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This is ../../gmp/doc/gmp.info, produced by makeinfo version 4.13 from
../../gmp/doc/gmp.texi.
This manual describes how to install and use the GNU multiple precision
arithmetic library, version 6.0.0.
Copyright 1991, 1993-2014 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.3 or any later version published by the Free Software Foundation;
with no Invariant Sections, with the Front-Cover Texts being "A GNU
Manual", and with the Back-Cover Texts being "You have freedom to copy
and modify this GNU Manual, like GNU software". A copy of the license
is included in *note GNU Free Documentation License::.
INFO-DIR-SECTION GNU libraries
START-INFO-DIR-ENTRY
* gmp: (gmp). GNU Multiple Precision Arithmetic Library.
END-INFO-DIR-ENTRY

File: gmp.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
GNU MP
******
This manual describes how to install and use the GNU multiple
precision arithmetic library, version 6.0.0.
Copyright 1991, 1993-2014 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.3 or any later version published by the Free Software Foundation;
with no Invariant Sections, with the Front-Cover Texts being "A GNU
Manual", and with the Back-Cover Texts being "You have freedom to copy
and modify this GNU Manual, like GNU software". A copy of the license
is included in *note GNU Free Documentation License::.
* Menu:
* Copying:: GMP Copying Conditions (LGPL).
* Introduction to GMP:: Brief introduction to GNU MP.
* Installing GMP:: How to configure and compile the GMP library.
* GMP Basics:: What every GMP user should know.
* Reporting Bugs:: How to usefully report bugs.
* Integer Functions:: Functions for arithmetic on signed integers.
* Rational Number Functions:: Functions for arithmetic on rational numbers.
* Floating-point Functions:: Functions for arithmetic on floats.
* Low-level Functions:: Fast functions for natural numbers.
* Random Number Functions:: Functions for generating random numbers.
* Formatted Output:: `printf' style output.
* Formatted Input:: `scanf' style input.
* C++ Class Interface:: Class wrappers around GMP types.
* Custom Allocation:: How to customize the internal allocation.
* Language Bindings:: Using GMP from other languages.
* Algorithms:: What happens behind the scenes.
* Internals:: How values are represented behind the scenes.
* Contributors:: Who brings you this library?
* References:: Some useful papers and books to read.
* GNU Free Documentation License::
* Concept Index::
* Function Index::

File: gmp.info, Node: Copying, Next: Introduction to GMP, Prev: Top, Up: Top
GNU MP Copying Conditions
*************************
This library is "free"; this means that everyone is free to use it and
free to redistribute it on a free basis. The library is not in the
public domain; it is copyrighted and there are restrictions on its
distribution, but these restrictions are designed to permit everything
that a good cooperating citizen would want to do. What is not allowed
is to try to prevent others from further sharing any version of this
library that they might get from you.
Specifically, we want to make sure that you have the right to give
away copies of the library, that you receive source code or else can
get it if you want it, that you can change this library or use pieces
of it in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to
deprive anyone else of these rights. For example, if you distribute
copies of the GNU MP library, you must give the recipients all the
rights that you have. You must make sure that they, too, receive or
can get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone
finds out that there is no warranty for the GNU MP library. If it is
modified by someone else and passed on, we want their recipients to
know that what they have is not what we distributed, so that any
problems introduced by others will not reflect on our reputation.
More precisely, the GNU MP library is dual licensed, under the
conditions of the GNU Lesser General Public License version 3 (see
`COPYING.LESSERv3'), or the GNU General Public License version 2 (see
`COPYINGv2'). This is the recipient's choice, and the recipient also has
the additional option of applying later versions of these licenses. (The
reason for this dual licensing is to make it possible to use the
library with programs which are licensed under GPL version 2, but which
for historical or other reasons do not allow use under later versions
of the GPL).
Programs which are not part of the library itself, such as
demonstration programs and the GMP testsuite, are licensed under the
terms of the GNU General Public License version 3 (see `COPYINGv3'), or
any later version.

File: gmp.info, Node: Introduction to GMP, Next: Installing GMP, Prev: Copying, Up: Top
1 Introduction to GNU MP
************************
GNU MP is a portable library written in C for arbitrary precision
arithmetic on integers, rational numbers, and floating-point numbers.
It aims to provide the fastest possible arithmetic for all applications
that need higher precision than is directly supported by the basic C
types.
Many applications use just a few hundred bits of precision; but some
applications may need thousands or even millions of bits. GMP is
designed to give good performance for both, by choosing algorithms
based on the sizes of the operands, and by carefully keeping the
overhead at a minimum.
The speed of GMP is achieved by using fullwords as the basic
arithmetic type, by using sophisticated algorithms, by including
carefully optimized assembly code for the most common inner loops for
many different CPUs, and by a general emphasis on speed (as opposed to
simplicity or elegance).
There is assembly code for these CPUs: ARM Cortex-A9, Cortex-A15,
and generic ARM, DEC Alpha 21064, 21164, and 21264, AMD K8 and K10
(sold under many brands, e.g. Athlon64, Phenom, Opteron) Bulldozer, and
Bobcat, Intel Pentium, Pentium Pro/II/III, Pentium 4, Core2, Nehalem,
Sandy bridge, Haswell, generic x86, Intel IA-64, Motorola/IBM PowerPC
32 and 64 such as POWER970, POWER5, POWER6, and POWER7, MIPS 32-bit and
64-bit, SPARC 32-bit ad 64-bit with special support for all UltraSPARC
models. There is also assembly code for many obsolete CPUs.
For up-to-date information on GMP, please see the GMP web pages at
`https://gmplib.org/'
The latest version of the library is available at
`https://ftp.gnu.org/gnu/gmp/'
Many sites around the world mirror `ftp.gnu.org', please use a mirror
near you, see `https://www.gnu.org/order/ftp.html' for a full list.
There are three public mailing lists of interest. One for release
announcements, one for general questions and discussions about usage of
the GMP library and one for bug reports. For more information, see
`https://gmplib.org/mailman/listinfo/'.
The proper place for bug reports is <gmp-bugs@gmplib.org>. See
*note Reporting Bugs:: for information about reporting bugs.
1.1 How to use this Manual
==========================
Everyone should read *note GMP Basics::. If you need to install the
library yourself, then read *note Installing GMP::. If you have a
system with multiple ABIs, then read *note ABI and ISA::, for the
compiler options that must be used on applications.
The rest of the manual can be used for later reference, although it
is probably a good idea to glance through it.

File: gmp.info, Node: Installing GMP, Next: GMP Basics, Prev: Introduction to GMP, Up: Top
2 Installing GMP
****************
GMP has an autoconf/automake/libtool based configuration system. On a
Unix-like system a basic build can be done with
./configure
make
Some self-tests can be run with
make check
And you can install (under `/usr/local' by default) with
make install
If you experience problems, please report them to
<gmp-bugs@gmplib.org>. See *note Reporting Bugs::, for information on
what to include in useful bug reports.
* Menu:
* Build Options::
* ABI and ISA::
* Notes for Package Builds::
* Notes for Particular Systems::
* Known Build Problems::
* Performance optimization::

File: gmp.info, Node: Build Options, Next: ABI and ISA, Prev: Installing GMP, Up: Installing GMP
2.1 Build Options
=================
All the usual autoconf configure options are available, run `./configure
--help' for a summary. The file `INSTALL.autoconf' has some generic
installation information too.
Tools
`configure' requires various Unix-like tools. See *note Notes for
Particular Systems::, for some options on non-Unix systems.
It might be possible to build without the help of `configure',
certainly all the code is there, but unfortunately you'll be on
your own.
Build Directory
To compile in a separate build directory, `cd' to that directory,
and prefix the configure command with the path to the GMP source
directory. For example
cd /my/build/dir
/my/sources/gmp-6.0.0/configure
Not all `make' programs have the necessary features (`VPATH') to
support this. In particular, SunOS and Slowaris `make' have bugs
that make them unable to build in a separate directory. Use GNU
`make' instead.
`--prefix' and `--exec-prefix'
The `--prefix' option can be used in the normal way to direct GMP
to install under a particular tree. The default is `/usr/local'.
`--exec-prefix' can be used to direct architecture-dependent files
like `libgmp.a' to a different location. This can be used to share
architecture-independent parts like the documentation, but
separate the dependent parts. Note however that `gmp.h' and
`mp.h' are architecture-dependent since they encode certain
aspects of `libgmp', so it will be necessary to ensure both
`$prefix/include' and `$exec_prefix/include' are available to the
compiler.
`--disable-shared', `--disable-static'
By default both shared and static libraries are built (where
possible), but one or other can be disabled. Shared libraries
result in smaller executables and permit code sharing between
separate running processes, but on some CPUs are slightly slower,
having a small cost on each function call.
Native Compilation, `--build=CPU-VENDOR-OS'
For normal native compilation, the system can be specified with
`--build'. By default `./configure' uses the output from running
`./config.guess'. On some systems `./config.guess' can determine
the exact CPU type, on others it will be necessary to give it
explicitly. For example,
./configure --build=ultrasparc-sun-solaris2.7
In all cases the `OS' part is important, since it controls how
libtool generates shared libraries. Running `./config.guess' is
the simplest way to see what it should be, if you don't know
already.
Cross Compilation, `--host=CPU-VENDOR-OS'
When cross-compiling, the system used for compiling is given by
`--build' and the system where the library will run is given by
`--host'. For example when using a FreeBSD Athlon system to build
GNU/Linux m68k binaries,
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnu
Compiler tools are sought first with the host system type as a
prefix. For example `m68k-mac-linux-gnu-ranlib' is tried, then
plain `ranlib'. This makes it possible for a set of
cross-compiling tools to co-exist with native tools. The prefix
is the argument to `--host', and this can be an alias, such as
`m68k-linux'. But note that tools don't have to be setup this
way, it's enough to just have a `PATH' with a suitable
cross-compiling `cc' etc.
Compiling for a different CPU in the same family as the build
system is a form of cross-compilation, though very possibly this
would merely be special options on a native compiler. In any case
`./configure' avoids depending on being able to run code on the
build system, which is important when creating binaries for a
newer CPU since they very possibly won't run on the build system.
In all cases the compiler must be able to produce an executable
(of whatever format) from a standard C `main'. Although only
object files will go to make up `libgmp', `./configure' uses
linking tests for various purposes, such as determining what
functions are available on the host system.
Currently a warning is given unless an explicit `--build' is used
when cross-compiling, because it may not be possible to correctly
guess the build system type if the `PATH' has only a
cross-compiling `cc'.
Note that the `--target' option is not appropriate for GMP. It's
for use when building compiler tools, with `--host' being where
they will run, and `--target' what they'll produce code for.
Ordinary programs or libraries like GMP are only interested in the
`--host' part, being where they'll run. (Some past versions of
GMP used `--target' incorrectly.)
CPU types
In general, if you want a library that runs as fast as possible,
you should configure GMP for the exact CPU type your system uses.
However, this may mean the binaries won't run on older members of
the family, and might run slower on other members, older or newer.
The best idea is always to build GMP for the exact machine type
you intend to run it on.
The following CPUs have specific support. See `configure.ac' for
details of what code and compiler options they select.
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
alphaev6, alphaev67, alphaev68 alphaev7
* Cray: c90, j90, t90, sv1
* HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64
* IA-64: ia64, itanium, itanium2
* MIPS: mips, mips3, mips64
* Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
m68060, m68302, m68360, m88k, m88110
* POWER: power, power1, power2, power2sc
* PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
powerpc821, powerpc823, powerpc860, powerpc970
* SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
* x86 family: i386, i486, i586, pentium, pentiummmx, pentiumpro,
pentium2, pentium3, pentium4, k6, k62, k63, athlon, amd64,
viac3, viac32
* Other: arm, sh, sh2, vax,
CPUs not listed will use generic C code.
Generic C Build
If some of the assembly code causes problems, or if otherwise
desired, the generic C code can be selected with the configure
`--disable-assembly'.
Note that this will run quite slowly, but it should be portable
and should at least make it possible to get something running if
all else fails.
Fat binary, `--enable-fat'
Using `--enable-fat' selects a "fat binary" build on x86, where
optimized low level subroutines are chosen at runtime according to
the CPU detected. This means more code, but gives good
performance on all x86 chips. (This option might become available
for more architectures in the future.)
`ABI'
On some systems GMP supports multiple ABIs (application binary
interfaces), meaning data type sizes and calling conventions. By
default GMP chooses the best ABI available, but a particular ABI
can be selected. For example
./configure --host=mips64-sgi-irix6 ABI=n32
See *note ABI and ISA::, for the available choices on relevant
CPUs, and what applications need to do.
`CC', `CFLAGS'
By default the C compiler used is chosen from among some likely
candidates, with `gcc' normally preferred if it's present. The
usual `CC=whatever' can be passed to `./configure' to choose
something different.
For various systems, default compiler flags are set based on the
CPU and compiler. The usual `CFLAGS="-whatever"' can be passed to
`./configure' to use something different or to set good flags for
systems GMP doesn't otherwise know.
The `CC' and `CFLAGS' used are printed during `./configure', and
can be found in each generated `Makefile'. This is the easiest way
to check the defaults when considering changing or adding
something.
Note that when `CC' and `CFLAGS' are specified on a system
supporting multiple ABIs it's important to give an explicit
`ABI=whatever', since GMP can't determine the ABI just from the
flags and won't be able to select the correct assembly code.
If just `CC' is selected then normal default `CFLAGS' for that
compiler will be used (if GMP recognises it). For example
`CC=gcc' can be used to force the use of GCC, with default flags
(and default ABI).
`CPPFLAGS'
Any flags like `-D' defines or `-I' includes required by the
preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
Compiling is done with both `CPPFLAGS' and `CFLAGS', but
preprocessing uses just `CPPFLAGS'. This distinction is because
most preprocessors won't accept all the flags the compiler does.
Preprocessing is done separately in some configure tests.
`CC_FOR_BUILD'
Some build-time programs are compiled and run to generate
host-specific data tables. `CC_FOR_BUILD' is the compiler used
for this. It doesn't need to be in any particular ABI or mode, it
merely needs to generate executables that can run. The default is
to try the selected `CC' and some likely candidates such as `cc'
and `gcc', looking for something that works.
No flags are used with `CC_FOR_BUILD' because a simple invocation
like `cc foo.c' should be enough. If some particular options are
required they can be included as for instance `CC_FOR_BUILD="cc
-whatever"'.
C++ Support, `--enable-cxx'
C++ support in GMP can be enabled with `--enable-cxx', in which
case a C++ compiler will be required. As a convenience
`--enable-cxx=detect' can be used to enable C++ support only if a
compiler can be found. The C++ support consists of a library
`libgmpxx.la' and header file `gmpxx.h' (*note Headers and
Libraries::).
A separate `libgmpxx.la' has been adopted rather than having C++
objects within `libgmp.la' in order to ensure dynamic linked C
programs aren't bloated by a dependency on the C++ standard
library, and to avoid any chance that the C++ compiler could be
required when linking plain C programs.
`libgmpxx.la' will use certain internals from `libgmp.la' and can
only be expected to work with `libgmp.la' from the same GMP
version. Future changes to the relevant internals will be
accompanied by renaming, so a mismatch will cause unresolved
symbols rather than perhaps mysterious misbehaviour.
In general `libgmpxx.la' will be usable only with the C++ compiler
that built it, since name mangling and runtime support are usually
incompatible between different compilers.
`CXX', `CXXFLAGS'
When C++ support is enabled, the C++ compiler and its flags can be
set with variables `CXX' and `CXXFLAGS' in the usual way. The
default for `CXX' is the first compiler that works from a list of
likely candidates, with `g++' normally preferred when available.
The default for `CXXFLAGS' is to try `CFLAGS', `CFLAGS' without
`-g', then for `g++' either `-g -O2' or `-O2', or for other
compilers `-g' or nothing. Trying `CFLAGS' this way is convenient
when using `gcc' and `g++' together, since the flags for `gcc' will
usually suit `g++'.
It's important that the C and C++ compilers match, meaning their
startup and runtime support routines are compatible and that they
generate code in the same ABI (if there's a choice of ABIs on the
system). `./configure' isn't currently able to check these things
very well itself, so for that reason `--disable-cxx' is the
default, to avoid a build failure due to a compiler mismatch.
Perhaps this will change in the future.
Incidentally, it's normally not good enough to set `CXX' to the
same as `CC'. Although `gcc' for instance recognises `foo.cc' as
C++ code, only `g++' will invoke the linker the right way when
building an executable or shared library from C++ object files.
Temporary Memory, `--enable-alloca=<choice>'
GMP allocates temporary workspace using one of the following three
methods, which can be selected with for instance
`--enable-alloca=malloc-reentrant'.
* `alloca' - C library or compiler builtin.
* `malloc-reentrant' - the heap, in a re-entrant fashion.
* `malloc-notreentrant' - the heap, with global variables.
For convenience, the following choices are also available.
`--disable-alloca' is the same as `no'.
* `yes' - a synonym for `alloca'.
* `no' - a synonym for `malloc-reentrant'.
* `reentrant' - `alloca' if available, otherwise
`malloc-reentrant'. This is the default.
* `notreentrant' - `alloca' if available, otherwise
`malloc-notreentrant'.
`alloca' is reentrant and fast, and is recommended. It actually
allocates just small blocks on the stack; larger ones use
malloc-reentrant.
`malloc-reentrant' is, as the name suggests, reentrant and thread
safe, but `malloc-notreentrant' is faster and should be used if
reentrancy is not required.
The two malloc methods in fact use the memory allocation functions
selected by `mp_set_memory_functions', these being `malloc' and
friends by default. *Note Custom Allocation::.
An additional choice `--enable-alloca=debug' is available, to help
when debugging memory related problems (*note Debugging::).
FFT Multiplication, `--disable-fft'
By default multiplications are done using Karatsuba, 3-way Toom,
higher degree Toom, and Fermat FFT. The FFT is only used on large
to very large operands and can be disabled to save code size if
desired.
Assertion Checking, `--enable-assert'
This option enables some consistency checking within the library.
This can be of use while debugging, *note Debugging::.
Execution Profiling, `--enable-profiling=prof/gprof/instrument'
Enable profiling support, in one of various styles, *note
Profiling::.
`MPN_PATH'
Various assembly versions of each mpn subroutines are provided.
For a given CPU, a search is made though a path to choose a
version of each. For example `sparcv8' has
MPN_PATH="sparc32/v8 sparc32 generic"
which means look first for v8 code, then plain sparc32 (which is
v7), and finally fall back on generic C. Knowledgeable users with
special requirements can specify a different path. Normally this
is completely unnecessary.
Documentation
The source for the document you're now reading is `doc/gmp.texi',
in Texinfo format, see *note Texinfo: (texinfo)Top.
Info format `doc/gmp.info' is included in the distribution. The
usual automake targets are available to make PostScript, DVI, PDF
and HTML (these will require various TeX and Texinfo tools).
DocBook and XML can be generated by the Texinfo `makeinfo' program
too, see *note Options for `makeinfo': (texinfo)makeinfo options.
Some supplementary notes can also be found in the `doc'
subdirectory.

