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This is mpir.info, produced by makeinfo version 4.11 from mpir.texi.
This manual describes how to install and use MPIR, the Multiple
Precision Integers and Rationals library, version 1.3.0.
Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
Copyright 2008 William Hart
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.2 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
* mpir: (mpir). MPIR Multiple Precision Integers and Rationals Library.
END-INFO-DIR-ENTRY

File: mpir.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
MPIR
****
This manual describes how to install and use MPIR, the Multiple
Precision Integers and Rationals library, version 1.3.0.
Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
2001, 2002, 2003, 2004, 2005, 2006 Free Software Foundation, Inc.
Copyright 2008 William Hart
Permission is granted to copy, distribute and/or modify this
document under the terms of the GNU Free Documentation License, Version
1.2 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:: MPIR Copying Conditions (LGPL).
* Introduction to MPIR:: Brief introduction to MPIR.
* Installing MPIR:: How to configure and compile the MPIR library.
* MPIR Basics:: What every MPIR 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 MPIR types.
* Custom Allocation:: How to customize the internal allocation.
* Language Bindings:: Using MPIR 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: mpir.info, Node: Copying, Next: Introduction to MPIR, Prev: Top, Up: Top
MPIR 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 MPIR 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 MPIR library. If it is
modified by someone else and passed on, we want their recipients to
know that what they have is not what we distributed, so that any
problems introduced by others will not reflect on our reputation.
The precise conditions of the license for the MPIR library are found
in the Lesser General Public License version 2.1 that accompanies the
source code, see `COPYING.LIB'. Certain demonstration programs are
provided under the terms of the plain General Public License version 2,
see `COPYING'.

File: mpir.info, Node: Introduction to MPIR, Next: Installing MPIR, Prev: Copying, Up: Top
1 Introduction to MPIR
**********************
MPIR 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. MPIR 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 MPIR is achieved by using fullwords as the basic
arithmetic type, by using sophisticated algorithms, by including
carefully optimized assembly code for the most common inner loops for
many different CPUs, and by a general emphasis on speed (as opposed to
simplicity or elegance).
There is assembly code for these CPUs: ARM, DEC Alpha 21064, 21164,
and 21264, AMD 29000, AMD K6, K6-2, Athlon, K8 and K10, Hitachi SuperH
and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III,
Pentium 4, generic x86, Intel IA-64, i960, Core 2 Motorola MC68000,
MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National
NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic
SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also
for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP.
For up-to-date information on, and latest version of, MPIR, please see
the MPIR web pages at
`http://www.mpir.org/'
There are a number of public mailing lists of interest. The
development list is
`http://groups.google.com/group/mpir-devel/'.
The proper place for bug reports is
`http://groups.google.com/group/mpir-devel'. See *note Reporting
Bugs:: for information about reporting bugs.
1.1 How to use this Manual
==========================
Everyone should read *note MPIR Basics::. If you need to install the
library yourself, then read *note Installing MPIR::. 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: mpir.info, Node: Installing MPIR, Next: MPIR Basics, Prev: Introduction to MPIR, Up: Top
2 Installing MPIR
*****************
MPIR 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
Important note: by default MPIR produces libraries named libmpir,
etc., and the header file mpir.h. If you wish to have MPIR to build a
library named libgmp as well, etc., and a gmp.h header file, so that
you can use mpir with programs designed to only work with GMP, then use
the `--enable-gmpcompat' option when invoking configure:
./configure --enable-gmpcompat
If you experience problems, please report them to
`http://groups.google.com/group/mpir-devel'. 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: mpir.info, Node: Build Options, Next: ABI and ISA, Prev: Installing MPIR, Up: Installing MPIR
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 MPIR source
directory. For example
cd /my/build/dir
/my/sources/mpir-1.3.0/configure
Not all `make' programs have the necessary features (`VPATH') to
support this. In particular, SunOS and Solaris `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 MPIR
to install under a particular tree. The default is `/usr/local'.
`--exec-prefix' can be used to direct architecture-dependent files
like `libmpir.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 `mpir.h' and
`mp.h' are architecture-dependent since they encode certain
aspects of `libmpir', so it will be necessary to ensure both
`$prefix/include' and `$exec_prefix/include' are available to the
compiler.
`--enable-gmpcompat'
By default make builds libmpir library files (and libmpirxx if C++
headers are requested) and the mpir.h header file. This option
allows you to specify that you want additional libraries created
called libgmp (and libgmpxx), etc., for libraries and gmp.h for
compatibility with GMP.
`--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 `libmpir', `./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 MPIR. 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 MPIR are only interested in
the `--host' part, being where they'll run.
CPU types
In general, if you want a library that runs as fast as possible,
you should configure MPIR 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 MPIR for the exact machine type
you intend to run it on.
The following CPUs have specific support. See `configure.in' for
details of what code and compiler options they select.
* Alpha: alpha, alphaev5, alphaev56, alphapca56, alphapca57,
alphaev6, alphaev67, alphaev68 alphaev7
* Cray: c90, j90, t90, sv1
* HPPA: hppa1.0, hppa1.1, hppa2.0, hppa2.0n, hppa2.0w, hppa64
* IA-64: ia64, itanium, itanium2
* MIPS: mips, mips3, mips64
* Motorola: m68k, m68000, m68010, m68020, m68030, m68040,
m68060, m68302, m68360, m88k, m88110
* POWER: power, power1, power2, power2sc
* PowerPC: powerpc, powerpc64, powerpc401, powerpc403,
powerpc405, powerpc505, powerpc601, powerpc602, powerpc603,
powerpc603e, powerpc604, powerpc604e, powerpc620, powerpc630,
powerpc740, powerpc7400, powerpc7450, powerpc750, powerpc801,
powerpc821, powerpc823, powerpc860, powerpc970
* SPARC: sparc, sparcv8, microsparc, supersparc, sparcv9,
ultrasparc, ultrasparc2, ultrasparc2i, ultrasparc3, sparc64
* x86 family: pentium, pentiummmx, pentiumpro, pentium2,
pentium3, pentium4, netburst, netburstlahf, prescott, core,
core2, penryn, nehalem, atom, k5, k6, k62, k63, k7, k8, k10
viac3, viac32
* Other: a29k, arm, clipper, i960, ns32k, pyramid, sh, sh2, vax,
z8k
CPUs not listed will use generic C code.
Generic C Build
If some of the assembly code causes problems, or if otherwise
desired, the generic C code can be selected with CPU `none'. For
example,
./configure --host=none-unknown-freebsd3.5
Note that this will run quite slowly, but it should be portable
and should at least make it possible to get something running if
all else fails.
Fat binary, `--enable-fat'
Using `--enable-fat' selects a "fat binary" build on x86 or x86_64
systems, where optimized low level subroutines are chosen at
runtime according to the CPU detected. This means more code, but
gives reasonable performance from a single binary for all x86
chips, or simililarly for all x86_64 chips. (This option might
become available for more architectures in the future.)
`ABI'
On some systems MPIR supports multiple ABIs (application binary
interfaces), meaning data type sizes and calling conventions. By
default MPIR 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 MPIR 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 MPIR can't determine the ABI just from the
flags and won't be able to select the correct assembler code.
If just `CC' is selected then normal default `CFLAGS' for that
compiler will be used (if MPIR recognises it). For example
`CC=gcc' can be used to force the use of GCC, with default flags
(and default ABI).
`CPPFLAGS'
Any flags like `-D' defines or `-I' includes required by the
preprocessor should be set in `CPPFLAGS' rather than `CFLAGS'.
Compiling is done with both `CPPFLAGS' and `CFLAGS', but
preprocessing uses just `CPPFLAGS'. This distinction is because
most preprocessors won't accept all the flags the compiler does.
Preprocessing is done separately in some configure tests, and in
the `ansi2knr' support for K&R compilers.
`CC_FOR_BUILD'
Some build-time programs are compiled and run to generate
host-specific data tables. `CC_FOR_BUILD' is the compiler used
for this. It doesn't need to be in any particular ABI or mode, it
merely needs to generate executables that can run. The default is
to try the selected `CC' and some likely candidates such as `cc'
and `gcc', looking for something that works.
No flags are used with `CC_FOR_BUILD' because a simple invocation
like `cc foo.c' should be enough. If some particular options are
required they can be included as for instance `CC_FOR_BUILD="cc
-whatever"'.
C++ Support, `--enable-cxx'
C++ support in MPIR 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
`libmpirxx.la' and header file `mpirxx.h' (*note Headers and
Libraries::).
A separate `libmpirxx.la' has been adopted rather than having C++
objects within `libmpir.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.
`libmpirxx.la' will use certain internals from `libmpir.la' and can
only be expected to work with `libmpir.la' from the same MPIR
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 `libmpirxx.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>'
MPIR allocates temporary workspace using one of the following
three methods, which can be selected with for instance
`--enable-alloca=malloc-reentrant'.
* `alloca' - C library or compiler builtin.
* `malloc-reentrant' - the heap, in a re-entrant fashion.
* `malloc-notreentrant' - the heap, with global variables.
For convenience, the following choices are also available.
`--disable-alloca' is the same as `no'.
* `yes' - a synonym for `alloca'.
* `no' - a synonym for `malloc-reentrant'.
* `reentrant' - `alloca' if available, otherwise
`malloc-reentrant'. This is the default.
* `notreentrant' - `alloca' if available, otherwise
`malloc-notreentrant'.
`alloca' is reentrant and fast, and is recommended. It actually
allocates just small blocks on the stack; larger ones use
malloc-reentrant.
`malloc-reentrant' is, as the name suggests, reentrant and thread
safe, but `malloc-notreentrant' is faster and should be used if
reentrancy is not required.
The two malloc methods in fact use the memory allocation functions
selected by `mp_set_memory_functions', these being `malloc' and
friends by default. *Note Custom Allocation::.
An additional choice `--enable-alloca=debug' is available, to help
when debugging memory related problems (*note Debugging::).
FFT Multiplication, `--disable-fft'
By default multiplications are done using Karatsuba, 3-way Toom,
and Fermat FFT. The FFT is only used on large to very large
operands and can be disabled to save code size if desired.
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 assembler 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/mpir.texi',
in Texinfo format, see *note Texinfo: (texinfo)Top.
Info format `doc/mpir.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: mpir.info, Node: ABI and ISA, Next: Notes for Package Builds, Prev: Build Options, Up: Installing MPIR
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.
MPIR supports some CPUs like this in both ABIs. In fact within MPIR
`ABI' means a combination of chip ABI, plus how MPIR chooses to use it.
For example in some 32-bit ABIs, MPIR may support a limb as either a
32-bit `long' or a 64-bit `long long'.
By default MPIR 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 MPIR 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 `mpir.h'. This is convenient
for applications, but it does mean that `mpir.h' will vary, and can't
be just copied around. `mpir.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 MPIR,
and it might be as simple as configuring with a special `libdir', or it
might require more than that. Note that builds for different ABIs need
to done separately, with a fresh `./configure' and `make' each.
AMD64 (`x86_64')
On AMD64 systems supporting both 32-bit and 64-bit modes for
applications, the following ABI choices are available.
`ABI=64'
The 64-bit ABI uses 64-bit limbs and pointers and makes full
use of the chip architecture. This is the default.
Applications will usually not need special compiler flags,
but for reference the option is
gcc -m64
`ABI=32'
The 32-bit ABI is the usual i386 conventions. This will be
slower, and is not recommended except for inter-operating
with other code not yet 64-bit capable. Applications must be
compiled with
gcc -m32
(In GCC 2.95 and earlier there's no `-m32' option, it's the
only mode.)
HPPA 2.0 (`hppa2.0*', `hppa64')
`ABI=2.0w'
The 2.0w ABI uses 64-bit limbs and pointers and is available
on HP-UX 11 or up. Applications must be compiled with
gcc [built for 2.0w]
cc +DD64
`ABI=2.0n'
The 2.0n ABI means the 32-bit HPPA 1.0 ABI and all its normal
calling conventions, but with 64-bit instructions permitted
within functions. MPIR 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 MPIR assembler 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. MPIR 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. MPIR performance is the same
in both.
`ABI=32'
In the 32-bit ABI, pointers, `int's and `long's are 32 bits
and MPIR 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 MPIR
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 MPIR 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. MPIR 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')
`ABI=aix64'
The AIX 64 ABI uses 64-bit limbs and pointers and is the
default on PowerPC 64 `*-*-aix*' systems. Applications must
be compiled with
gcc -maix64
xlc -q64
`ABI=mode32'
The `mode32' ABI uses a 64-bit `long long' limb but with the
chip still in 32-bit mode and using 32-bit calling
conventions. This is the default on PowerPC 64 `*-*-darwin*'
systems. No special compiler options are needed for
applications.
`ABI=32'
This is the basic 32-bit PowerPC ABI, with a 32-bit limb. No
special compiler options are needed for applications.
MPIR speed is greatest in `aix64' and `mode32'. In `ABI=32' only
the 32-bit ISA is used and this doesn't make full use of a 64-bit
chip. On a suitable system we could perhaps use more of the ISA,
but there are no plans to do so.
Sparc V9 (`sparc64', `sparcv9', `ultrasparc*')
`ABI=64'
The 64-bit V9 ABI is available on the various BSD sparc64
ports, recent versions of Sparc64 GNU/Linux, and Solaris 2.7
and up (when the kernel is in 64-bit mode). GCC 3.2 or
higher, or Sun `cc' is required. On GNU/Linux, depending on
the default `gcc' mode, applications must be compiled with
gcc -m64
On Solaris applications must be compiled with
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9
cc -xarch=v9
On the BSD sparc64 systems no special options are required,
since 64-bits is the only ABI available.
`ABI=32'
For the basic 32-bit ABI, MPIR 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
MPIR 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: mpir.info, Node: Notes for Package Builds, Next: Notes for Particular Systems, Prev: ABI and ISA, Up: Installing MPIR
2.3 Notes for Package Builds
============================
MPIR 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 and MPIR 1 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 MPIR.
An auxiliary mechanism may also be needed to express that
`libmpirxx.la' (from `--enable-cxx', *note Build Options::) requires
`libmpir.la' from the same MPIR 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 MPIR 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 `mpir.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 `mpir.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/mpir.h'
which tests preprocessor symbols and chooses the correct actual
`mpir.h'.

