz_fdiv_qr: Integer Division
mpz_fdiv_qr_ui: Integer Division
mpz_fdiv_r: Integer Division
mpz_fdiv_r_2exp: Integer Division
mpz_fdiv_r_ui: Integer Division
mpz_fdiv_ui: Integer Division
mpz_fib2_ui: Number Theoretic Functions
mpz_fib_ui: Number Theoretic Functions
mpz_fits_sint_p: Miscellaneous Integer Functions
mpz_fits_slong_p: Miscellaneous Integer Functions
mpz_fits_sshort_p: Miscellaneous Integer Functions
mpz_fits_uint_p: Miscellaneous Integer Functions
mpz_fits_ulong_p: Miscellaneous Integer Functions
mpz_fits_ushort_p: Miscellaneous Integer Functions
mpz_gcd: Number Theoretic Functions
mpz_gcd_ui: Number Theoretic Functions
mpz_gcdext: Number Theoretic Functions
mpz_get_d: Converting Integers
mpz_get_d_2exp: Converting Integers
mpz_get_si: Converting Integers
mpz_get_str: Converting Integers
mpz_get_ui: Converting Integers
mpz_getlimbn: Converting Integers
mpz_hamdist: Integer Logic and Bit Fiddling
mpz_import: Integer Import and Export
mpz_init: Initializing Integers
mpz_init2: Initializing Integers
mpz_init_set: Simultaneous Integer Init & Assign
mpz_init_set_d: Simultaneous Integer Init & Assign
mpz_init_set_si: Simultaneous Integer Init & Assign
mpz_init_set_str: Simultaneous Integer Init & Assign
mpz_init_set_ui: Simultaneous Integer Init & Assign
mpz_inp_raw: I/O of Integers
mpz_inp_str: I/O of Integers
mpz_invert: Number Theoretic Functions
mpz_ior: Integer Logic and Bit Fiddling
mpz_jacobi: Number Theoretic Functions
mpz_kronecker: Number Theoretic Functions
mpz_kronecker_si: Number Theoretic Functions
mpz_kronecker_ui: Number Theoretic Functions
mpz_lcm: Number Theoretic Functions
mpz_lcm_ui: Number Theoretic Functions
mpz_legendre: Number Theoretic Functions
mpz_lucnum2_ui: Number Theoretic Functions
mpz_lucnum_ui: Number Theoretic Functions
mpz_mod: Integer Division
mpz_mod_ui: Integer Division
mpz_mul: Integer Arithmetic
mpz_mul_2exp: Integer Arithmetic
mpz_mul_si: Integer Arithmetic
mpz_mul_ui: Integer Arithmetic
mpz_neg: Integer Arithmetic
mpz_nextprime: Number Theoretic Functions
mpz_odd_p: Miscellaneous Integer Functions
mpz_out_raw: I/O of Integers
mpz_out_str: I/O of Integers
mpz_perfect_power_p: Integer Roots
mpz_perfect_square_p: Integer Roots
mpz_popcount: Integer Logic and Bit Fiddling
mpz_pow_ui: Integer Exponentiation
mpz_powm: Integer Exponentiation
mpz_powm_ui: Integer Exponentiation
mpz_probab_prime_p: Number Theoretic Functions
mpz_random: Integer Random Numbers
mpz_random2: Integer Random Numbers
mpz_realloc2: Initializing Integers
mpz_remove: Number Theoretic Functions
mpz_root: Integer Roots
mpz_rrandomb: Integer Random Numbers
mpz_scan0: Integer Logic and Bit Fiddling
mpz_scan1: Integer Logic and Bit Fiddling
mpz_set: Assigning Integers
mpz_set_d: Assigning Integers
mpz_set_f: Assigning Integers
mpz_set_q: Assigning Integers
mpz_set_si: Assigning Integers
mpz_set_str: Assigning Integers
mpz_set_ui: Assigning Integers
mpz_setbit: Integer Logic and Bit Fiddling
mpz_sgn: Integer Comparisons
mpz_si_kronecker: Number Theoretic Functions
mpz_size: Miscellaneous Integer Functions
mpz_sizeinbase: Miscellaneous Integer Functions
mpz_sqrt: Integer Roots
mpz_sqrtrem: Integer Roots
mpz_sub: Integer Arithmetic
mpz_sub_ui: Integer Arithmetic
mpz_submul: Integer Arithmetic
mpz_submul_ui: Integer Arithmetic
mpz_swap: Assigning Integers
mpz_tmpz_tdiv_q: Integer Division
mpz_tdiv_q_2exp: Integer Division
mpz_tdiv_q_ui: Integer Division
mpz_tdiv_qr: Integer Division
mpz_tdiv_qr_ui: Integer Division
mpz_tdiv_r: Integer Division
mpz_tdiv_r_2exp: Integer Division
mpz_tdiv_r_ui: Integer Division
mpz_tdiv_ui: Integer Division
mpz_tstbit: Integer Logic and Bit Fiddling
mpz_ui_kronecker: Number Theoretic Functions
mpz_ui_pow_ui: Integer Exponentiation
mpz_ui_sub: Integer Arithmetic
mpz_urandomb: Integer Random Numbers
mpz_urandomm: Integer Random Numbers
mpz_xor: Integer Logic and Bit Fiddling
msqrt: BSD Compatible Functions
msub: BSD Compatible Functions
mtox: BSD Compatible Functions
mult: BSD Compatible Functions
operator%: C++ Interface Integers
operator/: C++ Interface Integers
operator<<: C++ Formatted Output
operator>>: C++ Interface Rationals, C++ Formatted Input
pow: BSD Compatible Functions
reallocate_function: Custom Allocation
rpow: BSD Compatible Functions
sdiv: BSD Compatible Functions
sgn: C++ Interface Floats, C++ Interface Rationals, C++ Interface Integers
sqrt: C++ Interface Floats, C++ Interface Integers
trunc: C++ Interface Floats
xtom: BSD Compatible Functions