File: gmp.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing GMP
2.2 ABI and ISA
===============
ABI (Application Binary Interface) refers to the calling conventions
between functions, meaning what registers are used and what sizes the
various C data types are. ISA (Instruction Set Architecture) refers to
the instructions and registers a CPU has available.
Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI
defined, the latter for compatibility with older CPUs in the family.
GMP supports some CPUs like this in both ABIs. In fact within GMP
`ABI' means a combination of chip ABI, plus how GMP chooses to use it.
For example in some 32-bit ABIs, GMP may support a limb as either a
32-bit `long' or a 64-bit `long long'.
By default GMP chooses the best ABI available for a given system,
and this generally gives significantly greater speed. But an ABI can
be chosen explicitly to make GMP compatible with other libraries, or
particular application requirements. For example,
./configure ABI=32
In all cases it's vital that all object code used in a given program
is compiled for the same ABI.
Usually a limb is implemented as a `long'. When a `long long' limb
is used this is encoded in the generated `gmp.h'. This is convenient
for applications, but it does mean that `gmp.h' will vary, and can't be
just copied around. `gmp.h' remains compiler independent though, since
all compilers for a particular ABI will be expected to use the same
limb type.
Currently no attempt is made to follow whatever conventions a system
has for installing library or header files built for a particular ABI.
This will probably only matter when installing multiple builds of GMP,
and it might be as simple as configuring with a special `libdir', or it
might require more than that. Note that builds for different ABIs need
to done separately, with a fresh `./configure' and `make' each.
AMD64 (`x86_64')
On AMD64 systems supporting both 32-bit and 64-bit modes for
applications, the following ABI choices are available.
`ABI=64'
The 64-bit ABI uses 64-bit limbs and pointers and makes full
use of the chip architecture. This is the default.
Applications will usually not need special compiler flags,
but for reference the option is
gcc -m64
`ABI=32'
The 32-bit ABI is the usual i386 conventions. This will be
slower, and is not recommended except for inter-operating
with other code not yet 64-bit capable. Applications must be
compiled with
gcc -m32
(In GCC 2.95 and earlier there's no `-m32' option, it's the
only mode.)
`ABI=x32'
The x32 ABI uses 64-bit limbs but 32-bit pointers. Like the
64-bit ABI, it makes full use of the chip's arithmetic
capabilities. This ABI is not supported by all operating
systems.
gcc -mx32
HPPA 2.0 (`hppa2.0*', `hppa64')
`ABI=2.0w'
The 2.0w ABI uses 64-bit limbs and pointers and is available
on HP-UX 11 or up. Applications must be compiled with
gcc [built for 2.0w]
cc +DD64
`ABI=2.0n'
The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal
calling conventions, but with 64-bit instructions permitted
within functions. GMP uses a 64-bit `long long' for a limb.
This ABI is available on hppa64 GNU/Linux and on HP-UX 10 or
higher. Applications must be compiled with
gcc [built for 2.0n]
cc +DA2.0 +e
Note that current versions of GCC (eg. 3.2) don't generate
64-bit instructions for `long long' operations and so may be
slower than for 2.0w. (The GMP assembly code is the same
though.)
`ABI=1.0'
HPPA 2.0 CPUs can run all HPPA 1.0 and 1.1 code in the 32-bit
HPPA 1.0 ABI. No special compiler options are needed for
applications.
All three ABIs are available for CPU types `hppa2.0w', `hppa2.0'
and `hppa64', but for CPU type `hppa2.0n' only 2.0n or 1.0 are
considered.
Note that GCC on HP-UX has no options to choose between 2.0n and
2.0w modes, unlike HP `cc'. Instead it must be built for one or
the other ABI. GMP will detect how it was built, and skip to the
corresponding `ABI'.
IA-64 under HP-UX (`ia64*-*-hpux*', `itanium*-*-hpux*')
HP-UX supports two ABIs for IA-64. GMP performance is the same in
both.
`ABI=32'
In the 32-bit ABI, pointers, `int's and `long's are 32 bits
and GMP uses a 64 bit `long long' for a limb. Applications
can be compiled without any special flags since this ABI is
the default in both HP C and GCC, but for reference the flags
are
gcc -milp32
cc +DD32
`ABI=64'
In the 64-bit ABI, `long's and pointers are 64 bits and GMP
uses a `long' for a limb. Applications must be compiled with
gcc -mlp64
cc +DD64
On other IA-64 systems, GNU/Linux for instance, `ABI=64' is the
only choice.
MIPS under IRIX 6 (`mips*-*-irix[6789]')
IRIX 6 always has a 64-bit MIPS 3 or better CPU, and supports ABIs
o32, n32, and 64. n32 or 64 are recommended, and GMP performance
will be the same in each. The default is n32.
`ABI=o32'
The o32 ABI is 32-bit pointers and integers, and no 64-bit
operations. GMP will be slower than in n32 or 64, this
option only exists to support old compilers, eg. GCC 2.7.2.
Applications can be compiled with no special flags on an old
compiler, or on a newer compiler with
gcc -mabi=32
cc -32
`ABI=n32'
The n32 ABI is 32-bit pointers and integers, but with a
64-bit limb using a `long long'. Applications must be
compiled with
gcc -mabi=n32
cc -n32
`ABI=64'
The 64-bit ABI is 64-bit pointers and integers. Applications
must be compiled with
gcc -mabi=64
cc -64
Note that MIPS GNU/Linux, as of kernel version 2.2, doesn't have
the necessary support for n32 or 64 and so only gets a 32-bit limb
and the MIPS 2 code.
PowerPC 64 (`powerpc64', `powerpc620', `powerpc630', `powerpc970', `power4', `power5')
`ABI=mode64'
The AIX 64 ABI uses 64-bit limbs and pointers and is the
default on PowerPC 64 `*-*-aix*' systems. Applications must
be compiled with
gcc -maix64
xlc -q64
On 64-bit GNU/Linux, BSD, and Mac OS X/Darwin systems, the
applications must be compiled with
gcc -m64
`ABI=mode32'
The `mode32' ABI uses a 64-bit `long long' limb but with the
chip still in 32-bit mode and using 32-bit calling
conventions. This is the default for systems where the true
64-bit ABI is unavailable. No special compiler options are
typically needed for applications. This ABI is not available
under AIX.
`ABI=32'
This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
special compiler options are needed for applications.
GMP's speed is greatest for the `mode64' ABI, the `mode32' ABI is
2nd best. In `ABI=32' only the 32-bit ISA is used and this
doesn't make full use of a 64-bit chip.
Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
`ABI=64'
The 64-bit V9 ABI is available on the various BSD sparc64
ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
and up (when the kernel is in 64-bit mode). GCC 3.2 or
higher, or Sun `cc' is required. On GNU/Linux, depending on
the default `gcc' mode, applications must be compiled with
gcc -m64
On Solaris applications must be compiled with
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc -xarch=v9
On the BSD sparc64 systems no special options are required,
since 64-bits is the only ABI available.
`ABI=32'
For the basic 32-bit ABI, GMP still uses as much of the V9
ISA as it can. In the Sun documentation this combination is
known as "v8plus". On GNU/Linux, depending on the default
`gcc' mode, applications may need to be compiled with
gcc -m32
On Solaris, no special compiler options are required for
applications, though using something like the following is
recommended. (`gcc' 2.8 and earlier only support `-mv8'
though.)
gcc -mv8plus
cc -xarch=v8plus
GMP speed is greatest in `ABI=64', so it's the default where
available. The speed is partly because there are extra registers
available and partly because 64-bits is considered the more
important case and has therefore had better code written for it.
Don't be confused by the names of the `-m' and `-x' compiler
options, they're called `arch' but effectively control both ABI
and ISA.
On Solaris 2.6 and earlier, only `ABI=32' is available since the
kernel doesn't save all registers.
On Solaris 2.7 with the kernel in 32-bit mode, a normal native
build will reject `ABI=64' because the resulting executables won't
run. `ABI=64' can still be built if desired by making it look
like a cross-compile, for example
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64

File: gmp.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing GMP
2.3 Notes for Package Builds
============================
GMP should present no great difficulties for packaging in a binary
distribution.
Libtool is used to build the library and `-version-info' is set
appropriately, having started from `3:0:0' in GMP 3.0 (*note Library
interface versions: (libtool)Versioning.).
The GMP 4 series will be upwardly binary compatible in each release
and will be upwardly binary compatible with all of the GMP 3 series.
Additional function interfaces may be added in each release, so on
systems where libtool versioning is not fully checked by the loader an
auxiliary mechanism may be needed to express that a dynamic linked
application depends on a new enough GMP.
An auxiliary mechanism may also be needed to express that
`libgmpxx.la' (from `--enable-cxx', *note Build Options::) requires
`libgmp.la' from the same GMP version, since this is not done by the
libtool versioning, nor otherwise. A mismatch will result in
unresolved symbols from the linker, or perhaps the loader.
When building a package for a CPU family, care should be taken to use
`--host' (or `--build') to choose the least common denominator among
the CPUs which might use the package. For example this might mean plain
`sparc' (meaning V7) for SPARCs.
For x86s, `--enable-fat' sets things up for a fat binary build,
making a runtime selection of optimized low level routines. This is a
good choice for packaging to run on a range of x86 chips.
Users who care about speed will want GMP built for their exact CPU
type, to make best use of the available optimizations. Providing a way
to suitably rebuild a package may be useful. This could be as simple
as making it possible for a user to omit `--build' (and `--host') so
`./config.guess' will detect the CPU. But a way to manually specify a
`--build' will be wanted for systems where `./config.guess' is inexact.
On systems with multiple ABIs, a packaged build will need to decide
which among the choices is to be provided, see *note ABI and ISA::. A
given run of `./configure' etc will only build one ABI. If a second
ABI is also required then a second run of `./configure' etc must be
made, starting from a clean directory tree (`make distclean').
As noted under "ABI and ISA", currently no attempt is made to follow
system conventions for install locations that vary with ABI, such as
`/usr/lib/sparcv9' for `ABI=64' as opposed to `/usr/lib' for `ABI=32'.
A package build can override `libdir' and other standard variables as
necessary.
Note that `gmp.h' is a generated file, and will be architecture and
ABI dependent. When attempting to install two ABIs simultaneously it
will be important that an application compile gets the correct `gmp.h'
for its desired ABI. If compiler include paths don't vary with ABI
options then it might be necessary to create a `/usr/include/gmp.h'
which tests preprocessor symbols and chooses the correct actual `gmp.h'.

File: gmp.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing GMP
2.4 Notes for Particular Systems
================================
AIX 3 and 4
On systems `*-*-aix[34]*' shared libraries are disabled by
default, since some versions of the native `ar' fail on the
convenience libraries used. A shared build can be attempted with
./configure --enable-shared --disable-static
Note that the `--disable-static' is necessary because in a shared
build libtool makes `libgmp.a' a symlink to `libgmp.so',
apparently for the benefit of old versions of `ld' which only
recognise `.a', but unfortunately this is done even if a fully
functional `ld' is available.
ARM
On systems `arm*-*-*', versions of GCC up to and including 2.95.3
have a bug in unsigned division, giving wrong results for some
operands. GMP `./configure' will demand GCC 2.95.4 or later.
Compaq C++
Compaq C++ on OSF 5.1 has two flavours of `iostream', a standard
one and an old pre-standard one (see `man iostream_intro'). GMP
can only use the standard one, which unfortunately is not the
default but must be selected by defining `__USE_STD_IOSTREAM'.
Configure with for instance
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
Floating Point Mode
On some systems, the hardware floating point has a control mode
which can set all operations to be done in a particular precision,
for instance single, double or extended on x86 systems (x87
floating point). The GMP functions involving a `double' cannot be
expected to operate to their full precision when the hardware is
in single precision mode. Of course this affects all code,
including application code, not just GMP.
FreeBSD 7.x, 8.x, 9.0, 9.1, 9.2
`m4' in these releases of FreeBSD has an eval function which
ignores its 2nd and 3rd arguments, which makes it unsuitable for
`.asm' file processing. `./configure' will detect the problem and
either abort or choose another m4 in the `PATH'. The bug is fixed
in FreeBSD 9.3 and 10.0, so either upgrade or use GNU m4. Note
that the FreeBSD package system installs GNU m4 under the name
`gm4', which GMP cannot guess.
FreeBSD 7.x, 8.x, 9.x
GMP releases starting with 6.0 do not support `ABI=32' on
FreeBSD/amd64 prior to release 10.0 of the system. The cause is a
broken `limits.h', which GMP no longer works around.
MS-DOS and MS Windows
On an MS-DOS system DJGPP can be used to build GMP, and on an MS
Windows system Cygwin, DJGPP and MINGW can be used. All three are
excellent ports of GCC and the various GNU tools.
`http://www.cygwin.com/'
`http://www.delorie.com/djgpp/'
`http://www.mingw.org/'
Microsoft also publishes an Interix "Services for Unix" which can
be used to build GMP on Windows (with a normal `./configure'), but
it's not free software.
MS Windows DLLs
On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
GMP builds only a static library, but a DLL can be built instead
using
./configure --disable-static --enable-shared
Static and DLL libraries can't both be built, since certain export
directives in `gmp.h' must be different.
A MINGW DLL build of GMP can be used with Microsoft C. Libtool
doesn't install a `.lib' format import library, but it can be
created with MS `lib' as follows, and copied to the install
directory. Similarly for `libmp' and `libgmpxx'.
cd .libs
lib /def:libgmp-3.dll.def /out:libgmp-3.lib
MINGW uses the C runtime library `msvcrt.dll' for I/O, so
applications wanting to use the GMP I/O routines must be compiled
with `cl /MD' to do the same. If one of the other C runtime
library choices provided by MS C is desired then the suggestion is
to use the GMP string functions and confine I/O to the application.
Motorola 68k CPU Types
`m68k' is taken to mean 68000. `m68020' or higher will give a
performance boost on applicable CPUs. `m68360' can be used for
CPU32 series chips. `m68302' can be used for "Dragonball" series
chips, though this is merely a synonym for `m68000'.
NetBSD 5.x
`m4' in these releases of NetBSD has an eval function which
ignores its 2nd and 3rd arguments, which makes it unsuitable for
`.asm' file processing. `./configure' will detect the problem and
either abort or choose another m4 in the `PATH'. The bug is fixed
in NetBSD 6, so either upgrade or use GNU m4. Note that the
NetBSD package system installs GNU m4 under the name `gm4', which
GMP cannot guess.
OpenBSD 2.6
`m4' in this release of OpenBSD has a bug in `eval' that makes it
unsuitable for `.asm' file processing. `./configure' will detect
the problem and either abort or choose another m4 in the `PATH'.
The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
Power CPU Types
In GMP, CPU types `power*' and `powerpc*' will each use
instructions not available on the other, so it's important to
choose the right one for the CPU that will be used. Currently GMP
has no assembly code support for using just the common instruction
subset. To get executables that run on both, the current
suggestion is to use the generic C code (`--disable-assembly'),
possibly with appropriate compiler options (like `-mcpu=common' for
`gcc'). CPU `rs6000' (which is not a CPU but a family of
workstations) is accepted by `config.sub', but is currently
equivalent to `--disable-assembly'.
Sparc CPU Types
`sparcv8' or `supersparc' on relevant systems will give a
significant performance increase over the V7 code selected by plain
`sparc'.
Sparc App Regs
The GMP assembly code for both 32-bit and 64-bit Sparc clobbers the
"application registers" `g2', `g3' and `g4', the same way that the
GCC default `-mapp-regs' does (*note SPARC Options: (gcc)SPARC
Options.).
This makes that code unsuitable for use with the special V9
`-mcmodel=embmedany' (which uses `g4' as a data segment pointer),
and for applications wanting to use those registers for special
purposes. In these cases the only suggestion currently is to
build GMP with `--disable-assembly' to avoid the assembly code.
SunOS 4
`/usr/bin/m4' lacks various features needed to process `.asm'
files, and instead `./configure' will automatically use
`/usr/5bin/m4', which we believe is always available (if not then
use GNU m4).
x86 CPU Types
`i586', `pentium' or `pentiummmx' code is good for its intended P5
Pentium chips, but quite slow when run on Intel P6 class chips
(PPro, P-II, P-III). `i386' is a better choice when making
binaries that must run on both.
x86 MMX and SSE2 Code
If the CPU selected has MMX code but the assembler doesn't support
it, a warning is given and non-MMX code is used instead. This
will be an inferior build, since the MMX code that's present is
there because it's faster than the corresponding plain integer
code. The same applies to SSE2.
Old versions of `gas' don't support MMX instructions, in particular
version 1.92.3 that comes with FreeBSD 2.2.8 or the more recent
OpenBSD 3.1 doesn't.
Solaris 2.6 and 2.7 `as' generate incorrect object code for
register to register `movq' instructions, and so can't be used for
MMX code. Install a recent `gas' if MMX code is wanted on these
systems.