File: mpir.info, Node: Notes for Particular Systems, Next: Known Build Problems, Prev: Notes for Package Builds, Up: Installing MPIR
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 `libmpir.a' a symlink to `libmpir.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. MPIR `./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'). MPIR
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 MPIR 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 MPIR.
MS-DOS and MS Windows
On an MS-DOS system DJGPP can be used to build MPIR, and on an MS
Windows system Cygwin, DJGPP and MINGW can all be used. All three
are excellent ports of GCC and the various GNU tools.
`http://www.cygwin.com/'
`http://www.delorie.com/djgpp/'
`http://www.mingw.org/'
In addition, project files for MSVC are provided, allowing MPIR to
build on Microsoft's compiler. This support is provided by Brian
Gladman.
Microsoft also publishes an Interix "Services for Unix" which can
be used to build MPIR on Windows (with a normal `./configure'),
but it's not free software.
MS Windows DLLs
On systems `*-*-cygwin*', `*-*-mingw*' and `*-*-pw32*' by default
MPIR 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 `mpir.h' must be different.
A MINGW DLL build of MPIR 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 `libmpir' and `libmpirxx'.
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 MPIR 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 MPIR string functions and confine I/O to the
application.
Motorola 68k CPU Types
`m68k' is taken to mean 68000. `m68020' or higher will give a
performance boost on applicable CPUs. `m68360' can be used for
CPU32 series chips. `m68302' can be used for "Dragonball" series
chips, though this is merely a synonym for `m68000'.
OpenBSD 2.6
`m4' in this release of OpenBSD has a bug in `eval' that makes it
unsuitable for `.asm' file processing. `./configure' will detect
the problem and either abort or choose another m4 in the `PATH'.
The bug is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
Power CPU Types
In MPIR, 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
MPIR has no assembler code support for using just the common
instruction subset. To get executables that run on both, the
current suggestion is to use the generic C code (CPU `none'),
possibly with appropriate compiler options (like `-mcpu=common' for
`gcc'). CPU `rs6000' (which is not a CPU but a family of
workstations) is accepted by `config.sub', but is currently
equivalent to `none'.
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 MPIR assembler 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 MPIR with CPU `none' to avoid the assembler code.
Solaris 10 First Release on SPARC
MPIR fails to build with Solaris 10 first release. Patch 123647-01
for SPARC, released by Sun in August 2006 fixes this problem.
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: mpir.info, Node: Known Build Problems, Next: Performance optimization, Prev: Notes for Particular Systems, Up: Installing MPIR
2.5 Known Build Problems
========================
You might find more up-to-date information at `http://www.mpir.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 `libmpir.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 `libmpir.a'.
Binutils 2.12 or higher can be used successfully.
The shared libraries `libmpir.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
`libmpir.la'. The symptom is a "syntax error" on the following
line of the top-level `Makefile'.
libmpir.la: $(libmpir_la_OBJECTS) $(libmpir_la_DEPENDENCIES)
Either use GNU Make, or as a workaround remove
`$(libmpir_la_DEPENDENCIES)' from that line (which will make the
initial build work, but if any recompiling is done `libmpir.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 MPIR, 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
MPIR 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 `libmpir.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 MPIR 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: mpir.info, Node: Performance optimization, Prev: Known Build Problems, Up: Installing MPIR
2.6 Performance optimization
============================
For optimal performance, build MPIR 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 MPIR uses assembly language for
the most critical operation.
In particular for long-running MPIR 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 MPIR for extremely large numbers, you may want to
run `tuneup' with a large NNN value.

File: mpir.info, Node: MPIR Basics, Next: Reporting Bugs, Prev: Installing MPIR, Up: Top
3 MPIR 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 MPIR.*
* 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: mpir.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPIR Basics, Up: MPIR Basics
3.1 Headers and Libraries
=========================
All declarations needed to use MPIR are collected in the include file
`mpir.h'. It is designed to work with both C and C++ compilers.
#include <mpir.h>
Note however that prototypes for MPIR functions with `FILE *'
parameters are only provided if `<stdio.h>' is included too.
#include <stdio.h>
#include <mpir.h>
Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes
with `va_list' parameters, such as `gmp_vprintf'. And `<obstack.h>'
for prototypes with `struct obstack' parameters, such as
`gmp_obstack_printf', when available.
All programs using MPIR must link against the `libmpir' library. On
a typical Unix-like system this can be done with `-lmpir' respectively,
for example
gcc myprogram.c -lmpir
MPIR C++ functions are in a separate `libmpirxx' library. This is
built and installed if C++ support has been enabled (*note Build
Options::). For example,
g++ mycxxprog.cc -lmpirxx -lmpir
MPIR is built using Libtool and an application can use that to link
if desired, *note GNU Libtool: (libtool)Top.
If MPIR 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: mpir.info, Node: Nomenclature and Types, Next: Function Classes, Prev: Headers and Libraries, Up: MPIR Basics
3.2 Nomenclature and Types
==========================
In this manual, "integer" usually means a multiple precision integer, as
defined by the MPIR 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 are represented in the C type `mp_size_t'. Currently
this is normally a `long', but on some systems it's an `int' for
efficiency.
"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 `unsigned long' is used for bit counts and ranges,
and `size_t' is used for byte or character counts.

File: mpir.info, Node: Function Classes, Next: Variable Conventions, Prev: Nomenclature and Types, Up: MPIR Basics
3.3 Function Classes
====================
There are five classes of functions in the MPIR library:
1. Functions for signed integer arithmetic, with names beginning with
`mpz_'. The associated type is `mpz_t'. There are about 150
functions in this class. (*note Integer Functions::)
2. Functions for rational number arithmetic, with names beginning with
`mpq_'. The associated type is `mpq_t'. There are about 40
functions in this class, but the integer functions can be used for
arithmetic on the numerator and denominator separately. (*note
Rational Number Functions::)
3. Functions for floating-point arithmetic, with names beginning with
`mpf_'. The associated type is `mpf_t'. There are about 60
functions is this class. (*note Floating-point Functions::)
4. Fast low-level functions that operate on natural numbers. These
are used by the functions in the preceding groups, and you can
also call them directly from very time-critical user programs.
These functions' names begin with `mpn_'. The associated type is
array of `mp_limb_t'. There are about 30 (hard-to-use) functions
in this class. (*note Low-level Functions::)
5. Miscellaneous functions. Functions for setting up custom
allocation and functions for generating random numbers. (*note
Custom Allocation::, and *note Random Number Functions::)

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

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

File: mpir.info, Node: Memory Management, Next: Reentrancy, Prev: Parameter Conventions, Up: MPIR Basics
3.6 Memory Management
=====================
The MPIR types like `mpz_t' are small, containing only a couple of
sizes, and pointers to allocated data. Once a variable is initialized,
MPIR 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: mpir.info, Node: Reentrancy, Next: Useful Macros and Constants, Prev: Memory Management, Up: MPIR Basics
3.7 Reentrancy
==============
MPIR 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 MPIR 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.
* `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 MPIR will not be reentrant either.
* If the standard I/O functions such as `fwrite' are not reentrant
then the MPIR I/O functions using them will not be reentrant
either.
* It's safe for two threads to read from the same MPIR variable
simultaneously, but it's not safe for one to read while the
another might be writing, nor for two threads to write
simultaneously. It's not safe for two threads to generate a
random number from the same `gmp_randstate_t' simultaneously,
since this involves an update of that variable.

File: mpir.info, Node: Useful Macros and Constants, Next: Compatibility with older versions, Prev: Reentrancy, Up: MPIR 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. These numbers represent the version of GMP fully
supported by this version of MPIR.
-- Macro: __MPIR_VERSION
-- Macro: __MPIR_VERSION_MINOR
-- Macro: __MPIR_VERSION_PATCHLEVEL
The major and minor MPIR version, and patch level, respectively,
as integers. For MPIR i.j, these numbers will be i, j, and 0,
respectively. For MPIR i.j.k, these numbers will be i, j, and k,
respectively.
-- Global Constant: const char * const gmp_version
The GNU MP version number, as a null-terminated string, in the
form "i.j" or "i.j.k".
-- Global Constant: const char * const mpir_version
The MPIR version number, as a null-terminated string, in the form
"i.j" or "i.j.k". This release is "1.3.0".

File: mpir.info, Node: Compatibility with older versions, Next: Demonstration Programs, Prev: Useful Macros and Constants, Up: MPIR Basics
3.9 Compatibility with older versions
=====================================
This version of MPIR is upwardly binary compatible with all GMP 4.x and
3.x versions, and upwardly compatible at the source level with all 2.x
versions, with the following exceptions.
* `mpn_gcd' had its source arguments swapped as of GMP 3.0, for
consistency with other `mpn' functions.
* `mpf_get_prec' counted precision slightly differently in GMP 3.0
and 3.0.1, but in 3.1 reverted to the 2.x style.
There are a number of compatibility issues between GMP 1 and GMP 2
that of course also apply when porting applications from GMP 1 to GMP 4
and MPIR 1. Please see the GMP 2 manual for details.

