Add libgmp 6.2.0 to third_party
Don't build it yet. That will come in the next review.
Change-Id: Idf3266558165e5ab45f4a41c98cc8c838c8244d5
diff --git a/third_party/gmp/doc/gmp.info-1 b/third_party/gmp/doc/gmp.info-1
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+This is gmp.info, produced by makeinfo version 6.6 from gmp.texi.
+
+This manual describes how to install and use the GNU multiple precision
+arithmetic library, version 6.2.0.
+
+ Copyright 1991, 1993-2016, 2018 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 6.2.0.
+
+ Copyright 1991, 1993-2016, 2018 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-6.2.0/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' is
+ architecture-dependent since it encodes certain aspects of
+ 'libgmp', so it will be necessary to ensure both '$prefix/include'
+ and '$exec_prefix/include' are available to the compiler.
+
+'--disable-shared', '--disable-static'
+ By default both shared and static libraries are built (where
+ possible), but one or other can be disabled. Shared libraries
+ result in smaller executables and permit code sharing between
+ separate running processes, but on some CPUs are slightly slower,
+ having a small cost on each function call.
+
+Native Compilation, '--build=CPU-VENDOR-OS'
+ 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.
+
+ <https://www.cygwin.com/>
+ <http://www.delorie.com/djgpp/>
+ <http://www.mingw.org/>
+
+ Microsoft also publishes an Interix "Services for Unix" which can
+ be used to build GMP on Windows (with a normal './configure'), but
+ 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.
+
+ 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);
+ }
+
+ GMP types like 'mpz_t' are implemented as one-element arrays of
+certain structures. Declaring a variable creates an object with the
+fields GMP needs, but variables are normally manipulated by using the
+pointer to the object. For both behavior and efficiency reasons, it is
+discouraged to make copies of the GMP object itself (either directly or
+via aggregate objects containing such GMP objects). If copies are done,
+all of them must be used read-only; using a copy as the output of some
+function will invalidate all the other copies. Note that the actual
+fields in each 'mpz_t' etc are for internal use only and should not be
+accessed directly by code that expects to be compatible with future GMP
+releases.
+
+
+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 that when the function stores a value there
+it will change the original in the caller. Parameters which are
+input-only can be designated 'const' to provoke a compiler error or
+warning on attempting to modify them.
+
+ When a function is going to return a GMP result, it should designate
+a parameter that it sets, like the library functions do. More than one
+value can be returned by having more than one output parameter, again
+like the library functions. A 'return' of an 'mpz_t' etc doesn't return
+the object, only a pointer, and this is almost certainly not what's
+wanted.
+
+ 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;
+ }
+
+ Our function 'foo' works even if its caller passes the same variable
+for 'param' and 'result', just like the library functions. But
+sometimes it's tricky to make that work, and an application might not
+want to bother supporting that sort of thing.
+
+ Since GMP types are implemented as one-element arrays, using a GMP
+variable as a parameter passes a pointer to the object. Hence the
+call-by-reference.
+
+
+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 "6.2.0". 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://cs.ecs.baylor.edu/~donahoo/tools/ccmalloc/>
+ <http://dmalloc.com/>
+ <https://wiki.gnome.org/Apps/MemProf>
+
+ The GMP default allocation routines in 'memory.c' also have a
+ simple sentinel scheme which can be enabled with '#define DEBUG' in
+ that file. This is mainly designed for detecting buffer overruns
+ during GMP development, but might find other uses.
+
+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-6.2.0/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
+
+ <https://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
+ (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 the inverse BASE^(-1) mod MOD exists
+ (see 'mpz_invert' in *note Number Theoretic Functions::). If an
+ inverse doesn't exist then a divide by zero is raised.
+
+ -- Function: void mpz_powm_sec (mpz_t ROP, 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 non-prime.
+
+ This function performs some trial divisions, a Baillie-PSW probable
+ prime test, then REPS-24 Miller-Rabin probabilistic primality
+ tests. A higher REPS value will reduce the chances of a non-prime
+ being identified as "probably prime". A composite number will be
+ identified as a prime with an asymptotic probability of less than
+ 4^(-REPS). Reasonable values of REPS are between 15 and 50.
+
+ GMP versions up to and including 6.1.2 did not use the Baillie-PSW
+ primality test. In those older versions of GMP, this function
+ performed REPS Miller-Rabin tests.
+
+ -- Function: void mpz_nextprime (mpz_t ROP, 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 or G is 'NULL' then that value is not computed.
+
+ -- Function: void mpz_lcm (mpz_t ROP, const mpz_t OP1, const mpz_t OP2)
+ -- Function: void mpz_lcm_ui (mpz_t ROP, const mpz_t OP1, unsigned long
+ OP2)
+ Set ROP to the least common multiple of OP1 and OP2. ROP is always
+ positive, irrespective of the signs of OP1 and OP2. ROP will be
+ zero if either OP1 or OP2 is zero.
+
+ -- Function: int mpz_invert (mpz_t ROP, 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) (with ROP = 0 possible only when
+ abs(OP2) = 1, i.e., in the somewhat degenerate zero ring). If an
+ inverse doesn't exist the return value is zero and ROP is
+ undefined. The behaviour of this function is undefined when OP2 is
+ zero.
+
+ -- Function: int mpz_jacobi (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 2^(N-1) to 2^N-1, inclusive.
+
+ The variable STATE must be initialized by calling one of the
+ 'gmp_randinit' functions (*note Random State Initialization::)
+ 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 form means that the
+denominator and the numerator have no common factors, and that the
+denominator is positive. Zero has the unique representation 0/1.
+
+ 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 argument
+ may vary from 2 to 62 or from -2 to -36. The string will be of the
+ form 'num/den', or if the denominator is 1 then just 'num'.
+
+ For BASE in the range 2..36, digits and lower-case letters are
+ used; for -2..-36, digits and upper-case letters are used; for
+ 37..62, digits, upper-case letters, and lower-case letters (in that
+ significance order) are used.
+
+ If STR is 'NULL', the result string is allocated using the current
+ 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)
+ -- Function: int mpq_cmp_z (const mpq_t OP1, const mpz_t OP2)
+ Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if
+ OP1 = OP2, and a negative value if OP1 < OP2.
+
+ To determine if two rationals are equal, 'mpq_equal' is faster than
+ 'mpq_cmp'.
+
+ -- 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 argument may vary from 2 to 62 or from -2 to -36.
+ Output is in the form 'num/den' or if the denominator is 1 then
+ just 'num'.
+
+ For BASE in the range 2..36, digits and lower-case letters are
+ used; for -2..-36, digits and upper-case letters are used; for
+ 37..62, digits, upper-case letters, and lower-case letters (in that
+ significance order) are used.
+
+ Return the number of bytes written, or if an error occurred, return
+ 0.
+
+ -- 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 62, or can be 0 in which case the
+ leading characters of the string determine the base, '0x' or '0X'
+ for hexadecimal, '0b' and '0B' for binary, '0' for octal, or
+ decimal otherwise. The leading characters are examined separately
+ for the numerator and denominator of a fraction, so for instance
+ '0x10/11' is 16/11, whereas '0x10/0x11' is 16/17.
+
+
+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, in
+practice only limited by available memory. Each variable has its own
+precision, and that can be increased or decreased at any time. This
+selectable precision is a minimum value, GMP rounds it up to a whole
+limb.
+
+ The accuracy of a calculation is determined by the priorly set
+precision of the destination variable and the numeric values of the
+input variables. Input variables' set precisions do not affect
+calculations (except indirectly as their values might have been affected
+when they were assigned).
+
+ The exponent of each float has fixed precision, one machine word on
+most systems. In the current implementation the exponent is a count of
+limbs, so for example on a 32-bit system this means a range of roughly
+2^-68719476768 to 2^68719476736, or on a 64-bit system this will be much
+greater. Note however that 'mpf_get_str' can only return an exponent
+which fits an 'mp_exp_t' and currently 'mpf_set_str' doesn't accept
+exponents bigger than a 'long'.
+
+ Each variable keeps track of the mantissa data actually in use. This
+means that if a float is exactly represented in only a few bits then
+only those bits will be used in a calculation, even if the variable's
+selected precision is high. This is a performance optimization; it does
+not affect the numeric results.
+
+ Internally, GMP sometimes calculates with higher precision than that
+of the destination variable in order to limit errors. Final results are
+always truncated to the destination variable's precision.
+
+ The mantissa is stored in binary. One consequence of this is that
+decimal fractions like 0.1 cannot be represented exactly. The same is
+true of plain IEEE 'double' floats. This makes both highly unsuitable
+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.
+
+ Note that the 'mpf' functions are _not_ intended as a smooth
+extension to IEEE P754 arithmetic. In particular results obtained on
+one computer often differ from the results on a computer with a
+different word size.
+
+ New projects should consider using the GMP extension library MPFR
+(<http://mpfr.org>) instead. MPFR provides well-defined precision and
+accurate rounding, and thereby naturally extends IEEE P754.
+
+* Menu:
+
+* Initializing Floats::
+* 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
+ (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_z (const mpf_t OP1, const mpz_t OP2)
+ -- Function: int mpf_cmp_d (const mpf_t OP1, double OP2)
+ -- Function: int mpf_cmp_ui (const mpf_t OP1, unsigned long int OP2)
+ -- Function: int mpf_cmp_si (const mpf_t OP1, signed long int OP2)
+ Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if
+ OP1 = OP2, and a negative value if OP1 < OP2.
+
+ 'mpf_cmp_d' can be called with an infinity, but results are
+ undefined for a NaN.
+
+ -- Function: int mpf_eq (const mpf_t OP1, const mpf_t OP2, mp_bitcnt_t
+ op3)
+ *This function is mathematically ill-defined and should not be
+ used.*
+
+ Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
+ zero otherwise. Note that numbers like e.g., 256 (binary
+ 100000000) and 255 (binary 11111111) will never be equal by this
+ function's measure, and furthermore that 0 will only be equal to
+ itself.
+
+ -- Function: void mpf_reldiff (mpf_t ROP, 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 0
+ or 1.
+
+ -- Function: void mpn_mul_n (mp_limb_t *RP, const mp_limb_t *S1P, const
+ mp_limb_t *S2P, mp_size_t N)
+ Multiply {S1P, N} and {S2P, N}, and write the 2*N-limb result to
+ RP.
+
+ The destination has to have space for 2*N limbs, even if the
+ product's most significant limb is zero. No overlap is permitted
+ between the destination and either source.
+
+ If the two input operands are the same, use 'mpn_sqr'.
+
+ -- Function: mp_limb_t mpn_mul (mp_limb_t *RP, const mp_limb_t *S1P,
+ mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
+ Multiply {S1P, S1N} and {S2P, S2N}, and write the (S1N+S2N)-limb
+ result to RP. Return the most significant limb of the result.
+
+ The destination has to have space for S1N + S2N limbs, even if the
+ product's most significant limb is zero. No overlap is permitted
+ between the destination and either source.
+
+ This function requires that S1N is greater than or equal to S2N.
+
+ -- Function: void mpn_sqr (mp_limb_t *RP, const mp_limb_t *S1P,
+ mp_size_t N)
+ Compute the square of {S1P, N} and write the 2*N-limb result to RP.
+
+ The destination has to have space for 2N limbs, even if the
+ result's most significant limb is zero. No overlap is permitted
+ between the destination and the source.
+
+ -- Function: mp_limb_t mpn_mul_1 (mp_limb_t *RP, const mp_limb_t *S1P,
+ mp_size_t N, mp_limb_t S2LIMB)
+ Multiply {S1P, N} by S2LIMB, and write the N least significant
+ limbs of the product to RP. Return the most significant limb of
+ the product. {S1P, N} and {RP, N} are allowed to overlap provided
+ RP <= S1P.
+
+ This is a low-level function that is a building block for general
+ multiplication as well as other operations in GMP. It is written
+ in assembly for most CPUs.
+
+ Don't call this function if S2LIMB is a power of 2; use
+ 'mpn_lshift' with a count equal to the logarithm of S2LIMB instead,
+ for optimal speed.
+
+ -- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
+ *S1P, mp_size_t N, mp_limb_t S2LIMB)
+ Multiply {S1P, N} and S2LIMB, and add the N least significant limbs
+ of the product to {RP, N} and write the result to RP. Return the
+ most significant limb of the product, plus carry-out from the
+ addition. {S1P, N} and {RP, N} are allowed to overlap provided RP
+ <= S1P.
+
+ This is a low-level function that is a building block for general
+ multiplication as well as other operations in GMP. It is written
+ in assembly for most CPUs.
+
+ -- Function: mp_limb_t mpn_submul_1 (mp_limb_t *RP, const mp_limb_t
+ *S1P, mp_size_t N, mp_limb_t S2LIMB)
+ Multiply {S1P, N} and S2LIMB, and subtract the N least significant
+ limbs of the product from {RP, N} and write the result to RP.
+ Return the most significant limb of the product, plus borrow-out
+ from the subtraction. {S1P, N} and {RP, N} are allowed to overlap
+ provided RP <= S1P.
+
+ This is a low-level function that is a building block for general
+ multiplication and division as well as other operations in GMP. It
+ is written in assembly for most CPUs.
+
+ -- Function: void mpn_tdiv_qr (mp_limb_t *QP, mp_limb_t *RP, mp_size_t
+ QXN, const mp_limb_t *NP, mp_size_t NN, const mp_limb_t *DP,
+ mp_size_t DN)
+ Divide {NP, NN} by {DP, DN} and put the quotient at {QP, NN-DN+1}
+ and the remainder at {RP, DN}. The quotient is rounded towards 0.
+
+ No overlap is permitted between arguments, except that NP might
+ equal RP. The dividend size NN must be greater than or equal to
+ divisor size DN. The most significant limb of the divisor must be
+ non-zero. The QXN operand must be zero.
+
+ -- Function: mp_limb_t mpn_divrem (mp_limb_t *R1P, mp_size_t QXN,
+ mp_limb_t *RS2P, mp_size_t RS2N, const mp_limb_t *S3P,
+ mp_size_t S3N)
+ [This function is obsolete. Please call 'mpn_tdiv_qr' instead for
+ best performance.]
+
+ Divide {RS2P, RS2N} by {S3P, S3N}, and write the quotient at R1P,
+ with the exception of the most significant limb, which is returned.
+ The remainder replaces the dividend at RS2P; it will be S3N limbs
+ long (i.e., as many limbs as the divisor).
+
+ In addition to an integer quotient, QXN fraction limbs are
+ developed, and stored after the integral limbs. For most usages,
+ QXN will be zero.
+
+ It is required that RS2N is greater than or equal to S3N. It is
+ required that the most significant bit of the divisor is set.
+
+ If the quotient is not needed, pass RS2P + S3N as R1P. Aside from
+ that special case, no overlap between arguments is permitted.
+
+ Return the most significant limb of the quotient, either 0 or 1.
+
+ The area at R1P needs to be RS2N - S3N + QXN limbs large.
+
+ -- Function: mp_limb_t mpn_divrem_1 (mp_limb_t *R1P, mp_size_t QXN,
+ mp_limb_t *S2P, mp_size_t S2N, mp_limb_t S3LIMB)
+ -- Macro: mp_limb_t mpn_divmod_1 (mp_limb_t *R1P, mp_limb_t *S2P,
+ mp_size_t S2N, mp_limb_t S3LIMB)
+ Divide {S2P, S2N} by S3LIMB, and write the quotient at R1P. Return
+ the remainder.
+
+ The integer quotient is written to {R1P+QXN, S2N} and in addition
+ QXN fraction limbs are developed and written to {R1P, QXN}. Either
+ or both S2N and QXN can be zero. For most usages, QXN will be
+ zero.
+
+ 'mpn_divmod_1' exists for upward source compatibility and is simply
+ a macro calling 'mpn_divrem_1' with a QXN of 0.
+
+ The areas at R1P and S2P have to be identical or completely
+ separate, not partially overlapping.
+
+ -- Function: mp_limb_t mpn_divmod (mp_limb_t *R1P, mp_limb_t *RS2P,
+ mp_size_t RS2N, const mp_limb_t *S3P, mp_size_t S3N)
+ [This function is obsolete. Please call 'mpn_tdiv_qr' instead for
+ best performance.]
+
+ -- Function: void mpn_divexact_1 (mp_limb_t * RP, const mp_limb_t * SP,
+ mp_size_t N, mp_limb_t D)
+ Divide {SP, N} by D, expecting it to divide exactly, and writing
+ the result to {RP, N}. If D doesn't divide exactly, the value
+ written to {RP, N} is undefined. The areas at RP and SP have to be
+ identical or completely separate, not partially overlapping.