File: gmp.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing GMP
2.5 Known Build Problems
========================
You might find more up-to-date information at `https://gmplib.org/'.
Compiler link options
The version of libtool currently in use rather aggressively strips
compiler options when linking a shared library. This will
hopefully be relaxed in the future, but for now if this is a
problem the suggestion is to create a little script to hide them,
and for instance configure with
./configure CC=gcc-with-my-options
DJGPP (`*-*-msdosdjgpp*')
The DJGPP port of `bash' 2.03 is unable to run the `configure'
script, it exits silently, having died writing a preamble to
`config.log'. Use `bash' 2.04 or higher.
`make all' was found to run out of memory during the final
`libgmp.la' link on one system tested, despite having 64Mb
available. Running `make libgmp.la' directly helped, perhaps
recursing into the various subdirectories uses up memory.
GNU binutils `strip' prior to 2.12
`strip' from GNU binutils 2.11 and earlier should not be used on
the static libraries `libgmp.a' and `libmp.a' since it will
discard all but the last of multiple archive members with the same
name, like the three versions of `init.o' in `libgmp.a'. Binutils
2.12 or higher can be used successfully.
The shared libraries `libgmp.so' and `libmp.so' are not affected by
this and any version of `strip' can be used on them.
`make' syntax error
On certain versions of SCO OpenServer 5 and IRIX 6.5 the native
`make' is unable to handle the long dependencies list for
`libgmp.la'. The symptom is a "syntax error" on the following
line of the top-level `Makefile'.
libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)
Either use GNU Make, or as a workaround remove
`$(libgmp_la_DEPENDENCIES)' from that line (which will make the
initial build work, but if any recompiling is done `libgmp.la'
might not be rebuilt).
MacOS X (`*-*-darwin*')
Libtool currently only knows how to create shared libraries on
MacOS X using the native `cc' (which is a modified GCC), not a
plain GCC. A static-only build should work though
(`--disable-shared').
NeXT prior to 3.3
The system compiler on old versions of NeXT was a massacred and
old GCC, even if it called itself `cc'. This compiler cannot be
used to build GMP, you need to get a real GCC, and install that.
(NeXT may have fixed this in release 3.3 of their system.)
POWER and PowerPC
Bugs in GCC 2.7.2 (and 2.6.3) mean it can't be used to compile GMP
on POWER or PowerPC. If you want to use GCC for these machines,
get GCC 2.7.2.1 (or later).
Sequent Symmetry
Use the GNU assembler instead of the system assembler, since the
latter has serious bugs.
Solaris 2.6
The system `sed' prints an error "Output line too long" when
libtool builds `libgmp.la'. This doesn't seem to cause any
obvious ill effects, but GNU `sed' is recommended, to avoid any
doubt.
Sparc Solaris 2.7 with gcc 2.95.2 in `ABI=32'
A shared library build of GMP seems to fail in this combination,
it builds but then fails the tests, apparently due to some
incorrect data relocations within `gmp_randinit_lc_2exp_size'.
The exact cause is unknown, `--disable-shared' is recommended.

File: gmp.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing GMP
2.6 Performance optimization
============================
For optimal performance, build GMP for the exact CPU type of the target
computer, see *note Build Options::.
Unlike what is the case for most other programs, the compiler
typically doesn't matter much, since GMP uses assembly language for the
most critical operation.
In particular for long-running GMP applications, and applications
demanding extremely large numbers, building and running the `tuneup'
program in the `tune' subdirectory, can be important. For example,
cd tune
make tuneup
./tuneup
will generate better contents for the `gmp-mparam.h' parameter file.
To use the results, put the output in the file indicated in the
`Parameters for ...' header. Then recompile from scratch.
The `tuneup' program takes one useful parameter, `-f NNN', which
instructs the program how long to check FFT multiply parameters. If
you're going to use GMP for extremely large numbers, you may want to
run `tuneup' with a large NNN value.

File: gmp.info, Node: GMP Basics, Next: Reporting Bugs, Prev: Installing GMP, Up: Top
3 GMP Basics
************
*Using functions, macros, data types, etc. not documented in this
manual is strongly discouraged. If you do so your application is
guaranteed to be incompatible with future versions of GMP.*
* Menu:
* Headers and Libraries::
* Nomenclature and Types::
* Function Classes::
* Variable Conventions::
* Parameter Conventions::
* Memory Management::
* Reentrancy::
* Useful Macros and Constants::
* Compatibility with older versions::
* Demonstration Programs::
* Efficiency::
* Debugging::
* Profiling::
* Autoconf::
* Emacs::

File: gmp.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: GMP Basics, Up: GMP Basics
3.1 Headers and Libraries
=========================
All declarations needed to use GMP are collected in the include file
`gmp.h'. It is designed to work with both C and C++ compilers.
#include <gmp.h>
Note however that prototypes for GMP functions with `FILE *'
parameters are only provided if `<stdio.h>' is included too.
#include <stdio.h>
#include <gmp.h>
Likewise `<stdarg.h>' is required for prototypes with `va_list'
parameters, such as `gmp_vprintf'. And `<obstack.h>' for prototypes
with `struct obstack' parameters, such as `gmp_obstack_printf', when
available.
All programs using GMP must link against the `libgmp' library. On a
typical Unix-like system this can be done with `-lgmp', for example
gcc myprogram.c -lgmp
GMP C++ functions are in a separate `libgmpxx' library. This is
built and installed if C++ support has been enabled (*note Build
Options::). For example,
g++ mycxxprog.cc -lgmpxx -lgmp
GMP is built using Libtool and an application can use that to link
if desired, *note GNU Libtool: (libtool)Top.
If GMP has been installed to a non-standard location then it may be
necessary to use `-I' and `-L' compiler options to point to the right
directories, and some sort of run-time path for a shared library.

File: gmp.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: GMP Basics
3.2 Nomenclature and Types
==========================
In this manual, "integer" usually means a multiple precision integer, as
defined by the GMP library. The C data type for such integers is
`mpz_t'. Here are some examples of how to declare such integers:
mpz_t sum;
struct foo { mpz_t x, y; };
mpz_t vec[20];
"Rational number" means a multiple precision fraction. The C data
type for these fractions is `mpq_t'. For example:
mpq_t quotient;
"Floating point number" or "Float" for short, is an arbitrary
precision mantissa with a limited precision exponent. The C data type
for such objects is `mpf_t'. For example:
mpf_t fp;
The floating point functions accept and return exponents in the C
type `mp_exp_t'. Currently this is usually a `long', but on some
systems it's an `int' for efficiency.
A "limb" means the part of a multi-precision number that fits in a
single machine word. (We chose this word because a limb of the human
body is analogous to a digit, only larger, and containing several
digits.) Normally a limb is 32 or 64 bits. The C data type for a limb
is `mp_limb_t'.
Counts of limbs of a multi-precision number represented in the C type
`mp_size_t'. Currently this is normally a `long', but on some systems
it's an `int' for efficiency, and on some systems it will be `long
long' in the future.
Counts of bits of a multi-precision number are represented in the C
type `mp_bitcnt_t'. Currently this is always an `unsigned long', but on
some systems it will be an `unsigned long long' in the future.
"Random state" means an algorithm selection and current state data.
The C data type for such objects is `gmp_randstate_t'. For example:
gmp_randstate_t rstate;
Also, in general `mp_bitcnt_t' is used for bit counts and ranges, and
`size_t' is used for byte or character counts.

File: gmp.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: GMP Basics
3.3 Function Classes
====================
There are six classes of functions in the GMP library:
1. Functions for signed integer arithmetic, with names beginning with
`mpz_'. The associated type is `mpz_t'. There are about 150
functions in this class. (*note Integer Functions::)
2. Functions for rational number arithmetic, with names beginning with
`mpq_'. The associated type is `mpq_t'. There are about 35
functions in this class, but the integer functions can be used for
arithmetic on the numerator and denominator separately. (*note
Rational Number Functions::)
3. Functions for floating-point arithmetic, with names beginning with
`mpf_'. The associated type is `mpf_t'. There are about 70
functions is this class. (*note Floating-point Functions::)
4. Fast low-level functions that operate on natural numbers. These
are used by the functions in the preceding groups, and you can
also call them directly from very time-critical user programs.
These functions' names begin with `mpn_'. The associated type is
array of `mp_limb_t'. There are about 60 (hard-to-use) functions
in this class. (*note Low-level Functions::)
5. Miscellaneous functions. Functions for setting up custom
allocation and functions for generating random numbers. (*note
Custom Allocation::, and *note Random Number Functions::)

File: gmp.info, Node: Variable Conventions, Next: Parameter Conventions, Prev: Function Classes, Up: GMP Basics
3.4 Variable Conventions
========================
GMP functions generally have output arguments before input arguments.
This notation is by analogy with the assignment operator. The BSD MP
compatibility functions are exceptions, having the output arguments
last.
GMP lets you use the same variable for both input and output in one
call. For example, the main function for integer multiplication,
`mpz_mul', can be used to square `x' and put the result back in `x' with
mpz_mul (x, x, x);
Before you can assign to a GMP variable, you need to initialize it
by calling one of the special initialization functions. When you're
done with a variable, you need to clear it out, using one of the
functions for that purpose. Which function to use depends on the type
of variable. See the chapters on integer functions, rational number
functions, and floating-point functions for details.
A variable should only be initialized once, or at least cleared
between each initialization. After a variable has been initialized, it
may be assigned to any number of times.
For efficiency reasons, avoid excessive initializing and clearing.
In general, initialize near the start of a function and clear near the
end. For example,
void
foo (void)
{
mpz_t n;
int i;
mpz_init (n);
for (i = 1; i < 100; i++)
{
mpz_mul (n, ...);
mpz_fdiv_q (n, ...);
...
}
mpz_clear (n);
}

File: gmp.info, Node: Parameter Conventions, Next: Memory Management, Prev: Variable Conventions, Up: GMP Basics
3.5 Parameter Conventions
=========================
When a GMP variable is used as a function parameter, it's effectively a
call-by-reference, meaning if the function stores a value there it will
change the original in the caller. Parameters which are input-only can
be designated `const' to provoke a compiler error or warning on
attempting to modify them.
When a function is going to return a GMP result, it should designate
a parameter that it sets, like the library functions do. More than one
value can be returned by having more than one output parameter, again
like the library functions. A `return' of an `mpz_t' etc doesn't
return the object, only a pointer, and this is almost certainly not
what's wanted.
Here's an example accepting an `mpz_t' parameter, doing a
calculation, and storing the result to the indicated parameter.
void
foo (mpz_t result, const mpz_t param, unsigned long n)
{
unsigned long i;
mpz_mul_ui (result, param, n);
for (i = 1; i < n; i++)
mpz_add_ui (result, result, i*7);
}
int
main (void)
{
mpz_t r, n;
mpz_init (r);
mpz_init_set_str (n, "123456", 0);
foo (r, n, 20L);
gmp_printf ("%Zd\n", r);
return 0;
}
`foo' works even if the mainline passes the same variable for
`param' and `result', just like the library functions. But sometimes
it's tricky to make that work, and an application might not want to
bother supporting that sort of thing.
For interest, the GMP types `mpz_t' etc are implemented as
one-element arrays of certain structures. This is why declaring a
variable creates an object with the fields GMP needs, but then using it
as a parameter passes a pointer to the object. Note that the actual
fields in each `mpz_t' etc are for internal use only and should not be
accessed directly by code that expects to be compatible with future GMP
releases.

File: gmp.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: GMP Basics
3.6 Memory Management
=====================
The GMP types like `mpz_t' are small, containing only a couple of sizes,
and pointers to allocated data. Once a variable is initialized, GMP
takes care of all space allocation. Additional space is allocated
whenever a variable doesn't have enough.
`mpz_t' and `mpq_t' variables never reduce their allocated space.
Normally this is the best policy, since it avoids frequent reallocation.
Applications that need to return memory to the heap at some particular
point can use `mpz_realloc2', or clear variables no longer needed.
`mpf_t' variables, in the current implementation, use a fixed amount
of space, determined by the chosen precision and allocated at
initialization, so their size doesn't change.
All memory is allocated using `malloc' and friends by default, but
this can be changed, see *note Custom Allocation::. Temporary memory
on the stack is also used (via `alloca'), but this can be changed at
build-time if desired, see *note Build Options::.

File: gmp.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: GMP Basics
3.7 Reentrancy
==============
GMP is reentrant and thread-safe, with some exceptions:
* If configured with `--enable-alloca=malloc-notreentrant' (or with
`--enable-alloca=notreentrant' when `alloca' is not available),
then naturally GMP is not reentrant.
* `mpf_set_default_prec' and `mpf_init' use a global variable for the
selected precision. `mpf_init2' can be used instead, and in the
C++ interface an explicit precision to the `mpf_class' constructor.
* `mpz_random' and the other old random number functions use a global
random state and are hence not reentrant. The newer random number
functions that accept a `gmp_randstate_t' parameter can be used
instead.
* `gmp_randinit' (obsolete) returns an error indication through a
global variable, which is not thread safe. Applications are
advised to use `gmp_randinit_default' or `gmp_randinit_lc_2exp'
instead.
* `mp_set_memory_functions' uses global variables to store the
selected memory allocation functions.
* If the memory allocation functions set by a call to
`mp_set_memory_functions' (or `malloc' and friends by default) are
not reentrant, then GMP will not be reentrant either.
* If the standard I/O functions such as `fwrite' are not reentrant
then the GMP I/O functions using them will not be reentrant either.
* It's safe for two threads to read from the same GMP variable
simultaneously, but it's not safe for one to read while another
might be writing, nor for two threads to write simultaneously.
It's not safe for two threads to generate a random number from the
same `gmp_randstate_t' simultaneously, since this involves an
update of that variable.

File: gmp.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: GMP Basics
3.8 Useful Macros and Constants
===============================
-- Global Constant: const int mp_bits_per_limb
The number of bits per limb.
-- Macro: __GNU_MP_VERSION
-- Macro: __GNU_MP_VERSION_MINOR
-- Macro: __GNU_MP_VERSION_PATCHLEVEL
The major and minor GMP version, and patch level, respectively, as
integers. For GMP i.j, these numbers will be i, j, and 0,
respectively. For GMP i.j.k, these numbers will be i, j, and k,
respectively.
-- Global Constant: const char * const gmp_version
The GMP version number, as a null-terminated string, in the form
"i.j.k". This release is "6.0.0". Note that the format "i.j" was
used, before version 4.3.0, when k was zero.
-- Macro: __GMP_CC
-- Macro: __GMP_CFLAGS
The compiler and compiler flags, respectively, used when compiling
GMP, as strings.

File: gmp.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: GMP Basics
3.9 Compatibility with older versions
=====================================
This version of GMP is upwardly binary compatible with all 5.x, 4.x,
and 3.x versions, and upwardly compatible at the source level with all
2.x versions, with the following exceptions.
* `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
consistency with other `mpn' functions.
* `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.
* `mpn_bdivmod', documented as preliminary in GMP 4, has been
removed.
There are a number of compatibility issues between GMP 1 and GMP 2
that of course also apply when porting applications from GMP 1 to GMP
5. Please see the GMP 2 manual for details.

File: gmp.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: GMP Basics
3.10 Demonstration programs
===========================
The `demos' subdirectory has some sample programs using GMP. These
aren't built or installed, but there's a `Makefile' with rules for them.
For instance,
make pexpr
./pexpr 68^975+10
The following programs are provided
* `pexpr' is an expression evaluator, the program used on the GMP
web page.
* The `calc' subdirectory has a similar but simpler evaluator using
`lex' and `yacc'.
* The `expr' subdirectory is yet another expression evaluator, a
library designed for ease of use within a C program. See
`demos/expr/README' for more information.
* `factorize' is a Pollard-Rho factorization program.
* `isprime' is a command-line interface to the `mpz_probab_prime_p'
function.
* `primes' counts or lists primes in an interval, using a sieve.
* `qcn' is an example use of `mpz_kronecker_ui' to estimate quadratic
class numbers.
* The `perl' subdirectory is a comprehensive perl interface to GMP.
See `demos/perl/INSTALL' for more information. Documentation is
in POD format in `demos/perl/GMP.pm'.
As an aside, consideration has been given at various times to some
sort of expression evaluation within the main GMP library. Going
beyond something minimal quickly leads to matters like user-defined
functions, looping, fixnums for control variables, etc, which are
considered outside the scope of GMP (much closer to language
interpreters or compilers, *Note Language Bindings::.) Something
simple for program input convenience may yet be a possibility, a
combination of the `expr' demo and the `pexpr' tree back-end perhaps.
But for now the above evaluators are offered as illustrations.