File: mpir.info, Node: Demonstration Programs, Next: Efficiency, Prev: Compatibility with older versions, Up: MPIR Basics
3.10 Demonstration programs
===========================
The `demos' subdirectory has some sample programs using MPIR. 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 MPIR.
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 MPIR 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 MPIR (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: mpir.info, Node: Efficiency, Next: Debugging, Prev: Demonstration Programs, Up: MPIR 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 MPIR 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 `libmpir.a'
should be used for maximum speed, since the PIC code in the shared
`libmpir.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 MPIR 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: mpir.info, Node: Debugging, Next: Profiling, Prev: Efficiency, Up: MPIR 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 MPIR 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 MPIR is heap
corruption. Failing to `init' MPIR variables will have
unpredictable effects, and corruption arising elsewhere in a
program may well affect MPIR. Initializing MPIR variables more
than once or failing to clear them will cause memory leaks.
In all such cases a `malloc' debugger is recommended. On a GNU or
BSD system the standard C library `malloc' has some diagnostic
facilities, see *note Allocation Debugging: (libc)Allocation
Debugging, or `man 3 malloc'. Other possibilities, in no
particular order, include
`http://www.inf.ethz.ch/personal/biere/projects/ccmalloc/'
`http://dmalloc.com/'
`http://www.perens.com/FreeSoftware/' (electric fence)
`http://packages.debian.org/fda'
`http://www.gnupdate.org/components/leakbug/'
`http://people.redhat.com/~otaylor/memprof/'
`http://www.cbmamiga.demon.co.uk/mpatrol/'
The MPIR 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 MPIR development, but might find other uses.
Stack Backtraces
On some systems the compiler options MPIR 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 MPIR 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 MPIR.
Source File Paths
MPIR 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-1.3.0/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 assembler code, so CPU `none' should be
used for maximum checking.
Temporary Memory Checking
The build option `--enable-alloca=debug' arranges that each block
of temporary memory in MPIR 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 MPIR 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, an MPIR build for maximum debuggability
would be
./configure --disable-shared --enable-assert \
--enable-alloca=debug --host=none CFLAGS=-g
For C++, add `--enable-cxx CXXFLAGS=-g'.
Checker
The GCC checker (`http://savannah.gnu.org/projects/checker/') can
be used with MPIR. It contains a stub library which means MPIR
applications compiled with checker can use a normal MPIR build.
A build of MPIR with checking within MPIR itself can be made.
This will run very very slowly. On GNU/Linux for example,
./configure --host=none-pc-linux-gnu CC=checkergcc
`--host=none' must be used, since the MPIR assembler code doesn't
support the checking scheme. The MPIR C++ features cannot be
used, since current versions of checker (0.9.9.1) don't yet
support the standard C++ library.
Valgrind
The valgrind program (`http://valgrind.org/') is a memory checker
for x86s. It translates and emulates machine instructions to do
strong checks for uninitialized data (at the level of individual
bits), memory accesses through bad pointers, and memory leaks.
Recent versions of Valgrind are getting support for MMX and
SSE/SSE2 instructions, for past versions MPIR will need to be
configured not to use those, ie. for an x86 without them (for
instance plain `i486').
Other Problems
Any suspected bug in MPIR itself should be isolated to make sure
it's not an application problem, see *note Reporting Bugs::.

File: mpir.info, Node: Profiling, Next: Autoconf, Prev: Debugging, Up: MPIR 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 MPIR 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 MPIR
assembler 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 assembler code has support for this option, but on other
processors the assembler 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' (*note GNU gprof: (gprof)Top.),
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 assembler code has support for this option, but on other
processors the assembler routines will be as if compiled without
`-pg' and therefore not be included in the call counts.
On x86 and m68k systems `-pg' and `-fomit-frame-pointer' are
incompatible, so the latter is omitted from the default flags in
that case, which might result in poorer code generation.
Incidentally, it should be possible to use the `gprof' program
with a plain `--enable-profiling=prof' build. But in that case
only the `gprof -p' flat profile and call counts can be expected
to be valid, not the `gprof -q' call graph.
`--enable-profiling=instrument'
Build with the GCC option `-finstrument-functions' added to the
`CFLAGS' (*note Options for Code Generation: (gcc)Code Gen
Options.).
This inserts special instrumenting calls at the start and end of
each function, allowing exact timing and full call graph
construction.
This instrumenting is not normally a standard system feature and
will require support from an external library, such as
`http://sourceforge.net/projects/fnccheck/'
This should be included in `LIBS' during the MPIR 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 assembler code supports this option, but on other
processors the assembler routines will be as if compiled without
`-finstrument-functions' meaning time spent in them will
effectively be attributed to their caller.

File: mpir.info, Node: Autoconf, Next: Emacs, Prev: Profiling, Up: MPIR Basics
3.14 Autoconf
=============
Autoconf based applications can easily check whether MPIR is installed.
The only thing to be noted is that GMP/MPIR library symbols from
version 3 of GMP and version 1 of MPIR onwards have prefixes like
`__gmpz'. The following therefore would be a simple test,
AC_CHECK_LIB(mpir, __gmpz_init)
This just uses the default `AC_CHECK_LIB' actions for found or not
found, but an application that must have MPIR would want to generate an
error if not found. For example,
AC_CHECK_LIB(mpir, __gmpz_init, ,
[AC_MSG_ERROR([MPIR not found, see http://www.mpir.org/])])
If functions added in some particular version of GMP/MPIR 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(mpir, __gmpz_mul_si, ,
[AC_MSG_ERROR(
[GMP/MPIR not found, or not GMP 3.1 or up or MPIR 1.0 or up, see http://www.mpir.org/])])
An alternative would be to test the version number in `mpir.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 MPIR 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 and MPIR 1.0 and 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 <mpir.h>])

File: mpir.info, Node: Emacs, Prev: Autoconf, Up: MPIR 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 MPIR 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: mpir.info, Node: Reporting Bugs, Next: Integer Functions, Prev: MPIR Basics, Up: Top
4 Reporting Bugs
****************
If you think you have found a bug in the MPIR library, please
investigate it and report it. We have made this library available to
you, and it is not too much to ask you to report the bugs you find.
Before you report a bug, check it's not already addressed in *note
Known Build Problems::, or perhaps *note Notes for Particular
Systems::. You may also want to check `http://www.mpir.org/' for
patches for this release.
Please include the following in any report,
* The MPIR version number, and if pre-packaged or patched then say
so.
* A test program that makes it possible for us to reproduce the bug.
Include instructions on how to run the program.
* A description of what is wrong. If the results are incorrect, in
what way. If you get a crash, say so.
* If you get a crash, include a stack backtrace from the debugger if
it's informative (`where' in `gdb', or `$C' in `adb').
* Please do not send core dumps, executables or `strace's.
* The configuration options you used when building MPIR, if any.
* The name of the compiler and its version. For `gcc', get the
version with `gcc -v', otherwise perhaps `what `which cc`', or
similar.
* The output from running `uname -a'.
* The output from running `./config.guess', and from running
`./configfsf.guess' (might be the same).
* If the bug is related to `configure', then the 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 MPIR 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: `http://groups.google.com/group/mpir-devel'.
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: mpir.info, Node: Integer Functions, Next: Rational Number Functions, Prev: Reporting Bugs, Up: Top
5 Integer Functions
*******************
This chapter describes the MPIR functions for performing integer
arithmetic. These functions start with the prefix `mpz_'.
MPIR 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: mpir.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 INTEGER)
Initialize INTEGER, and set its value to 0.
-- Function: void mpz_init2 (mpz_t INTEGER, unsigned long N)
Initialize INTEGER, with space for N bits, and set its value to 0.
N is only the initial space, INTEGER will grow automatically in
the normal way, if necessary, for subsequent values stored.
`mpz_init2' makes it possible to avoid such reallocations if a
maximum size is known in advance.
-- Function: void mpz_clear (mpz_t INTEGER)
Free the space occupied by INTEGER. Call this function for all
`mpz_t' variables when you are done with them.
-- Function: void mpz_realloc2 (mpz_t INTEGER, unsigned long N)
Change the space allocated for INTEGER to N bits. The value in
INTEGER is preserved if it fits, or is set to 0 if not.
This function can be used to increase the space for a variable in
order to avoid repeated automatic reallocations, or to decrease it
to give memory back to the heap.

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

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

File: mpir.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 MPIR integers to
standard C types. Functions for converting _to_ MPIR integers are
described in *note Assigning Integers:: and *note I/O of Integers::.
-- Function: unsigned long int mpz_get_ui (mpz_t OP)
Return the value of OP as an `unsigned long'.
If OP is too big to fit an `unsigned long' then just the least
significant bits that do fit are returned. The sign of OP is
ignored, only the absolute value is used.
-- Function: signed long int mpz_get_si (mpz_t OP)
If OP fits into a `signed long int' return the value of OP.
Otherwise return the least significant part of OP, with the same
sign as OP.
If OP is too big to fit in a `signed long int', the returned
result is probably not very useful. To find out if the value will
fit, use the function `mpz_fits_slong_p'.
-- Function: double mpz_get_d (mpz_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero).
If the exponent from the conversion is too big, the result is
system dependent. An infinity is returned where available. A
hardware overflow trap may or may not occur.
-- Function: double mpz_get_d_2exp (signed long int *EXP, mpz_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero), and returning the exponent separately.
The return value is in the range 0.5<=abs(D)<1 and the exponent is
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
is zero, the return is 0.0 and 0 is stored to `*EXP'.
This is similar to the standard C `frexp' function (*note
Normalization Functions: (libc)Normalization Functions.).
-- Function: char * mpz_get_str (char *STR, int BASE, mpz_t OP)
Convert OP to a string of digits in base BASE. The base may vary
from 2 to 36.
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: mpir.info, Node: Integer Arithmetic, Next: Integer Division, Prev: Converting Integers, Up: Integer Functions
5.5 Arithmetic Functions
========================
-- Function: void mpz_add (mpz_t ROP, mpz_t OP1, mpz_t OP2)
-- Function: void mpz_add_ui (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
Set ROP to OP1 + OP2.
-- Function: void mpz_sub (mpz_t ROP, mpz_t OP1, mpz_t OP2)
-- Function: void mpz_sub_ui (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
-- Function: void mpz_ui_sub (mpz_t ROP, unsigned long int OP1, mpz_t
OP2)
Set ROP to OP1 - OP2.
-- Function: void mpz_mul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
-- Function: void mpz_mul_si (mpz_t ROP, mpz_t OP1, long int OP2)
-- Function: void mpz_mul_ui (mpz_t ROP, mpz_t OP1, unsigned long int
OP2)
Set ROP to OP1 times OP2.
-- Function: void mpz_addmul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
-- Function: void mpz_addmul_ui (mpz_t ROP, mpz_t OP1, unsigned long
int OP2)
Set ROP to ROP + OP1 times OP2.
-- Function: void mpz_submul (mpz_t ROP, mpz_t OP1, mpz_t OP2)
-- Function: void mpz_submul_ui (mpz_t ROP, mpz_t OP1, unsigned long
int OP2)
Set ROP to ROP - OP1 times OP2.
-- Function: void mpz_mul_2exp (mpz_t ROP, mpz_t OP1, unsigned long
int OP2)
Set ROP to OP1 times 2 raised to OP2. This operation can also be
defined as a left shift by OP2 bits.
-- Function: void mpz_neg (mpz_t ROP, mpz_t OP)
Set ROP to -OP.
-- Function: void mpz_abs (mpz_t ROP, mpz_t OP)
Set ROP to the absolute value of OP.