+
+ -- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
+ mp_size_t N)
+ -- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t *SP,
+ mp_size_t N, mp_limb_t CARRY)
+ Divide {SP, N} by 3, expecting it to divide exactly, and writing
+ the result to {RP, N}. If 3 divides exactly, the return value is
+ zero and the result is the quotient. If not, the return value is
+ non-zero and the result won't be anything useful.
+
+ 'mpn_divexact_by3c' takes an initial carry parameter, which can be
+ the return value from a previous call, so a large calculation can
+ be done piece by piece from low to high. 'mpn_divexact_by3' is
+ simply a macro calling 'mpn_divexact_by3c' with a 0 carry
+ parameter.
+
+ These routines use a multiply-by-inverse and will be faster than
+ 'mpn_divrem_1' on CPUs with fast multiplication but slow division.
+
+ The source a, result q, size n, initial carry i, and return value c
+ satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return c is
+ always 0, 1 or 2, and the initial carry i must also be 0, 1 or 2
+ (these are both borrows really). When c=0 clearly q=(a-i)/3. When
+ c!=0, the remainder (a-i) mod 3 is given by 3-c, because b == 1 mod
+ 3 (when 'mp_bits_per_limb' is even, which is always so currently).
+
+ -- Function: mp_limb_t mpn_mod_1 (const mp_limb_t *S1P, mp_size_t S1N,
+ mp_limb_t S2LIMB)
+ Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
+ zero.
+
+ -- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
+ mp_size_t N, unsigned int COUNT)
+ Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
+ The bits shifted out at the left are returned in the least
+ significant COUNT bits of the return value (the rest of the return
+ value is zero).
+
+ COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
+ {SP, N} and {RP, N} may overlap, provided RP >= SP.
+
+ This function is written in assembly for most CPUs.
+
+ -- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
+ mp_size_t N, unsigned int COUNT)
+ Shift {SP, N} right by COUNT bits, and write the result to {RP, N}.
+ The bits shifted out at the right are returned in the most
+ significant COUNT bits of the return value (the rest of the return
+ value is zero).
+
+ COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
+ {SP, N} and {RP, N} may overlap, provided RP <= SP.
+
+ This function is written in assembly for most CPUs.
+
+ -- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
+ mp_size_t N)
+ Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
+ S2, 0 if they are equal, or a negative value if S1 < S2.
+
+ -- Function: int mpn_zero_p (const mp_limb_t *SP, mp_size_t N)
+ Test {SP, N} and return 1 if the operand is zero, 0 otherwise.
+
+ -- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *XP, mp_size_t
+ XN, mp_limb_t *YP, mp_size_t YN)
+ Set {RP, RETVAL} to the greatest common divisor of {XP, XN} and
+ {YP, YN}. The result can be up to YN limbs, the return value is
+ the actual number produced. Both source operands are destroyed.
+
+ It is required that XN >= YN > 0, the most significant limb of {YP,
+ YN} must be non-zero, and at least one of the two operands must be
+ odd. No overlap is permitted between {XP, XN} and {YP, YN}.
+
+ -- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *XP, mp_size_t XN,
+ mp_limb_t YLIMB)
+ Return the greatest common divisor of {XP, XN} and YLIMB. Both
+ operands must be non-zero.
+
+ -- Function: mp_size_t mpn_gcdext (mp_limb_t *GP, mp_limb_t *SP,
+ mp_size_t *SN, mp_limb_t *UP, mp_size_t UN, mp_limb_t *VP,
+ mp_size_t VN)
+ Let U be defined by {UP, UN} and let V be defined by {VP, VN}.
+
+ Compute the greatest common divisor G of U and V. Compute a
+ cofactor S such that G = US + VT. The second cofactor T is not
+ computed but can easily be obtained from (G - U*S) / V (the
+ division will be exact). It is required that UN >= VN > 0, and the
+ most significant limb of {VP, VN} must be non-zero.
+
+ S satisfies S = 1 or abs(S) < V / (2 G). S = 0 if and only if V
+ divides U (i.e., G = V).
+
+ Store G at GP and let the return value define its limb count.
+ Store S at SP and let |*SN| define its limb count. S can be
+ negative; when this happens *SN will be negative. The area at GP
+ should have room for VN limbs and the area at SP should have room
+ for VN+1 limbs.
+
+ Both source operands are destroyed.
+
+ Compatibility notes: GMP 4.3.0 and 4.3.1 defined S less strictly.
+ Earlier as well as later GMP releases define S as described here.
+ GMP releases before GMP 4.3.0 required additional space for both
+ input and output areas. More precisely, the areas {UP, UN+1} and
+ {VP, VN+1} were destroyed (i.e. the operands plus an extra limb
+ past the end of each), and the areas pointed to by GP and SP should
+ each have room for UN+1 limbs.
+
+ -- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
+ const mp_limb_t *SP, mp_size_t N)
+ Compute the square root of {SP, N} and put the result at {R1P,
+ ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space for
+ N limbs, but the return value indicates how many are produced.
+
+ The most significant limb of {SP, N} must be non-zero. The areas
+ {R1P, ceil(N/2)} and {SP, N} must be completely separate. The
+ areas {R2P, N} and {SP, N} must be either identical or completely
+ separate.
+
+ If the remainder is not wanted then R2P can be 'NULL', and in this
+ case the return value is zero or non-zero according to whether the
+ remainder would have been zero or non-zero.
+
+ A return value of zero indicates a perfect square. See also
+ 'mpn_perfect_square_p'.
+
+ -- Function: size_t mpn_sizeinbase (const mp_limb_t *XP, mp_size_t N,
+ int BASE)
+ Return the size of {XP,N} measured in number of digits in the given
+ BASE. BASE can vary from 2 to 62. Requires N > 0 and XP[N-1] > 0.
+ The result will be either exact or 1 too big. If BASE is a power
+ of 2, the result is always exact.
+
+ -- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
+ mp_limb_t *S1P, mp_size_t S1N)
+ Convert {S1P, S1N} to a raw unsigned char array at STR in base
+ BASE, and return the number of characters produced. There may be
+ leading zeros in the string. The string is not in ASCII; to
+ convert it to printable format, add the ASCII codes for '0' or 'A',
+ depending on the base and range. BASE can vary from 2 to 256.
+
+ The most significant limb of the input {S1P, S1N} must be non-zero.
+ The input {S1P, S1N} is clobbered, except when BASE is a power of
+ 2, in which case it's unchanged.
+
+ The area at STR has to have space for the largest possible number
+ represented by a S1N long limb array, plus one extra character.
+
+ -- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
+ *STR, size_t STRSIZE, int BASE)
+ Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
+
+ STR[0] is the most significant input byte and STR[STRSIZE-1] is the
+ least significant input byte. Each byte should be a value in the
+ range 0 to BASE-1, not an ASCII character. BASE can vary from 2 to
+ 256.
+
+ The converted value is {RP,RN} where RN is the return value. If
+ the most significant input byte STR[0] is non-zero, then RP[RN-1]
+ will be non-zero, else RP[RN-1] and some number of subsequent limbs
+ may be zero.
+
+ The area at RP has to have space for the largest possible number
+ with STRSIZE digits in the chosen base, plus one extra limb.
+
+ The input must have at least one byte, and no overlap is permitted
+ between {STR,STRSIZE} and the result at RP.
+
+ -- Function: mp_bitcnt_t mpn_scan0 (const mp_limb_t *S1P, mp_bitcnt_t
+ BIT)
+ Scan S1P from bit position BIT for the next clear bit.
+
+ It is required that there be a clear bit within the area at S1P at
+ or beyond bit position BIT, so that the function has something to
+ return.
+
+ -- Function: mp_bitcnt_t mpn_scan1 (const mp_limb_t *S1P, mp_bitcnt_t
+ BIT)
+ Scan S1P from bit position BIT for the next set bit.
+
+ It is required that there be a set bit within the area at S1P at or
+ beyond bit position BIT, so that the function has something to
+ return.
+
+ -- Function: void mpn_random (mp_limb_t *R1P, mp_size_t R1N)
+ -- Function: void mpn_random2 (mp_limb_t *R1P, mp_size_t R1N)
+ Generate a random number of length R1N and store it at R1P. The
+ most significant limb is always non-zero. 'mpn_random' generates
+ uniformly distributed limb data, 'mpn_random2' generates long
+ strings of zeros and ones in the binary representation.
+
+ 'mpn_random2' is intended for testing the correctness of the 'mpn'
+ routines.
+
+ -- Function: mp_bitcnt_t mpn_popcount (const mp_limb_t *S1P, mp_size_t
+ N)
+ Count the number of set bits in {S1P, N}.
+
+ -- Function: mp_bitcnt_t mpn_hamdist (const mp_limb_t *S1P, const
+ mp_limb_t *S2P, mp_size_t N)
+ Compute the hamming distance between {S1P, N} and {S2P, N}, which
+ is the number of bit positions where the two operands have
+ different bit values.
+
+ -- Function: int mpn_perfect_square_p (const mp_limb_t *S1P, mp_size_t
+ N)
+ Return non-zero iff {S1P, N} is a perfect square. The most
+ significant limb of the input {S1P, N} must be non-zero.
+
+ -- Function: void mpn_and_n (mp_limb_t *RP, const mp_limb_t *S1P, const
+ mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical and of {S1P, N} and {S2P, N}, and write
+ the result to {RP, N}.
+
+ -- Function: void mpn_ior_n (mp_limb_t *RP, const mp_limb_t *S1P, const
+ mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
+ and write the result to {RP, N}.
+
+ -- Function: void mpn_xor_n (mp_limb_t *RP, const mp_limb_t *S1P, const
+ mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
+ and write the result to {RP, N}.
+
+ -- Function: void mpn_andn_n (mp_limb_t *RP, const mp_limb_t *S1P,
+ const mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical and of {S1P, N} and the bitwise
+ complement of {S2P, N}, and write the result to {RP, N}.
+
+ -- Function: void mpn_iorn_n (mp_limb_t *RP, const mp_limb_t *S1P,
+ const mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical inclusive or of {S1P, N} and the
+ bitwise complement of {S2P, N}, and write the result to {RP, N}.
+
+ -- Function: void mpn_nand_n (mp_limb_t *RP, const mp_limb_t *S1P,
+ const mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical and of {S1P, N} and {S2P, N}, and write
+ the bitwise complement of the result to {RP, N}.
+
+ -- Function: void mpn_nior_n (mp_limb_t *RP, const mp_limb_t *S1P,
+ const mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical inclusive or of {S1P, N} and {S2P, N},
+ and write the bitwise complement of the result to {RP, N}.
+
+ -- Function: void mpn_xnor_n (mp_limb_t *RP, const mp_limb_t *S1P,
+ const mp_limb_t *S2P, mp_size_t N)
+ Perform the bitwise logical exclusive or of {S1P, N} and {S2P, N},
+ and write the bitwise complement of the result to {RP, N}.
+
+ -- Function: void mpn_com (mp_limb_t *RP, const mp_limb_t *SP,
+ mp_size_t N)
+ Perform the bitwise complement of {SP, N}, and write the result to
+ {RP, N}.
+
+ -- Function: void mpn_copyi (mp_limb_t *RP, const mp_limb_t *S1P,
+ mp_size_t N)
+ Copy from {S1P, N} to {RP, N}, increasingly.
+
+ -- Function: void mpn_copyd (mp_limb_t *RP, const mp_limb_t *S1P,
+ mp_size_t N)
+ Copy from {S1P, N} to {RP, N}, decreasingly.
+
+ -- Function: void mpn_zero (mp_limb_t *RP, mp_size_t N)
+ Zero {RP, N}.
+
+
+8.1 Low-level functions for cryptography
+========================================
+
+The functions prefixed with 'mpn_sec_' and 'mpn_cnd_' are designed to
+perform the exact same low-level operations and have the same cache
+access patterns for any two same-size arguments, assuming that function
+arguments are placed at the same position and that the machine state is
+identical upon function entry. These functions are intended for
+cryptographic purposes, where resilience to side-channel attacks is
+desired.
+
+ These functions are less efficient than their "leaky" counterparts;
+their performance for operands of the sizes typically used for
+cryptographic applications is between 15% and 100% worse. For larger
+operands, these functions might be inadequate, since they rely on
+asymptotically elementary algorithms.
+
+ These functions do not make any explicit allocations. Those of these
+functions that need scratch space accept a scratch space operand. This
+convention allows callers to keep sensitive data in designated memory
+areas. Note however that compilers may choose to spill scalar values
+used within these functions to their stack frame and that such scalars
+may contain sensitive data.
+
+ In addition to these specially crafted functions, the following 'mpn'
+functions are naturally side-channel resistant: 'mpn_add_n',
+'mpn_sub_n', 'mpn_lshift', 'mpn_rshift', 'mpn_zero', 'mpn_copyi',
+'mpn_copyd', 'mpn_com', and the logical function ('mpn_and_n', etc).
+
+ There are some exceptions from the side-channel resilience: (1) Some
+assembly implementations of 'mpn_lshift' identify shift-by-one as a
+special case. This is a problem iff the shift count is a function of
+sensitive data. (2) Alpha ev6 and Pentium4 using 64-bit limbs have
+leaky 'mpn_add_n' and 'mpn_sub_n'. (3) Alpha ev6 has a leaky
+'mpn_mul_1' which also makes 'mpn_sec_mul' on those systems unsafe.
+
+ -- Function: mp_limb_t mpn_cnd_add_n (mp_limb_t CND, mp_limb_t *RP,
+ const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N)
+ -- Function: mp_limb_t mpn_cnd_sub_n (mp_limb_t CND, mp_limb_t *RP,
+ const mp_limb_t *S1P, const mp_limb_t *S2P, mp_size_t N)
+ These functions do conditional addition and subtraction. If CND is
+ non-zero, they produce the same result as a regular 'mpn_add_n' or
+ 'mpn_sub_n', and if CND is zero, they copy {S1P,N} to the result
+ area and return zero. The functions are designed to have timing
+ and memory access patterns depending only on size and location of
+ the data areas, but independent of the condition CND. Like for
+ 'mpn_add_n' and 'mpn_sub_n', on most machines, the timing will also
+ be independent of the actual limb values.
+
+ -- Function: mp_limb_t mpn_sec_add_1 (mp_limb_t *RP, const mp_limb_t
+ *AP, mp_size_t N, mp_limb_t B, mp_limb_t *TP)
+ -- Function: mp_limb_t mpn_sec_sub_1 (mp_limb_t *RP, const mp_limb_t
+ *AP, mp_size_t N, mp_limb_t B, mp_limb_t *TP)
+ Set R to A + B or A - B, respectively, where R = {RP,N}, A =
+ {AP,N}, and B is a single limb. Returns carry.
+
+ These functions take O(N) time, unlike the leaky functions
+ 'mpn_add_1' which are O(1) on average. They require scratch space
+ of 'mpn_sec_add_1_itch(N)' and 'mpn_sec_sub_1_itch(N)' limbs,
+ respectively, to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to be at most N limbs, and increase
+ monotonously in the operand size.
+
+ -- Function: void mpn_cnd_swap (mp_limb_t CND, volatile mp_limb_t *AP,
+ volatile mp_limb_t *BP, mp_size_t N)
+ If CND is non-zero, swaps the contents of the areas {AP,N} and
+ {BP,N}. Otherwise, the areas are left unmodified. Implemented
+ using logical operations on the limbs, with the same memory
+ accesses independent of the value of CND.
+
+ -- Function: void mpn_sec_mul (mp_limb_t *RP, const mp_limb_t *AP,
+ mp_size_t AN, const mp_limb_t *BP, mp_size_t BN, mp_limb_t
+ *TP)
+ -- Function: mp_size_t mpn_sec_mul_itch (mp_size_t AN, mp_size_t BN)
+ Set R to A * B, where A = {AP,AN}, B = {BP,BN}, and R = {RP,AN+BN}.
+
+ It is required that AN >= BN > 0.
+
+ No overlapping between R and the input operands is allowed. For A
+ = B, use 'mpn_sec_sqr' for optimal performance.
+
+ This function requires scratch space of 'mpn_sec_mul_itch(AN, BN)'
+ limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ sizes.
+
+ -- Function: void mpn_sec_sqr (mp_limb_t *RP, const mp_limb_t *AP,
+ mp_size_t AN, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_sqr_itch (mp_size_t AN)
+ Set R to A^2, where A = {AP,AN}, and R = {RP,2AN}.