File: gmp.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: GMP Basics
3.11 Efficiency
===============
Small Operands
On small operands, the time for function call overheads and memory
allocation can be significant in comparison to actual calculation.
This is unavoidable in a general purpose variable precision
library, although GMP attempts to be as efficient as it can on
both large and small operands.
Static Linking
On some CPUs, in particular the x86s, the static `libgmp.a' should
be used for maximum speed, since the PIC code in the shared
`libgmp.so' will have a small overhead on each function call and
global data address. For many programs this will be
insignificant, but for long calculations there's a gain to be had.
Initializing and Clearing
Avoid excessive initializing and clearing of variables, since this
can be quite time consuming, especially in comparison to otherwise
fast operations like addition.
A language interpreter might want to keep a free list or stack of
initialized variables ready for use. It should be possible to
integrate something like that with a garbage collector too.
Reallocations
An `mpz_t' or `mpq_t' variable used to hold successively increasing
values will have its memory repeatedly `realloc'ed, which could be
quite slow or could fragment memory, depending on the C library.
If an application can estimate the final size then `mpz_init2' or
`mpz_realloc2' can be called to allocate the necessary space from
the beginning (*note Initializing Integers::).
It doesn't matter if a size set with `mpz_init2' or `mpz_realloc2'
is too small, since all functions will do a further reallocation
if necessary. Badly overestimating memory required will waste
space though.
`2exp' Functions
It's up to an application to call functions like `mpz_mul_2exp'
when appropriate. General purpose functions like `mpz_mul' make
no attempt to identify powers of two or other special forms,
because such inputs will usually be very rare and testing every
time would be wasteful.
`ui' and `si' Functions
The `ui' functions and the small number of `si' functions exist for
convenience and should be used where applicable. But if for
example an `mpz_t' contains a value that fits in an `unsigned
long' there's no need extract it and call a `ui' function, just
use the regular `mpz' function.
In-Place Operations
`mpz_abs', `mpq_abs', `mpf_abs', `mpz_neg', `mpq_neg' and
`mpf_neg' are fast when used for in-place operations like
`mpz_abs(x,x)', since in the current implementation only a single
field of `x' needs changing. On suitable compilers (GCC for
instance) this is inlined too.
`mpz_add_ui', `mpz_sub_ui', `mpf_add_ui' and `mpf_sub_ui' benefit
from an in-place operation like `mpz_add_ui(x,x,y)', since usually
only one or two limbs of `x' will need to be changed. The same
applies to the full precision `mpz_add' etc if `y' is small. If
`y' is big then cache locality may be helped, but that's all.
`mpz_mul' is currently the opposite, a separate destination is
slightly better. A call like `mpz_mul(x,x,y)' will, unless `y' is
only one limb, make a temporary copy of `x' before forming the
result. Normally that copying will only be a tiny fraction of the
time for the multiply, so this is not a particularly important
consideration.
`mpz_set', `mpq_set', `mpq_set_num', `mpf_set', etc, make no
attempt to recognise a copy of something to itself, so a call like
`mpz_set(x,x)' will be wasteful. Naturally that would never be
written deliberately, but if it might arise from two pointers to
the same object then a test to avoid it might be desirable.
if (x != y)
mpz_set (x, y);
Note that it's never worth introducing extra `mpz_set' calls just
to get in-place operations. If a result should go to a particular
variable then just direct it there and let GMP take care of data
movement.
Divisibility Testing (Small Integers)
`mpz_divisible_ui_p' and `mpz_congruent_ui_p' are the best
functions for testing whether an `mpz_t' is divisible by an
individual small integer. They use an algorithm which is faster
than `mpz_tdiv_ui', but which gives no useful information about
the actual remainder, only whether it's zero (or a particular
value).
However when testing divisibility by several small integers, it's
best to take a remainder modulo their product, to save
multi-precision operations. For instance to test whether a number
is divisible by any of 23, 29 or 31 take a remainder modulo
23*29*31 = 20677 and then test that.
The division functions like `mpz_tdiv_q_ui' which give a quotient
as well as a remainder are generally a little slower than the
remainder-only functions like `mpz_tdiv_ui'. If the quotient is
only rarely wanted then it's probably best to just take a
remainder and then go back and calculate the quotient if and when
it's wanted (`mpz_divexact_ui' can be used if the remainder is
zero).
Rational Arithmetic
The `mpq' functions operate on `mpq_t' values with no common
factors in the numerator and denominator. Common factors are
checked-for and cast out as necessary. In general, cancelling
factors every time is the best approach since it minimizes the
sizes for subsequent operations.
However, applications that know something about the factorization
of the values they're working with might be able to avoid some of
the GCDs used for canonicalization, or swap them for divisions.
For example when multiplying by a prime it's enough to check for
factors of it in the denominator instead of doing a full GCD. Or
when forming a big product it might be known that very little
cancellation will be possible, and so canonicalization can be left
to the end.
The `mpq_numref' and `mpq_denref' macros give access to the
numerator and denominator to do things outside the scope of the
supplied `mpq' functions. *Note Applying Integer Functions::.
The canonical form for rationals allows mixed-type `mpq_t' and
integer additions or subtractions to be done directly with
multiples of the denominator. This will be somewhat faster than
`mpq_add'. For example,
/* mpq increment */
mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q));
/* mpq += unsigned long */
mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL);
/* mpq -= mpz */
mpz_submul (mpq_numref(q), mpq_denref(q), z);
Number Sequences
Functions like `mpz_fac_ui', `mpz_fib_ui' and `mpz_bin_uiui' are
designed for calculating isolated values. If a range of values is
wanted it's probably best to call to get a starting point and
iterate from there.
Text Input/Output
Hexadecimal or octal are suggested for input or output in text
form. Power-of-2 bases like these can be converted much more
efficiently than other bases, like decimal. For big numbers
there's usually nothing of particular interest to be seen in the
digits, so the base doesn't matter much.
Maybe we can hope octal will one day become the normal base for
everyday use, as proposed by King Charles XII of Sweden and later
reformers.

File: gmp.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: GMP Basics
3.12 Debugging
==============
Stack Overflow
Depending on the system, a segmentation violation or bus error
might be the only indication of stack overflow. See
`--enable-alloca' choices in *note Build Options::, for how to
address this.
In new enough versions of GCC, `-fstack-check' may be able to
ensure an overflow is recognised by the system before too much
damage is done, or `-fstack-limit-symbol' or
`-fstack-limit-register' may be able to add checking if the system
itself doesn't do any (*note Options for Code Generation:
(gcc)Code Gen Options.). These options must be added to the
`CFLAGS' used in the GMP build (*note Build Options::), adding
them just to an application will have no effect. Note also
they're a slowdown, adding overhead to each function call and each
stack allocation.
Heap Problems
The most likely cause of application problems with GMP is heap
corruption. Failing to `init' GMP variables will have
unpredictable effects, and corruption arising elsewhere in a
program may well affect GMP. Initializing GMP variables more than
once or failing to clear them will cause memory leaks.
In all such cases a `malloc' debugger is recommended. On a GNU or
BSD system the standard C library `malloc' has some diagnostic
facilities, see *note Allocation Debugging: (libc)Allocation
Debugging, or `man 3 malloc'. Other possibilities, in no
particular order, include
`http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/'
`http://dmalloc.com/'
`http://www.perens.com/FreeSoftware/' (electric fence)
`http://packages.debian.org/stable/devel/fda'
`http://www.gnupdate.org/components/leakbug/'
`http://people.redhat.com/~otaylor/memprof/'
`http://www.cbmamiga.demon.co.uk/mpatrol/'
The GMP default allocation routines in `memory.c' also have a
simple sentinel scheme which can be enabled with `#define DEBUG'
in that file. This is mainly designed for detecting buffer
overruns during GMP development, but might find other uses.
Stack Backtraces
On some systems the compiler options GMP uses by default can
interfere with debugging. In particular on x86 and 68k systems
`-fomit-frame-pointer' is used and this generally inhibits stack
backtracing. Recompiling without such options may help while
debugging, though the usual caveats about it potentially moving a
memory problem or hiding a compiler bug will apply.
GDB, the GNU Debugger
A sample `.gdbinit' is included in the distribution, showing how
to call some undocumented dump functions to print GMP variables
from within GDB. Note that these functions shouldn't be used in
final application code since they're undocumented and may be
subject to incompatible changes in future versions of GMP.
Source File Paths
GMP has multiple source files with the same name, in different
directories. For example `mpz', `mpq' and `mpf' each have an
`init.c'. If the debugger can't already determine the right one
it may help to build with absolute paths on each C file. One way
to do that is to use a separate object directory with an absolute
path to the source directory.
cd /my/build/dir
/my/source/dir/gmp-6.0.0/configure
This works via `VPATH', and might require GNU `make'. Alternately
it might be possible to change the `.c.lo' rules appropriately.
Assertion Checking
The build option `--enable-assert' is available to add some
consistency checks to the library (see *note Build Options::).
These are likely to be of limited value to most applications.
Assertion failures are just as likely to indicate memory
corruption as a library or compiler bug.
Applications using the low-level `mpn' functions, however, will
benefit from `--enable-assert' since it adds checks on the
parameters of most such functions, many of which have subtle
restrictions on their usage. Note however that only the generic C
code has checks, not the assembly code, so `--disable-assembly'
should be used for maximum checking.
Temporary Memory Checking
The build option `--enable-alloca=debug' arranges that each block
of temporary memory in GMP is allocated with a separate call to
`malloc' (or the allocation function set with
`mp_set_memory_functions').
This can help a malloc debugger detect accesses outside the
intended bounds, or detect memory not released. In a normal
build, on the other hand, temporary memory is allocated in blocks
which GMP divides up for its own use, or may be allocated with a
compiler builtin `alloca' which will go nowhere near any malloc
debugger hooks.
Maximum Debuggability
To summarize the above, a GMP build for maximum debuggability
would be
./configure --disable-shared --enable-assert \
--enable-alloca=debug --disable-assembly CFLAGS=-g
For C++, add `--enable-cxx CXXFLAGS=-g'.
Checker
The GCC checker (`https://savannah.nongnu.org/projects/checker/')
can be used with GMP. It contains a stub library which means GMP
applications compiled with checker can use a normal GMP build.
A build of GMP with checking within GMP itself can be made. This
will run very very slowly. On GNU/Linux for example,
./configure --disable-assembly CC=checkergcc
`--disable-assembly' must be used, since the GMP assembly code
doesn't support the checking scheme. The GMP C++ features cannot
be used, since current versions of checker (0.9.9.1) don't yet
support the standard C++ library.
Valgrind
Valgrind (`http://valgrind.org/') is a memory checker for x86,
ARM, MIPS, PowerPC, and S/390. It translates and emulates machine
instructions to do strong checks for uninitialized data (at the
level of individual bits), memory accesses through bad pointers,
and memory leaks.
Valgrind does not always support every possible instruction, in
particular ones recently added to an ISA. Valgrind might
therefore be incompatible with a recent GMP or even a less recent
GMP which is compiled using a recent GCC.
GMP's assembly code sometimes promotes a read of the limbs to some
larger size, for efficiency. GMP will do this even at the start
and end of a multilimb operand, using naturally aligned operations
on the larger type. This may lead to benign reads outside of
allocated areas, triggering complaints from Valgrind. Valgrind's
option `--partial-loads-ok=yes' should help.
Other Problems
Any suspected bug in GMP itself should be isolated to make sure
it's not an application problem, see *note Reporting Bugs::.

File: gmp.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: GMP Basics
3.13 Profiling
==============
Running a program under a profiler is a good way to find where it's
spending most time and where improvements can be best sought. The
profiling choices for a GMP build are as follows.
`--disable-profiling'
The default is to add nothing special for profiling.
It should be possible to just compile the mainline of a program
with `-p' and use `prof' to get a profile consisting of
timer-based sampling of the program counter. Most of the GMP
assembly code has the necessary symbol information.
This approach has the advantage of minimizing interference with
normal program operation, but on most systems the resolution of
the sampling is quite low (10 milliseconds for instance),
requiring long runs to get accurate information.
`--enable-profiling=prof'
Build with support for the system `prof', which means `-p' added
to the `CFLAGS'.
This provides call counting in addition to program counter
sampling, which allows the most frequently called routines to be
identified, and an average time spent in each routine to be
determined.
The x86 assembly code has support for this option, but on other
processors the assembly routines will be as if compiled without
`-p' and therefore won't appear in the call counts.
On some systems, such as GNU/Linux, `-p' in fact means `-pg' and in
this case `--enable-profiling=gprof' described below should be used
instead.
`--enable-profiling=gprof'
Build with support for `gprof', which means `-pg' added to the
`CFLAGS'.
This provides call graph construction in addition to call counting
and program counter sampling, which makes it possible to count
calls coming from different locations. For example the number of
calls to `mpn_mul' from `mpz_mul' versus the number from
`mpf_mul'. The program counter sampling is still flat though, so
only a total time in `mpn_mul' would be accumulated, not a
separate amount for each call site.
The x86 assembly code has support for this option, but on other
processors the assembly routines will be as if compiled without
`-pg' and therefore not be included in the call counts.
On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
incompatible, so the latter is omitted from the default flags in
that case, which might result in poorer code generation.
Incidentally, it should be possible to use the `gprof' program
with a plain `--enable-profiling=prof' build. But in that case
only the `gprof -p' flat profile and call counts can be expected
to be valid, not the `gprof -q' call graph.
`--enable-profiling=instrument'
Build with the GCC option `-finstrument-functions' added to the
`CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
Options.).
This inserts special instrumenting calls at the start and end of
each function, allowing exact timing and full call graph
construction.
This instrumenting is not normally a standard system feature and
will require support from an external library, such as
`http://sourceforge.net/projects/fnccheck/'
This should be included in `LIBS' during the GMP configure so that
test programs will link. For example,
./configure --enable-profiling=instrument LIBS=-lfc
On a GNU system the C library provides dummy instrumenting
functions, so programs compiled with this option will link. In
this case it's only necessary to ensure the correct library is
added when linking an application.
The x86 assembly code supports this option, but on other
processors the assembly routines will be as if compiled without
`-finstrument-functions' meaning time spent in them will
effectively be attributed to their caller.

File: gmp.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: GMP Basics
3.14 Autoconf
=============
Autoconf based applications can easily check whether GMP is installed.
The only thing to be noted is that GMP library symbols from version 3
onwards have prefixes like `__gmpz'. The following therefore would be
a simple test,
AC_CHECK_LIB(gmp, __gmpz_init)
This just uses the default `AC_CHECK_LIB' actions for found or not
found, but an application that must have GMP would want to generate an
error if not found. For example,
AC_CHECK_LIB(gmp, __gmpz_init, ,
[AC_MSG_ERROR([GNU MP not found, see https://gmplib.org/])])
If functions added in some particular version of GMP are required,
then one of those can be used when checking. For example `mpz_mul_si'
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 https://gmplib.org/])])
An alternative would be to test the version number in `gmp.h' using
say `AC_EGREP_CPP'. That would make it possible to test the exact
version, if some particular sub-minor release is known to be necessary.
In general it's recommended that applications should simply demand a
new enough GMP rather than trying to provide supplements for features
not available in past versions.
Occasionally an application will need or want to know the size of a
type at configuration or preprocessing time, not just with `sizeof' in
the code. This can be done in the normal way with `mp_limb_t' etc, but
GMP 4.0 or up is best for this, since prior versions needed certain
`-D' defines on systems using a `long long' limb. The following would
suit Autoconf 2.50 or up,
AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])

File: gmp.info, Node: Emacs, Prev: Autoconf, Up: GMP Basics
3.15 Emacs
==========
<C-h C-i> (`info-lookup-symbol') is a good way to find documentation on
C functions while editing (*note Info Documentation Lookup: (emacs)Info
Lookup.).
The GMP manual can be included in such lookups by putting the
following in your `.emacs',
(eval-after-load "info-look"
'(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist))))
(setcar (nthcdr 3 mode-value)
(cons '("(gmp)Function Index" nil "^ -.* " "\\>")
(nth 3 mode-value)))))

File: gmp.info, Node: Reporting Bugs, Next: Integer Functions, Prev: GMP Basics, Up: Top
4 Reporting Bugs
****************
If you think you have found a bug in the GMP library, please
investigate it and report it. We have made this library available to
you, and it is not too much to ask you to report the bugs you find.
Before you report a bug, check it's not already addressed in *note
Known Build Problems::, or perhaps *note Notes for Particular
Systems::. You may also want to check `https://gmplib.org/' for
patches for this release.
Please include the following in any report,
* The GMP version number, and if pre-packaged or patched then say so.
* A test program that makes it possible for us to reproduce the bug.
Include instructions on how to run the program.
* A description of what is wrong. If the results are incorrect, in
what way. If you get a crash, say so.
* If you get a crash, include a stack backtrace from the debugger if
it's informative (`where' in `gdb', or `$C' in `adb').
* Please do not send core dumps, executables or `strace's.
* The `configure' options you used when building GMP, if any.
* The output from `configure', as printed to stdout, with any
options used.
* The name of the compiler and its version. For `gcc', get the
version with `gcc -v', otherwise perhaps `what `which cc`', or
similar.
* The output from running `uname -a'.
* The output from running `./config.guess', and from running
`./configfsf.guess' (might be the same).
* If the bug is related to `configure', then the compressed contents
of `config.log'.
* If the bug is related to an `asm' file not assembling, then the
contents of `config.m4' and the offending line or lines from the
temporary `mpn/tmp-<file>.s'.
Please make an effort to produce a self-contained report, with
something definite that can be tested or debugged. Vague queries or
piecemeal messages are difficult to act on and don't help the
development effort.
It is not uncommon that an observed problem is actually due to a bug
in the compiler; the GMP code tends to explore interesting corners in
compilers.
If your bug report is good, we will do our best to help you get a
corrected version of the library; if the bug report is poor, we won't
do anything about it (except maybe ask you to send a better report).
Send your report to: <gmp-bugs@gmplib.org>.
If you think something in this manual is unclear, or downright
incorrect, or if the language needs to be improved, please send a note
to the same address.