File: mpir.info, Node: Integer Division, Next: Integer Exponentiation, Prev: Integer Arithmetic, Up: Integer Functions
5.6 Division Functions
======================
Division is undefined if the divisor is zero. Passing a zero divisor
to the division or modulo functions (including the modular powering
functions `mpz_powm' and `mpz_powm_ui'), will cause an intentional
division by zero. This lets a program handle arithmetic exceptions in
these functions the same way as for normal C `int' arithmetic.
-- Function: void mpz_cdiv_q (mpz_t Q, mpz_t N, mpz_t D)
-- Function: void mpz_cdiv_r (mpz_t R, mpz_t N, mpz_t D)
-- Function: void mpz_cdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
-- Function: unsigned long int mpz_cdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_cdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_cdiv_qr_ui (mpz_t Q, mpz_t R,
mpz_t N, unsigned long int D)
-- Function: unsigned long int mpz_cdiv_ui (mpz_t N,
unsigned long int D)
-- Function: void mpz_cdiv_q_2exp (mpz_t Q, mpz_t N,
unsigned long int B)
-- Function: void mpz_cdiv_r_2exp (mpz_t R, mpz_t N,
unsigned long int B)
-- Function: void mpz_fdiv_q (mpz_t Q, mpz_t N, mpz_t D)
-- Function: void mpz_fdiv_r (mpz_t R, mpz_t N, mpz_t D)
-- Function: void mpz_fdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
-- Function: unsigned long int mpz_fdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_fdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_fdiv_qr_ui (mpz_t Q, mpz_t R,
mpz_t N, unsigned long int D)
-- Function: unsigned long int mpz_fdiv_ui (mpz_t N,
unsigned long int D)
-- Function: void mpz_fdiv_q_2exp (mpz_t Q, mpz_t N,
unsigned long int B)
-- Function: void mpz_fdiv_r_2exp (mpz_t R, mpz_t N,
unsigned long int B)
-- Function: void mpz_tdiv_q (mpz_t Q, mpz_t N, mpz_t D)
-- Function: void mpz_tdiv_r (mpz_t R, mpz_t N, mpz_t D)
-- Function: void mpz_tdiv_qr (mpz_t Q, mpz_t R, mpz_t N, mpz_t D)
-- Function: unsigned long int mpz_tdiv_q_ui (mpz_t Q, mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_tdiv_r_ui (mpz_t R, mpz_t N,
unsigned long int D)
-- Function: unsigned long int mpz_tdiv_qr_ui (mpz_t Q, mpz_t R,
mpz_t N, unsigned long int D)
-- Function: unsigned long int mpz_tdiv_ui (mpz_t N,
unsigned long int D)
-- Function: void mpz_tdiv_q_2exp (mpz_t Q, mpz_t N,
unsigned long int B)
-- Function: void mpz_tdiv_r_2exp (mpz_t R, mpz_t N,
unsigned long int B)
Divide N by D, forming a quotient Q and/or remainder R. For the
`2exp' functions, D=2^B. The rounding is in three styles, each
suiting different applications.
* `cdiv' rounds Q up towards +infinity, and R will have the
opposite sign to D. The `c' stands for "ceil".
* `fdiv' rounds Q down towards -infinity, and R will have the
same sign as D. The `f' stands for "floor".
* `tdiv' rounds Q towards zero, and R will have the same sign
as N. The `t' stands for "truncate".
In all cases Q and R will satisfy N=Q*D+R, and R will satisfy
0<=abs(R)<abs(D).
The `q' functions calculate only the quotient, the `r' functions
only the remainder, and the `qr' functions calculate both. Note
that for `qr' the same variable cannot be passed for both Q and R,
or results will be unpredictable.
For the `ui' variants the return value is the remainder, and in
fact returning the remainder is all the `div_ui' functions do. For
`tdiv' and `cdiv' the remainder can be negative, so for those the
return value is the absolute value of the remainder.
For the `2exp' variants the divisor is 2^B. These functions are
implemented as right shifts and bit masks, but of course they
round the same as the other functions.
For positive N both `mpz_fdiv_q_2exp' and `mpz_tdiv_q_2exp' are
simple bitwise right shifts. For negative N, `mpz_fdiv_q_2exp' is
effectively an arithmetic right shift treating N as twos complement
the same as the bitwise logical functions do, whereas
`mpz_tdiv_q_2exp' effectively treats N as sign and magnitude.
-- Function: void mpz_mod (mpz_t R, mpz_t N, mpz_t D)
-- Function: unsigned long int mpz_mod_ui (mpz_t R, mpz_t N,
unsigned long int D)
Set R to N `mod' D. The sign of the divisor is ignored; the
result is always non-negative.
`mpz_mod_ui' is identical to `mpz_fdiv_r_ui' above, returning the
remainder as well as setting R. See `mpz_fdiv_ui' above if only
the return value is wanted.
-- Function: void mpz_divexact (mpz_t Q, mpz_t N, mpz_t D)
-- Function: void mpz_divexact_ui (mpz_t Q, mpz_t N, unsigned long D)
Set Q to N/D. These functions produce correct results only when
it is known in advance that D divides N.
These routines are much faster than the other division functions,
and are the best choice when exact division is known to occur, for
example reducing a rational to lowest terms.
-- Function: int mpz_divisible_p (mpz_t N, mpz_t D)
-- Function: int mpz_divisible_ui_p (mpz_t N, unsigned long int D)
-- Function: int mpz_divisible_2exp_p (mpz_t N, unsigned long int B)
Return non-zero if N is exactly divisible by D, or in the case of
`mpz_divisible_2exp_p' by 2^B.
N is divisible by D if there exists an integer Q satisfying N =
Q*D. Unlike the other division functions, D=0 is accepted and
following the rule it can be seen that only 0 is considered
divisible by 0.
-- Function: int mpz_congruent_p (mpz_t N, mpz_t C, mpz_t D)
-- Function: int mpz_congruent_ui_p (mpz_t N, unsigned long int C,
unsigned long int D)
-- Function: int mpz_congruent_2exp_p (mpz_t N, mpz_t C, unsigned long
int 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: mpir.info, Node: Integer Exponentiation, Next: Integer Roots, Prev: Integer Division, Up: Integer Functions
5.7 Exponentiation Functions
============================
-- Function: void mpz_powm (mpz_t ROP, mpz_t BASE, mpz_t EXP, mpz_t
MOD)
-- Function: void mpz_powm_ui (mpz_t ROP, mpz_t BASE, unsigned long
int EXP, mpz_t MOD)
Set ROP to (BASE raised to EXP) modulo MOD.
Negative EXP is supported if an inverse BASE^-1 mod MOD exists
(see `mpz_invert' in *note Number Theoretic Functions::). If an
inverse doesn't exist then a divide by zero is raised.
-- Function: void mpz_pow_ui (mpz_t ROP, mpz_t BASE, unsigned long int
EXP)
-- Function: void mpz_ui_pow_ui (mpz_t ROP, unsigned long int BASE,
unsigned long int EXP)
Set ROP to BASE raised to EXP. The case 0^0 yields 1.

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

File: mpir.info, Node: Number Theoretic Functions, Next: Integer Comparisons, Prev: Integer Roots, Up: Integer Functions
5.9 Number Theoretic Functions
==============================
-- Function: int mpz_probable_prime_p (mpz_t N, gmp_randstate_t STATE,
int PROB, unsigned long DIV)
Determine whether N is a probable prime with the chance of error
being at most 1 in 2^prob. return value is 1 if N is probably
prime, or 0 if N is definitely composite.
This function does some trial divisions to speed up the average
case, then some probabilistic primality tests to achieve the
desired level of error.
DIV can be used to inform the function that trial division up to
DIV has already been performed on N and so N has NO divisors <=
DIV.
*This function interface is preliminary and may change in the
future.*
-- Function: int mpz_likely_prime_p (mpz_t N, gmp_randstate_t STATE,
unsigned long DIV)
Determine whether N is likely a prime, i.e. you can consider it a
prime for practical purposes. return value is 1 if N can be
considered prime, or 0 if N is definitely composite.
This function does some trial divisions to speed up the average
case, then some probabilistic primality tests. The term "likely"
refers to the fact that the number will not have small factors.
DIV can be used to inform the function that trial division up to
DIV has already been performed on N and so N has NO divisors <= DIV
*This function interface is preliminary and may change in the
future.*
-- Function: int mpz_probab_prime_p (mpz_t N, int REPS)
Determine whether N is prime. Return 2 if N is definitely prime,
return 1 if N is probably prime (without being certain), or return
0 if N is definitely composite.
This function does some trial divisions, then some Miller-Rabin
probabilistic primality tests. REPS controls how many such tests
are done, 5 to 10 is a reasonable number, more will reduce the
chances of a composite being returned as "probably prime".
Miller-Rabin and similar tests can be more properly called
compositeness tests. Numbers which fail are known to be composite
but those which pass might be prime or might be composite. Only a
few composites pass, hence those which pass are considered
probably prime.
*This function is obsolete. It will disappear from future MPIR
releases.*
-- Function: void mpz_nextprime (mpz_t ROP, mpz_t OP)
Set ROP to the next prime greater than OP.
This function uses a probabilistic algorithm to identify primes.
For practical purposes it's adequate, the chance of a composite
passing will be extremely small.
*This function is obsolete. It will disappear from future MPIR
releases.*
-- Function: void mpz_next_likely_prime (mpz_t ROP, mpz_t OP,
gmp_randstate_t STATE)
Set ROP to the next likely 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.
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_gcd (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to the greatest common divisor of OP1 and OP2. The result
is always positive even if one or both input operands are negative.
-- Function: unsigned long int mpz_gcd_ui (mpz_t ROP, mpz_t OP1,
unsigned long int OP2)
Compute the greatest common divisor of OP1 and OP2. If ROP is not
`NULL', store the result there.
If the result is small enough to fit in an `unsigned long int', it
is returned. If the result does not fit, 0 is returned, and the
result is equal to the argument OP1. Note that the result will
always fit if OP2 is non-zero.
-- Function: void mpz_gcdext (mpz_t G, mpz_t S, mpz_t T, mpz_t A,
mpz_t B)
Set G to the greatest common divisor of A and B, and in addition
set S and T to coefficients satisfying A*S + B*T = G. G is always
positive, even if one or both of A and B are negative.
If T is `NULL' then that value is not computed.
-- Function: void mpz_lcm (mpz_t ROP, mpz_t OP1, mpz_t OP2)
-- Function: void mpz_lcm_ui (mpz_t ROP, mpz_t OP1, unsigned long OP2)
Set ROP to the least common multiple of OP1 and OP2. ROP is
always positive, irrespective of the signs of OP1 and OP2. ROP
will be zero if either OP1 or OP2 is zero.
-- Function: int mpz_invert (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Compute the inverse of OP1 modulo OP2 and put the result in ROP.
If the inverse exists, the return value is non-zero and ROP will
satisfy 0 <= ROP < OP2. If an inverse doesn't exist the return
value is zero and ROP is undefined.
-- Function: int mpz_jacobi (mpz_t A, mpz_t B)
Calculate the Jacobi symbol (A/B). This is defined only for B odd.
-- Function: int mpz_legendre (mpz_t A, mpz_t P)
Calculate the Legendre symbol (A/P). This is defined only for P
an odd positive prime, and for such P it's identical to the Jacobi
symbol.
-- Function: int mpz_kronecker (mpz_t A, mpz_t B)
-- Function: int mpz_kronecker_si (mpz_t A, long B)
-- Function: int mpz_kronecker_ui (mpz_t A, unsigned long B)
-- Function: int mpz_si_kronecker (long A, mpz_t B)
-- Function: int mpz_ui_kronecker (unsigned long A, mpz_t B)
Calculate the Jacobi symbol (A/B) with the Kronecker extension
(a/2)=(2/a) when a odd, or (a/2)=0 when a even.
When B is odd the Jacobi symbol and Kronecker symbol are
identical, so `mpz_kronecker_ui' etc can be used for mixed
precision Jacobi symbols too.
For more information see Henri Cohen section 1.4.2 (*note
References::), or any number theory textbook. See also the
example program `demos/qcn.c' which uses `mpz_kronecker_ui'.
-- Function: unsigned long int mpz_remove (mpz_t ROP, mpz_t OP, mpz_t
F)
Remove all occurrences of the factor F from OP and store the
result in ROP. The return value is how many such occurrences were
removed.
-- Function: void mpz_fac_ui (mpz_t ROP, unsigned long int OP)
Set ROP to OP!, the factorial of OP.
-- Function: void mpz_bin_ui (mpz_t ROP, mpz_t N, unsigned long int K)
-- Function: void mpz_bin_uiui (mpz_t ROP, unsigned long int N,
unsigned long int K)
Compute the binomial coefficient N over K and store the result in
ROP. Negative values of N are supported by `mpz_bin_ui', using
the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1
section 1.2.6 part G.
-- Function: void mpz_fib_ui (mpz_t FN, unsigned long int N)
-- Function: void mpz_fib2_ui (mpz_t FN, mpz_t FNSUB1, unsigned long
int N)
`mpz_fib_ui' sets FN to to F[n], the N'th Fibonacci number.
`mpz_fib2_ui' sets FN to F[n], and FNSUB1 to F[n-1].
These functions are designed for calculating isolated Fibonacci
numbers. When a sequence of values is wanted it's best to start
with `mpz_fib2_ui' and iterate the defining F[n+1]=F[n]+F[n-1] or
similar.
-- Function: void mpz_lucnum_ui (mpz_t LN, unsigned long int N)
-- Function: void mpz_lucnum2_ui (mpz_t LN, mpz_t LNSUB1, unsigned
long int N)
`mpz_lucnum_ui' sets LN to to L[n], the N'th Lucas number.
`mpz_lucnum2_ui' sets LN to L[n], and LNSUB1 to L[n-1].
These functions are designed for calculating isolated Lucas
numbers. When a sequence of values is wanted it's best to start
with `mpz_lucnum2_ui' and iterate the defining L[n+1]=L[n]+L[n-1]
or similar.
The Fibonacci numbers and Lucas numbers are related sequences, so
it's never necessary to call both `mpz_fib2_ui' and
`mpz_lucnum2_ui'. The formulas for going from Fibonacci to Lucas
can be found in *note Lucas Numbers Algorithm::, the reverse is
straightforward too.