+
+ It is required that AN > 0.
+
+ No overlapping between R and the input operands is allowed.
+
+ This function requires scratch space of 'mpn_sec_sqr_itch(AN)'
+ limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ size.
+
+ -- Function: void mpn_sec_powm (mp_limb_t *RP, const mp_limb_t *BP,
+ mp_size_t BN, const mp_limb_t *EP, mp_bitcnt_t ENB, const
+ mp_limb_t *MP, mp_size_t N, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_powm_itch (mp_size_t BN, mp_bitcnt_t
+ ENB, size_t N)
+ Set R to (B raised to E) modulo M, where R = {RP,N}, M = {MP,N},
+ and E = {EP,ceil(ENB / 'GMP\_NUMB\_BITS')}.
+
+ It is required that B > 0, that M > 0 is odd, and that E < 2^ENB,
+ with ENB > 0.
+
+ No overlapping between R and the input operands is allowed.
+
+ This function requires scratch space of 'mpn_sec_powm_itch(BN, ENB,
+ N)' limbs to be passed in the TP parameter. The scratch space
+ requirements are guaranteed to increase monotonously in the operand
+ sizes.
+
+ -- Function: void mpn_sec_tabselect (mp_limb_t *RP, const mp_limb_t
+ *TAB, mp_size_t N, mp_size_t NENTS, mp_size_t WHICH)
+ Select entry WHICH from table TAB, which has NENTS entries, each N
+ limbs. Store the selected entry at RP.
+
+ This function reads the entire table to avoid side-channel
+ information leaks.
+
+ -- Function: mp_limb_t mpn_sec_div_qr (mp_limb_t *QP, mp_limb_t *NP,
+ mp_size_t NN, const mp_limb_t *DP, mp_size_t DN, mp_limb_t
+ *TP)
+ -- Function: mp_size_t mpn_sec_div_qr_itch (mp_size_t NN, mp_size_t DN)
+
+ Set Q to the truncated quotient N / D and R to N modulo D, where N
+ = {NP,NN}, D = {DP,DN}, Q's most significant limb is the function
+ return value and the remaining limbs are {QP,NN-DN}, and R =
+ {NP,DN}.
+
+ It is required that NN >= DN >= 1, and that DP[DN-1] != 0. This
+ does not imply that N >= D since N might be zero-padded.
+
+ Note the overlapping between N and R. No other operand overlapping
+ is allowed. The entire space occupied by N is overwritten.
+
+ This function requires scratch space of 'mpn_sec_div_qr_itch(NN,
+ DN)' limbs to be passed in the TP parameter.
+
+ -- Function: void mpn_sec_div_r (mp_limb_t *NP, mp_size_t NN, const
+ mp_limb_t *DP, mp_size_t DN, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_div_r_itch (mp_size_t NN, mp_size_t DN)
+
+ Set R to N modulo D, where N = {NP,NN}, D = {DP,DN}, and R =
+ {NP,DN}.
+
+ It is required that NN >= DN >= 1, and that DP[DN-1] != 0. This
+ does not imply that N >= D since N might be zero-padded.
+
+ Note the overlapping between N and R. No other operand overlapping
+ is allowed. The entire space occupied by N is overwritten.
+
+ This function requires scratch space of 'mpn_sec_div_r_itch(NN,
+ DN)' limbs to be passed in the TP parameter.
+
+ -- Function: int mpn_sec_invert (mp_limb_t *RP, mp_limb_t *AP, const
+ mp_limb_t *MP, mp_size_t N, mp_bitcnt_t NBCNT, mp_limb_t *TP)
+ -- Function: mp_size_t mpn_sec_invert_itch (mp_size_t N)
+ Set R to the inverse of A modulo M, where R = {RP,N}, A = {AP,N},
+ and M = {MP,N}. *This function's interface is preliminary.*
+
+ If an inverse exists, return 1, otherwise return 0 and leave R
+ undefined. In either case, the input A is destroyed.
+
+ It is required that M is odd, and that NBCNT >= ceil(\log(A+1)) +
+ ceil(\log(M+1)). A safe choice is NBCNT = 2 * N * GMP_NUMB_BITS,
+ but a smaller value might improve performance if M or A are known
+ to have leading zero bits.
+
+ This function requires scratch space of 'mpn_sec_invert_itch(N)'
+ limbs to be passed in the TP parameter.
+
+
+8.2 Nails
+=========
+
+*Everything in this section is highly experimental and may disappear or
+be subject to incompatible changes in a future version of GMP.*
+
+ Nails are an experimental feature whereby a few bits are left unused
+at the top of each 'mp_limb_t'. This can significantly improve carry
+handling on some processors.
+
+ All the 'mpn' functions accepting limb data will expect the nail bits
+to be zero on entry, and will return data with the nails similarly all
+zero. This applies both to limb vectors and to single limb arguments.
+
+ Nails can be enabled by configuring with '--enable-nails'. By
+default the number of bits will be chosen according to what suits the
+host processor, but a particular number can be selected with
+'--enable-nails=N'.
+
+ At the mpn level, a nail build is neither source nor binary
+compatible with a non-nail build, strictly speaking. But programs
+acting on limbs only through the mpn functions are likely to work
+equally well with either build, and judicious use of the definitions
+below should make any program compatible with either build, at the
+source level.
+
+ For the higher level routines, meaning 'mpz' etc, a nail build should
+be fully source and binary compatible with a non-nail build.
+
+ -- Macro: GMP_NAIL_BITS
+ -- Macro: GMP_NUMB_BITS
+ -- Macro: GMP_LIMB_BITS
+ 'GMP_NAIL_BITS' is the number of nail bits, or 0 when nails are not
+ in use. 'GMP_NUMB_BITS' is the number of data bits in a limb.
+ 'GMP_LIMB_BITS' is the total number of bits in an 'mp_limb_t'. In
+ all cases
+
+ GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
+
+ -- Macro: GMP_NAIL_MASK
+ -- Macro: GMP_NUMB_MASK
+ Bit masks for the nail and number parts of a limb. 'GMP_NAIL_MASK'
+ is 0 when nails are not in use.
+
+ 'GMP_NAIL_MASK' is not often needed, since the nail part can be
+ obtained with 'x >> GMP_NUMB_BITS', and that means one less large
+ constant, which can help various RISC chips.
+
+ -- Macro: GMP_NUMB_MAX
+ The maximum value that can be stored in the number part of a limb.
+ This is the same as 'GMP_NUMB_MASK', but can be used for clarity
+ when doing comparisons rather than bit-wise operations.
+
+ The term "nails" comes from finger or toe nails, which are at the
+ends of a limb (arm or leg). "numb" is short for number, but is also
+how the developers felt after trying for a long time to come up with
+sensible names for these things.
+
+ In the future (the distant future most likely) a non-zero nail might
+be permitted, giving non-unique representations for numbers in a limb
+vector. This would help vector processors since carries would only ever
+need to propagate one or two limbs.
+
+
+File: gmp.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
+
+9 Random Number Functions
+*************************
+
+Sequences of pseudo-random numbers in GMP are generated using a variable
+of type 'gmp_randstate_t', which holds an algorithm selection and a
+current state. Such a variable must be initialized by a call to one of
+the 'gmp_randinit' functions, and can be seeded with one of the
+'gmp_randseed' functions.
+
+ The functions actually generating random numbers are described in
+*note Integer Random Numbers::, and *note Miscellaneous Float
+Functions::.
+
+ The older style random number functions don't accept a
+'gmp_randstate_t' parameter but instead share a global variable of that
+type. They use a default algorithm and are currently not seeded (though
+perhaps that will change in the future). The new functions accepting a
+'gmp_randstate_t' are recommended for applications that care about
+randomness.
+
+* Menu:
+
+* Random State Initialization::
+* Random State Seeding::
+* Random State Miscellaneous::
+
+
+File: gmp.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
+
+9.1 Random State Initialization
+===============================
+
+ -- Function: void gmp_randinit_default (gmp_randstate_t STATE)
+ Initialize STATE with a default algorithm. This will be a
+ compromise between speed and randomness, and is recommended for
+ applications with no special requirements. Currently this is
+ 'gmp_randinit_mt'.
+
+ -- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
+ Initialize STATE for a Mersenne Twister algorithm. This algorithm
+ is fast and has good randomness properties.
+
+ -- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, const
+ mpz_t A, unsigned long C, mp_bitcnt_t M2EXP)
+ Initialize STATE with a linear congruential algorithm X = (A*X + C)
+ mod 2^M2EXP.
+
+ The low bits of X in this algorithm are not very random. The least
+ significant bit will have a period no more than 2, and the second
+ bit no more than 4, etc. For this reason only the high half of
+ each X is actually used.
+
+ When a random number of more than M2EXP/2 bits is to be generated,
+ multiple iterations of the recurrence are used and the results
+ concatenated.
+
+ -- Function: int gmp_randinit_lc_2exp_size (gmp_randstate_t STATE,
+ mp_bitcnt_t SIZE)
+ Initialize STATE for a linear congruential algorithm as per
+ 'gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
+ chosen so that SIZE bits (or more) of each X will be used, i.e.
+ M2EXP/2 >= SIZE.
+
+ If successful the return value is non-zero. If SIZE is bigger than
+ the table data provides then the return value is zero. The maximum
+ SIZE currently supported is 128.
+
+ -- Function: void gmp_randinit_set (gmp_randstate_t ROP,
+ gmp_randstate_t OP)
+ Initialize ROP with a copy of the algorithm and state from OP.
+
+ -- Function: void gmp_randinit (gmp_randstate_t STATE,
+ gmp_randalg_t ALG, ...)
+ *This function is obsolete.*
+
+ Initialize STATE with an algorithm selected by ALG. The only
+ choice is 'GMP_RAND_ALG_LC', which is 'gmp_randinit_lc_2exp_size'
+ described above. A third parameter of type 'unsigned long' is
+ required, this is the SIZE for that function.
+ 'GMP_RAND_ALG_DEFAULT' or 0 are the same as 'GMP_RAND_ALG_LC'.
+
+ 'gmp_randinit' sets bits in the global variable 'gmp_errno' to
+ indicate an error. 'GMP_ERROR_UNSUPPORTED_ARGUMENT' if ALG is
+ unsupported, or 'GMP_ERROR_INVALID_ARGUMENT' if the SIZE parameter
+ is too big. It may be noted this error reporting is not thread
+ safe (a good reason to use 'gmp_randinit_lc_2exp_size' instead).
+
+ -- Function: void gmp_randclear (gmp_randstate_t STATE)
+ Free all memory occupied by STATE.
+
+
+File: gmp.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
+
+9.2 Random State Seeding
+========================
+
+ -- Function: void gmp_randseed (gmp_randstate_t STATE, const mpz_t
+ SEED)
+ -- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
+ unsigned long int SEED)
+ Set an initial seed value into STATE.
+
+ The size of a seed determines how many different sequences of
+ random numbers that it's possible to generate. The "quality" of
+ the seed is the randomness of a given seed compared to the previous
+ seed used, and this affects the randomness of separate number
+ sequences. The method for choosing a seed is critical if the
+ generated numbers are to be used for important applications, such
+ as generating cryptographic keys.
+
+ Traditionally the system time has been used to seed, but care needs
+ to be taken with this. If an application seeds often and the
+ resolution of the system clock is low, then the same sequence of
+ numbers might be repeated. Also, the system time is quite easy to
+ guess, so if unpredictability is required then it should definitely
+ not be the only source for the seed value. On some systems there's
+ a special device '/dev/random' which provides random data better
+ suited for use as a seed.
+
+
+File: gmp.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
+
+9.3 Random State Miscellaneous
+==============================
+
+ -- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
+ unsigned long N)
+ Return a uniformly distributed random number of N bits, i.e. in the
+ range 0 to 2^N-1 inclusive. N must be less than or equal to the
+ number of bits in an 'unsigned long'.
+
+ -- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE,
+ unsigned long N)
+ Return a uniformly distributed random number in the range 0 to N-1,
+ inclusive.
+
+
+File: gmp.info, Node: Formatted Output, Next: Formatted Input, Prev: Random Number Functions, Up: Top
+
+10 Formatted Output
+*******************
+
+* Menu:
+
+* Formatted Output Strings::
+* Formatted Output Functions::
+* C++ Formatted Output::
+
+
+File: gmp.info, Node: Formatted Output Strings, Next: Formatted Output Functions, Prev: Formatted Output, Up: Formatted Output
+
+10.1 Format Strings
+===================
+
+'gmp_printf' and friends accept format strings similar to the standard C
+'printf' (*note Formatted Output: (libc)Formatted Output.). A format
+specification is of the form
+
+ % [flags] [width] [.[precision]] [type] conv
+
+ GMP adds types 'Z', 'Q' and 'F' for 'mpz_t', 'mpq_t' and 'mpf_t'
+respectively, 'M' for 'mp_limb_t', and 'N' for an 'mp_limb_t' array.
+'Z', 'Q', 'M' and 'N' behave like integers. 'Q' will print a '/' and a
+denominator, if needed. 'F' behaves like a float. For example,
+
+ mpz_t z;
+ gmp_printf ("%s is an mpz %Zd\n", "here", z);
+
+ mpq_t q;
+ gmp_printf ("a hex rational: %#40Qx\n", q);
+
+ mpf_t f;
+ int n;
+ gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n);
+
+ mp_limb_t l;
+ gmp_printf ("limb %Mu\n", l);
+
+ const mp_limb_t *ptr;
+ mp_size_t size;
+ gmp_printf ("limb array %Nx\n", ptr, size);
+
+ For 'N' the limbs are expected least significant first, as per the
+'mpn' functions (*note Low-level Functions::). A negative size can be
+given to print the value as a negative.
+
+ All the standard C 'printf' types behave the same as the C library
+'printf', and can be freely intermixed with the GMP extensions. In the
+current implementation the standard parts of the format string are
+simply handed to 'printf' and only the GMP extensions handled directly.
+
+ The flags accepted are as follows. GLIBC style ' is only for the
+standard C types (not the GMP types), and only if the C library supports
+it.
+
+ 0 pad with zeros (rather than spaces)
+ # show the base with '0x', '0X' or '0'
+ + always show a sign
+ (space) show a space or a '-' sign
+ ' group digits, GLIBC style (not GMP
+ types)
+
+ The optional width and precision can be given as a number within the
+format string, or as a '*' to take an extra parameter of type 'int', the
+same as the standard 'printf'.
+
+ The standard types accepted are as follows. 'h' and 'l' are
+portable, the rest will depend on the compiler (or include files) for
+the type and the C library for the output.
+
+ h short
+ hh char
+ j intmax_t or uintmax_t
+ l long or wchar_t
+ ll long long
+ L long double
+ q quad_t or u_quad_t
+ t ptrdiff_t
+ z size_t
+
+The GMP types are
+
+ F mpf_t, float conversions
+ Q mpq_t, integer conversions
+ M mp_limb_t, integer conversions
+ N mp_limb_t array, integer conversions
+ Z mpz_t, integer conversions
+
+ The conversions accepted are as follows. 'a' and 'A' are always
+supported for 'mpf_t' but depend on the C library for standard C float
+types. 'm' and 'p' depend on the C library.
+
+ a A hex floats, C99 style
+ c character
+ d decimal integer
+ e E scientific format float
+ f fixed point float
+ i same as d
+ g G fixed or scientific float
+ m 'strerror' string, GLIBC style
+ n store characters written so far
+ o octal integer
+ p pointer
+ s string
+ u unsigned integer
+ x X hex integer
+
+ 'o', 'x' and 'X' are unsigned for the standard C types, but for types
+'Z', 'Q' and 'N' they are signed. 'u' is not meaningful for 'Z', 'Q'
+and 'N'.
+
+ 'M' is a proxy for the C library 'l' or 'L', according to the size of
+'mp_limb_t'. Unsigned conversions will be usual, but a signed
+conversion can be used and will interpret the value as a twos complement
+negative.
+
+ 'n' can be used with any type, even the GMP types.
+
+ Other types or conversions that might be accepted by the C library
+'printf' cannot be used through 'gmp_printf', this includes for instance
+extensions registered with GLIBC 'register_printf_function'. Also
+currently there's no support for POSIX '$' style numbered arguments
+(perhaps this will be added in the future).
+
+ The precision field has its usual meaning for integer 'Z' and float
+'F' types, but is currently undefined for 'Q' and should not be used
+with that.
+
+ 'mpf_t' conversions only ever generate as many digits as can be
+accurately represented by the operand, the same as 'mpf_get_str' does.