File: gmp.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
5 Integer Functions
*******************
This chapter describes the GMP functions for performing integer
arithmetic. These functions start with the prefix `mpz_'.
GMP integers are stored in objects of type `mpz_t'.
* Menu:
* Initializing Integers::
* Assigning Integers::
* Simultaneous Integer Init & Assign::
* Converting Integers::
* Integer Arithmetic::
* Integer Division::
* Integer Exponentiation::
* Integer Roots::
* Number Theoretic Functions::
* Integer Comparisons::
* Integer Logic and Bit Fiddling::
* I/O of Integers::
* Integer Random Numbers::
* Integer Import and Export::
* Miscellaneous Integer Functions::
* Integer Special Functions::

File: gmp.info, Node: Initializing Integers, Next: Assigning Integers, Prev: Integer Functions, Up: Integer Functions
5.1 Initialization Functions
============================
The functions for integer arithmetic assume that all integer objects are
initialized. You do that by calling the function `mpz_init'. For
example,
{
mpz_t integ;
mpz_init (integ);
...
mpz_add (integ, ...);
...
mpz_sub (integ, ...);
/* Unless the program is about to exit, do ... */
mpz_clear (integ);
}
As you can see, you can store new values any number of times, once an
object is initialized.
-- Function: void mpz_init (mpz_t X)
Initialize X, and set its value to 0.
-- Function: void mpz_inits (mpz_t X, ...)
Initialize a NULL-terminated list of `mpz_t' variables, and set
their values to 0.
-- Function: void mpz_init2 (mpz_t X, mp_bitcnt_t N)
Initialize X, with space for N-bit numbers, and set its value to 0.
Calling this function instead of `mpz_init' or `mpz_inits' is never
necessary; reallocation is handled automatically by GMP when
needed.
While N defines the initial space, X will grow automatically in the
normal way, if necessary, for subsequent values stored.
`mpz_init2' makes it possible to avoid such reallocations if a
maximum size is known in advance.
In preparation for an operation, GMP often allocates one limb more
than ultimately needed. To make sure GMP will not perform
reallocation for X, you need to add the number of bits in
`mp_limb_t' to N.
-- Function: void mpz_clear (mpz_t X)
Free the space occupied by X. Call this function for all `mpz_t'
variables when you are done with them.
-- Function: void mpz_clears (mpz_t X, ...)
Free the space occupied by a NULL-terminated list of `mpz_t'
variables.
-- Function: void mpz_realloc2 (mpz_t X, mp_bitcnt_t N)
Change the space allocated for X to N bits. The value in X is
preserved if it fits, or is set to 0 if not.
Calling this function is never necessary; reallocation is handled
automatically by GMP when needed. But this function can be used
to increase the space for a variable in order to avoid repeated
automatic reallocations, or to decrease it to give memory back to
the heap.

File: gmp.info, Node: Assigning Integers, Next: Simultaneous Integer Init & Assign, Prev: Initializing Integers, Up: Integer Functions
5.2 Assignment Functions
========================
These functions assign new values to already initialized integers
(*note Initializing Integers::).
-- Function: void mpz_set (mpz_t ROP, const mpz_t OP)
-- Function: void mpz_set_ui (mpz_t ROP, unsigned long int OP)
-- Function: void mpz_set_si (mpz_t ROP, signed long int OP)
-- Function: void mpz_set_d (mpz_t ROP, double OP)
-- Function: void mpz_set_q (mpz_t ROP, const mpq_t OP)
-- Function: void mpz_set_f (mpz_t ROP, const mpf_t OP)
Set the value of ROP from OP.
`mpz_set_d', `mpz_set_q' and `mpz_set_f' truncate OP to make it an
integer.
-- Function: int mpz_set_str (mpz_t ROP, const char *STR, int BASE)
Set the value of ROP from STR, a null-terminated C string in base
BASE. White space is allowed in the string, and is simply ignored.
The BASE may vary from 2 to 62, or if BASE is 0, then the leading
characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
for binary, `0' for octal, or decimal otherwise.
For bases up to 36, case is ignored; upper-case and lower-case
letters have the same value. For bases 37 to 62, upper-case
letter represent the usual 10..35 while lower-case letter
represent 36..61.
This function returns 0 if the entire string is a valid number in
base BASE. Otherwise it returns -1.
-- Function: void mpz_swap (mpz_t ROP1, mpz_t ROP2)
Swap the values ROP1 and ROP2 efficiently.

File: gmp.info, Node: Simultaneous Integer Init & Assign, Next: Converting Integers, Prev: Assigning Integers, Up: Integer Functions
5.3 Combined Initialization and Assignment Functions
====================================================
For convenience, GMP provides a parallel series of initialize-and-set
functions which initialize the output and then store the value there.
These functions' names have the form `mpz_init_set...'
Here is an example of using one:
{
mpz_t pie;
mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10);
...
mpz_sub (pie, ...);
...
mpz_clear (pie);
}
Once the integer has been initialized by any of the `mpz_init_set...'
functions, it can be used as the source or destination operand for the
ordinary integer functions. Don't use an initialize-and-set function
on a variable already initialized!
-- Function: void mpz_init_set (mpz_t ROP, const mpz_t OP)
-- Function: void mpz_init_set_ui (mpz_t ROP, unsigned long int OP)
-- Function: void mpz_init_set_si (mpz_t ROP, signed long int OP)
-- Function: void mpz_init_set_d (mpz_t ROP, double OP)
Initialize ROP with limb space and set the initial numeric value
from OP.
-- Function: int mpz_init_set_str (mpz_t ROP, const char *STR, int
BASE)
Initialize ROP and set its value like `mpz_set_str' (see its
documentation above for details).
If the string is a correct base BASE number, the function returns
0; if an error occurs it returns -1. ROP is initialized even if
an error occurs. (I.e., you have to call `mpz_clear' for it.)

File: gmp.info, Node: Converting Integers, Next: Integer Arithmetic, Prev: Simultaneous Integer Init & Assign, Up: Integer Functions
5.4 Conversion Functions
========================
This section describes functions for converting GMP integers to
standard C types. Functions for converting _to_ GMP integers are
described in *note Assigning Integers:: and *note I/O of Integers::.
-- Function: unsigned long int mpz_get_ui (const mpz_t OP)
Return the value of OP as an `unsigned long'.
If OP is too big to fit an `unsigned long' then just the least
significant bits that do fit are returned. The sign of OP is
ignored, only the absolute value is used.
-- Function: signed long int mpz_get_si (const mpz_t OP)
If OP fits into a `signed long int' return the value of OP.
Otherwise return the least significant part of OP, with the same
sign as OP.
If OP is too big to fit in a `signed long int', the returned
result is probably not very useful. To find out if the value will
fit, use the function `mpz_fits_slong_p'.
-- Function: double mpz_get_d (const mpz_t OP)
Convert OP to a `double', truncating if necessary (i.e. rounding
towards zero).
If the exponent from the conversion is too big, the result is
system dependent. An infinity is returned where available. A
hardware overflow trap may or may not occur.
-- Function: double mpz_get_d_2exp (signed long int *EXP, const mpz_t
OP)
Convert OP to a `double', truncating if necessary (i.e. rounding
towards zero), and returning the exponent separately.
The return value is in the range 0.5<=abs(D)<1 and the exponent is
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
is zero, the return is 0.0 and 0 is stored to `*EXP'.
This is similar to the standard C `frexp' function (*note
Normalization Functions: (libc)Normalization Functions.).
-- Function: char * mpz_get_str (char *STR, int BASE, const mpz_t OP)
Convert OP to a string of digits in base BASE. The base argument
may vary from 2 to 62 or from -2 to -36.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of storage large
enough for the result, that being `mpz_sizeinbase (OP, BASE) + 2'.
The two extra bytes are for a possible minus sign, and the
null-terminator.
A pointer to the result string is returned, being either the
allocated block, or the given STR.

File: gmp.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
5.5 Arithmetic Functions
========================
-- Function: void mpz_add (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
-- Function: void mpz_add_ui (mpz_t ROP, const mpz_t OP1, unsigned
long int OP2)
Set ROP to OP1 + OP2.
-- Function: void mpz_sub (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
-- Function: void mpz_sub_ui (mpz_t ROP, const mpz_t OP1, unsigned
long int OP2)
-- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, const
mpz_t OP2)
Set ROP to OP1 - OP2.
-- Function: void mpz_mul (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
-- Function: void mpz_mul_si (mpz_t ROP, const mpz_t OP1, long int OP2)
-- Function: void mpz_mul_ui (mpz_t ROP, const mpz_t OP1, unsigned
long int OP2)
Set ROP to OP1 times OP2.
-- Function: void mpz_addmul (mpz_t ROP, const mpz_t OP1, const mpz_t
OP2)
-- Function: void mpz_addmul_ui (mpz_t ROP, const mpz_t OP1, unsigned
long int OP2)
Set ROP to ROP + OP1 times OP2.
-- Function: void mpz_submul (mpz_t ROP, const mpz_t OP1, const mpz_t
OP2)
-- Function: void mpz_submul_ui (mpz_t ROP, const mpz_t OP1, unsigned
long int OP2)
Set ROP to ROP - OP1 times OP2.
-- Function: void mpz_mul_2exp (mpz_t ROP, const mpz_t OP1,
mp_bitcnt_t OP2)
Set ROP to OP1 times 2 raised to OP2. This operation can also be
defined as a left shift by OP2 bits.
-- Function: void mpz_neg (mpz_t ROP, const mpz_t OP)
Set ROP to -OP.
-- Function: void mpz_abs (mpz_t ROP, const mpz_t OP)
Set ROP to the absolute value of OP.

File: gmp.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
5.6 Division Functions
======================
Division is undefined if the divisor is zero. Passing a zero divisor
to the division or modulo functions (including the modular powering
functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
division by zero. This lets a program handle arithmetic exceptions in
these functions the same way as for normal C `int' arithmetic.
-- Function: void mpz_cdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
-- Function: void mpz_cdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
-- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
mpz_t D)
-- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
const mpz_t N, unsigned long int D)
-- Function: unsigned long int mpz_cdiv_ui (const mpz_t N,
unsigned long int D)
-- Function: void mpz_cdiv_q_2exp (mpz_t Q, const mpz_t N,
mp_bitcnt_t B)
-- Function: void mpz_cdiv_r_2exp (mpz_t R, const mpz_t N,
mp_bitcnt_t B)
-- Function: void mpz_fdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
-- Function: void mpz_fdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
-- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
mpz_t D)
-- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
const mpz_t N, unsigned long int D)
-- Function: unsigned long int mpz_fdiv_ui (const mpz_t N,
unsigned long int D)
-- Function: void mpz_fdiv_q_2exp (mpz_t Q, const mpz_t N,
mp_bitcnt_t B)
-- Function: void mpz_fdiv_r_2exp (mpz_t R, const mpz_t N,
mp_bitcnt_t B)
-- Function: void mpz_tdiv_q (mpz_t Q, const mpz_t N, const mpz_t D)
-- Function: void mpz_tdiv_r (mpz_t R, const mpz_t N, const mpz_t D)
-- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, const mpz_t N, const
mpz_t D)
-- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, const mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
const mpz_t N, unsigned long int D)
-- Function: unsigned long int mpz_tdiv_ui (const mpz_t N,
unsigned long int D)
-- Function: void mpz_tdiv_q_2exp (mpz_t Q, const mpz_t N,
mp_bitcnt_t B)
-- Function: void mpz_tdiv_r_2exp (mpz_t R, const mpz_t N,
mp_bitcnt_t B)
Divide N by D, forming a quotient Q and/or remainder R. For the
`2exp' functions, D=2^B. The rounding is in three styles, each
suiting different applications.
* `cdiv' rounds Q up towards +infinity, and R will have the
opposite sign to D. The `c' stands for "ceil".
* `fdiv' rounds Q down towards -infinity, and R will have the
same sign as D. The `f' stands for "floor".
* `tdiv' rounds Q towards zero, and R will have the same sign
as N. The `t' stands for "truncate".
In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
0<=abs(R)<abs(D).
The `q' functions calculate only the quotient, the `r' functions
only the remainder, and the `qr' functions calculate both. Note
that for `qr' the same variable cannot be passed for both Q and R,
or results will be unpredictable.
For the `ui' variants the return value is the remainder, and in
fact returning the remainder is all the `div_ui' functions do. For
`tdiv' and `cdiv' the remainder can be negative, so for those the
return value is the absolute value of the remainder.
For the `2exp' variants the divisor is 2^B. These functions are
implemented as right shifts and bit masks, but of course they
round the same as the other functions.
For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
effectively an arithmetic right shift treating N as twos complement
the same as the bitwise logical functions do, whereas
`mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
-- Function: void mpz_mod (mpz_t R, const mpz_t N, const mpz_t D)
-- Function: unsigned long int mpz_mod_ui (mpz_t R, const mpz_t N,
unsigned long int D)
Set R to N `mod' D. The sign of the divisor is ignored; the
result is always non-negative.
`mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
remainder as well as setting R. See `mpz_fdiv_ui' above if only
the return value is wanted.
-- Function: void mpz_divexact (mpz_t Q, const mpz_t N, const mpz_t D)
-- Function: void mpz_divexact_ui (mpz_t Q, const mpz_t N, unsigned
long D)
Set Q to N/D. These functions produce correct results only when
it is known in advance that D divides N.
These routines are much faster than the other division functions,
and are the best choice when exact division is known to occur, for
example reducing a rational to lowest terms.
-- Function: int mpz_divisible_p (const mpz_t N, const mpz_t D)
-- Function: int mpz_divisible_ui_p (const mpz_t N, unsigned long int
D)
-- Function: int mpz_divisible_2exp_p (const mpz_t N, mp_bitcnt_t B)
Return non-zero if N is exactly divisible by D, or in the case of
`mpz_divisible_2exp_p' by 2^B.
N is divisible by D if there exists an integer Q satisfying N =
Q*D. Unlike the other division functions, D=0 is accepted and
following the rule it can be seen that only 0 is considered
divisible by 0.
-- Function: int mpz_congruent_p (const mpz_t N, const mpz_t C, const
mpz_t D)
-- Function: int mpz_congruent_ui_p (const mpz_t N, unsigned long int
C, unsigned long int D)
-- Function: int mpz_congruent_2exp_p (const mpz_t N, const mpz_t C,
mp_bitcnt_t B)
Return non-zero if N is congruent to C modulo D, or in the case of
`mpz_congruent_2exp_p' modulo 2^B.
N is congruent to C mod D if there exists an integer Q satisfying
N = C + Q*D. Unlike the other division functions, D=0 is accepted
and following the rule it can be seen that N and C are considered
congruent mod 0 only when exactly equal.

File: gmp.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
5.7 Exponentiation Functions
============================
-- Function: void mpz_powm (mpz_t ROP, const mpz_t BASE, const mpz_t
EXP, const mpz_t MOD)
-- Function: void mpz_powm_ui (mpz_t ROP, const mpz_t BASE, unsigned
long int EXP, const mpz_t MOD)
Set ROP to (BASE raised to EXP) modulo MOD.
Negative EXP is supported if an inverse BASE^-1 mod MOD exists
(see `mpz_invert' in *note Number Theoretic Functions::). If an
inverse doesn't exist then a divide by zero is raised.
-- Function: void mpz_powm_sec (mpz_t ROP, const mpz_t BASE, const
mpz_t EXP, const mpz_t MOD)
Set ROP to (BASE raised to EXP) modulo MOD.
It is required that EXP > 0 and that MOD is odd.
This function is designed to take the same time and have the same
cache access patterns for any two same-size arguments, assuming
that function arguments are placed at the same position and that
the machine state is identical upon function entry. This function
is intended for cryptographic purposes, where resilience to
side-channel attacks is desired.
-- Function: void mpz_pow_ui (mpz_t ROP, const mpz_t BASE, unsigned
long int EXP)
-- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
unsigned long int EXP)
Set ROP to BASE raised to EXP. The case 0^0 yields 1.

File: gmp.info, Node: Integer Roots, Next: Number Theoretic Functions, Prev: Integer Exponentiation, Up: Integer Functions
5.8 Root Extraction Functions
=============================
-- Function: int mpz_root (mpz_t ROP, const mpz_t OP, unsigned long
int N)
Set ROP to the truncated integer part of the Nth root of OP.
Return non-zero if the computation was exact, i.e., if OP is ROP
to the Nth power.
-- Function: void mpz_rootrem (mpz_t ROOT, mpz_t REM, const mpz_t U,
unsigned long int N)
Set ROOT to the truncated integer part of the Nth root of U. Set
REM to the remainder, U-ROOT**N.
-- Function: void mpz_sqrt (mpz_t ROP, const mpz_t OP)
Set ROP to the truncated integer part of the square root of OP.
-- Function: void mpz_sqrtrem (mpz_t ROP1, mpz_t ROP2, const mpz_t OP)
Set ROP1 to the truncated integer part of the square root of OP,
like `mpz_sqrt'. Set ROP2 to the remainder OP-ROP1*ROP1, which
will be zero if OP is a perfect square.
If ROP1 and ROP2 are the same variable, the results are undefined.
-- Function: int mpz_perfect_power_p (const mpz_t OP)
Return non-zero if OP is a perfect power, i.e., if there exist
integers A and B, with B>1, such that OP equals A raised to the
power B.
Under this definition both 0 and 1 are considered to be perfect
powers. Negative values of OP are accepted, but of course can
only be odd perfect powers.
-- Function: int mpz_perfect_square_p (const mpz_t OP)
Return non-zero if OP is a perfect square, i.e., if the square
root of OP is an integer. Under this definition both 0 and 1 are
considered to be perfect squares.