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

File: mpir.info, Node: Integer Logic and Bit Fiddling, Next: I/O of Integers, Prev: Integer Comparisons, Up: Integer Functions
5.11 Logical and Bit Manipulation Functions
===========================================
These functions behave as if twos complement arithmetic were used
(although sign-magnitude is the actual implementation). The least
significant bit is number 0.
-- Function: void mpz_and (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 bitwise-and OP2.
-- Function: void mpz_ior (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 bitwise inclusive-or OP2.
-- Function: void mpz_xor (mpz_t ROP, mpz_t OP1, mpz_t OP2)
Set ROP to OP1 bitwise exclusive-or OP2.
-- Function: void mpz_com (mpz_t ROP, mpz_t OP)
Set ROP to the one's complement of OP.
-- Function: unsigned long int mpz_popcount (mpz_t OP)
If OP>=0, return the population count of OP, which is the number
of 1 bits in the binary representation. If OP<0, the number of 1s
is infinite, and the return value is ULONG_MAX, the largest
possible `unsigned long'.
-- Function: unsigned long int mpz_hamdist (mpz_t OP1, mpz_t OP2)
If OP1 and OP2 are both >=0 or both <0, return the hamming
distance between the two operands, which is the number of bit
positions where OP1 and OP2 have different bit values. If one
operand is >=0 and the other <0 then the number of bits different
is infinite, and the return value is ULONG_MAX, the largest
possible `unsigned long'.
-- Function: unsigned long int mpz_scan0 (mpz_t OP, unsigned long int
STARTING_BIT)
-- Function: unsigned long int mpz_scan1 (mpz_t OP, unsigned long int
STARTING_BIT)
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 ULONG_MAX is returned. This will
happen in `mpz_scan0' past the end of a positive number, or
`mpz_scan1' past the end of a negative.
-- Function: void mpz_setbit (mpz_t ROP, unsigned long int BIT_INDEX)
Set bit BIT_INDEX in ROP.
-- Function: void mpz_clrbit (mpz_t ROP, unsigned long int BIT_INDEX)
Clear bit BIT_INDEX in ROP.
-- Function: void mpz_combit (mpz_t ROP, unsigned long int BIT_INDEX)
Complement bit BIT_INDEX in ROP.
-- Function: int mpz_tstbit (mpz_t OP, unsigned long int BIT_INDEX)
Test bit BIT_INDEX in OP and return 0 or 1 accordingly.

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

File: mpir.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 MPIR 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,
unsigned long int N)
Generate a uniformly distributed random integer in the range 0 to
2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_urandomm (mpz_t ROP, gmp_randstate_t STATE,
mpz_t N)
Generate a uniform random integer in the range 0 to N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpz_rrandomb (mpz_t ROP, gmp_randstate_t STATE,
unsigned long int N)
Generate a random integer with long strings of zeros and ones in
the binary representation. Useful for testing functions and
algorithms, since this kind of random numbers have proven to be
more likely to trigger corner-case bugs. The random number will
be in the range 0 to 2^N-1, inclusive.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.

File: mpir.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];
mpz_t z;
mpz_import (z, 20, 1, sizeof(a[0]), 0, 0, a);
This example assumes the full `sizeof' bytes are used for data in
the given type, which is usually true, and certainly true for
`unsigned long' everywhere we know of. However on Cray vector
systems it may be noted that `short' and `int' are always stored
in 8 bytes (and with `sizeof' indicating that) but use only 32 or
46 bits. The NAILS feature can account for this, by passing for
instance `8*sizeof(int)-INT_BIT'.
-- Function: void * mpz_export (void *ROP, size_t *COUNTP, int ORDER,
size_t SIZE, int ENDIAN, size_t NAILS, mpz_t OP)
Fill ROP with word data from OP.
The parameters specify the format of the data produced. Each word
will be SIZE bytes and ORDER can be 1 for most significant word
first or -1 for least significant first. Within each word ENDIAN
can be 1 for most significant byte first, -1 for least significant
first, or 0 for the native endianness of the host CPU. The most
significant NAILS bits of each word are unused and set to zero,
this can be 0 to produce full words.
The number of words produced is written to `*COUNTP', or COUNTP
can be `NULL' to discard the count. ROP must have enough space
for the data, or if ROP is `NULL' then a result array of the
necessary size is allocated using the current MPIR 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: mpir.info, Node: Miscellaneous Integer Functions, Next: Integer Special Functions, Prev: Integer Import and Export, Up: Integer Functions
5.15 Miscellaneous Functions
============================
-- Function: int mpz_fits_ulong_p (mpz_t OP)
-- Function: int mpz_fits_slong_p (mpz_t OP)
-- Function: int mpz_fits_uint_p (mpz_t OP)
-- Function: int mpz_fits_sint_p (mpz_t OP)
-- Function: int mpz_fits_ushort_p (mpz_t OP)
-- Function: int mpz_fits_sshort_p (mpz_t OP)
Return non-zero iff the value of OP fits in an `unsigned long int',
`signed long int', `unsigned int', `signed int', `unsigned short
int', or `signed short int', respectively. Otherwise, return zero.
-- Macro: int mpz_odd_p (mpz_t OP)
-- Macro: int mpz_even_p (mpz_t OP)
Determine whether OP is odd or even, respectively. Return
non-zero if yes, zero if no. These macros evaluate their argument
more than once.
-- Function: size_t mpz_sizeinbase (mpz_t OP, int BASE)
Return the size of OP measured in number of digits in the given
BASE. BASE can vary from 2 to 36. 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: mpir.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, size_t
ARRAY_SIZE, mp_size_t FIXED_NUM_BITS)
This is a special type of initialization. *Fixed* space of
FIXED_NUM_BITS is allocated to each of the ARRAY_SIZE integers in
INTEGER_ARRAY. There is no way to free the storage allocated by
this function. Don't call `mpz_clear'!
The INTEGER_ARRAY parameter is the first `mpz_t' in the array. For
example,
mpz_t arr[20000];
mpz_array_init (arr[0], 20000, 512);
This function is only intended for programs that create a large
number of integers and need to reduce memory usage by avoiding the
overheads of allocating and reallocating lots of small blocks. In
normal programs this function is not recommended.
The space allocated to each integer by this function will not be
automatically increased, unlike the normal `mpz_init', so an
application must ensure it is sufficient for any value stored.
The following space requirements apply to various routines,
* `mpz_abs', `mpz_neg', `mpz_set', `mpz_set_si' and
`mpz_set_ui' need room for the value they store.
* `mpz_add', `mpz_add_ui', `mpz_sub' and `mpz_sub_ui' need room
for the larger of the two operands, plus an extra
`mp_bits_per_limb'.
* `mpz_mul', `mpz_mul_ui' and `mpz_mul_ui' need room for the sum
of the number of bits in their operands, but each rounded up
to a multiple of `mp_bits_per_limb'.
* `mpz_swap' can be used between two array variables, but not
between an array and a normal variable.
For other functions, or if in doubt, the suggestion is to
calculate in a regular `mpz_init' variable and copy the result to
an array variable with `mpz_set'.
*This function is obsolete. It will disappear from future MPIR
releases.*
-- Function: void * _mpz_realloc (mpz_t INTEGER, mp_size_t NEW_ALLOC)
Change the space for INTEGER to NEW_ALLOC limbs. The value in
INTEGER is preserved if it fits, or is set to 0 if not. The return
value is not useful to applications and should be ignored.
`mpz_realloc2' is the preferred way to accomplish allocation
changes like this. `mpz_realloc2' and `_mpz_realloc' are the same
except that `_mpz_realloc' takes its size in limbs.
-- Function: mp_limb_t mpz_getlimbn (mpz_t OP, mp_size_t N)
Return limb number N from OP. The sign of OP is ignored, just the
absolute value is used. The least significant limb is number 0.
`mpz_size' can be used to find how many limbs make up OP.
`mpz_getlimbn' returns zero if N is outside the range 0 to
`mpz_size(OP)-1'.
-- Function: size_t mpz_size (mpz_t OP)
Return the size of OP measured in number of limbs. If OP is zero,
the returned value will be zero.