+Zeros will be used if necessary to pad to the requested precision. This
+happens even for an 'f' conversion of an 'mpf_t' which is an integer,
+for instance 2^1024 in an 'mpf_t' of 128 bits precision will only
+produce about 40 digits, then pad with zeros to the decimal point. An
+empty precision field like '%.Fe' or '%.Ff' can be used to specifically
+request just the significant digits. Without any dot and thus no
+precision field, a precision value of 6 will be used. Note that these
+rules mean that '%Ff', '%.Ff', and '%.0Ff' will all be different.
+
+ The decimal point character (or string) is taken from the current
+locale settings on systems which provide 'localeconv' (*note Locales and
+Internationalization: (libc)Locales.). The C library will normally do
+the same for standard float output.
+
+ The format string is only interpreted as plain 'char's, multibyte
+characters are not recognised. Perhaps this will change in the future.
+
+
+File: gmp.info, Node: Formatted Output Functions, Next: C++ Formatted Output, Prev: Formatted Output Strings, Up: Formatted Output
+
+10.2 Functions
+==============
+
+Each of the following functions is similar to the corresponding C
+library function. The basic 'printf' forms take a variable argument
+list. The 'vprintf' forms take an argument pointer, see *note Variadic
+Functions: (libc)Variadic Functions, or 'man 3 va_start'.
+
+ It should be emphasised that if a format string is invalid, or the
+arguments don't match what the format specifies, then the behaviour of
+any of these functions will be unpredictable. GCC format string
+checking is not available, since it doesn't recognise the GMP
+extensions.
+
+ The file based functions 'gmp_printf' and 'gmp_fprintf' will return
+-1 to indicate a write error. Output is not "atomic", so partial output
+may be produced if a write error occurs. All the functions can return
+-1 if the C library 'printf' variant in use returns -1, but this
+shouldn't normally occur.
+
+ -- Function: int gmp_printf (const char *FMT, ...)
+ -- Function: int gmp_vprintf (const char *FMT, va_list AP)
+ Print to the standard output 'stdout'. Return the number of
+ characters written, or -1 if an error occurred.
+
+ -- Function: int gmp_fprintf (FILE *FP, const char *FMT, ...)
+ -- Function: int gmp_vfprintf (FILE *FP, const char *FMT, va_list AP)
+ Print to the stream FP. Return the number of characters written,
+ or -1 if an error occurred.
+
+ -- Function: int gmp_sprintf (char *BUF, const char *FMT, ...)
+ -- Function: int gmp_vsprintf (char *BUF, const char *FMT, va_list AP)
+ Form a null-terminated string in BUF. Return the number of
+ characters written, excluding the terminating null.
+
+ No overlap is permitted between the space at BUF and the string
+ FMT.
+
+ These functions are not recommended, since there's no protection
+ against exceeding the space available at BUF.
+
+ -- Function: int gmp_snprintf (char *BUF, size_t SIZE, const char *FMT,
+ ...)
+ -- Function: int gmp_vsnprintf (char *BUF, size_t SIZE, const char
+ *FMT, va_list AP)
+ Form a null-terminated string in BUF. No more than SIZE bytes will
+ be written. To get the full output, SIZE must be enough for the
+ string and null-terminator.
+
+ The return value is the total number of characters which ought to
+ have been produced, excluding the terminating null. If RETVAL >=
+ SIZE then the actual output has been truncated to the first SIZE-1
+ characters, and a null appended.
+
+ No overlap is permitted between the region {BUF,SIZE} and the FMT
+ string.
+
+ Notice the return value is in ISO C99 'snprintf' style. This is so
+ even if the C library 'vsnprintf' is the older GLIBC 2.0.x style.
+
+ -- Function: int gmp_asprintf (char **PP, const char *FMT, ...)
+ -- Function: int gmp_vasprintf (char **PP, const char *FMT, va_list AP)
+ Form a null-terminated string in a block of memory obtained from
+ the current memory allocation function (*note Custom Allocation::).
+ The block will be the size of the string and null-terminator. The
+ address of the block in stored to *PP. The return value is the
+ number of characters produced, excluding the null-terminator.
+
+ Unlike the C library 'asprintf', 'gmp_asprintf' doesn't return -1
+ if there's no more memory available, it lets the current allocation
+ function handle that.
+
+ -- Function: int gmp_obstack_printf (struct obstack *OB, const char
+ *FMT, ...)
+ -- Function: int gmp_obstack_vprintf (struct obstack *OB, const char
+ *FMT, va_list AP)
+ Append to the current object in OB. The return value is the number
+ of characters written. A null-terminator is not written.
+
+ FMT cannot be within the current object in OB, since that object
+ might move as it grows.
+
+ These functions are available only when the C library provides the
+ obstack feature, which probably means only on GNU systems, see
+ *note Obstacks: (libc)Obstacks.
+
+
+File: gmp.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
+
+10.3 C++ Formatted Output
+=========================
+
+The following functions are provided in 'libgmpxx' (*note Headers and
+Libraries::), which is built if C++ support is enabled (*note Build
+Options::). Prototypes are available from '<gmp.h>'.
+
+ -- Function: ostream& operator<< (ostream& STREAM, const mpz_t OP)
+ Print OP to STREAM, using its 'ios' formatting settings.
+ 'ios::width' is reset to 0 after output, the same as the standard
+ 'ostream operator<<' routines do.
+
+ In hex or octal, OP is printed as a signed number, the same as for
+ decimal. This is unlike the standard 'operator<<' routines on
+ 'int' etc, which instead give twos complement.
+
+ -- Function: ostream& operator<< (ostream& STREAM, const mpq_t OP)
+ Print OP to STREAM, using its 'ios' formatting settings.
+ 'ios::width' is reset to 0 after output, the same as the standard
+ 'ostream operator<<' routines do.
+
+ Output will be a fraction like '5/9', or if the denominator is 1
+ then just a plain integer like '123'.
+
+ In hex or octal, OP is printed as a signed value, the same as for
+ decimal. If 'ios::showbase' is set then a base indicator is shown
+ on both the numerator and denominator (if the denominator is
+ required).
+
+ -- Function: ostream& operator<< (ostream& STREAM, const mpf_t OP)
+ Print OP to STREAM, using its 'ios' formatting settings.
+ 'ios::width' is reset to 0 after output, the same as the standard
+ 'ostream operator<<' routines do.
+
+ The decimal point follows the standard library float 'operator<<',
+ which on recent systems means the 'std::locale' imbued on STREAM.
+
+ Hex and octal are supported, unlike the standard 'operator<<' on
+ 'double'. The mantissa will be in hex or octal, the exponent will
+ be in decimal. For hex the exponent delimiter is an '@'. This is
+ as per 'mpf_out_str'.
+
+ 'ios::showbase' is supported, and will put a base on the mantissa,
+ for example hex '0x1.8' or '0x0.8', or octal '01.4' or '00.4'.
+ This last form is slightly strange, but at least differentiates
+ itself from decimal.
+
+ These operators mean that GMP types can be printed in the usual C++
+way, for example,
+
+ mpz_t z;
+ int n;
+ ...
+ cout << "iteration " << n << " value " << z << "\n";
+
+ But note that 'ostream' output (and 'istream' input, *note C++
+Formatted Input::) is the only overloading available for the GMP types
+and that for instance using '+' with an 'mpz_t' will have unpredictable
+results. For classes with overloading, see *note C++ Class Interface::.
+
+
+File: gmp.info, Node: Formatted Input, Next: C++ Class Interface, Prev: Formatted Output, Up: Top
+
+11 Formatted Input
+******************
+
+* Menu:
+
+* Formatted Input Strings::
+* Formatted Input Functions::
+* C++ Formatted Input::
+
+
+File: gmp.info, Node: Formatted Input Strings, Next: Formatted Input Functions, Prev: Formatted Input, Up: Formatted Input
+
+11.1 Formatted Input Strings
+============================
+
+'gmp_scanf' and friends accept format strings similar to the standard C
+'scanf' (*note Formatted Input: (libc)Formatted Input.). A format
+specification is of the form
+
+ % [flags] [width] [type] conv
+
+ GMP adds types 'Z', 'Q' and 'F' for 'mpz_t', 'mpq_t' and 'mpf_t'
+respectively. 'Z' and 'Q' behave like integers. 'Q' will read a '/'
+and a denominator, if present. 'F' behaves like a float.
+
+ GMP variables don't require an '&' when passed to 'gmp_scanf', since
+they're already "call-by-reference". For example,
+
+ /* to read say "a(5) = 1234" */
+ int n;
+ mpz_t z;
+ gmp_scanf ("a(%d) = %Zd\n", &n, z);
+
+ mpq_t q1, q2;
+ gmp_sscanf ("0377 + 0x10/0x11", "%Qi + %Qi", q1, q2);
+
+ /* to read say "topleft (1.55,-2.66)" */
+ mpf_t x, y;
+ char buf[32];
+ gmp_scanf ("%31s (%Ff,%Ff)", buf, x, y);
+
+ All the standard C 'scanf' types behave the same as in the C library
+'scanf', and can be freely intermixed with the GMP extensions. In the
+current implementation the standard parts of the format string are
+simply handed to 'scanf' and only the GMP extensions handled directly.
+
+ The flags accepted are as follows. 'a' and ''' will depend on
+support from the C library, and ''' cannot be used with GMP types.
+
+ * read but don't store
+ a allocate a buffer (string conversions)
+ ' grouped digits, GLIBC style (not GMP
+ types)
+
+ The standard types accepted are as follows. 'h' and 'l' are
+portable, the rest will depend on the compiler (or include files) for
+the type and the C library for the input.
+
+ h short
+ hh char
+ j intmax_t or uintmax_t
+ l long int, double or wchar_t
+ ll long long
+ L long double
+ q quad_t or u_quad_t
+ t ptrdiff_t
+ z size_t
+
+The GMP types are
+
+ F mpf_t, float conversions
+ Q mpq_t, integer conversions
+ Z mpz_t, integer conversions
+
+ The conversions accepted are as follows. 'p' and '[' will depend on
+support from the C library, the rest are standard.
+
+ c character or characters
+ d decimal integer
+ e E f g float
+ G
+ i integer with base indicator
+ n characters read so far
+ o octal integer
+ p pointer
+ s string of non-whitespace characters
+ u decimal integer
+ x X hex integer
+ [ string of characters in a set
+
+ 'e', 'E', 'f', 'g' and 'G' are identical, they all read either fixed
+point or scientific format, and either upper or lower case 'e' for the
+exponent in scientific format.
+
+ C99 style hex float format ('printf %a', *note Formatted Output
+Strings::) is always accepted for 'mpf_t', but for the standard float
+types it will depend on the C library.
+
+ 'x' and 'X' are identical, both accept both upper and lower case
+hexadecimal.
+
+ 'o', 'u', 'x' and 'X' all read positive or negative values. For the
+standard C types these are described as "unsigned" conversions, but that
+merely affects certain overflow handling, negatives are still allowed
+(per 'strtoul', *note Parsing of Integers: (libc)Parsing of Integers.).
+For GMP types there are no overflows, so 'd' and 'u' are identical.
+
+ 'Q' type reads the numerator and (optional) denominator as given. If
+the value might not be in canonical form then 'mpq_canonicalize' must be
+called before using it in any calculations (*note Rational Number
+Functions::).
+
+ 'Qi' will read a base specification separately for the numerator and
+denominator. For example '0x10/11' would be 16/11, whereas '0x10/0x11'
+would be 16/17.
+
+ 'n' can be used with any of the types above, even the GMP types. '*'
+to suppress assignment is allowed, though in that case it would do
+nothing at all.
+
+ Other conversions or types that might be accepted by the C library
+'scanf' cannot be used through 'gmp_scanf'.
+
+ Whitespace is read and discarded before a field, except for 'c' and
+'[' conversions.
+
+ For float conversions, the decimal point character (or string)
+expected is taken from the current locale settings on systems which
+provide 'localeconv' (*note Locales and Internationalization:
+(libc)Locales.). The C library will normally do the same for standard
+float input.
+
+ The format string is only interpreted as plain 'char's, multibyte
+characters are not recognised. Perhaps this will change in the future.
+
+
+File: gmp.info, Node: Formatted Input Functions, Next: C++ Formatted Input, Prev: Formatted Input Strings, Up: Formatted Input
+
+11.2 Formatted Input Functions
+==============================
+
+Each of the following functions is similar to the corresponding C
+library function. The plain 'scanf' forms take a variable argument
+list. The 'vscanf' forms take an argument pointer, see *note Variadic
+Functions: (libc)Variadic Functions, or 'man 3 va_start'.
+
+ It should be emphasised that if a format string is invalid, or the
+arguments don't match what the format specifies, then the behaviour of
+any of these functions will be unpredictable. GCC format string
+checking is not available, since it doesn't recognise the GMP
+extensions.
+
+ No overlap is permitted between the FMT string and any of the results
+produced.
+
+ -- Function: int gmp_scanf (const char *FMT, ...)
+ -- Function: int gmp_vscanf (const char *FMT, va_list AP)
+ Read from the standard input 'stdin'.
+
+ -- Function: int gmp_fscanf (FILE *FP, const char *FMT, ...)
+ -- Function: int gmp_vfscanf (FILE *FP, const char *FMT, va_list AP)
+ Read from the stream FP.
+
+ -- Function: int gmp_sscanf (const char *S, const char *FMT, ...)
+ -- Function: int gmp_vsscanf (const char *S, const char *FMT, va_list
+ AP)
+ Read from a null-terminated string S.
+
+ The return value from each of these functions is the same as the
+standard C99 'scanf', namely the number of fields successfully parsed
+and stored. '%n' fields and fields read but suppressed by '*' don't
+count towards the return value.
+
+ If end of input (or a file error) is reached before a character for a
+field or a literal, and if no previous non-suppressed fields have
+matched, then the return value is 'EOF' instead of 0. A whitespace
+character in the format string is only an optional match and doesn't
+induce an 'EOF' in this fashion. Leading whitespace read and discarded
+for a field don't count as characters for that field.
+
+ For the GMP types, input parsing follows C99 rules, namely one
+character of lookahead is used and characters are read while they
+continue to meet the format requirements. If this doesn't provide a
+complete number then the function terminates, with that field not stored
+nor counted towards the return value. For instance with 'mpf_t' an
+input '1.23e-XYZ' would be read up to the 'X' and that character pushed
+back since it's not a digit. The string '1.23e-' would then be
+considered invalid since an 'e' must be followed by at least one digit.
+
+ For the standard C types, in the current implementation GMP calls the
+C library 'scanf' functions, which might have looser rules about what
+constitutes a valid input.
+
+ Note that 'gmp_sscanf' is the same as 'gmp_fscanf' and only does one
+character of lookahead when parsing. Although clearly it could look at
+its entire input, it is deliberately made identical to 'gmp_fscanf', the
+same way C99 'sscanf' is the same as 'fscanf'.
+
+
+File: gmp.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
+
+11.3 C++ Formatted Input
+========================
+
+The following functions are provided in 'libgmpxx' (*note Headers and
+Libraries::), which is built only if C++ support is enabled (*note Build
+Options::). Prototypes are available from '<gmp.h>'.
+
+ -- Function: istream& operator>> (istream& STREAM, mpz_t ROP)
+ Read ROP from STREAM, using its 'ios' formatting settings.
+
+ -- Function: istream& operator>> (istream& STREAM, mpq_t ROP)
+ An integer like '123' will be read, or a fraction like '5/9'. No
+ whitespace is allowed around the '/'. If the fraction is not in
+ canonical form then 'mpq_canonicalize' must be called (*note
+ Rational Number Functions::) before operating on it.
+
+ As per integer input, an '0' or '0x' base indicator is read when
+ none of 'ios::dec', 'ios::oct' or 'ios::hex' are set. This is done
+ separately for numerator and denominator, so that for instance
+ '0x10/11' is 16/11 and '0x10/0x11' is 16/17.
+
+ -- Function: istream& operator>> (istream& STREAM, mpf_t ROP)
+ Read ROP from STREAM, using its 'ios' formatting settings.
+
+ Hex or octal floats are not supported, but might be in the future,
+ or perhaps it's best to accept only what the standard float
+ 'operator>>' does.
+
+ Note that digit grouping specified by the 'istream' locale is
+currently not accepted. Perhaps this will change in the future.
+
+
+ These operators mean that GMP types can be read in the usual C++ way,
+for example,
+
+ mpz_t z;
+ ...