File: gmp.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
5.9 Number Theoretic Functions
==============================
-- Function: int mpz_probab_prime_p (const mpz_t N, int REPS)
Determine whether N is prime. Return 2 if N is definitely prime,
return 1 if N is probably prime (without being certain), or return
0 if N is definitely composite.
This function does some trial divisions, then some Miller-Rabin
probabilistic primality tests. The argument REPS controls how
many such tests are done; a higher value will reduce the chances
of a composite being returned as "probably prime". 25 is a
reasonable number; a composite number will then be identified as a
prime with a probability of less than 2^(-50).
Miller-Rabin and similar tests can be more properly called
compositeness tests. Numbers which fail are known to be composite
but those which pass might be prime or might be composite. Only a
few composites pass, hence those which pass are considered
probably prime.
-- Function: void mpz_nextprime (mpz_t ROP, const mpz_t OP)
Set ROP to the next prime greater than OP.
This function uses a probabilistic algorithm to identify primes.
For practical purposes it's adequate, the chance of a composite
passing will be extremely small.
-- Function: void mpz_gcd (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to the greatest common divisor of OP1 and OP2. The result
is always positive even if one or both input operands are negative.
Except if both inputs are zero; then this function defines
gcd(0,0) = 0.
-- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, const mpz_t OP1,
unsigned long int OP2)
Compute the greatest common divisor of OP1 and OP2. If ROP is not
`NULL', store the result there.
If the result is small enough to fit in an `unsigned long int', it
is returned. If the result does not fit, 0 is returned, and the
result is equal to the argument OP1. Note that the result will
always fit if OP2 is non-zero.
-- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, const mpz_t
A, const mpz_t B)
Set G to the greatest common divisor of A and B, and in addition
set S and T to coefficients satisfying A*S + B*T = G. The value
in G is always positive, even if one or both of A and B are
negative (or zero if both inputs are zero). The values in S and T
are chosen such that normally, abs(S) < abs(B) / (2 G) and abs(T)
< abs(A) / (2 G), and these relations define S and T uniquely.
There are a few exceptional cases:
If abs(A) = abs(B), then S = 0, T = sgn(B).
Otherwise, S = sgn(A) if B = 0 or abs(B) = 2 G, and T = sgn(B) if
A = 0 or abs(A) = 2 G.
In all cases, S = 0 if and only if G = abs(B), i.e., if B divides
A or A = B = 0.
If T is `NULL' then that value is not computed.
-- Function: void mpz_lcm (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
-- Function: void mpz_lcm_ui (mpz_t ROP, const mpz_t OP1, unsigned
long OP2)
Set ROP to the least common multiple of OP1 and OP2. ROP is
always positive, irrespective of the signs of OP1 and OP2. ROP
will be zero if either OP1 or OP2 is zero.
-- Function: int mpz_invert (mpz_t ROP, const mpz_t OP1, const mpz_t
OP2)
Compute the inverse of OP1 modulo OP2 and put the result in ROP.
If the inverse exists, the return value is non-zero and ROP will
satisfy 0 < ROP < abs(OP2). If an inverse doesn't exist the
return value is zero and ROP is undefined. The behaviour of this
function is undefined when OP2 is zero.
-- Function: int mpz_jacobi (const mpz_t A, const mpz_t B)
Calculate the Jacobi symbol (A/B). This is defined only for B odd.
-- Function: int mpz_legendre (const mpz_t A, const mpz_t P)
Calculate the Legendre symbol (A/P). This is defined only for P
an odd positive prime, and for such P it's identical to the Jacobi
symbol.
-- Function: int mpz_kronecker (const mpz_t A, const mpz_t B)
-- Function: int mpz_kronecker_si (const mpz_t A, long B)
-- Function: int mpz_kronecker_ui (const mpz_t A, unsigned long B)
-- Function: int mpz_si_kronecker (long A, const mpz_t B)
-- Function: int mpz_ui_kronecker (unsigned long A, const mpz_t B)
Calculate the Jacobi symbol (A/B) with the Kronecker extension
(a/2)=(2/a) when a odd, or (a/2)=0 when a even.
When B is odd the Jacobi symbol and Kronecker symbol are
identical, so `mpz_kronecker_ui' etc can be used for mixed
precision Jacobi symbols too.
For more information see Henri Cohen section 1.4.2 (*note
References::), or any number theory textbook. See also the
example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
-- Function: mp_bitcnt_t mpz_remove (mpz_t ROP, const mpz_t OP, const
mpz_t F)
Remove all occurrences of the factor F from OP and store the
result in ROP. The return value is how many such occurrences were
removed.
-- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int N)
-- Function: void mpz_2fac_ui (mpz_t ROP, unsigned long int N)
-- Function: void mpz_mfac_uiui (mpz_t ROP, unsigned long int N,
unsigned long int M)
Set ROP to the factorial of N: `mpz_fac_ui' computes the plain
factorial N!, `mpz_2fac_ui' computes the double-factorial N!!, and
`mpz_mfac_uiui' the M-multi-factorial N!^(M).
-- Function: void mpz_primorial_ui (mpz_t ROP, unsigned long int N)
Set ROP to the primorial of N, i.e. the product of all positive
prime numbers <=N.
-- Function: void mpz_bin_ui (mpz_t ROP, const mpz_t N, unsigned long
int K)
-- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
unsigned long int K)
Compute the binomial coefficient N over K and store the result in
ROP. Negative values of N are supported by `mpz_bin_ui', using
the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
section 1.2.6 part G.
-- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
-- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
int N)
`mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
`mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
These functions are designed for calculating isolated Fibonacci
numbers. When a sequence of values is wanted it's best to start
with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
similar.
-- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
-- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
long int N)
`mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
`mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
These functions are designed for calculating isolated Lucas
numbers. When a sequence of values is wanted it's best to start
with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
or similar.
The Fibonacci numbers and Lucas numbers are related sequences, so
it's never necessary to call both `mpz_fib2_ui' and
`mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
can be found in *note Lucas Numbers Algorithm::, the reverse is
straightforward too.

File: gmp.info, Node: Integer Comparisons, Next: Integer Logic and Bit Fiddling, Prev: Number Theoretic Functions, Up: Integer Functions
5.10 Comparison Functions
=========================
-- Function: int mpz_cmp (const mpz_t OP1, const mpz_t OP2)
-- Function: int mpz_cmp_d (const mpz_t OP1, double OP2)
-- Macro: int mpz_cmp_si (const mpz_t OP1, signed long int OP2)
-- Macro: int mpz_cmp_ui (const mpz_t OP1, unsigned long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, or a negative value if OP1 < OP2.
`mpz_cmp_ui' and `mpz_cmp_si' are macros and will evaluate their
arguments more than once. `mpz_cmp_d' can be called with an
infinity, but results are undefined for a NaN.
-- Function: int mpz_cmpabs (const mpz_t OP1, const mpz_t OP2)
-- Function: int mpz_cmpabs_d (const mpz_t OP1, double OP2)
-- Function: int mpz_cmpabs_ui (const mpz_t OP1, unsigned long int OP2)
Compare the absolute values of OP1 and OP2. Return a positive
value if abs(OP1) > abs(OP2), zero if abs(OP1) = abs(OP2), or a
negative value if abs(OP1) < abs(OP2).
`mpz_cmpabs_d' can be called with an infinity, but results are
undefined for a NaN.
-- Macro: int mpz_sgn (const mpz_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its argument multiple times.

File: gmp.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
5.11 Logical and Bit Manipulation Functions
===========================================
These functions behave as if twos complement arithmetic were used
(although sign-magnitude is the actual implementation). The least
significant bit is number 0.
-- Function: void mpz_and (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise-and OP2.
-- Function: void mpz_ior (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise inclusive-or OP2.
-- Function: void mpz_xor (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
Set ROP to OP1 bitwise exclusive-or OP2.
-- Function: void mpz_com (mpz_t ROP, const mpz_t OP)
Set ROP to the one's complement of OP.
-- Function: mp_bitcnt_t mpz_popcount (const mpz_t OP)
If OP>=0, return the population count of OP, which is the number
of 1 bits in the binary representation. If OP<0, the number of 1s
is infinite, and the return value is the largest possible
`mp_bitcnt_t'.
-- Function: mp_bitcnt_t mpz_hamdist (const mpz_t OP1, const mpz_t OP2)
If OP1 and OP2 are both >=0 or both <0, return the hamming
distance between the two operands, which is the number of bit
positions where OP1 and OP2 have different bit values. If one
operand is >=0 and the other <0 then the number of bits different
is infinite, and the return value is the largest possible
`mp_bitcnt_t'.
-- Function: mp_bitcnt_t mpz_scan0 (const mpz_t OP, mp_bitcnt_t
STARTING_BIT)
-- Function: mp_bitcnt_t mpz_scan1 (const mpz_t OP, mp_bitcnt_t
STARTING_BIT)
Scan OP, starting from bit STARTING_BIT, towards more significant
bits, until the first 0 or 1 bit (respectively) is found. Return
the index of the found bit.
If the bit at STARTING_BIT is already what's sought, then
STARTING_BIT is returned.
If there's no bit found, then the largest possible `mp_bitcnt_t' is
returned. This will happen in `mpz_scan0' past the end of a
negative number, or `mpz_scan1' past the end of a nonnegative
number.
-- Function: void mpz_setbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
Set bit BIT_INDEX in ROP.
-- Function: void mpz_clrbit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
Clear bit BIT_INDEX in ROP.
-- Function: void mpz_combit (mpz_t ROP, mp_bitcnt_t BIT_INDEX)
Complement bit BIT_INDEX in ROP.
-- Function: int mpz_tstbit (const mpz_t OP, mp_bitcnt_t BIT_INDEX)
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.

File: gmp.info, Node: I/O of Integers, Next: Integer Random Numbers, Prev: Integer Logic and Bit Fiddling, Up: Integer Functions
5.12 Input and Output Functions
===============================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream, of `mpz' numbers. Passing a `NULL' pointer
for a STREAM argument to any of these functions will make them read from
`stdin' and write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
See also *note Formatted Output:: and *note Formatted Input::.
-- Function: size_t mpz_out_str (FILE *STREAM, int BASE, const mpz_t
OP)
Output OP on stdio stream STREAM, as a string of digits in base
BASE. The base argument may vary from 2 to 62 or from -2 to -36.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
Return the number of bytes written, or if an error occurred,
return 0.
-- Function: size_t mpz_inp_str (mpz_t ROP, FILE *STREAM, int BASE)
Input a possibly white-space preceded string in base BASE from
stdio stream STREAM, and put the read integer in ROP.
The BASE may vary from 2 to 62, or if BASE is 0, then the leading
characters are used: `0x' and `0X' for hexadecimal, `0b' and `0B'
for binary, `0' for octal, or decimal otherwise.
For bases up to 36, case is ignored; upper-case and lower-case
letters have the same value. For bases 37 to 62, upper-case
letter represent the usual 10..35 while lower-case letter
represent 36..61.
Return the number of bytes read, or if an error occurred, return 0.
-- Function: size_t mpz_out_raw (FILE *STREAM, const mpz_t OP)
Output OP on stdio stream STREAM, in raw binary format. The
integer is written in a portable format, with 4 bytes of size
information, and that many bytes of limbs. Both the size and the
limbs are written in decreasing significance order (i.e., in
big-endian).
The output can be read with `mpz_inp_raw'.
Return the number of bytes written, or if an error occurred,
return 0.
The output of this can not be read by `mpz_inp_raw' from GMP 1,
because of changes necessary for compatibility between 32-bit and
64-bit machines.
-- Function: size_t mpz_inp_raw (mpz_t ROP, FILE *STREAM)
Input from stdio stream STREAM in the format written by
`mpz_out_raw', and put the result in ROP. Return the number of
bytes read, or if an error occurred, return 0.
This routine can read the output from `mpz_out_raw' also from GMP
1, in spite of changes necessary for compatibility between 32-bit
and 64-bit machines.

File: gmp.info, Node: Integer Random Numbers, Next: Integer Import and Export, Prev: I/O of Integers, Up: Integer Functions
5.13 Random Number Functions
============================
The random number functions of GMP come in two groups; older function
that rely on a global state, and newer functions that accept a state
parameter that is read and modified. Please see the *note Random
Number Functions:: for more information on how to use and not to use
random number functions.
-- Function: void mpz_urandomb (mpz_t ROP, gmp_randstate_t STATE,
mp_bitcnt_t N)
Generate a uniformly distributed random integer in the range 0 to
2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
const mpz_t N)
Generate a uniform random integer in the range 0 to N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
mp_bitcnt_t N)
Generate a random integer with long strings of zeros and ones in
the binary representation. Useful for testing functions and
algorithms, since this kind of random numbers have proven to be
more likely to trigger corner-case bugs. The random number will
be in the range 0 to 2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_random (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs. The generated
random number doesn't satisfy any particular requirements of
randomness. Negative random numbers are generated when MAX_SIZE
is negative.
This function is obsolete. Use `mpz_urandomb' or `mpz_urandomm'
instead.
-- Function: void mpz_random2 (mpz_t ROP, mp_size_t MAX_SIZE)
Generate a random integer of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. Useful
for testing functions and algorithms, since this kind of random
numbers have proven to be more likely to trigger corner-case bugs.
Negative random numbers are generated when MAX_SIZE is negative.
This function is obsolete. Use `mpz_rrandomb' instead.

File: gmp.info, Node: Integer Import and Export, Next: Miscellaneous Integer Functions, Prev: Integer Random Numbers, Up: Integer Functions
5.14 Integer Import and Export
==============================
`mpz_t' variables can be converted to and from arbitrary words of binary
data with the following functions.
-- Function: void mpz_import (mpz_t ROP, size_t COUNT, int ORDER,
size_t SIZE, int ENDIAN, size_t NAILS, const void *OP)
Set ROP from an array of word data at OP.
The parameters specify the format of the data. COUNT many words
are read, each SIZE bytes. ORDER can be 1 for most significant
word first or -1 for least significant first. Within each word
ENDIAN can be 1 for most significant byte first, -1 for least
significant first, or 0 for the native endianness of the host CPU.
The most significant NAILS bits of each word are skipped, this can
be 0 to use the full words.
There is no sign taken from the data, ROP will simply be a positive
integer. An application can handle any sign itself, and apply it
for instance with `mpz_neg'.
There are no data alignment restrictions on OP, any address is
allowed.
Here's an example converting an array of `unsigned long' data, most
significant element first, and host byte order within each value.
unsigned long a[20];
/* Initialize Z and A */
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
This example assumes the full `sizeof' bytes are used for data in
the given type, which is usually true, and certainly true for
`unsigned long' everywhere we know of. However on Cray vector
systems it may be noted that `short' and `int' are always stored
in 8 bytes (and with `sizeof' indicating that) but use only 32 or
46 bits. The NAILS feature can account for this, by passing for
instance `8*sizeof(int)-INT_BIT'.
-- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
size_t SIZE, int ENDIAN, size_t NAILS, const mpz_t OP)
Fill ROP with word data from OP.
The parameters specify the format of the data produced. Each word
will be SIZE bytes and ORDER can be 1 for most significant word
first or -1 for least significant first. Within each word ENDIAN
can be 1 for most significant byte first, -1 for least significant
first, or 0 for the native endianness of the host CPU. The most
significant NAILS bits of each word are unused and set to zero,
this can be 0 to produce full words.
The number of words produced is written to `*COUNTP', or COUNTP
can be `NULL' to discard the count. ROP must have enough space
for the data, or if ROP is `NULL' then a result array of the
necessary size is allocated using the current GMP allocation
function (*note Custom Allocation::). In either case the return
value is the destination used, either ROP or the allocated block.
If OP is non-zero then the most significant word produced will be
non-zero. If OP is zero then the count returned will be zero and
nothing written to ROP. If ROP is `NULL' in this case, no block
is allocated, just `NULL' is returned.
The sign of OP is ignored, just the absolute value is exported. An
application can use `mpz_sgn' to get the sign and handle it as
desired. (*note Integer Comparisons::)
There are no data alignment restrictions on ROP, any address is
allowed.
When an application is allocating space itself the required size
can be determined with a calculation like the following. Since
`mpz_sizeinbase' always returns at least 1, `count' here will be
at least one, which avoids any portability problems with
`malloc(0)', though if `z' is zero no space at all is actually
needed (or written).
numb = 8*size - nail;
count = (mpz_sizeinbase (z, 2) + numb-1) / numb;
p = malloc (count * size);

File: gmp.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
5.15 Miscellaneous Functions
============================
-- Function: int mpz_fits_ulong_p (const mpz_t OP)
-- Function: int mpz_fits_slong_p (const mpz_t OP)
-- Function: int mpz_fits_uint_p (const mpz_t OP)
-- Function: int mpz_fits_sint_p (const mpz_t OP)
-- Function: int mpz_fits_ushort_p (const mpz_t OP)
-- Function: int mpz_fits_sshort_p (const mpz_t OP)
Return non-zero iff the value of OP fits in an `unsigned long int',
`signed long int', `unsigned int', `signed int', `unsigned short
int', or `signed short int', respectively. Otherwise, return zero.
-- Macro: int mpz_odd_p (const mpz_t OP)
-- Macro: int mpz_even_p (const mpz_t OP)
Determine whether OP is odd or even, respectively. Return
non-zero if yes, zero if no. These macros evaluate their argument
more than once.
-- Function: size_t mpz_sizeinbase (const mpz_t OP, int BASE)
Return the size of OP measured in number of digits in the given
BASE. BASE can vary from 2 to 62. The sign of OP is ignored,
just the absolute value is used. The result will be either exact
or 1 too big. If BASE is a power of 2, the result is always
exact. If OP is zero the return value is always 1.
This function can be used to determine the space required when
converting OP to a string. The right amount of allocation is
normally two more than the value returned by `mpz_sizeinbase', one
extra for a minus sign and one for the null-terminator.
It will be noted that `mpz_sizeinbase(OP,2)' can be used to locate
the most significant 1 bit in OP, counting from 1. (Unlike the
bitwise functions which start from 0, *Note Logical and Bit
Manipulation Functions: Integer Logic and Bit Fiddling.)