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

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

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

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

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

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

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

File: mpir.info, Node: Floating-point Functions, Next: Low-level Functions, Prev: Rational Number Functions, Up: Top
7 Floating-point Functions
**************************
MPIR floating point numbers are stored in objects of type `mpf_t' and
functions operating on them have an `mpf_' prefix.
The mantissa of each float has a user-selectable precision, limited
only by available memory. Each variable has its own precision, and
that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on
most systems. In the current implementation the exponent is a count of
limbs, so for example on a 32-bit system this means a range of roughly
2^-68719476768 to 2^68719476736, or on a 64-bit system this will be
greater. Note however `mpf_get_str' can only return an exponent which
fits an `mp_exp_t' and currently `mpf_set_str' doesn't accept exponents
bigger than a `long'.
Each variable keeps a size for the mantissa data actually in use.
This means that if a float is exactly represented in only a few bits
then only those bits will be used in a calculation, even if the
selected precision is high.
All calculations are performed to the precision of the destination
variable. Each function is defined to calculate with "infinite
precision" followed by a truncation to the destination precision, but
of course the work done is only what's needed to determine a result
under that definition.
The precision selected for a variable is a minimum value, MPIR may
increase it a little to facilitate efficient calculation. Currently
this means rounding up to a whole limb, and then sometimes having a
further partial limb, depending on the high limb of the mantissa. But
applications shouldn't be concerned by such details.
The mantissa in stored in binary, as might be imagined from the fact
precisions are expressed in bits. One consequence of this is that
decimal fractions like 0.1 cannot be represented exactly. The same is
true of plain IEEE `double' floats. This makes both highly unsuitable
for calculations involving money or other values that should be exact
decimal fractions. (Suitably scaled integers, or perhaps rationals,
are better choices.)
`mpf' functions and variables have no special notion of infinity or
not-a-number, and applications must take care not to overflow the
exponent or results will be unpredictable. This might change in a
future release.
Note that the `mpf' functions are _not_ intended as a smooth
extension to IEEE P754 arithmetic. In particular results obtained on
one computer often differ from the results on a computer with a
different word size.
* Menu:
* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::

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

File: mpir.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, MPIR provides a parallel series of initialize-and-set
functions which initialize the output and then store the value there.
These functions' names have the form `mpf_init_set...'
Once the float has been initialized by any of the `mpf_init_set...'
functions, it can be used as the source or destination operand for the
ordinary float functions. Don't use an initialize-and-set function on
a variable already initialized!
-- Function: void mpf_init_set (mpf_t ROP, mpf_t OP)
-- Function: void mpf_init_set_ui (mpf_t ROP, unsigned long int OP)
-- Function: void mpf_init_set_si (mpf_t ROP, signed long int OP)
-- Function: void mpf_init_set_d (mpf_t ROP, double OP)
Initialize ROP and set its value from OP.
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.
-- Function: int mpf_init_set_str (mpf_t ROP, char *STR, int BASE)
Initialize ROP and set its value from the string in STR. See
`mpf_set_str' above for details on the assignment operation.
Note that ROP is initialized even if an error occurs. (I.e., you
have to call `mpf_clear' for it.)
The precision of ROP will be taken from the active default
precision, as set by `mpf_set_default_prec'.

File: mpir.info, Node: Converting Floats, Next: Float Arithmetic, Prev: Simultaneous Float Init & Assign, Up: Floating-point Functions
7.4 Conversion Functions
========================
-- Function: double mpf_get_d (mpf_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero).
If the exponent in OP is too big or too small to fit a `double'
then the result is system dependent. For too big an infinity is
returned when available. For too small 0.0 is normally returned.
Hardware overflow, underflow and denorm traps may or may not occur.
-- Function: double mpf_get_d_2exp (signed long int *EXP, mpf_t OP)
Convert OP to a `double', truncating if necessary (ie. rounding
towards zero), and with an exponent returned separately.
The return value is in the range 0.5<=abs(D)<1 and the exponent is
stored to `*EXP'. D * 2^EXP is the (truncated) OP value. If OP
is zero, the return is 0.0 and 0 is stored to `*EXP'.
This is similar to the standard C `frexp' function (*note
Normalization Functions: (libc)Normalization Functions.).
-- Function: long mpf_get_si (mpf_t OP)
-- Function: unsigned long mpf_get_ui (mpf_t OP)
Convert OP to a `long' or `unsigned long', truncating any fraction
part. If OP is too big for the return type, the result is
undefined.
See also `mpf_fits_slong_p' and `mpf_fits_ulong_p' (*note
Miscellaneous Float Functions::).
-- Function: char * mpf_get_str (char *STR, mp_exp_t *EXPPTR, int
BASE, size_t N_DIGITS, mpf_t OP)
Convert OP to a string of digits in base BASE. BASE can be 2 to
36. Up to N_DIGITS digits will be generated. Trailing zeros are
not returned. No more digits than can be accurately represented
by OP are ever generated. If N_DIGITS is 0 then that accurate
maximum number of digits are generated.
If STR is `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: mpir.info, Node: Float Arithmetic, Next: Float Comparison, Prev: Converting Floats, Up: Floating-point Functions
7.5 Arithmetic Functions
========================
-- Function: void mpf_add (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_add_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 + OP2.
-- Function: void mpf_sub (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_ui_sub (mpf_t ROP, unsigned long int OP1, mpf_t
OP2)
-- Function: void mpf_sub_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 - OP2.
-- Function: void mpf_mul (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_mul_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 times OP2.
Division is undefined if the divisor is zero, and passing a zero
divisor to the divide functions will make these functions intentionally
divide by zero. This lets the user handle arithmetic exceptions in
these functions in the same manner as other arithmetic exceptions.
-- Function: void mpf_div (mpf_t ROP, mpf_t OP1, mpf_t OP2)
-- Function: void mpf_ui_div (mpf_t ROP, unsigned long int OP1, mpf_t
OP2)
-- Function: void mpf_div_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1/OP2.
-- Function: void mpf_sqrt (mpf_t ROP, mpf_t OP)
-- Function: void mpf_sqrt_ui (mpf_t ROP, unsigned long int OP)
Set ROP to the square root of OP.
-- Function: void mpf_pow_ui (mpf_t ROP, mpf_t OP1, unsigned long int
OP2)
Set ROP to OP1 raised to the power OP2.
-- Function: void mpf_neg (mpf_t ROP, mpf_t OP)
Set ROP to -OP.
-- Function: void mpf_abs (mpf_t ROP, mpf_t OP)
Set ROP to the absolute value of OP.
-- Function: void mpf_mul_2exp (mpf_t ROP, mpf_t OP1, unsigned long
int OP2)
Set ROP to OP1 times 2 raised to OP2.
-- Function: void mpf_div_2exp (mpf_t ROP, mpf_t OP1, unsigned long
int OP2)
Set ROP to OP1 divided by 2 raised to OP2.

File: mpir.info, Node: Float Comparison, Next: I/O of Floats, Prev: Float Arithmetic, Up: Floating-point Functions
7.6 Comparison Functions
========================
-- Function: int mpf_cmp (mpf_t OP1, mpf_t OP2)
-- Function: int mpf_cmp_d (mpf_t OP1, double OP2)
-- Function: int mpf_cmp_ui (mpf_t OP1, unsigned long int OP2)
-- Function: int mpf_cmp_si (mpf_t OP1, signed long int OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
if OP1 = OP2, and a negative value if OP1 < OP2.
`mpf_cmp_d' can be called with an infinity, but results are
undefined for a NaN.
-- Function: int mpf_eq (mpf_t OP1, mpf_t OP2, unsigned long int op3)
Return non-zero if the first OP3 bits of OP1 and OP2 are equal,
zero otherwise. I.e., test if OP1 and OP2 are approximately equal.
In the future values like 1000 and 0111 may be considered the same
to 3 bits (on the basis that their difference is that small).
-- Function: void mpf_reldiff (mpf_t ROP, mpf_t OP1, mpf_t OP2)
Compute the relative difference between OP1 and OP2 and store the
result in ROP. This is abs(OP1-OP2)/OP1.
-- Macro: int mpf_sgn (mpf_t OP)
Return +1 if OP > 0, 0 if OP = 0, and -1 if OP < 0.
This function is actually implemented as a macro. It evaluates
its arguments multiple times.

File: mpir.info, Node: I/O of Floats, Next: Miscellaneous Float Functions, Prev: Float Comparison, Up: Floating-point Functions
7.7 Input and Output Functions
==============================
Functions that perform input from a stdio stream, and functions that
output to a stdio stream. Passing a `NULL' pointer for a STREAM
argument to any of these functions will make them read from `stdin' and
write to `stdout', respectively.
When using any of these functions, it is a good idea to include
`stdio.h' before `mpir.h', since that will allow `mpir.h' to define
prototypes for these functions.
-- Function: size_t mpf_out_str (FILE *STREAM, int BASE, size_t
N_DIGITS, mpf_t OP)
Print OP to STREAM, as a string of digits. Return the number of
bytes written, or if an error occurred, return 0.
The mantissa is prefixed with an `0.' and is in the given BASE,
which may vary from 2 to 36. An exponent then printed, separated
by an `e', or if 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'.
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: mpir.info, Node: Miscellaneous Float Functions, Prev: I/O of Floats, Up: Floating-point Functions
7.8 Miscellaneous Functions
===========================
-- Function: void mpf_ceil (mpf_t ROP, mpf_t OP)
-- Function: void mpf_floor (mpf_t ROP, mpf_t OP)
-- Function: void mpf_trunc (mpf_t ROP, mpf_t OP)
Set ROP to OP rounded to an integer. `mpf_ceil' rounds to the
next higher integer, `mpf_floor' to the next lower, and `mpf_trunc'
to the integer towards zero.
-- Function: int mpf_integer_p (mpf_t OP)
Return non-zero if OP is an integer.
-- Function: int mpf_fits_ulong_p (mpf_t OP)
-- Function: int mpf_fits_slong_p (mpf_t OP)
-- Function: int mpf_fits_uint_p (mpf_t OP)
-- Function: int mpf_fits_sint_p (mpf_t OP)
-- Function: int mpf_fits_ushort_p (mpf_t OP)
-- Function: int mpf_fits_sshort_p (mpf_t OP)
Return non-zero if OP would fit in the respective C data type, when
truncated to an integer.
-- Function: void mpf_urandomb (mpf_t ROP, gmp_randstate_t STATE,
unsigned long int NBITS)
Generate a uniformly distributed random float in ROP, such that 0
<= ROP < 1, with NBITS significant bits in the mantissa.
The variable STATE must be initialized by calling one of the
`gmp_randinit' functions (*note Random State Initialization::)
before invoking this function.
-- Function: void mpf_rrandomb (mpf_t ROP, gmp_randstate_t STATE,
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.
*This interface is preliminary. It might change incompatibly in
future revisions.*
-- 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.
*This function is obsolete. It will disappear from future MPIR
releases.*