+ cin >> z;
+
+ But note that 'istream' input (and 'ostream' output, *note C++
+Formatted Output::) is the only overloading available for the GMP types
+and that for instance using '+' with an 'mpz_t' will have unpredictable
+results. For classes with overloading, see *note C++ Class Interface::.
+
+
+File: gmp.info, Node: C++ Class Interface, Next: Custom Allocation, Prev: Formatted Input, Up: Top
+
+12 C++ Class Interface
+**********************
+
+This chapter describes the C++ class based interface to GMP.
+
+ All GMP C language types and functions can be used in C++ programs,
+since 'gmp.h' has 'extern "C"' qualifiers, but the class interface
+offers overloaded functions and operators which may be more convenient.
+
+ Due to the implementation of this interface, a reasonably recent C++
+compiler is required, one supporting namespaces, partial specialization
+of templates and member templates.
+
+ *Everything described in this chapter is to be considered preliminary
+and might be subject to incompatible changes if some unforeseen
+difficulty reveals itself.*
+
+* Menu:
+
+* C++ Interface General::
+* C++ Interface Integers::
+* C++ Interface Rationals::
+* C++ Interface Floats::
+* C++ Interface Random Numbers::
+* C++ Interface Limitations::
+
+
+File: gmp.info, Node: C++ Interface General, Next: C++ Interface Integers, Prev: C++ Class Interface, Up: C++ Class Interface
+
+12.1 C++ Interface General
+==========================
+
+All the C++ classes and functions are available with
+
+ #include <gmpxx.h>
+
+ Programs should be linked with the 'libgmpxx' and 'libgmp' libraries.
+For example,
+
+ g++ mycxxprog.cc -lgmpxx -lgmp
+
+The classes defined are
+
+ -- Class: mpz_class
+ -- Class: mpq_class
+ -- Class: mpf_class
+
+ The standard operators and various standard functions are overloaded
+to allow arithmetic with these classes. For example,
+
+ int
+ main (void)
+ {
+ mpz_class a, b, c;
+
+ a = 1234;
+ b = "-5678";
+ c = a+b;
+ cout << "sum is " << c << "\n";
+ cout << "absolute value is " << abs(c) << "\n";
+
+ return 0;
+ }
+
+ An important feature of the implementation is that an expression like
+'a=b+c' results in a single call to the corresponding 'mpz_add', without
+using a temporary for the 'b+c' part. Expressions which by their nature
+imply intermediate values, like 'a=b*c+d*e', still use temporaries
+though.
+
+ The classes can be freely intermixed in expressions, as can the
+classes and the standard types 'long', 'unsigned long' and 'double'.
+Smaller types like 'int' or 'float' can also be intermixed, since C++
+will promote them.
+
+ Note that 'bool' is not accepted directly, but must be explicitly
+cast to an 'int' first. This is because C++ will automatically convert
+any pointer to a 'bool', so if GMP accepted 'bool' it would make all
+sorts of invalid class and pointer combinations compile but almost
+certainly not do anything sensible.
+
+ Conversions back from the classes to standard C++ types aren't done
+automatically, instead member functions like 'get_si' are provided (see
+the following sections for details).
+
+ Also there are no automatic conversions from the classes to the
+corresponding GMP C types, instead a reference to the underlying C
+object can be obtained with the following functions,
+
+ -- Function: mpz_t mpz_class::get_mpz_t ()
+ -- Function: mpq_t mpq_class::get_mpq_t ()
+ -- Function: mpf_t mpf_class::get_mpf_t ()
+
+ These can be used to call a C function which doesn't have a C++ class
+interface. For example to set 'a' to the GCD of 'b' and 'c',
+
+ mpz_class a, b, c;
+ ...
+ mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());
+
+ In the other direction, a class can be initialized from the
+corresponding GMP C type, or assigned to if an explicit constructor is
+used. In both cases this makes a copy of the value, it doesn't create
+any sort of association. For example,
+
+ mpz_t z;
+ // ... init and calculate z ...
+ mpz_class x(z);
+ mpz_class y;
+ y = mpz_class (z);
+
+ There are no namespace setups in 'gmpxx.h', all types and functions
+are simply put into the global namespace. This is what 'gmp.h' has done
+in the past, and continues to do for compatibility. The extras provided
+by 'gmpxx.h' follow GMP naming conventions and are unlikely to clash
+with anything.
+
+
+File: gmp.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface
+
+12.2 C++ Interface Integers
+===========================
+
+ -- Function: mpz_class::mpz_class (type N)
+ Construct an 'mpz_class'. All the standard C++ types may be used,
+ except 'long long' and 'long double', and all the GMP C++ classes
+ can be used, although conversions from 'mpq_class' and 'mpf_class'
+ are 'explicit'. Any necessary conversion follows the corresponding
+ C function, for example 'double' follows 'mpz_set_d' (*note
+ Assigning Integers::).
+
+ -- Function: explicit mpz_class::mpz_class (const mpz_t Z)
+ Construct an 'mpz_class' from an 'mpz_t'. The value in Z is copied
+ into the new 'mpz_class', there won't be any permanent association
+ between it and Z.
+
+ -- Function: explicit mpz_class::mpz_class (const char *S, int BASE =
+ 0)
+ -- Function: explicit mpz_class::mpz_class (const string& S, int BASE =
+ 0)
+ Construct an 'mpz_class' converted from a string using
+ 'mpz_set_str' (*note Assigning Integers::).
+
+ If the string is not a valid integer, an 'std::invalid_argument'
+ exception is thrown. The same applies to 'operator='.
+
+ -- Function: mpz_class operator"" _mpz (const char *STR)
+ With C++11 compilers, integers can be constructed with the syntax
+ '123_mpz' which is equivalent to 'mpz_class("123")'.
+
+ -- Function: mpz_class operator/ (mpz_class A, mpz_class D)
+ -- Function: mpz_class operator% (mpz_class A, mpz_class D)
+ Divisions involving 'mpz_class' round towards zero, as per the
+ 'mpz_tdiv_q' and 'mpz_tdiv_r' functions (*note Integer Division::).
+ This is the same as the C99 '/' and '%' operators.
+
+ The 'mpz_fdiv...' or 'mpz_cdiv...' functions can always be called
+ directly if desired. For example,
+
+ mpz_class q, a, d;
+ ...
+ mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
+
+ -- Function: mpz_class abs (mpz_class OP)
+ -- Function: int cmp (mpz_class OP1, type OP2)
+ -- Function: int cmp (type OP1, mpz_class OP2)
+
+ -- Function: bool mpz_class::fits_sint_p (void)
+ -- Function: bool mpz_class::fits_slong_p (void)
+ -- Function: bool mpz_class::fits_sshort_p (void)
+
+ -- Function: bool mpz_class::fits_uint_p (void)
+ -- Function: bool mpz_class::fits_ulong_p (void)
+ -- Function: bool mpz_class::fits_ushort_p (void)
+
+ -- Function: double mpz_class::get_d (void)
+ -- Function: long mpz_class::get_si (void)
+ -- Function: string mpz_class::get_str (int BASE = 10)
+ -- Function: unsigned long mpz_class::get_ui (void)
+
+ -- Function: int mpz_class::set_str (const char *STR, int BASE)
+ -- Function: int mpz_class::set_str (const string& STR, int BASE)
+ -- Function: int sgn (mpz_class OP)
+ -- Function: mpz_class sqrt (mpz_class OP)
+
+ -- Function: mpz_class gcd (mpz_class OP1, mpz_class OP2)
+ -- Function: mpz_class lcm (mpz_class OP1, mpz_class OP2)
+ -- Function: mpz_class mpz_class::factorial (type OP)
+ -- Function: mpz_class factorial (mpz_class OP)
+ -- Function: mpz_class mpz_class::primorial (type OP)
+ -- Function: mpz_class primorial (mpz_class OP)
+ -- Function: mpz_class mpz_class::fibonacci (type OP)
+ -- Function: mpz_class fibonacci (mpz_class OP)
+
+ -- Function: void mpz_class::swap (mpz_class& OP)
+ -- Function: void swap (mpz_class& OP1, mpz_class& OP2)
+ These functions provide a C++ class interface to the corresponding
+ GMP C routines. Calling 'factorial' or 'primorial' on a negative
+ number is undefined.
+
+ 'cmp' can be used with any of the classes or the standard C++
+ types, except 'long long' and 'long double'.
+
+
+ Overloaded operators for combinations of 'mpz_class' and 'double' are
+provided for completeness, but it should be noted that if the given
+'double' is not an integer then the way any rounding is done is
+currently unspecified. The rounding might take place at the start, in
+the middle, or at the end of the operation, and it might change in the
+future.
+
+ Conversions between 'mpz_class' and 'double', however, are defined to
+follow the corresponding C functions 'mpz_get_d' and 'mpz_set_d'. And
+comparisons are always made exactly, as per 'mpz_cmp_d'.
+
+
+File: gmp.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface
+
+12.3 C++ Interface Rationals
+============================
+
+In all the following constructors, if a fraction is given then it should
+be in canonical form, or if not then 'mpq_class::canonicalize' called.
+
+ -- Function: mpq_class::mpq_class (type OP)
+ -- Function: mpq_class::mpq_class (integer NUM, integer DEN)
+ Construct an 'mpq_class'. The initial value can be a single value
+ of any type (conversion from 'mpf_class' is 'explicit'), or a pair
+ of integers ('mpz_class' or standard C++ integer types)
+ representing a fraction, except that 'long long' and 'long double'
+ are not supported. For example,
+
+ mpq_class q (99);
+ mpq_class q (1.75);
+ mpq_class q (1, 3);
+
+ -- Function: explicit mpq_class::mpq_class (const mpq_t Q)
+ Construct an 'mpq_class' from an 'mpq_t'. The value in Q is copied
+ into the new 'mpq_class', there won't be any permanent association
+ between it and Q.
+
+ -- Function: explicit mpq_class::mpq_class (const char *S, int BASE =
+ 0)
+ -- Function: explicit mpq_class::mpq_class (const string& S, int BASE =
+ 0)
+ Construct an 'mpq_class' converted from a string using
+ 'mpq_set_str' (*note Initializing Rationals::).
+
+ If the string is not a valid rational, an 'std::invalid_argument'
+ exception is thrown. The same applies to 'operator='.
+
+ -- Function: mpq_class operator"" _mpq (const char *STR)
+ With C++11 compilers, integral rationals can be constructed with
+ the syntax '123_mpq' which is equivalent to 'mpq_class(123_mpz)'.
+ Other rationals can be built as '-1_mpq/2' or '0xb_mpq/123456_mpz'.
+
+ -- Function: void mpq_class::canonicalize ()
+ Put an 'mpq_class' into canonical form, as per *note Rational
+ Number Functions::. All arithmetic operators require their
+ operands in canonical form, and will return results in canonical
+ form.
+
+ -- Function: mpq_class abs (mpq_class OP)
+ -- Function: int cmp (mpq_class OP1, type OP2)
+ -- Function: int cmp (type OP1, mpq_class OP2)
+
+ -- Function: double mpq_class::get_d (void)
+ -- Function: string mpq_class::get_str (int BASE = 10)
+
+ -- Function: int mpq_class::set_str (const char *STR, int BASE)
+ -- Function: int mpq_class::set_str (const string& STR, int BASE)
+ -- Function: int sgn (mpq_class OP)
+
+ -- Function: void mpq_class::swap (mpq_class& OP)
+ -- Function: void swap (mpq_class& OP1, mpq_class& OP2)
+ These functions provide a C++ class interface to the corresponding
+ GMP C routines.
+
+ 'cmp' can be used with any of the classes or the standard C++
+ types, except 'long long' and 'long double'.
+
+ -- Function: mpz_class& mpq_class::get_num ()
+ -- Function: mpz_class& mpq_class::get_den ()
+ Get a reference to an 'mpz_class' which is the numerator or
+ denominator of an 'mpq_class'. This can be used both for read and
+ write access. If the object returned is modified, it modifies the
+ original 'mpq_class'.
+
+ If direct manipulation might produce a non-canonical value, then
+ 'mpq_class::canonicalize' must be called before further operations.
+
+ -- Function: mpz_t mpq_class::get_num_mpz_t ()
+ -- Function: mpz_t mpq_class::get_den_mpz_t ()
+ Get a reference to the underlying 'mpz_t' numerator or denominator
+ of an 'mpq_class'. This can be passed to C functions expecting an
+ 'mpz_t'. Any modifications made to the 'mpz_t' will modify the
+ original 'mpq_class'.
+
+ If direct manipulation might produce a non-canonical value, then
+ 'mpq_class::canonicalize' must be called before further operations.
+
+ -- Function: istream& operator>> (istream& STREAM, mpq_class& ROP);
+ Read ROP from STREAM, using its 'ios' formatting settings, the same
+ as 'mpq_t operator>>' (*note C++ Formatted Input::).
+
+ If the ROP read might not be in canonical form then
+ 'mpq_class::canonicalize' must be called.
+
+
+File: gmp.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
+
+12.4 C++ Interface Floats
+=========================
+
+When an expression requires the use of temporary intermediate
+'mpf_class' values, like 'f=g*h+x*y', those temporaries will have the
+same precision as the destination 'f'. Explicit constructors can be
+used if this doesn't suit.
+
+ -- Function: mpf_class::mpf_class (type OP)
+ -- Function: mpf_class::mpf_class (type OP, mp_bitcnt_t PREC)
+ Construct an 'mpf_class'. Any standard C++ type can be used,
+ except 'long long' and 'long double', and any of the GMP C++
+ classes can be used.
+
+ If PREC is given, the initial precision is that value, in bits. If
+ PREC is not given, then the initial precision is determined by the
+ type of OP given. An 'mpz_class', 'mpq_class', or C++ builtin type
+ will give the default 'mpf' precision (*note Initializing
+ Floats::). An 'mpf_class' or expression will give the precision of
+ that value. The precision of a binary expression is the higher of
+ the two operands.
+
+ mpf_class f(1.5); // default precision
+ mpf_class f(1.5, 500); // 500 bits (at least)
+ mpf_class f(x); // precision of x
+ mpf_class f(abs(x)); // precision of x
+ mpf_class f(-g, 1000); // 1000 bits (at least)
+ mpf_class f(x+y); // greater of precisions of x and y
+
+ -- Function: explicit mpf_class::mpf_class (const mpf_t F)
+ -- Function: mpf_class::mpf_class (const mpf_t F, mp_bitcnt_t PREC)
+ Construct an 'mpf_class' from an 'mpf_t'. The value in F is copied
+ into the new 'mpf_class', there won't be any permanent association
+ between it and F.
+
+ If PREC is given, the initial precision is that value, in bits. If
+ PREC is not given, then the initial precision is that of F.
+
+ -- Function: explicit mpf_class::mpf_class (const char *S)
+ -- Function: mpf_class::mpf_class (const char *S, mp_bitcnt_t PREC, int
+ BASE = 0)
+ -- Function: explicit mpf_class::mpf_class (const string& S)
+ -- Function: mpf_class::mpf_class (const string& S, mp_bitcnt_t PREC,
+ int BASE = 0)
+ Construct an 'mpf_class' converted from a string using
+ 'mpf_set_str' (*note Assigning Floats::). If PREC is given, the
+ initial precision is that value, in bits. If not, the default
+ 'mpf' precision (*note Initializing Floats::) is used.
+
+ If the string is not a valid float, an 'std::invalid_argument'
+ exception is thrown. The same applies to 'operator='.
+
+ -- Function: mpf_class operator"" _mpf (const char *STR)
+ With C++11 compilers, floats can be constructed with the syntax
+ '1.23e-1_mpf' which is equivalent to 'mpf_class("1.23e-1")'.
+
+ -- Function: mpf_class& mpf_class::operator= (type OP)
+ Convert and store the given OP value to an 'mpf_class' object. The
+ same types are accepted as for the constructors above.
+
+ Note that 'operator=' only stores a new value, it doesn't copy or
+ change the precision of the destination, instead the value is
+ truncated if necessary. This is the same as 'mpf_set' etc. Note
+ in particular this means for 'mpf_class' a copy constructor is not
+ the same as a default constructor plus assignment.
+
+ mpf_class x (y); // x created with precision of y
+
+ mpf_class x; // x created with default precision
+ x = y; // value truncated to that precision
+
+ Applications using templated code may need to be careful about the
+ assumptions the code makes in this area, when working with
+ 'mpf_class' values of various different or non-default precisions.
+ For instance implementations of the standard 'complex' template
+ have been seen in both styles above, though of course 'complex' is
+ normally only actually specified for use with the builtin float
+ types.