File: gmp.info, Node: Integer Special Functions, Prev: Miscellaneous Integer Functions, Up: Integer Functions
5.16 Special Functions
======================
The functions in this section are for various special purposes. Most
applications will not need them.
-- Function: void mpz_array_init (mpz_t INTEGER_ARRAY, mp_size_t
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
*This is an obsolete function. Do not use it.*
-- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
Change the space for INTEGER to NEW_ALLOC limbs. The value in
INTEGER is preserved if it fits, or is set to 0 if not. The return
value is not useful to applications and should be ignored.
`mpz_realloc2' is the preferred way to accomplish allocation
changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
except that `_mpz_realloc' takes its size in limbs.
-- Function: mp_limb_t mpz_getlimbn (const mpz_t OP, mp_size_t N)
Return limb number N from OP. The sign of OP is ignored, just the
absolute value is used. The least significant limb is number 0.
`mpz_size' can be used to find how many limbs make up OP.
`mpz_getlimbn' returns zero if N is outside the range 0 to
`mpz_size(OP)-1'.
-- Function: size_t mpz_size (const mpz_t OP)
Return the size of OP measured in number of limbs. If OP is zero,
the returned value will be zero.
-- Function: const mp_limb_t * mpz_limbs_read (const mpz_t X)
Return a pointer to the limb array representing the absolute value
of X. The size of the array is `mpz_size(X)'. Intended for read
access only.
-- Function: mp_limb_t * mpz_limbs_write (mpz_t X, mp_size_t N)
-- Function: mp_limb_t * mpz_limbs_modify (mpz_t X, mp_size_t N)
Return a pointer to the limb array, intended for write access. The
array is reallocated as needed, to make room for N limbs. Requires
N > 0. The `mpz_limbs_modify' function returns an array that holds
the old absolute value of X, while `mpz_limbs_write' may destroy
the old value and return an array with unspecified contents.
-- Function: void mpz_limbs_finish (mpz_t X, mp_size_t S)
Updates the internal size field of X. Used after writing to the
limb array pointer returned by `mpz_limbs_write' or
`mpz_limbs_modify' is completed. The array should contain abs(S)
valid limbs, representing the new absolute value for X, and the
sign of X is taken from the sign of S. This function never
reallocates X, so the limb pointer remains valid.
void foo (mpz_t x)
{
mp_size_t n, i;
mp_limb_t *xp;
n = mpz_size (x);
xp = mpz_limbs_modify(x, 2*n);
for (i = 0; i < n; i++)
xp[n+i] = xp[n-1-i];
mpz_limbs_finish (x, mpz_sgn (x) < 0 ? - 2*n : 2*n);
}
-- Function: mpz_srcptr mpz_roinit_n (mpz_t X, const mp_limb_t *XP,
mp_size_t XS)
Special initialization of X, using the given limb array and size.
X should be treated as read-only: it can be passed safely as input
to any mpz function, but not as an output. The array XP must point
to at least a readable limb, its size is abs(XS), and the sign of
X is the sign of XS. For convenience, the function returns X, but
cast to a const pointer type.
void foo (mpz_t x)
{
static const mp_limb_t y[3] = { 0x1, 0x2, 0x3 };
mpz_t tmp;
mpz_add (x, x, mpz_roinit_n (tmp, y, 3));
}
-- Macro: mpz_t MPZ_ROINIT_N (mp_limb_t *XP, mp_size_t XS)
This macro expands to an initializer which can be assigned to an
mpz_t variable. The limb array XP must point to at least a
readable limb, moreover, unlike the `mpz_roinit_n' function, the
array must be normalized: if XS is non-zero, then `XP[abs(XS)-1]'
must be non-zero. Intended primarily for constant values. Using it
for non-constant values requires a C compiler supporting C99.
void foo (mpz_t x)
{
static const mp_limb_t ya[3] = { 0x1, 0x2, 0x3 };
static const mpz_t y = MPZ_ROINIT_N ((mp_limb_t *) ya, 3);
mpz_add (x, x, y);
}

File: gmp.info, Node: Rational Number Functions, Next: Floating-point Functions, Prev: Integer Functions, Up: Top
6 Rational Number Functions
***************************
This chapter describes the GMP functions for performing arithmetic on
rational numbers. These functions start with the prefix `mpq_'.
Rational numbers are stored in objects of type `mpq_t'.
All rational arithmetic functions assume operands have a canonical
form, and canonicalize their result. The canonical from means that the
denominator and the numerator have no common factors, and that the
denominator is positive. Zero has the unique representation 0/1.
Pure assignment functions do not canonicalize the assigned variable.
It is the responsibility of the user to canonicalize the assigned
variable before any arithmetic operations are performed on that
variable.
-- Function: void mpq_canonicalize (mpq_t OP)
Remove any factors that are common to the numerator and
denominator of OP, and make the denominator positive.
* Menu:
* Initializing Rationals::
* Rational Conversions::
* Rational Arithmetic::
* Comparing Rationals::
* Applying Integer Functions::
* I/O of Rationals::

File: gmp.info, Node: Initializing Rationals, Next: Rational Conversions, Prev: Rational Number Functions, Up: Rational Number Functions
6.1 Initialization and Assignment Functions
===========================================
-- Function: void mpq_init (mpq_t X)
Initialize X and set it to 0/1. Each variable should normally
only be initialized once, or at least cleared out (using the
function `mpq_clear') between each initialization.
-- Function: void mpq_inits (mpq_t X, ...)
Initialize a NULL-terminated list of `mpq_t' variables, and set
their values to 0/1.
-- Function: void mpq_clear (mpq_t X)
Free the space occupied by X. Make sure to call this function for
all `mpq_t' variables when you are done with them.
-- Function: void mpq_clears (mpq_t X, ...)
Free the space occupied by a NULL-terminated list of `mpq_t'
variables.
-- Function: void mpq_set (mpq_t ROP, const mpq_t OP)
-- Function: void mpq_set_z (mpq_t ROP, const mpz_t OP)
Assign ROP from OP.
-- Function: void mpq_set_ui (mpq_t ROP, unsigned long int OP1,
unsigned long int OP2)
-- Function: void mpq_set_si (mpq_t ROP, signed long int OP1, unsigned
long int OP2)
Set the value of ROP to OP1/OP2. Note that if OP1 and OP2 have
common factors, ROP has to be passed to `mpq_canonicalize' before
any operations are performed on ROP.
-- Function: int mpq_set_str (mpq_t ROP, const char *STR, int BASE)
Set ROP from a null-terminated string STR in the given BASE.
The string can be an integer like "41" or a fraction like
"41/152". The fraction must be in canonical form (*note Rational
Number Functions::), or if not then `mpq_canonicalize' must be
called.
The numerator and optional denominator are parsed the same as in
`mpz_set_str' (*note Assigning Integers::). White space is
allowed in the string, and is simply ignored. The BASE can vary
from 2 to 62, or if BASE is 0 then the leading characters are
used: `0x' or `0X' for hex, `0b' or `0B' for binary, `0' for
octal, or decimal otherwise. Note that this is done separately
for the numerator and denominator, so for instance `0xEF/100' is
239/100, whereas `0xEF/0x100' is 239/256.
The return value is 0 if the entire string is a valid number, or
-1 if not.
-- Function: void mpq_swap (mpq_t ROP1, mpq_t ROP2)
Swap the values ROP1 and ROP2 efficiently.

File: gmp.info, Node: Rational Conversions, Next: Rational Arithmetic, Prev: Initializing Rationals, Up: Rational Number Functions
6.2 Conversion Functions
========================
-- Function: double mpq_get_d (const mpq_t OP)
Convert OP to a `double', truncating if necessary (i.e. rounding
towards zero).
If the exponent from the conversion is too big or too small to fit
a `double' then the result is system dependent. For too big an
infinity is returned when available. For too small 0.0 is
normally returned. Hardware overflow, underflow and denorm traps
may or may not occur.
-- Function: void mpq_set_d (mpq_t ROP, double OP)
-- Function: void mpq_set_f (mpq_t ROP, const mpf_t OP)
Set ROP to the value of OP. There is no rounding, this conversion
is exact.
-- Function: char * mpq_get_str (char *STR, int BASE, const mpq_t OP)
Convert OP to a string of digits in base BASE. The base may vary
from 2 to 36. The string will be of the form `num/den', or if the
denominator is 1 then just `num'.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of storage large
enough for the result, that being
mpz_sizeinbase (mpq_numref(OP), BASE)
+ mpz_sizeinbase (mpq_denref(OP), BASE) + 3
The three extra bytes are for a possible minus sign, possible
slash, and the null-terminator.
A pointer to the result string is returned, being either the
allocated block, or the given STR.

File: gmp.info, Node: Rational Arithmetic, Next: Comparing Rationals, Prev: Rational Conversions, Up: Rational Number Functions
6.3 Arithmetic Functions
========================
-- Function: void mpq_add (mpq_t SUM, const mpq_t ADDEND1, const mpq_t
ADDEND2)
Set SUM to ADDEND1 + ADDEND2.
-- Function: void mpq_sub (mpq_t DIFFERENCE, const mpq_t MINUEND,
const mpq_t SUBTRAHEND)
Set DIFFERENCE to MINUEND - SUBTRAHEND.
-- Function: void mpq_mul (mpq_t PRODUCT, const mpq_t MULTIPLIER,
const mpq_t MULTIPLICAND)
Set PRODUCT to MULTIPLIER times MULTIPLICAND.
-- Function: void mpq_mul_2exp (mpq_t ROP, const mpq_t OP1,
mp_bitcnt_t OP2)
Set ROP to OP1 times 2 raised to OP2.
-- Function: void mpq_div (mpq_t QUOTIENT, const mpq_t DIVIDEND, const
mpq_t DIVISOR)
Set QUOTIENT to DIVIDEND/DIVISOR.
-- Function: void mpq_div_2exp (mpq_t ROP, const mpq_t OP1,
mp_bitcnt_t OP2)
Set ROP to OP1 divided by 2 raised to OP2.
-- Function: void mpq_neg (mpq_t NEGATED_OPERAND, const mpq_t OPERAND)
Set NEGATED_OPERAND to -OPERAND.
-- Function: void mpq_abs (mpq_t ROP, const mpq_t OP)
Set ROP to the absolute value of OP.
-- Function: void mpq_inv (mpq_t INVERTED_NUMBER, const mpq_t NUMBER)
Set INVERTED_NUMBER to 1/NUMBER. If the new denominator is zero,
this routine will divide by zero.

File: gmp.info, Node: Comparing Rationals, Next: Applying Integer Functions, Prev: Rational Arithmetic, Up: Rational Number Functions
6.4 Comparison Functions
========================
-- Function: int mpq_cmp (const mpq_t OP1, const mpq_t OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
To determine if two rationals are equal, `mpq_equal' is faster than
`mpq_cmp'.
-- Macro: int mpq_cmp_ui (const mpq_t OP1, unsigned long int NUM2,
unsigned long int DEN2)
-- Macro: int mpq_cmp_si (const mpq_t OP1, long int NUM2, unsigned
long int DEN2)
Compare OP1 and NUM2/DEN2. Return a positive value if OP1 >
NUM2/DEN2, zero if OP1 = NUM2/DEN2, and a negative value if OP1 <
NUM2/DEN2.
NUM2 and DEN2 are allowed to have common factors.
These functions are implemented as a macros and evaluate their
arguments multiple times.
-- Macro: int mpq_sgn (const mpq_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates its
argument multiple times.
-- Function: int mpq_equal (const mpq_t OP1, const mpq_t OP2)
Return non-zero if OP1 and OP2 are equal, zero if they are
non-equal. Although `mpq_cmp' can be used for the same purpose,
this function is much faster.

File: gmp.info, Node: Applying Integer Functions, Next: I/O of Rationals, Prev: Comparing Rationals, Up: Rational Number Functions
6.5 Applying Integer Functions to Rationals
===========================================
The set of `mpq' functions is quite small. In particular, there are few
functions for either input or output. The following functions give
direct access to the numerator and denominator of an `mpq_t'.
Note that if an assignment to the numerator and/or denominator could
take an `mpq_t' out of the canonical form described at the start of
this chapter (*note Rational Number Functions::) then
`mpq_canonicalize' must be called before any other `mpq' functions are
applied to that `mpq_t'.
-- Macro: mpz_t mpq_numref (const mpq_t OP)
-- Macro: mpz_t mpq_denref (const mpq_t OP)
Return a reference to the numerator and denominator of OP,
respectively. The `mpz' functions can be used on the result of
these macros.
-- Function: void mpq_get_num (mpz_t NUMERATOR, const mpq_t RATIONAL)
-- Function: void mpq_get_den (mpz_t DENOMINATOR, const mpq_t RATIONAL)
-- Function: void mpq_set_num (mpq_t RATIONAL, const mpz_t NUMERATOR)
-- Function: void mpq_set_den (mpq_t RATIONAL, const mpz_t DENOMINATOR)
Get or set the numerator or denominator of a rational. These
functions are equivalent to calling `mpz_set' with an appropriate
`mpq_numref' or `mpq_denref'. Direct use of `mpq_numref' or
`mpq_denref' is recommended instead of these functions.

File: gmp.info, Node: I/O of Rationals, Prev: Applying Integer Functions, Up: Rational Number Functions
6.6 Input and Output Functions
==============================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream, of `mpq' numbers. Passing a `NULL' pointer
for a STREAM argument to any of these functions will make them read from
`stdin' and write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
See also *note Formatted Output:: and *note Formatted Input::.
-- Function: size_t mpq_out_str (FILE *STREAM, int BASE, const mpq_t
OP)
Output OP on stdio stream STREAM, as a string of digits in base
BASE. The base may vary from 2 to 36. Output is in the form
`num/den' or if the denominator is 1 then just `num'.
Return the number of bytes written, or if an error occurred,
return 0.
-- Function: size_t mpq_inp_str (mpq_t ROP, FILE *STREAM, int BASE)
Read a string of digits from STREAM and convert them to a rational
in ROP. Any initial white-space characters are read and
discarded. Return the number of characters read (including white
space), or 0 if a rational could not be read.
The input can be a fraction like `17/63' or just an integer like
`123'. Reading stops at the first character not in this form, and
white space is not permitted within the string. If the input
might not be in canonical form, then `mpq_canonicalize' must be
called (*note Rational Number Functions::).
The BASE can be between 2 and 36, or can be 0 in which case the
leading characters of the string determine the base, `0x' or `0X'
for hexadecimal, `0' for octal, or decimal otherwise. The leading
characters are examined separately for the numerator and
denominator of a fraction, so for instance `0x10/11' is 16/11,
whereas `0x10/0x11' is 16/17.