File: mpir.info, Node: Low-level Functions, Next: Random Number Functions, Prev: Floating-point Functions, Up: Top
8 Low-level Functions
*********************
This chapter describes low-level MPIR functions, used to implement the
high-level MPIR functions, but also intended for time-critical user
code.
These functions start with the prefix `mpn_'.
The `mpn' functions are designed to be as fast as possible, *not* to
provide a coherent calling interface. The different functions have
somewhat similar interfaces, but there are variations that make them
hard to use. These functions do as little as possible apart from the
real multiple precision computation, so that no time is spent on things
that not all callers need.
A source operand is specified by a pointer to the least significant
limb and a limb count. A destination operand is specified by just a
pointer. It is the responsibility of the caller to ensure that the
destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform
computations on subranges of an argument, and store the result into a
subrange of a destination.
A common requirement for all functions is that each source area
needs at least one limb. No size argument may be zero. Unless
otherwise stated, in-place operations are allowed where source and
destination are the same, but not where they only partly overlap.
The `mpn' functions are the base for the implementation of the
`mpz_', `mpf_', and `mpq_' functions.
This example adds the number beginning at S1P and the number
beginning at S2P and writes the sum at DESTP. All areas have N limbs.
cy = mpn_add_n (destp, s1p, s2p, n)
It should be noted that the `mpn' functions make no attempt to
identify high or low zero limbs on their operands, or other special
forms. On random data such cases will be unlikely and it'd be wasteful
for every function to check every time. An application knowing
something about its data can take steps to trim or perhaps split its
calculations.
In the notation used below, a source operand is identified by the
pointer to the least significant limb, and the limb count in braces.
For example, {S1P, S1N}.
-- Function: mp_limb_t mpn_add_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Add {S1P, N} and {S2P, N}, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
This is the lowest-level function for addition. It is the
preferred function for addition, since it is written in assembly
for most CPUs. For addition of a variable to itself (i.e., S1P
equals S2P, use `mpn_lshift' with a count of 1 for optimal speed.
-- Function: mp_limb_t mpn_add_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Add {S1P, N} and S2LIMB, and write the N least significant limbs
of the result to RP. Return carry, either 0 or 1.
-- Function: mp_limb_t mpn_add (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Add {S1P, S1N} and {S2P, S2N}, and write the S1N least significant
limbs of the result to RP. Return carry, either 0 or 1.
This function requires that S1N is greater than or equal to S2N.
-- Function: mp_limb_t mpn_sub_n (mp_limb_t *RP, const mp_limb_t *S1P,
const mp_limb_t *S2P, mp_size_t N)
Subtract {S2P, N} from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
This is the lowest-level function for subtraction. It is the
preferred function for subtraction, since it is written in
assembly for most CPUs.
-- Function: mp_limb_t mpn_sub_1 (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t N, mp_limb_t S2LIMB)
Subtract S2LIMB from {S1P, N}, and write the N least significant
limbs of the result to RP. Return borrow, either 0 or 1.
-- Function: mp_limb_t mpn_sub (mp_limb_t *RP, const mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N)
Subtract {S2P, S2N} from {S1P, S1N}, and write the S1N least
significant limbs of the result to RP. Return borrow, either 0 or
1.
This function requires that S1N is greater than or equal to S2N.
-- Function: void mpn_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.
-- 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 MPIR. It is written
in assembly for most CPUs.
Don't call this function if S2LIMB is a power of 2; use
`mpn_lshift' with a count equal to the logarithm of S2LIMB
instead, for optimal speed.
-- Function: mp_limb_t mpn_addmul_1 (mp_limb_t *RP, const mp_limb_t
*S1P, mp_size_t N, mp_limb_t S2LIMB)
Multiply {S1P, N} and S2LIMB, and add the N least significant
limbs of the product to {RP, N} and write the result to RP.
Return the most significant limb of the product, plus carry-out
from the addition.
This is a low-level function that is a building block for general
multiplication as well as other operations in MPIR. 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, minus borrow-out
from the subtraction.
This is a low-level function that is a building block for general
multiplication and division as well as other operations in MPIR.
It is written in assembly for most CPUs.
-- 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 result to RP.
Return the most significant limb of the result.
The destination has to have space for S1N + S2N limbs, even if the
result might be one limb smaller.
This function requires that S1N is greater than or equal to S2N.
The destination must be distinct from both input operands.
-- 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. NN must be greater
than or equal to DN. The most significant limb of DP must be
non-zero. The QXN operand must be zero.
-- 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.
-- Macro: mp_limb_t mpn_divexact_by3 (mp_limb_t *RP, mp_limb_t *SP,
mp_size_t N)
-- Function: mp_limb_t mpn_divexact_by3c (mp_limb_t *RP, mp_limb_t
*SP, mp_size_t N, mp_limb_t CARRY)
Divide {SP, N} by 3, expecting it to divide exactly, and writing
the result to {RP, N}. If 3 divides exactly, the return value is
zero and the result is the quotient. If not, the return value is
non-zero and the result won't be anything useful.
`mpn_divexact_by3c' takes an initial carry parameter, which can be
the return value from a previous call, so a large calculation can
be done piece by piece from low to high. `mpn_divexact_by3' is
simply a macro calling `mpn_divexact_by3c' with a 0 carry
parameter.
These routines use a multiply-by-inverse and will be faster than
`mpn_divrem_1' on CPUs with fast multiplication but slow division.
The source a, result q, size n, initial carry i, and return value
c satisfy c*b^n + a-i = 3*q, where b=2^GMP_NUMB_BITS. The return
c is always 0, 1 or 2, and the initial carry i must also be 0, 1
or 2 (these are both borrows really). When c=0 clearly q=(a-i)/3.
When c!=0, the remainder (a-i) mod 3 is given by 3-c, because b ==
1 mod 3 (when `mp_bits_per_limb' is even, which is always so
currently).
-- Function: mp_limb_t mpn_mod_1 (mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Divide {S1P, S1N} by S2LIMB, and return the remainder. S1N can be
zero.
-- Function: mp_limb_t mpn_bdivmod (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1N, const mp_limb_t *S2P, mp_size_t S2N, unsigned
long int D)
This function puts the low floor(D/mp_bits_per_limb) limbs of Q =
{S1P, S1N}/{S2P, S2N} mod 2^D at RP, and returns the high D mod
`mp_bits_per_limb' bits of Q.
{S1P, S1N} - Q * {S2P, S2N} mod 2^(S1N*mp_bits_per_limb) is placed
at S1P. Since the low floor(D/mp_bits_per_limb) limbs of this
difference are zero, it is possible to overwrite the low limbs at
S1P with this difference, provided RP <= S1P.
This function requires that S1N * mp_bits_per_limb >= D, and that
{S2P, S2N} is odd.
*This interface is preliminary. It might change incompatibly in
future revisions.*
-- Function: mp_limb_t mpn_lshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsigned int COUNT)
Shift {SP, N} left by COUNT bits, and write the result to {RP, N}.
The bits shifted out at the left are returned in the least
significant COUNT bits of the return value (the rest of the return
value is zero).
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
{SP, N} and {RP, N} may overlap, provided RP >= SP.
This function is written in assembly for most CPUs.
-- Function: mp_limb_t mpn_rshift (mp_limb_t *RP, const mp_limb_t *SP,
mp_size_t N, unsigned int COUNT)
Shift {SP, N} right by COUNT bits, and write the result to {RP,
N}. The bits shifted out at the right are returned in the most
significant COUNT bits of the return value (the rest of the return
value is zero).
COUNT must be in the range 1 to mp_bits_per_limb-1. The regions
{SP, N} and {RP, N} may overlap, provided RP <= SP.
This function is written in assembly for most CPUs.
-- Function: int mpn_cmp (const mp_limb_t *S1P, const mp_limb_t *S2P,
mp_size_t N)
Compare {S1P, N} and {S2P, N} and return a positive value if S1 >
S2, 0 if they are equal, or a negative value if S1 < S2.
-- Function: mp_size_t mpn_gcd (mp_limb_t *RP, mp_limb_t *S1P,
mp_size_t S1N, mp_limb_t *S2P, mp_size_t S2N)
Set {RP, RETVAL} to the greatest common divisor of {S1P, S1N} and
{S2P, S2N}. The result can be up to S2N limbs, the return value
is the actual number produced. Both source operands are destroyed.
{S1P, S1N} must have at least as many bits as {S2P, S2N}. {S2P,
S2N} must be odd. Both operands must have non-zero most
significant limbs. No overlap is permitted between {S1P, S1N} and
{S2P, S2N}.
-- Function: mp_limb_t mpn_gcd_1 (const mp_limb_t *S1P, mp_size_t S1N,
mp_limb_t S2LIMB)
Return the greatest common divisor of {S1P, S1N} and S2LIMB. Both
operands must be non-zero.
-- Function: mp_size_t mpn_gcdext (mp_limb_t *R1P, mp_limb_t *R2P,
mp_size_t *R2N, mp_limb_t *S1P, mp_size_t S1N, mp_limb_t
*S2P, mp_size_t S2N)
Calculate the greatest common divisor of {S1P, S1N} and {S2P,
S2N}. Store the gcd at {R1P, RETVAL} and the first cofactor at
{R2P, *R2N}, with *R2N negative if the cofactor is negative. R1P
and R2P should each have room for S1N+1 limbs, but the return
value and value stored through R2N indicate the actual number
produced.
{S1P, S1N} >= {S2P, S2N} is required, and both must be non-zero.
The regions {S1P, S1N+1} and {S2P, S2N+1} are destroyed (i.e. the
operands plus an extra limb past the end of each).
The cofactor R1 will satisfy R2*S1 + K*S2 = R1. The second
cofactor K is not calculated but can easily be obtained from (R1 -
R2*S1) / S2.
-- Function: mp_size_t mpn_sqrtrem (mp_limb_t *R1P, mp_limb_t *R2P,
const mp_limb_t *SP, mp_size_t N)
Compute the square root of {SP, N} and put the result at {R1P,
ceil(N/2)} and the remainder at {R2P, RETVAL}. R2P needs space
for N limbs, but the return value indicates how many are produced.
The most significant limb of {SP, N} must be non-zero. The areas
{R1P, ceil(N/2)} and {SP, N} must be completely separate. The
areas {R2P, N} and {SP, N} must be either identical or completely
separate.
If the remainder is not wanted then R2P can be `NULL', and in this
case the return value is zero or non-zero according to whether the
remainder would have been zero or non-zero.
A return value of zero indicates a perfect square. See also
`mpz_perfect_square_p'.
-- Function: mp_size_t mpn_get_str (unsigned char *STR, int BASE,
mp_limb_t *S1P, mp_size_t S1N)
Convert {S1P, S1N} to a raw unsigned char array at STR in base
BASE, and return the number of characters produced. There may be
leading zeros in the string. The string is not in ASCII; to
convert it to printable format, add the ASCII codes for `0' or
`A', depending on the base and range. BASE can vary from 2 to 256.
The most significant limb of the input {S1P, S1N} must be
non-zero. The input {S1P, S1N} is clobbered, except when BASE is
a power of 2, in which case it's unchanged.
The area at STR has to have space for the largest possible number
represented by a S1N long limb array, plus one extra character.
-- Function: mp_size_t mpn_set_str (mp_limb_t *RP, const unsigned char
*STR, size_t STRSIZE, int BASE)
Convert bytes {STR,STRSIZE} in the given BASE to limbs at RP.
STR[0] is the most significant byte and STR[STRSIZE-1] is the
least significant. Each byte should be a value in the range 0 to
BASE-1, not an ASCII character. BASE can vary from 2 to 256.
The return value is the number of limbs written to RP. If the most
significant input byte is non-zero then the high limb at RP will be
non-zero, and only that exact number of limbs will be required
there.
If the most significant input byte is zero then there may be high
zero limbs written to RP and included in the return value.
STRSIZE must be at least 1, and no overlap is permitted between
{STR,STRSIZE} and the result at RP.
-- Function: unsigned long int mpn_scan0 (const mp_limb_t *S1P,
unsigned long int 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: unsigned long int mpn_scan1 (const mp_limb_t *S1P,
unsigned long int 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.
*These functions are obsolete. They will disappear from future
MPIR releases.*
-- Function: void mpn_urandomb (mp_limb_t *RP, gmp_randstate_t STATE,
unsigned long N)
Generate a uniform random number of length N bits and store it at
RP.
*This function interface is preliminary and may change in the
future.*
-- Function: void mpn_urandomm (mp_limb_t *RP, gmp_randstate_t STATE,
const mp_limb_t *MP, mp_size_t N)
Generate a uniform random number modulo (MP,N) of length N limbs
and store it at RP.
*This function interface is preliminary and may change in the
future.*
-- Function: void mpn_randomb (mp_limb_t *RP, gmp_randstate_t STATE,
mp_size_t N)
Generate a random number of length N limbs and store it at RP.
The most significant limb is always non-zero.
*This function interface is preliminary and may change in the
future.*
-- Function: void mpn_rrandom (mp_limb_t *RP, gmp_randstate_t STATE,
mp_size_t N)
Generate a random number of length N limbs and store it at RP.
The most significant limb is always non-zero. Generates long
strings of zeros and ones in the binary representation and is
intended for testing the correctness of the `mpn' routines.
*This function interface is preliminary and may change in the
future.*
-- Function: unsigned long int mpn_popcount (const mp_limb_t *S1P,
mp_size_t N)
Count the number of set bits in {S1P, N}.
-- Function: unsigned long int 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.
8.1 Nails
=========
*Everything in this section is highly experimental and may disappear or
be subject to incompatible changes in a future version of MPIR.*
N.B: Nails are currently disabled and not supported in MPIR. They
may or may not return in a future version of MPIR.
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: mpir.info, Node: Random Number Functions, Next: Formatted Output, Prev: Low-level Functions, Up: Top
9 Random Number Functions
*************************
Sequences of pseudo-random numbers in MPIR 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: mpir.info, Node: Random State Initialization, Next: Random State Seeding, Prev: Random Number Functions, Up: Random Number Functions
9.1 Random State Initialization
===============================
-- Function: void gmp_randinit_default (gmp_randstate_t STATE)
Initialize STATE with a default algorithm. This will be a
compromise between speed and randomness, and is recommended for
applications with no special requirements. Currently this is
`gmp_randinit_mt'.
-- Function: void gmp_randinit_mt (gmp_randstate_t STATE)
Initialize STATE for a Mersenne Twister algorithm. This algorithm
is fast and has good randomness properties.
-- Function: void gmp_randinit_lc_2exp (gmp_randstate_t STATE, mpz_t
A, unsigned long C, unsigned long 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,
unsigned long SIZE)
Initialize STATE for a linear congruential algorithm as per
`gmp_randinit_lc_2exp'. A, C and M2EXP are selected from a table,
chosen so that SIZE bits (or more) of each X will be used, ie.
M2EXP/2 >= SIZE.
If successful the return value is non-zero. If SIZE is bigger
than the table data provides then the return value is zero. The
maximum SIZE currently supported is 128.
-- Function: int 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_randclear (gmp_randstate_t STATE)
Free all memory occupied by STATE.