+
+ -- Function: mpf_class abs (mpf_class OP)
+ -- Function: mpf_class ceil (mpf_class OP)
+ -- Function: int cmp (mpf_class OP1, type OP2)
+ -- Function: int cmp (type OP1, mpf_class OP2)
+
+ -- Function: bool mpf_class::fits_sint_p (void)
+ -- Function: bool mpf_class::fits_slong_p (void)
+ -- Function: bool mpf_class::fits_sshort_p (void)
+
+ -- Function: bool mpf_class::fits_uint_p (void)
+ -- Function: bool mpf_class::fits_ulong_p (void)
+ -- Function: bool mpf_class::fits_ushort_p (void)
+
+ -- Function: mpf_class floor (mpf_class OP)
+ -- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
+
+ -- Function: double mpf_class::get_d (void)
+ -- Function: long mpf_class::get_si (void)
+ -- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
+ size_t DIGITS = 0)
+ -- Function: unsigned long mpf_class::get_ui (void)
+
+ -- Function: int mpf_class::set_str (const char *STR, int BASE)
+ -- Function: int mpf_class::set_str (const string& STR, int BASE)
+ -- Function: int sgn (mpf_class OP)
+ -- Function: mpf_class sqrt (mpf_class OP)
+
+ -- Function: void mpf_class::swap (mpf_class& OP)
+ -- Function: void swap (mpf_class& OP1, mpf_class& OP2)
+ -- Function: mpf_class trunc (mpf_class OP)
+ These functions provide a C++ class interface to the corresponding
+ GMP C routines.
+
+ 'cmp' can be used with any of the classes or the standard C++
+ types, except 'long long' and 'long double'.
+
+ The accuracy provided by 'hypot' is not currently guaranteed.
+
+ -- Function: mp_bitcnt_t mpf_class::get_prec ()
+ -- Function: void mpf_class::set_prec (mp_bitcnt_t PREC)
+ -- Function: void mpf_class::set_prec_raw (mp_bitcnt_t PREC)
+ Get or set the current precision of an 'mpf_class'.
+
+ The restrictions described for 'mpf_set_prec_raw' (*note
+ Initializing Floats::) apply to 'mpf_class::set_prec_raw'. Note in
+ particular that the 'mpf_class' must be restored to it's allocated
+ precision before being destroyed. This must be done by application
+ code, there's no automatic mechanism for it.
+
+
+File: gmp.info, Node: C++ Interface Random Numbers, Next: C++ Interface Limitations, Prev: C++ Interface Floats, Up: C++ Class Interface
+
+12.5 C++ Interface Random Numbers
+=================================
+
+ -- Class: gmp_randclass
+ The C++ class interface to the GMP random number functions uses
+ 'gmp_randclass' to hold an algorithm selection and current state,
+ as per 'gmp_randstate_t'.
+
+ -- Function: gmp_randclass::gmp_randclass (void (*RANDINIT)
+ (gmp_randstate_t, ...), ...)
+ Construct a 'gmp_randclass', using a call to the given RANDINIT
+ function (*note Random State Initialization::). The arguments
+ expected are the same as RANDINIT, but with 'mpz_class' instead of
+ 'mpz_t'. For example,
+
+ gmp_randclass r1 (gmp_randinit_default);
+ gmp_randclass r2 (gmp_randinit_lc_2exp_size, 32);
+ gmp_randclass r3 (gmp_randinit_lc_2exp, a, c, m2exp);
+ gmp_randclass r4 (gmp_randinit_mt);
+
+ 'gmp_randinit_lc_2exp_size' will fail if the size requested is too
+ big, an 'std::length_error' exception is thrown in that case.
+
+ -- Function: gmp_randclass::gmp_randclass (gmp_randalg_t ALG, ...)
+ Construct a 'gmp_randclass' using the same parameters as
+ 'gmp_randinit' (*note Random State Initialization::). This
+ function is obsolete and the above RANDINIT style should be
+ preferred.
+
+ -- Function: void gmp_randclass::seed (unsigned long int S)
+ -- Function: void gmp_randclass::seed (mpz_class S)
+ Seed a random number generator. See *note Random Number
+ Functions::, for how to choose a good seed.
+
+ -- Function: mpz_class gmp_randclass::get_z_bits (mp_bitcnt_t BITS)
+ -- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
+ Generate a random integer with a specified number of bits.
+
+ -- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
+ Generate a random integer in the range 0 to N-1 inclusive.
+
+ -- Function: mpf_class gmp_randclass::get_f ()
+ -- Function: mpf_class gmp_randclass::get_f (mp_bitcnt_t PREC)
+ Generate a random float F in the range 0 <= F < 1. F will be to
+ PREC bits precision, or if PREC is not given then to the precision
+ of the destination. For example,
+
+ gmp_randclass r;
+ ...
+ mpf_class f (0, 512); // 512 bits precision
+ f = r.get_f(); // random number, 512 bits
+
+
+File: gmp.info, Node: C++ Interface Limitations, Prev: C++ Interface Random Numbers, Up: C++ Class Interface
+
+12.6 C++ Interface Limitations
+==============================
+
+'mpq_class' and Templated Reading
+ A generic piece of template code probably won't know that
+ 'mpq_class' requires a 'canonicalize' call if inputs read with
+ 'operator>>' might be non-canonical. This can lead to incorrect
+ results.
+
+ 'operator>>' behaves as it does for reasons of efficiency. A
+ canonicalize can be quite time consuming on large operands, and is
+ best avoided if it's not necessary.
+
+ But this potential difficulty reduces the usefulness of
+ 'mpq_class'. Perhaps a mechanism to tell 'operator>>' what to do
+ will be adopted in the future, maybe a preprocessor define, a
+ global flag, or an 'ios' flag pressed into service. Or maybe, at
+ the risk of inconsistency, the 'mpq_class' 'operator>>' could
+ canonicalize and leave 'mpq_t' 'operator>>' not doing so, for use
+ on those occasions when that's acceptable. Send feedback or
+ alternate ideas to <gmp-bugs@gmplib.org>.
+
+Subclassing
+ Subclassing the GMP C++ classes works, but is not currently
+ recommended.
+
+ Expressions involving subclasses resolve correctly (or seem to),
+ but in normal C++ fashion the subclass doesn't inherit constructors
+ and assignments. There's many of those in the GMP classes, and a
+ good way to reestablish them in a subclass is not yet provided.
+
+Templated Expressions
+ A subtle difficulty exists when using expressions together with
+ application-defined template functions. Consider the following,
+ with 'T' intended to be some numeric type,
+
+ template <class T>
+ T fun (const T &, const T &);
+
+ When used with, say, plain 'mpz_class' variables, it works fine:
+ 'T' is resolved as 'mpz_class'.
+
+ mpz_class f(1), g(2);
+ fun (f, g); // Good
+
+ But when one of the arguments is an expression, it doesn't work.
+
+ mpz_class f(1), g(2), h(3);
+ fun (f, g+h); // Bad
+
+ This is because 'g+h' ends up being a certain expression template
+ type internal to 'gmpxx.h', which the C++ template resolution rules
+ are unable to automatically convert to 'mpz_class'. The workaround
+ is simply to add an explicit cast.
+
+ mpz_class f(1), g(2), h(3);
+ fun (f, mpz_class(g+h)); // Good
+
+ Similarly, within 'fun' it may be necessary to cast an expression
+ to type 'T' when calling a templated 'fun2'.
+
+ template <class T>
+ void fun (T f, T g)
+ {
+ fun2 (f, f+g); // Bad
+ }
+
+ template <class T>
+ void fun (T f, T g)
+ {
+ fun2 (f, T(f+g)); // Good
+ }
+
+C++11
+ C++11 provides several new ways in which types can be inferred:
+ 'auto', 'decltype', etc. While they can be very convenient, they
+ don't mix well with expression templates. In this example, the
+ addition is performed twice, as if we had defined 'sum' as a macro.
+
+ mpz_class z = 33;
+ auto sum = z + z;
+ mpz_class prod = sum * sum;
+
+ This other example may crash, though some compilers might make it
+ look like it is working, because the expression 'z+z' goes out of
+ scope before it is evaluated.
+
+ mpz_class z = 33;
+ auto sum = z + z + z;
+ mpz_class prod = sum * 2;
+
+ It is thus strongly recommended to avoid 'auto' anywhere a GMP C++
+ expression may appear.
+
+
+File: gmp.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top
+
+13 Custom Allocation
+********************
+
+By default GMP uses 'malloc', 'realloc' and 'free' for memory
+allocation, and if they fail GMP prints a message to the standard error
+output and terminates the program.
+
+ Alternate functions can be specified, to allocate memory in a
+different way or to have a different error action on running out of
+memory.
+
+ -- Function: void mp_set_memory_functions (
+ void *(*ALLOC_FUNC_PTR) (size_t),
+ void *(*REALLOC_FUNC_PTR) (void *, size_t, size_t),
+ void (*FREE_FUNC_PTR) (void *, size_t))
+ Replace the current allocation functions from the arguments. If an
+ argument is 'NULL', the corresponding default function is used.
+
+ These functions will be used for all memory allocation done by GMP,
+ apart from temporary space from 'alloca' if that function is
+ available and GMP is configured to use it (*note Build Options::).
+
+ *Be sure to call 'mp_set_memory_functions' only when there are no
+ active GMP objects allocated using the previous memory functions!
+ Usually that means calling it before any other GMP function.*
+
+ The functions supplied should fit the following declarations:
+
+ -- Function: void * allocate_function (size_t ALLOC_SIZE)
+ Return a pointer to newly allocated space with at least ALLOC_SIZE
+ bytes.
+
+ -- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
+ size_t NEW_SIZE)
+ Resize a previously allocated block PTR of OLD_SIZE bytes to be
+ NEW_SIZE bytes.
+
+ The block may be moved if necessary or if desired, and in that case
+ the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to the
+ new location. The return value is a pointer to the resized block,
+ that being the new location if moved or just PTR if not.
+
+ PTR is never 'NULL', it's always a previously allocated block.
+ NEW_SIZE may be bigger or smaller than OLD_SIZE.
+
+ -- Function: void free_function (void *PTR, size_t SIZE)
+ De-allocate the space pointed to by PTR.
+
+ PTR is never 'NULL', it's always a previously allocated block of
+ SIZE bytes.
+
+ A "byte" here means the unit used by the 'sizeof' operator.
+
+ The REALLOCATE_FUNCTION parameter OLD_SIZE and the FREE_FUNCTION
+parameter SIZE are passed for convenience, but of course they can be
+ignored if not needed by an implementation. The default functions using
+'malloc' and friends for instance don't use them.
+
+ No error return is allowed from any of these functions, if they
+return then they must have performed the specified operation. In
+particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
+return 'NULL'.
+
+ Getting a different fatal error action is a good use for custom
+allocation functions, for example giving a graphical dialog rather than
+the default print to 'stderr'. How much is possible when genuinely out
+of memory is another question though.
+
+ There's currently no defined way for the allocation functions to
+recover from an error such as out of memory, they must terminate program
+execution. A 'longjmp' or throwing a C++ exception will have undefined
+results. This may change in the future.
+
+ GMP may use allocated blocks to hold pointers to other allocated
+blocks. This will limit the assumptions a conservative garbage
+collection scheme can make.
+
+ Since the default GMP allocation uses 'malloc' and friends, those
+functions will be linked in even if the first thing a program does is an
+'mp_set_memory_functions'. It's necessary to change the GMP sources if
+this is a problem.
+
+
+ -- Function: void mp_get_memory_functions (
+ void *(**ALLOC_FUNC_PTR) (size_t),
+ void *(**REALLOC_FUNC_PTR) (void *, size_t, size_t),
+ void (**FREE_FUNC_PTR) (void *, size_t))
+ Get the current allocation functions, storing function pointers to
+ the locations given by the arguments. If an argument is 'NULL',
+ that function pointer is not stored.
+
+ For example, to get just the current free function,
+
+ void (*freefunc) (void *, size_t);
+
+ mp_get_memory_functions (NULL, NULL, &freefunc);
+
+
+File: gmp.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
+
+14 Language Bindings
+********************
+
+The following packages and projects offer access to GMP from languages
+other than C, though perhaps with varying levels of functionality and
+efficiency.
+
+
+C++
+ * GMP C++ class interface, *note C++ Class Interface::
+ Straightforward interface, expression templates to eliminate
+ temporaries.
+ * ALP <https://www-sop.inria.fr/saga/logiciels/ALP/>
+ Linear algebra and polynomials using templates.
+ * CLN <https://www.ginac.de/CLN/>
+ High level classes for arithmetic.
+ * Linbox <http://www.linalg.org/>
+ Sparse vectors and matrices.
+ * NTL <http://www.shoup.net/ntl/>
+ A C++ number theory library.
+
+Eiffel
+ * Eiffelroom <http://www.eiffelroom.org/node/442>
+
+Haskell
+ * Glasgow Haskell Compiler <https://www.haskell.org/ghc/>
+
+Java
+ * Kaffe <https://github.com/kaffe/kaffe>
+
+Lisp
+ * GNU Common Lisp <https://www.gnu.org/software/gcl/gcl.html>
+ * Librep <http://librep.sourceforge.net/>
+ * XEmacs (21.5.18 beta and up) <https://www.xemacs.org>
+ Optional big integers, rationals and floats using GMP.
+
+ML
+ * MLton compiler <http://mlton.org/>
+
+Objective Caml
+ * MLGMP <https://opam.ocaml.org/packages/mlgmp/>
+ * Numerix <http://pauillac.inria.fr/~quercia/>
+ Optionally using GMP.
+
+Oz
+ * Mozart <https://mozart.github.io/>
+
+Pascal
+ * GNU Pascal Compiler <http://www.gnu-pascal.de/>
+ GMP unit.
+ * Numerix <http://pauillac.inria.fr/~quercia/>
+ For Free Pascal, optionally using GMP.
+
+Perl
+ * GMP module, see 'demos/perl' in the GMP sources (*note
+ Demonstration Programs::).
+ * Math::GMP <https://www.cpan.org/>
+ Compatible with Math::BigInt, but not as many functions as the
+ GMP module above.
+ * Math::BigInt::GMP <https://www.cpan.org/>
+ Plug Math::GMP into normal Math::BigInt operations.
+
+Pike
+ * pikempz module in the standard distribution,
+ <https://pike.lysator.liu.se/>
+
+Prolog
+ * SWI Prolog <http://www.swi-prolog.org/>
+ Arbitrary precision floats.
+
+Python
+ * GMPY <https://code.google.com/p/gmpy/>
+
+Ruby
+ * <https://rubygems.org/gems/gmp>
+
+Scheme
+ * GNU Guile <https://www.gnu.org/software/guile/guile.html>
+ * RScheme <https://www.rscheme.org/>
+ * STklos <http://www.stklos.net/>
+
+Smalltalk
+ * GNU Smalltalk <http://smalltalk.gnu.org/>
+
+Other
+ * Axiom <https://savannah.nongnu.org/projects/axiom>
+ Computer algebra using GCL.
+ * DrGenius <http://drgenius.seul.org/>
+ Geometry system and mathematical programming language.
+ * GiNaC <httsp://www.ginac.de/>
+ C++ computer algebra using CLN.
+ * GOO <https://www.eecs.berkeley.edu/~jrb/goo/>
+ Dynamic object oriented language.
+ * Maxima <https://www.ma.utexas.edu/users/wfs/maxima.html>
+ Macsyma computer algebra using GCL.
+ * Regina <http://regina.sourceforge.net/>
+ Topological calculator.
+ * Yacas <http://yacas.sourceforge.net>
+ Yet another computer algebra system.
+
+
+File: gmp.info, Node: Algorithms, Next: Internals, Prev: Language Bindings, Up: Top
+
+15 Algorithms
+*************
+
+This chapter is an introduction to some of the algorithms used for
+various GMP operations. The code is likely to be hard to understand
+without knowing something about the algorithms.
+
+ Some GMP internals are mentioned, but applications that expect to be
+compatible with future GMP releases should take care to use only the
+documented functions.
+
+* Menu:
+
+* Multiplication Algorithms::
+* Division Algorithms::
+* Greatest Common Divisor Algorithms::
+* Powering Algorithms::
+* Root Extraction Algorithms::
+* Radix Conversion Algorithms::
+* Other Algorithms::
+* Assembly Coding::
+
+
+File: gmp.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
+
+15.1 Multiplication
+===================
+
+NxN limb multiplications and squares are done using one of seven
+algorithms, as the size N increases.