File: gmp.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
7 Floating-point Functions
**************************
GMP floating point numbers are stored in objects of type `mpf_t' and
functions operating on them have an `mpf_' prefix.
The mantissa of each float has a user-selectable precision, limited
only by available memory. Each variable has its own precision, and
that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on
most systems. In the current implementation the exponent is a count of
limbs, so for example on a 32-bit system this means a range of roughly
2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
greater. Note however that `mpf_get_str' can only return an exponent
which fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept
exponents bigger than a `long'.
Each variable keeps 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 by the user for a variable is a minimum
value, GMP may increase it 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.
The mantissa is stored in binary. One consequence of this is that
decimal fractions like 0.1 cannot be represented exactly. The same is
true of plain IEEE `double' floats. This makes both highly unsuitable
for calculations involving money or other values that should be exact
decimal fractions. (Suitably scaled integers, or perhaps rationals,
are better choices.)
The `mpf' functions and variables have no special notion of infinity
or not-a-number, and applications must take care not to overflow the
exponent or results will be unpredictable. 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.
The GMP extension library MPFR (`http://mpfr.org') is an alternative
to GMP's `mpf' functions. MPFR provides well-defined precision and
accurate rounding, and thereby naturally extends IEEE P754.
* Menu:
* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::

File: gmp.info, Node: Initializing Floats, Next: Assigning Floats, Prev: Floating-point Functions, Up: Floating-point Functions
7.1 Initialization Functions
============================
-- Function: void mpf_set_default_prec (mp_bitcnt_t PREC)
Set the default precision to be *at least* PREC bits. All
subsequent calls to `mpf_init' will use this precision, but
previously initialized variables are unaffected.
-- Function: mp_bitcnt_t mpf_get_default_prec (void)
Return the default precision actually used.
An `mpf_t' object must be initialized before storing the first value
in it. The functions `mpf_init' and `mpf_init2' are used for that
purpose.
-- Function: void mpf_init (mpf_t X)
Initialize X to 0. Normally, a variable should be initialized
once only or at least be cleared, using `mpf_clear', between
initializations. The precision of X is undefined unless a default
precision has already been established by a call to
`mpf_set_default_prec'.
-- Function: void mpf_init2 (mpf_t X, mp_bitcnt_t PREC)
Initialize X to 0 and set its precision to be *at least* PREC
bits. Normally, a variable should be initialized once only or at
least be cleared, using `mpf_clear', between initializations.
-- Function: void mpf_inits (mpf_t X, ...)
Initialize a NULL-terminated list of `mpf_t' variables, and set
their values to 0. The precision of the initialized variables is
undefined unless a default precision has already been established
by a call to `mpf_set_default_prec'.
-- Function: void mpf_clear (mpf_t X)
Free the space occupied by X. Make sure to call this function for
all `mpf_t' variables when you are done with them.
-- Function: void mpf_clears (mpf_t X, ...)
Free the space occupied by a NULL-terminated list of `mpf_t'
variables.
Here is an example on how to initialize floating-point variables:
{
mpf_t x, y;
mpf_init (x); /* use default precision */
mpf_init2 (y, 256); /* precision _at least_ 256 bits */
...
/* Unless the program is about to exit, do ... */
mpf_clear (x);
mpf_clear (y);
}
The following three functions are useful for changing the precision
during a calculation. A typical use would be for adjusting the
precision gradually in iterative algorithms like Newton-Raphson, making
the computation precision closely match the actual accurate part of the
numbers.
-- Function: mp_bitcnt_t mpf_get_prec (const mpf_t OP)
Return the current precision of OP, in bits.
-- Function: void mpf_set_prec (mpf_t ROP, mp_bitcnt_t PREC)
Set the precision of ROP to be *at least* PREC bits. The value in
ROP will be truncated to the new precision.
This function requires a call to `realloc', and so should not be
used in a tight loop.
-- Function: void mpf_set_prec_raw (mpf_t ROP, mp_bitcnt_t PREC)
Set the precision of ROP to be *at least* PREC bits, without
changing the memory allocated.
PREC must be no more than the allocated precision for ROP, that
being the precision when ROP was initialized, or in the most recent
`mpf_set_prec'.
The value in ROP is unchanged, and in particular if it had a higher
precision than PREC it will retain that higher precision. New
values written to ROP will use the new PREC.
Before calling `mpf_clear' or the full `mpf_set_prec', another
`mpf_set_prec_raw' call must be made to restore ROP to its original
allocated precision. Failing to do so will have unpredictable
results.
`mpf_get_prec' can be used before `mpf_set_prec_raw' to get the
original allocated precision. After `mpf_set_prec_raw' it
reflects the PREC value set.
`mpf_set_prec_raw' is an efficient way to use an `mpf_t' variable
at different precisions during a calculation, perhaps to gradually
increase precision in an iteration, or just to use various
different precisions for different purposes during a calculation.

File: gmp.info, Node: Assigning Floats, Next: Simultaneous Float Init & Assign, Prev: Initializing Floats, Up: Floating-point Functions
7.2 Assignment Functions
========================
These functions assign new values to already initialized floats (*note
Initializing Floats::).
-- Function: void mpf_set (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_set_ui (mpf_t ROP, unsigned long int OP)
-- Function: void mpf_set_si (mpf_t ROP, signed long int OP)
-- Function: void mpf_set_d (mpf_t ROP, double OP)
-- Function: void mpf_set_z (mpf_t ROP, const mpz_t OP)
-- Function: void mpf_set_q (mpf_t ROP, const mpq_t OP)
Set the value of ROP from OP.
-- Function: int mpf_set_str (mpf_t ROP, const char *STR, int BASE)
Set the value of ROP from the string in STR. The string is of the
form `M@N' or, if the base is 10 or less, alternatively `MeN'.
`M' is the mantissa and `N' is the exponent. The mantissa is
always in the specified base. The exponent is either in the
specified base or, if BASE is negative, in decimal. The decimal
point expected is taken from the current locale, on systems
providing `localeconv'.
The argument BASE may be in the ranges 2 to 62, or -62 to -2.
Negative values are used to specify that the exponent is in
decimal.
For bases up to 36, case is ignored; upper-case and lower-case
letters have the same value; for bases 37 to 62, upper-case letter
represent the usual 10..35 while lower-case letter represent
36..61.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
White space is allowed in the string, and is simply ignored.
[This is not really true; white-space is ignored in the beginning
of the string and within the mantissa, but not in other places,
such as after a minus sign or in the exponent. We are considering
changing the definition of this function, making it fail when
there is any white-space in the input, since that makes a lot of
sense. Please tell us your opinion about this change. Do you
really want it to accept "3 14" as meaning 314 as it does now?]
This function returns 0 if the entire string is a valid number in
base BASE. Otherwise it returns -1.
-- Function: void mpf_swap (mpf_t ROP1, mpf_t ROP2)
Swap ROP1 and ROP2 efficiently. Both the values and the
precisions of the two variables are swapped.

File: gmp.info, Node: Simultaneous Float Init & Assign, Next: Converting Floats, Prev: Assigning Floats, Up: Floating-point Functions
7.3 Combined Initialization and Assignment Functions
====================================================
For convenience, GMP provides a parallel series of initialize-and-set
functions which initialize the output and then store the value there.
These functions' names have the form `mpf_init_set...'
Once the float has been initialized by any of the `mpf_init_set...'
functions, it can be used as the source or destination operand for the
ordinary float functions. Don't use an initialize-and-set function on
a variable already initialized!
-- Function: void mpf_init_set (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
-- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
-- Function: void mpf_init_set_d (mpf_t ROP, double OP)
Initialize ROP and set its value from OP.
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.
-- Function: int mpf_init_set_str (mpf_t ROP, const char *STR, int
BASE)
Initialize ROP and set its value from the string in STR. See
`mpf_set_str' above for details on the assignment operation.
Note that ROP is initialized even if an error occurs. (I.e., you
have to call `mpf_clear' for it.)
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.

File: gmp.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
7.4 Conversion Functions
========================
-- Function: double mpf_get_d (const mpf_t OP)
Convert OP to a `double', truncating if necessary (i.e. rounding
towards zero).
If the exponent in OP is too big or too small to fit a `double'
then the result is system dependent. For too big an infinity is
returned when available. For too small 0.0 is normally returned.
Hardware overflow, underflow and denorm traps may or may not occur.
-- Function: double mpf_get_d_2exp (signed long int *EXP, const mpf_t
OP)
Convert OP to a `double', truncating if necessary (i.e. rounding
towards zero), and with an exponent returned separately.
The return value is in the range 0.5<=abs(D)<1 and the exponent is
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
is zero, the return is 0.0 and 0 is stored to `*EXP'.
This is similar to the standard C `frexp' function (*note
Normalization Functions: (libc)Normalization Functions.).
-- Function: long mpf_get_si (const mpf_t OP)
-- Function: unsigned long mpf_get_ui (const mpf_t OP)
Convert OP to a `long' or `unsigned long', truncating any fraction
part. If OP is too big for the return type, the result is
undefined.
See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
Miscellaneous Float Functions::).
-- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
BASE, size_t N_DIGITS, const mpf_t OP)
Convert OP to a string of digits in base BASE. The base argument
may vary from 2 to 62 or from -2 to -36. Up to N_DIGITS digits
will be generated. Trailing zeros are not returned. No more
digits than can be accurately represented by OP are ever
generated. If N_DIGITS is 0 then that accurate maximum number of
digits are generated.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
If STR is `NULL', the result string is allocated using the current
allocation function (*note Custom Allocation::). The block will be
`strlen(str)+1' bytes, that being exactly enough for the string and
null-terminator.
If STR is not `NULL', it should point to a block of N_DIGITS + 2
bytes, that being enough for the mantissa, a possible minus sign,
and a null-terminator. When N_DIGITS is 0 to get all significant
digits, an application won't be able to know the space required,
and STR should be `NULL' in that case.
The generated string is a fraction, with an implicit radix point
immediately to the left of the first digit. The applicable
exponent is written through the EXPPTR pointer. For example, the
number 3.1416 would be returned as string "31416" and exponent 1.
When OP is zero, an empty string is produced and the exponent
returned is 0.
A pointer to the result string is returned, being either the
allocated block or the given STR.

File: gmp.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
7.5 Arithmetic Functions
========================
-- Function: void mpf_add (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
-- Function: void mpf_add_ui (mpf_t ROP, const mpf_t OP1, unsigned
long int OP2)
Set ROP to OP1 + OP2.
-- Function: void mpf_sub (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
-- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, const
mpf_t OP2)
-- Function: void mpf_sub_ui (mpf_t ROP, const mpf_t OP1, unsigned
long int OP2)
Set ROP to OP1 - OP2.
-- Function: void mpf_mul (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
-- Function: void mpf_mul_ui (mpf_t ROP, const mpf_t OP1, unsigned
long int OP2)
Set ROP to OP1 times OP2.
Division is undefined if the divisor is zero, and passing a zero
divisor to the divide functions will make these functions intentionally
divide by zero. This lets the user handle arithmetic exceptions in
these functions in the same manner as other arithmetic exceptions.
-- Function: void mpf_div (mpf_t ROP, const mpf_t OP1, const mpf_t OP2)
-- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, const
mpf_t OP2)
-- Function: void mpf_div_ui (mpf_t ROP, const mpf_t OP1, unsigned
long int OP2)
Set ROP to OP1/OP2.
-- Function: void mpf_sqrt (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
Set ROP to the square root of OP.
-- Function: void mpf_pow_ui (mpf_t ROP, const mpf_t OP1, unsigned
long int OP2)
Set ROP to OP1 raised to the power OP2.
-- Function: void mpf_neg (mpf_t ROP, const mpf_t OP)
Set ROP to -OP.
-- Function: void mpf_abs (mpf_t ROP, const mpf_t OP)
Set ROP to the absolute value of OP.
-- Function: void mpf_mul_2exp (mpf_t ROP, const mpf_t OP1,
mp_bitcnt_t OP2)
Set ROP to OP1 times 2 raised to OP2.
-- Function: void mpf_div_2exp (mpf_t ROP, const mpf_t OP1,
mp_bitcnt_t OP2)
Set ROP to OP1 divided by 2 raised to OP2.

File: gmp.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
7.6 Comparison Functions
========================
-- Function: int mpf_cmp (const mpf_t OP1, const mpf_t OP2)
-- Function: int mpf_cmp_d (const mpf_t OP1, double OP2)
-- Function: int mpf_cmp_ui (const mpf_t OP1, unsigned long int OP2)
-- Function: int mpf_cmp_si (const mpf_t OP1, signed long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
`mpf_cmp_d' can be called with an infinity, but results are
undefined for a NaN.
-- Function: int mpf_eq (const mpf_t OP1, const mpf_t OP2, mp_bitcnt_t
op3)
Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
Caution 1: All version of GMP up to version 4.2.4 compared just
whole limbs, meaning sometimes more than OP3 bits, sometimes fewer.
Caution 2: This function will consider XXX11...111 and XX100...000
different, even if ... is replaced by a semi-infinite number of
bits. Such numbers are really just one ulp off, and should be
considered equal.
-- Function: void mpf_reldiff (mpf_t ROP, const mpf_t OP1, const mpf_t
OP2)
Compute the relative difference between OP1 and OP2 and store the
result in ROP. This is abs(OP1-OP2)/OP1.
-- Macro: int mpf_sgn (const mpf_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its argument multiple times.

File: gmp.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
7.7 Input and Output Functions
==============================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream, of `mpf' numbers. Passing a `NULL' pointer
for a STREAM argument to any of these functions will make them read from
`stdin' and write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `gmp.h', since that will allow `gmp.h' to define
prototypes for these functions.
See also *note Formatted Output:: and *note Formatted Input::.
-- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
N_DIGITS, const mpf_t OP)
Print OP to STREAM, as a string of digits. Return the number of
bytes written, or if an error occurred, return 0.
The mantissa is prefixed with an `0.' and is in the given BASE,
which may vary from 2 to 62 or from -2 to -36. An exponent is
then printed, separated by an `e', or if the base is greater than
10 then by an `@'. The exponent is always in decimal. The
decimal point follows the current locale, on systems providing
`localeconv'.
For BASE in the range 2..36, digits and lower-case letters are
used; for -2..-36, digits and upper-case letters are used; for
37..62, digits, upper-case letters, and lower-case letters (in
that significance order) are used.
Up to N_DIGITS will be printed from the mantissa, except that no
more digits than are accurately representable by OP will be
printed. N_DIGITS can be 0 to select that accurate maximum.
-- Function: size_t mpf_inp_str (mpf_t ROP, FILE *STREAM, int BASE)
Read a string in base BASE from STREAM, and put the read float in
ROP. The string is of the form `M@N' or, if the base is 10 or
less, alternatively `MeN'. `M' is the mantissa and `N' is the
exponent. The mantissa is always in the specified base. The
exponent is either in the specified base or, if BASE is negative,
in decimal. The decimal point expected is taken from the current
locale, on systems providing `localeconv'.
The argument BASE may be in the ranges 2 to 36, or -36 to -2.
Negative values are used to specify that the exponent is in
decimal.
Unlike the corresponding `mpz' function, the base will not be
determined from the leading characters of the string if BASE is 0.
This is so that numbers like `0.23' are not interpreted as octal.
Return the number of bytes read, or if an error occurred, return 0.

File: gmp.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
7.8 Miscellaneous Functions
===========================
-- Function: void mpf_ceil (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_floor (mpf_t ROP, const mpf_t OP)
-- Function: void mpf_trunc (mpf_t ROP, const mpf_t OP)
Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
to the integer towards zero.
-- Function: int mpf_integer_p (const mpf_t OP)
Return non-zero if OP is an integer.
-- Function: int mpf_fits_ulong_p (const mpf_t OP)
-- Function: int mpf_fits_slong_p (const mpf_t OP)
-- Function: int mpf_fits_uint_p (const mpf_t OP)
-- Function: int mpf_fits_sint_p (const mpf_t OP)
-- Function: int mpf_fits_ushort_p (const mpf_t OP)
-- Function: int mpf_fits_sshort_p (const mpf_t OP)
Return non-zero if OP would fit in the respective C data type, when
truncated to an integer.
-- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
mp_bitcnt_t NBITS)
Generate a uniformly distributed random float in ROP, such that 0
<= ROP < 1, with NBITS significant bits in the mantissa or less if
the precision of ROP is smaller.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpf_random2 (mpf_t ROP, mp_size_t MAX_SIZE, mp_exp_t
EXP)
Generate a random float of at most MAX_SIZE limbs, with long
strings of zeros and ones in the binary representation. The
exponent of the number is in the interval -EXP to EXP (in limbs).
This function is useful for testing functions and algorithms,
since these kind of random numbers have proven to be more likely
to trigger corner-case bugs. Negative random numbers are
generated when MAX_SIZE is negative.

File: gmp.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
8 Low-level Functions
*********************
This chapter describes low-level GMP functions, used to implement the
high-level GMP functions, but also intended for time-critical user code.
These functions start with the prefix `mpn_'.
The `mpn' functions are designed to be as fast as possible, *not* to
provide a coherent calling interface. The different functions have
somewhat similar interfaces, but there are variations that make them
hard to use. These functions do as little as possible apart from the
real multiple precision computation, so that no time is spent on things
that not all callers need.
A source operand is specified by a pointer to the least significant
limb and a limb count. A destination operand is specified by just a
pointer. It is the responsibility of the caller to ensure that the
destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform
computations on subranges of an argument, and store the result into a
subrange of a destination.
A common requirement for all functions is that each source area
needs at least one limb. No size argument may be zero. Unless
otherwise stated, in-place operations are allowed where source and
destination are the same, but not where they only partly overlap.
The `mpn' functions are the base for the implementation of the
`mpz_', `mpf_', and `mpq_' functions.
This example adds the number beginning at S1P and the number
beginning at S2P and writes the sum at DESTP. All areas have N limbs.
cy = mpn_add_n (destp, s1p, s2p, n)
It should be noted that the `mpn' functions make no attempt to
identify high or low zero limbs on their operands, or other special
forms. On random data such cases will be unlikely and it'd be wasteful
for every function to check every time. An application knowing
something about its data can take steps to trim or perhaps split its
calculations.
In the notation used below, a source operand is identified by the
pointer to the least significant limb, and the limb count in braces.
For example, {S1P, S1N}.
-- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
This is the lowest-level function for addition. It is the
preferred function for addition, since it is written in assembly
for most CPUs. For addition of a variable to itself (i.e., S1P
equals S2P) use `mpn_lshift' with a count of 1 for optimal speed.
-- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Add {S1P, N} and S2LIMB, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
-- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
limbs of the result to RP. Return carry, either 0 or 1.
This function requires that S1N is greater than or equal to S2N.
-- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Subtract {S2P, N} from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
This is the lowest-level function for subtraction. It is the
preferred function for subtraction, since it is written in
assembly for most CPUs.
-- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Subtract S2LIMB from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
-- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
significant limbs of the result to RP. Return borrow, either 0 or
1.
This function requires that S1N is greater than or equal to S2N.
-- Function: mp_limb_t mpn_neg (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N)
Perform the negation of {SP, N}, and write the result to {RP, N}.
This is equivalent to calling `mpn_sub_n' with a N-limb zero
minuend and passing {SP, N} as subtrahend. Return borrow, either