+
+ Algorithm Threshold
+ Basecase (none)
+ Karatsuba 'MUL_TOOM22_THRESHOLD'
+ Toom-3 'MUL_TOOM33_THRESHOLD'
+ Toom-4 'MUL_TOOM44_THRESHOLD'
+ Toom-6.5 'MUL_TOOM6H_THRESHOLD'
+ Toom-8.5 'MUL_TOOM8H_THRESHOLD'
+ FFT 'MUL_FFT_THRESHOLD'
+
+ Similarly for squaring, with the 'SQR' thresholds.
+
+ NxM multiplications of operands with different sizes above
+'MUL_TOOM22_THRESHOLD' are currently done by special Toom-inspired
+algorithms or directly with FFT, depending on operand size (*note
+Unbalanced Multiplication::).
+
+* Menu:
+
+* Basecase Multiplication::
+* Karatsuba Multiplication::
+* Toom 3-Way Multiplication::
+* Toom 4-Way Multiplication::
+* Higher degree Toom'n'half::
+* FFT Multiplication::
+* Other Multiplication::
+* Unbalanced Multiplication::
+
+
+File: gmp.info, Node: Basecase Multiplication, Next: Karatsuba Multiplication, Prev: Multiplication Algorithms, Up: Multiplication Algorithms
+
+15.1.1 Basecase Multiplication
+------------------------------
+
+Basecase NxM multiplication is a straightforward rectangular set of
+cross-products, the same as long multiplication done by hand and for
+that reason sometimes known as the schoolbook or grammar school method.
+This is an O(N*M) algorithm. See Knuth section 4.3.1 algorithm M (*note
+References::), and the 'mpn/generic/mul_basecase.c' code.
+
+ Assembly implementations of 'mpn_mul_basecase' are essentially the
+same as the generic C code, but have all the usual assembly tricks and
+obscurities introduced for speed.
+
+ A square can be done in roughly half the time of a multiply, by using
+the fact that the cross products above and below the diagonal are the
+same. A triangle of products below the diagonal is formed, doubled
+(left shift by one bit), and then the products on the diagonal added.
+This can be seen in 'mpn/generic/sqr_basecase.c'. Again the assembly
+implementations take essentially the same approach.
+
+ u0 u1 u2 u3 u4
+ +---+---+---+---+---+
+ u0 | d | | | | |
+ +---+---+---+---+---+
+ u1 | | d | | | |
+ +---+---+---+---+---+
+ u2 | | | d | | |
+ +---+---+---+---+---+
+ u3 | | | | d | |
+ +---+---+---+---+---+
+ u4 | | | | | d |
+ +---+---+---+---+---+
+
+ In practice squaring isn't a full 2x faster than multiplying, it's
+usually around 1.5x. Less than 1.5x probably indicates
+'mpn_sqr_basecase' wants improving on that CPU.
+
+ On some CPUs 'mpn_mul_basecase' can be faster than the generic C
+'mpn_sqr_basecase' on some small sizes. 'SQR_BASECASE_THRESHOLD' is the
+size at which to use 'mpn_sqr_basecase', this will be zero if that
+routine should be used always.
+
+
+File: gmp.info, Node: Karatsuba Multiplication, Next: Toom 3-Way Multiplication, Prev: Basecase Multiplication, Up: Multiplication Algorithms
+
+15.1.2 Karatsuba Multiplication
+-------------------------------
+
+The Karatsuba multiplication algorithm is described in Knuth section
+4.3.3 part A, and various other textbooks. A brief description is given
+here.
+
+ The inputs x and y are treated as each split into two parts of equal
+length (or the most significant part one limb shorter if N is odd).
+
+ high low
+ +----------+----------+
+ | x1 | x0 |
+ +----------+----------+
+
+ +----------+----------+
+ | y1 | y0 |
+ +----------+----------+
+
+ Let b be the power of 2 where the split occurs, i.e. if x0 is k limbs
+(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
+y=y1*b+y0, and the following holds,
+
+ x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
+
+ This formula means doing only three multiplies of (N/2)x(N/2) limbs,
+whereas a basecase multiply of NxN limbs is equivalent to four
+multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
+positions where the three products must be added.
+
+ high low
+ +--------+--------+ +--------+--------+
+ | x1*y1 | | x0*y0 |
+ +--------+--------+ +--------+--------+
+ +--------+--------+
+ add | x1*y1 |
+ +--------+--------+
+ +--------+--------+
+ add | x0*y0 |
+ +--------+--------+
+ +--------+--------+
+ sub | (x1-x0)*(y1-y0) |
+ +--------+--------+
+
+ The term (x1-x0)*(y1-y0) is best calculated as an absolute value, and
+the sign used to choose to add or subtract. Notice the sum
+high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
+additions, rather than 6*k, but in GMP extra function call overheads
+outweigh the saving.
+
+ Squaring is similar to multiplying, but with x=y the formula reduces
+to an equivalent with three squares,
+
+ x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
+
+ The final result is accumulated from those three squares the same way
+as for the three multiplies above. The middle term (x1-x0)^2 is now
+always positive.
+
+ A similar formula for both multiplying and squaring can be
+constructed with a middle term (x1+x0)*(y1+y0). But those sums can
+exceed k limbs, leading to more carry handling and additions than the
+form above.
+
+ Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
+the exponent being log(3)/log(2), representing 3 multiplies each 1/2 the
+size of the inputs. This is a big improvement over the basecase
+multiply at O(N^2) and the advantage soon overcomes the extra additions
+Karatsuba performs. 'MUL_TOOM22_THRESHOLD' can be as little as 10
+limbs. The 'SQR' threshold is usually about twice the 'MUL'.
+
+ The basecase algorithm will take a time of the form M(N) = a*N^2 +
+b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, which
+expands to K(N) = 3/4*a*N^2 + 3/2*b*N + 3*c + d*N + e. The factor 3/4
+for a means per-crossproduct speedups in the basecase code will increase
+the threshold since they benefit M(N) more than K(N). And conversely the
+3/2 for b means linear style speedups of b will increase the threshold
+since they benefit K(N) more than M(N). The latter can be seen for
+instance when adding an optimized 'mpn_sqr_diagonal' to
+'mpn_sqr_basecase'. Of course all speedups reduce total time, and in
+that sense the algorithm thresholds are merely of academic interest.
+
+
+File: gmp.info, Node: Toom 3-Way Multiplication, Next: Toom 4-Way Multiplication, Prev: Karatsuba Multiplication, Up: Multiplication Algorithms
+
+15.1.3 Toom 3-Way Multiplication
+--------------------------------
+
+The Karatsuba formula is the simplest case of a general approach to
+splitting inputs that leads to both Toom and FFT algorithms. A
+description of Toom can be found in Knuth section 4.3.3, with an example
+3-way calculation after Theorem A. The 3-way form used in GMP is
+described here.
+
+ The operands are each considered split into 3 pieces of equal length
+(or the most significant part 1 or 2 limbs shorter than the other two).
+
+ high low
+ +----------+----------+----------+
+ | x2 | x1 | x0 |
+ +----------+----------+----------+
+
+ +----------+----------+----------+
+ | y2 | y1 | y0 |
+ +----------+----------+----------+
+
+These parts are treated as the coefficients of two polynomials
+
+ X(t) = x2*t^2 + x1*t + x0
+ Y(t) = y2*t^2 + y1*t + y0
+
+ Let b equal the power of 2 which is the size of the x0, x1, y0 and y1
+pieces, i.e. if they're k limbs each then b=2^(k*mp_bits_per_limb).
+With this x=X(b) and y=Y(b).
+
+ Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
+
+ W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
+
+ The w[i] are going to be determined, and when they are they'll give
+the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
+coefficients will be roughly b^2 each, and the final W(b) will be an
+addition like,
+
+ high low
+ +-------+-------+
+ | w4 |
+ +-------+-------+
+ +--------+-------+
+ | w3 |
+ +--------+-------+
+ +--------+-------+
+ | w2 |
+ +--------+-------+
+ +--------+-------+
+ | w1 |
+ +--------+-------+
+ +-------+-------+
+ | w0 |
+ +-------+-------+
+
+ The w[i] coefficients could be formed by a simple set of cross
+products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
+this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
+equivalent merely to a basecase multiply. Instead the following
+approach is used.
+
+ X(t) and Y(t) are evaluated and multiplied at 5 points, giving values
+of W(t) at those points. In GMP the following points are used,
+
+ Point Value
+ t=0 x0 * y0, which gives w0 immediately
+ t=1 (x2+x1+x0) * (y2+y1+y0)
+ t=-1 (x2-x1+x0) * (y2-y1+y0)
+ t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
+ t=inf x2 * y2, which gives w4 immediately
+
+ At t=-1 the values can be negative and that's handled using the
+absolute values and tracking the sign separately. At t=inf the value is
+actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but it's
+much easier to think of as simply x2*y2 giving w4 immediately (much like
+x0*y0 at t=0 gives w0 immediately).
+
+ Each of the points substituted into W(t)=w4*t^4+...+w0 gives a linear
+combination of the w[i] coefficients, and the value of those
+combinations has just been calculated.
+
+ W(0) = w0
+ W(1) = w4 + w3 + w2 + w1 + w0
+ W(-1) = w4 - w3 + w2 - w1 + w0
+ W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
+ W(inf) = w4
+
+ This is a set of five equations in five unknowns, and some elementary
+linear algebra quickly isolates each w[i]. This involves adding or
+subtracting one W(t) value from another, and a couple of divisions by
+powers of 2 and one division by 3, the latter using the special
+'mpn_divexact_by3' (*note Exact Division::).
+
+ The conversion of W(t) values to the coefficients is interpolation.
+A polynomial of degree 4 like W(t) is uniquely determined by values
+known at 5 different points. The points are arbitrary and can be chosen
+to make the linear equations come out with a convenient set of steps for
+quickly isolating the w[i].
+
+ Squaring follows the same procedure as multiplication, but there's
+only one X(t) and it's evaluated at the 5 points, and those values
+squared to give values of W(t). The interpolation is then identical,
+and in fact the same 'toom_interpolate_5pts' subroutine is used for both
+squaring and multiplying.
+
+ Toom-3 is asymptotically O(N^1.465), the exponent being
+log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
+size each. This is an improvement over Karatsuba at O(N^1.585), though
+Toom does more work in the evaluation and interpolation and so it only
+realizes its advantage above a certain size.
+
+ Near the crossover between Toom-3 and Karatsuba there's generally a
+range of sizes where the difference between the two is small.
+'MUL_TOOM33_THRESHOLD' is a somewhat arbitrary point in that range and
+successive runs of the tune program can give different values due to
+small variations in measuring. A graph of time versus size for the two
+shows the effect, see 'tune/README'.
+
+ At the fairly small sizes where the Toom-3 thresholds occur it's
+worth remembering that the asymptotic behaviour for Karatsuba and Toom-3
+can't be expected to make accurate predictions, due of course to the big
+influence of all sorts of overheads, and the fact that only a few
+recursions of each are being performed. Even at large sizes there's a
+good chance machine dependent effects like cache architecture will mean
+actual performance deviates from what might be predicted.
+
+ The formula given for the Karatsuba algorithm (*note Karatsuba
+Multiplication::) has an equivalent for Toom-3 involving only five
+multiplies, but this would be complicated and unenlightening.
+
+ An alternate view of Toom-3 can be found in Zuras (*note
+References::), using a vector to represent the x and y splits and a
+matrix multiplication for the evaluation and interpolation stages. The
+matrix inverses are not meant to be actually used, and they have
+elements with values much greater than in fact arise in the
+interpolation steps. The diagram shown for the 3-way is attractive, but
+again doesn't have to be implemented that way and for example with a bit
+of rearrangement just one division by 6 can be done.
+
+
+File: gmp.info, Node: Toom 4-Way Multiplication, Next: Higher degree Toom'n'half, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
+
+15.1.4 Toom 4-Way Multiplication
+--------------------------------
+
+Karatsuba and Toom-3 split the operands into 2 and 3 coefficients,
+respectively. Toom-4 analogously splits the operands into 4
+coefficients. Using the notation from the section on Toom-3
+multiplication, we form two polynomials:
+
+ X(t) = x3*t^3 + x2*t^2 + x1*t + x0
+ Y(t) = y3*t^3 + y2*t^2 + y1*t + y0
+
+ X(t) and Y(t) are evaluated and multiplied at 7 points, giving values
+of W(t) at those points. In GMP the following points are used,
+
+ Point Value
+ t=0 x0 * y0, which gives w0 immediately
+ t=1/2 (x3+2*x2+4*x1+8*x0) * (y3+2*y2+4*y1+8*y0)
+ t=-1/2 (-x3+2*x2-4*x1+8*x0) * (-y3+2*y2-4*y1+8*y0)
+ t=1 (x3+x2+x1+x0) * (y3+y2+y1+y0)
+ t=-1 (-x3+x2-x1+x0) * (-y3+y2-y1+y0)
+ t=2 (8*x3+4*x2+2*x1+x0) * (8*y3+4*y2+2*y1+y0)
+ t=inf x3 * y3, which gives w6 immediately
+
+ The number of additions and subtractions for Toom-4 is much larger
+than for Toom-3. But several subexpressions occur multiple times, for
+example x2+x0, occurs for both t=1 and t=-1.
+
+ Toom-4 is asymptotically O(N^1.404), the exponent being
+log(7)/log(4), representing 7 recursive multiplies of 1/4 the original
+size each.
+
+
+File: gmp.info, Node: Higher degree Toom'n'half, Next: FFT Multiplication, Prev: Toom 4-Way Multiplication, Up: Multiplication Algorithms
+
+15.1.5 Higher degree Toom'n'half
+--------------------------------
+
+The Toom algorithms described above (*note Toom 3-Way Multiplication::,
+*note Toom 4-Way Multiplication::) generalizes to split into an
+arbitrary number of pieces. In general a split of two equally long
+operands into r pieces leads to evaluations and pointwise
+multiplications done at 2*r-1 points. To fully exploit symmetries it
+would be better to have a multiple of 4 points, that's why for higher
+degree Toom'n'half is used.
+
+ Toom'n'half means that the existence of one more piece is considered
+for a single operand. It can be virtual, i.e. zero, or real, when the
+two operand are not exactly balanced. By choosing an even r, Toom-r+1/2
+requires 2r points, a multiple of four.
+
+ The quadruplets of points include 0, inf, +1, -1 and +-2^i, +-2^-i .
+Each of them giving shortcuts for the evaluation phase and for some
+steps in the interpolation phase. Further tricks are used to reduce the
+memory footprint of the whole multiplication algorithm to a memory
+buffer equal in size to the result of the product.
+
+ Current GMP uses both Toom-6'n'half and Toom-8'n'half.
+
+
+File: gmp.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Higher degree Toom'n'half, Up: Multiplication Algorithms
+
+15.1.6 FFT Multiplication
+-------------------------
+
+At large to very large sizes a Fermat style FFT multiplication is used,
+following Schönhage and Strassen (*note References::). Descriptions of
+FFTs in various forms can be found in many textbooks, for instance Knuth
+section 4.3.3 part C or Lipson chapter IX. A brief description of the
+form used in GMP is given here.
+
+ The multiplication done is x*y mod 2^N+1, for a given N. A full
+product x*y is obtained by choosing N>=bits(x)+bits(y) and padding x and
+y with high zero limbs. The modular product is the native form for the
+algorithm, so padding to get a full product is unavoidable.
+
+ The algorithm follows a split, evaluate, pointwise multiply,
+interpolate and combine similar to that described above for Karatsuba
+and Toom-3. A k parameter controls the split, with an FFT-k splitting
+into 2^k pieces of M=N/2^k bits each. N must be a multiple of
+(2^k)*mp_bits_per_limb so the split falls on limb boundaries, avoiding
+bit shifts in the split and combine stages.
+
+ The evaluations, pointwise multiplications, and interpolation, are
+all done modulo 2^N'+1 where N' is 2M+k+3 rounded up to a multiple of
+2^k and of 'mp_bits_per_limb'. The results of interpolation will be the
+following negacyclic convolution of the input pieces, and the choice of
+N' ensures these sums aren't truncated.
+
+ ---
+ \ b
+ w[n] = / (-1) * x[i] * y[j]
+ ---
+ i+j==b*2^k+n
+ b=0,1
+
+ The points used for the evaluation are g^i for i=0 to 2^k-1 where
+g=2^(2N'/2^k). g is a 2^k'th root of unity mod 2^N'+1, which produces
+necessary cancellations at the interpolation stage, and it's also a
+power of 2 so the fast Fourier transforms used for the evaluation and
+interpolation do only shifts, adds and negations.
+
+ The pointwise multiplications are done modulo 2^N'+1 and either
+recurse into a further FFT or use a plain multiplication (Toom-3,
+Karatsuba or basecase), whichever is optimal at the size N'. The
+interpolation is an inverse fast Fourier transform. The resulting set
+of sums of x[i]*y[j] are added at appropriate offsets to give the final
+result.
+
+ Squaring is the same, but x is the only input so it's one transform
+at the evaluate stage and the pointwise multiplies are squares. The
+interpolation is the same.
+
+ For a mod 2^N+1 product, an FFT-k is an O(N^(k/(k-1))) algorithm, the
+exponent representing 2^k recursed modular multiplies each 1/2^(k-1) the
+size of the original. Each successive k is an asymptotic improvement,
+but overheads mean each is only faster at bigger and bigger sizes. In
+the code, 'MUL_FFT_TABLE' and 'SQR_FFT_TABLE' are the thresholds where
+each k is used. Each new k effectively swaps some multiplying for some
+shifts, adds and overheads.
+
+ A mod 2^N+1 product can be formed with a normal NxN->2N bit multiply
+plus a subtraction, so an FFT and Toom-3 etc can be compared directly.
+A k=4 FFT at O(N^1.333) can be expected to be the first faster than
+Toom-3 at O(N^1.465). In practice this is what's found, with
+'MUL_FFT_MODF_THRESHOLD' and 'SQR_FFT_MODF_THRESHOLD' being between 300
+and 1000 limbs, depending on the CPU. So far it's been found that only
+very large FFTs recurse into pointwise multiplies above these sizes.
+
+ When an FFT is to give a full product, the change of N to 2N doesn't
+alter the theoretical complexity for a given k, but for the purposes of
+considering where an FFT might be first used it can be assumed that the
+FFT is recursing into a normal multiply and that on that basis it's
+doing 2^k recursed multiplies each 1/2^(k-2) the size of the inputs,
+making it O(N^(k/(k-2))). This would mean k=7 at O(N^1.4) would be the
+first FFT faster than Toom-3. In practice 'MUL_FFT_THRESHOLD' and
+'SQR_FFT_THRESHOLD' have been found to be in the k=8 range, somewhere
+between 3000 and 10000 limbs.
+
+ The way N is split into 2^k pieces and then 2M+k+3 is rounded up to a
+multiple of 2^k and 'mp_bits_per_limb' means that when
+2^k>=mp\_bits\_per\_limb the effective N is a multiple of 2^(2k-1) bits.
+The +k+3 means some values of N just under such a multiple will be
+rounded to the next. The complexity calculations above assume that a
+favourable size is used, meaning one which isn't padded through
+rounding, and it's also assumed that the extra +k+3 bits are negligible
+at typical FFT sizes.
+
+ The practical effect of the 2^(2k-1) constraint is to introduce a
+step-effect into measured speeds. For example k=8 will round N up to a
+multiple of 32768 bits, so for a 32-bit limb there'll be 512 limb groups
+of sizes for which 'mpn_mul_n' runs at the same speed. Or for k=9
+groups of 2048 limbs, k=10 groups of 8192 limbs, etc. In practice it's
+been found each k is used at quite small multiples of its size
+constraint and so the step effect is quite noticeable in a time versus
+size graph.
+
+ The threshold determinations currently measure at the mid-points of
+size steps, but this is sub-optimal since at the start of a new step it
+can happen that it's better to go back to the previous k for a while.
+Something more sophisticated for 'MUL_FFT_TABLE' and 'SQR_FFT_TABLE'
+will be needed.
+
+
+File: gmp.info, Node: Other Multiplication, Next: Unbalanced Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
+
+15.1.7 Other Multiplication
+---------------------------
+
+The Toom algorithms described above (*note Toom 3-Way Multiplication::,
+*note Toom 4-Way Multiplication::) generalizes to split into an
+arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C.
+This is not currently used. The notes here are merely for interest.
+
+ In general a split into r+1 pieces is made, and evaluations and
+pointwise multiplications done at 2*r+1 points. A 4-way split does 7
+pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way
+algorithm is O(N^(log(2*r+1)/log(r+1))). Only the pointwise
+multiplications count towards big-O complexity, but the time spent in
+the evaluate and interpolate stages grows with r and has a significant
+practical impact, with the asymptotic advantage of each r realized only
+at bigger and bigger sizes. The overheads grow as O(N*r), whereas in an
+r=2^k FFT they grow only as O(N*log(r)).
+
+ Knuth algorithm C evaluates at points 0,1,2,...,2*r, but exercise 4
+uses -r,...,0,...,r and the latter saves some small multiplies in the
+evaluate stage (or rather trades them for additions), and has a further
+saving of nearly half the interpolate steps. The idea is to separate
+odd and even final coefficients and then perform algorithm C steps C7
+and C8 on them separately. The divisors at step C7 become j^2 and the
+multipliers at C8 become 2*t*j-j^2.
+
+ Splitting odd and even parts through positive and negative points can
+be thought of as using -1 as a square root of unity. If a 4th root of
+unity was available then a further split and speedup would be possible,
+but no such root exists for plain integers. Going to complex integers
+with i=sqrt(-1) doesn't help, essentially because in Cartesian form it
+takes three real multiplies to do a complex multiply. The existence of
+2^k'th roots of unity in a suitable ring or field lets the fast Fourier
+transform keep splitting and get to O(N*log(r)).
+
+ Floating point FFTs use complex numbers approximating Nth roots of
+unity. Some processors have special support for such FFTs. But these
+are not used in GMP since it's very difficult to guarantee an exact
+result (to some number of bits). An occasional difference of 1 in the
+last bit might not matter to a typical signal processing algorithm, but
+is of course of vital importance to GMP.
+
+
+File: gmp.info, Node: Unbalanced Multiplication, Prev: Other Multiplication, Up: Multiplication Algorithms
+
+15.1.8 Unbalanced Multiplication
+--------------------------------
+
+Multiplication of operands with different sizes, both below
+'MUL_TOOM22_THRESHOLD' are done with plain schoolbook multiplication
+(*note Basecase Multiplication::).
+
+ For really large operands, we invoke FFT directly.
+
+ For operands between these sizes, we use Toom inspired algorithms
+suggested by Alberto Zanoni and Marco Bodrato. The idea is to split the
+operands into polynomials of different degree. GMP currently splits the
+smaller operand onto 2 coefficients, i.e., a polynomial of degree 1, but
+the larger operand can be split into 2, 3, or 4 coefficients, i.e., a
+polynomial of degree 1 to 3.
+
+
+File: gmp.info, Node: Division Algorithms, Next: Greatest Common Divisor Algorithms, Prev: Multiplication Algorithms, Up: Algorithms
+
+15.2 Division Algorithms
+========================
+
+* Menu:
+
+* Single Limb Division::
+* Basecase Division::
+* Divide and Conquer Division::
+* Block-Wise Barrett Division::
+* Exact Division::
+* Exact Remainder::
+* Small Quotient Division::
+
+
+File: gmp.info, Node: Single Limb Division, Next: Basecase Division, Prev: Division Algorithms, Up: Division Algorithms
+
+15.2.1 Single Limb Division
+---------------------------
+
+Nx1 division is implemented using repeated 2x1 divisions from high to
+low, either with a hardware divide instruction or a multiplication by
+inverse, whichever is best on a given CPU.
+
+ The multiply by inverse follows "Improved division by invariant
+integers" by Möller and Granlund (*note References::) and is implemented
+as 'udiv_qrnnd_preinv' in 'gmp-impl.h'. The idea is to have a
+fixed-point approximation to 1/d (see 'invert_limb') and then multiply
+by the high limb (plus one bit) of the dividend to get a quotient q.
+With d normalized (high bit set), q is no more than 1 too small.
+Subtracting q*d from the dividend gives a remainder, and reveals whether
+q or q-1 is correct.
+
+ The result is a division done with two multiplications and four or
+five arithmetic operations. On CPUs with low latency multipliers this
+can be much faster than a hardware divide, though the cost of
+calculating the inverse at the start may mean it's only better on inputs
+bigger than say 4 or 5 limbs.
+
+ When a divisor must be normalized, either for the generic C
+'__udiv_qrnnd_c' or the multiply by inverse, the division performed is
+actually a*2^k by d*2^k where a is the dividend and k is the power
+necessary to have the high bit of d*2^k set. The bit shifts for the
+dividend are usually accomplished "on the fly" meaning by extracting the
+appropriate bits at each step. Done this way the quotient limbs come
+out aligned ready to store. When only the remainder is wanted, an
+alternative is to take the dividend limbs unshifted and calculate r = a
+mod d*2^k followed by an extra final step r*2^k mod d*2^k. This can
+help on CPUs with poor bit shifts or few registers.
+
+ The multiply by inverse can be done two limbs at a time. The
+calculation is basically the same, but the inverse is two limbs and the
+divisor treated as if padded with a low zero limb. This means more
+work, since the inverse will need a 2x2 multiply, but the four 1x1s to
+do that are independent and can therefore be done partly or wholly in
+parallel. Likewise for a 2x1 calculating q*d. The net effect is to
+process two limbs with roughly the same two multiplies worth of latency
+that one limb at a time gives. This extends to 3 or 4 limbs at a time,
+though the extra work to apply the inverse will almost certainly soon
+reach the limits of multiplier throughput.
+
+ A similar approach in reverse can be taken to process just half a
+limb at a time if the divisor is only a half limb. In this case the 1x1
+multiply for the inverse effectively becomes two (1/2)x1 for each limb,
+which can be a saving on CPUs with a fast half limb multiply, or in fact
+if the only multiply is a half limb, and especially if it's not
+pipelined.
+
+
+File: gmp.info, Node: Basecase Division, Next: Divide and Conquer Division, Prev: Single Limb Division, Up: Division Algorithms
+
+15.2.2 Basecase Division
+------------------------
+
+Basecase NxM division is like long division done by hand, but in base
+2^mp_bits_per_limb. See Knuth section 4.3.1 algorithm D, and
+'mpn/generic/sb_divrem_mn.c'.
+
+ Briefly stated, while the dividend remains larger than the divisor, a
+high quotient limb is formed and the Nx1 product q*d subtracted at the
+top end of the dividend. With a normalized divisor (most significant
+bit set), each quotient limb can be formed with a 2x1 division and a 1x1
+multiplication plus some subtractions. The 2x1 division is by the high
+limb of the divisor and is done either with a hardware divide or a
+multiply by inverse (the same as in *note Single Limb Division::)
+whichever is faster. Such a quotient is sometimes one too big,
+requiring an addback of the divisor, but that happens rarely.
+
+ With Q=N-M being the number of quotient limbs, this is an O(Q*M)
+algorithm and will run at a speed similar to a basecase QxM
+multiplication, differing in fact only in the extra multiply and divide
+for each of the Q quotient limbs.
+
+
+File: gmp.info, Node: Divide and Conquer Division, Next: Block-Wise Barrett Division, Prev: Basecase Division, Up: Division Algorithms
+
+15.2.3 Divide and Conquer Division
+----------------------------------
+
+For divisors larger than 'DC_DIV_QR_THRESHOLD', division is done by
+dividing. Or to be precise by a recursive divide and conquer algorithm
+based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
+(*note References::).
+
+ The algorithm consists essentially of recognising that a 2NxN
+division can be done with the basecase division algorithm (*note
+Basecase Division::), but using N/2 limbs as a base, not just a single
+limb. This way the multiplications that arise are (N/2)x(N/2) and can
+take advantage of Karatsuba and higher multiplication algorithms (*note
+Multiplication Algorithms::). The two "digits" of the quotient are
+formed by recursive Nx(N/2) divisions.
+
+ If the (N/2)x(N/2) multiplies are done with a basecase multiplication
+then the work is about the same as a basecase division, but with more
+function call overheads and with some subtractions separated from the
+multiplies. These overheads mean that it's only when N/2 is above
+'MUL_TOOM22_THRESHOLD' that divide and conquer is of use.
+
+ 'DC_DIV_QR_THRESHOLD' is based on the divisor size N, so it will be
+somewhere above twice 'MUL_TOOM22_THRESHOLD', but how much above depends
+on the CPU. An optimized 'mpn_mul_basecase' can lower
+'DC_DIV_QR_THRESHOLD' a little by offering a ready-made advantage over
+repeated 'mpn_submul_1' calls.
+
+ Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is the
+time for an NxN multiplication done with FFTs. The actual time is a sum
+over multiplications of the recursed sizes, as can be seen near the end
+of section 2.2 of Burnikel and Ziegler. For example, within the Toom-3
+range, divide and conquer is 2.63*M(N). With higher algorithms the M(N)
+term improves and the multiplier tends to log(N). In practice, at
+moderate to large sizes, a 2NxN division is about 2 to 4 times slower
+than an NxN multiplication.
+
+
+File: gmp.info, Node: Block-Wise Barrett Division, Next: Exact Division, Prev: Divide and Conquer Division, Up: Division Algorithms
+
+15.2.4 Block-Wise Barrett Division
+----------------------------------
+
+For the largest divisions, a block-wise Barrett division algorithm is
+used. Here, the divisor is inverted to a precision determined by the
+relative size of the dividend and divisor. Blocks of quotient limbs are
+then generated by multiplying blocks from the dividend by the inverse.
+
+ Our block-wise algorithm computes a smaller inverse than in the plain
+Barrett algorithm. For a 2n/n division, the inverse will be just
+ceil(n/2) limbs.
+
+
+File: gmp.info, Node: Exact Division, Next: Exact Remainder, Prev: Block-Wise Barrett Division, Up: Division Algorithms
+
+15.2.5 Exact Division
+---------------------
+
+A so-called exact division is when the dividend is known to be an exact
+multiple of the divisor. Jebelean's exact division algorithm uses this
+knowledge to make some significant optimizations (*note References::).
+
+ The idea can be illustrated in decimal for example with 368154
+divided by 543. Because the low digit of the dividend is 4, the low
+digit of the quotient must be 8. This is arrived at from 4*7 mod 10,
+using the fact 7 is the modular inverse of 3 (the low digit of the
+divisor), since 3*7 == 1 mod 10. So 8*543=4344 can be subtracted from
+the dividend leaving 363810. Notice the low digit has become zero.
+
+ The procedure is repeated at the second digit, with the next quotient
+digit 7 (7 == 1*7 mod 10), subtracting 7*543=3801, leaving 325800. And
+finally at the third digit with quotient digit 6 (8*7 mod 10),
+subtracting 6*543=3258 leaving 0. So the quotient is 678.
+
+ Notice however that the multiplies and subtractions don't need to
+extend past the low three digits of the dividend, since that's enough to
+determine the three quotient digits. For the last quotient digit no
+subtraction is needed at all. On a 2NxN division like this one, only
+about half the work of a normal basecase division is necessary.
+
+ For an NxM exact division producing Q=N-M quotient limbs, the saving
+over a normal basecase division is in two parts. Firstly, each of the Q
+quotient limbs needs only one multiply, not a 2x1 divide and multiply.
+Secondly, the crossproducts are reduced when Q>M to Q*M-M*(M+1)/2, or
+when Q<=M to Q*(Q-1)/2. Notice the savings are complementary. If Q is
+big then many divisions are saved, or if Q is small then the
+crossproducts reduce to a small number.
+
+ The modular inverse used is calculated efficiently by 'binvert_limb'
+in 'gmp-impl.h'. This does four multiplies for a 32-bit limb, or six
+for a 64-bit limb. 'tune/modlinv.c' has some alternate implementations
+that might suit processors better at bit twiddling than multiplying.
+
+ The sub-quadratic exact division described by Jebelean in "Exact
+Division with Karatsuba Complexity" is not currently implemented. It
+uses a rearrangement similar to the divide and conquer for normal
+division (*note Divide and Conquer Division::), but operating from low
+to high. A further possibility not currently implemented is
+"Bidirectional Exact Integer Division" by Krandick and Jebelean which
+forms quotient limbs from both the high and low ends of the dividend,
+and can halve once more the number of crossproducts needed in a 2NxN
+division.
+
+ A special case exact division by 3 exists in 'mpn_divexact_by3',
+supporting Toom-3 multiplication and 'mpq' canonicalizations. It forms
+quotient digits with a multiply by the modular inverse of 3 (which is
+'0xAA..AAB') and uses two comparisons to determine a borrow for the next
+limb. The multiplications don't need to be on the dependent chain, as
+long as the effect of the borrows is applied, which can help chips with
+pipelined multipliers.
+