File: mpir.info, Node: Random State Seeding, Next: Random State Miscellaneous, Prev: Random State Initialization, Up: Random Number Functions
9.2 Random State Seeding
========================
-- Function: void gmp_randseed (gmp_randstate_t STATE, mpz_t SEED)
-- Function: void gmp_randseed_ui (gmp_randstate_t STATE,
unsigned long int SEED)
Set an initial seed value into STATE.
The size of a seed determines how many different sequences of
random numbers that it's possible to generate. The "quality" of
the seed is the randomness of a given seed compared to the
previous seed used, and this affects the randomness of separate
number sequences. The method for choosing a seed is critical if
the generated numbers are to be used for important applications,
such as generating cryptographic keys.
Traditionally the system time has been used to seed, but care
needs to be taken with this. If an application seeds often and
the resolution of the system clock is low, then the same sequence
of numbers might be repeated. Also, the system time is quite easy
to guess, so if unpredictability is required then it should
definitely not be the only source for the seed value. On some
systems there's a special device `/dev/random' which provides
random data better suited for use as a seed.

File: mpir.info, Node: Random State Miscellaneous, Prev: Random State Seeding, Up: Random Number Functions
9.3 Random State Miscellaneous
==============================
-- Function: unsigned long gmp_urandomb_ui (gmp_randstate_t STATE,
unsigned long N)
Return a uniformly distributed random number of N bits, ie. in the
range 0 to 2^N-1 inclusive. N must be less than or equal to the
number of bits in an `unsigned long'.
-- Function: unsigned long gmp_urandomm_ui (gmp_randstate_t STATE,
unsigned long N)
Return a uniformly distributed random number in the range 0 to
N-1, inclusive.

File: mpir.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: mpir.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
MPIR 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", limb);
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 MPIR extensions. In the
current implementation the standard parts of the format string are
simply handed to `printf' and only the MPIR extensions handled directly.
The flags accepted are as follows. GLIBC style ' is only for the
standard C types (not the MPIR 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 MPIR
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 MPIR 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 MPIR types.
Other types or conversions that might be accepted by the C library
`printf' cannot be used through `gmp_printf', this includes for
instance extensions registered with GLIBC `register_printf_function'.
Also currently there's no support for POSIX `$' style numbered arguments
(perhaps this will be added in the future).
The precision field has it's usual meaning for integer `Z' and float
`F' types, but is currently undefined for `Q' and should not be used
with that.
`mpf_t' conversions only ever generate as many digits as can be
accurately represented by the operand, the same as `mpf_get_str' does.
Zeros will be used if necessary to pad to the requested precision. This
happens even for an `f' conversion of an `mpf_t' which is an integer,
for instance 2^1024 in an `mpf_t' of 128 bits precision will only
produce about 40 digits, then pad with zeros to the decimal point. An
empty precision field like `%.Fe' or `%.Ff' can be used to specifically
request just the significant digits.
The decimal point character (or string) is taken from the current
locale settings on systems which provide `localeconv' (*note Locales
and Internationalization: (libc)Locales.). The C library will normally
do the same for standard float output.
The format string is only interpreted as plain `char's, multibyte
characters are not recognised. Perhaps this will change in the future.

File: mpir.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 MPIR
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: mpir.info, Node: C++ Formatted Output, Prev: Formatted Output Functions, Up: Formatted Output
10.3 C++ Formatted Output
=========================
The following functions are provided in `libmpirxx' (*note Headers and
Libraries::), which is built if C++ support is enabled (*note Build
Options::). Prototypes are available from `<mpir.h>'.
-- Function: ostream& operator<< (ostream& STREAM, mpz_t OP)
Print OP to STREAM, using its `ios' formatting settings.
`ios::width' is reset to 0 after output, the same as the standard
`ostream operator<<' routines do.
In hex or octal, OP is printed as a signed number, the same as for
decimal. This is unlike the standard `operator<<' routines on
`int' etc, which instead give twos complement.
-- Function: ostream& operator<< (ostream& STREAM, mpq_t OP)
Print OP to STREAM, using its `ios' formatting settings.
`ios::width' is reset to 0 after output, the same as the standard
`ostream operator<<' routines do.
Output will be a fraction like `5/9', or if the denominator is 1
then just a plain integer like `123'.
In hex or octal, OP is printed as a signed value, the same as for
decimal. If `ios::showbase' is set then a base indicator is shown
on both the numerator and denominator (if the denominator is
required).
-- Function: ostream& operator<< (ostream& STREAM, mpf_t OP)
Print OP to STREAM, using its `ios' formatting settings.
`ios::width' is reset to 0 after output, the same as the standard
`ostream operator<<' routines do.
The decimal point follows the standard library float `operator<<',
which on recent systems means the `std::locale' imbued on STREAM.
Hex and octal are supported, unlike the standard `operator<<' on
`double'. The mantissa will be in hex or octal, the exponent will
be in decimal. For hex the exponent delimiter is an `@'. This is
as per `mpf_out_str'.
`ios::showbase' is supported, and will put a base on the mantissa,
for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'.
This last form is slightly strange, but at least differentiates
itself from decimal.
These operators mean that MPIR 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 MPIR types
and that for instance using `+' with an `mpz_t' will have unpredictable
results. For classes with overloading, see *note C++ Class Interface::.

File: mpir.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: mpir.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
MPIR 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.
MPIR 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 MPIR extensions. In the
current implementation the standard parts of the format string are
simply handed to `scanf' and only the MPIR extensions handled directly.
The flags accepted are as follows. `a' and `'' will depend on
support from the C library, and `'' cannot be used with MPIR types.
* read but don't store
a allocate a buffer (string conversions)
' grouped digits, GLIBC style (not MPIR
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 MPIR types are
F mpf_t, float conversions
Q mpq_t, integer conversions
Z mpz_t, integer conversions
The conversions accepted are as follows. `p' and `[' will depend on
support from the C library, the rest are standard.
c character or characters
d decimal integer
e E f g G float
i integer with base indicator
n characters read so far
o octal integer
p pointer
s string of non-whitespace characters
u decimal integer
x X hex integer
[ string of characters in a set
`e', `E', `f', `g' and `G' are identical, they all read either fixed
point or scientific format, and either upper or lower case `e' for the
exponent in scientific format.
C99 style hex float format (`printf %a', *note Formatted Output
Strings::) is always accepted for `mpf_t', but for the standard float
types it will depend on the C library.
`x' and `X' are identical, both accept both upper and lower case
hexadecimal.
`o', `u', `x' and `X' all read positive or negative values. For the
standard C types these are described as "unsigned" conversions, but
that merely affects certain overflow handling, negatives are still
allowed (per `strtoul', *note Parsing of Integers: (libc)Parsing of
Integers.). For MPIR 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 MPIR 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: mpir.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 MPIR
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 MPIR 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 MPIR 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: mpir.info, Node: C++ Formatted Input, Prev: Formatted Input Functions, Up: Formatted Input
11.3 C++ Formatted Input
========================
The following functions are provided in `libmpirxx' (*note Headers and
Libraries::), which is built only if C++ support is enabled (*note
Build Options::). Prototypes are available from `<mpir.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 MPIR 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 MPIR
types and that for instance using `+' with an `mpz_t' will have
unpredictable results. For classes with overloading, see *note C++
Class Interface::.

File: mpir.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 MPIR.
All MPIR C language types and functions can be used in C++ programs,
since `mpir.h' has `extern "C"' qualifiers, but the class interface
offers overloaded functions and operators which may be more convenient.
Due to the implementation of this interface, a reasonably recent C++
compiler is required, one supporting namespaces, partial specialization
of templates and member templates. For GCC this means version 2.91 or
later.
*Everything described in this chapter is to be considered preliminary
and might be subject to incompatible changes if some unforeseen
difficulty reveals itself.*
* Menu:
* C++ Interface General::
* C++ Interface Integers::
* C++ Interface Rationals::
* C++ Interface Floats::
* C++ Interface Random Numbers::
* C++ Interface Limitations::

File: mpir.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 <mpirxx.h>
Programs should be linked with the `libmpirxx' and `libmpir'
libraries. For example,
g++ mycxxprog.cc -lmpirxx -lmpir
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 MPIR 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 MPIR 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 MPIR 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 `mpirxx.h', all types and functions
are simply put into the global namespace. This is what `mpir.h' has
done in the past, and continues to do for compatibility. The extras
provided by `mpirxx.h' follow MPIR naming conventions and are unlikely
to clash with anything.

File: mpir.info, Node: C++ Interface Integers, Next: C++ Interface Rationals, Prev: C++ Interface General, Up: C++ Class Interface
12.2 C++ Interface Integers
===========================
-- Function: void mpz_class::mpz_class (type N)
Construct an `mpz_class'. All the standard C++ types may be used,
except `long long' and `long double', and all the MPIR C++ classes
can be used. Any necessary conversion follows the corresponding C
function, for example `double' follows `mpz_set_d' (*note
Assigning Integers::).
-- Function: void mpz_class::mpz_class (mpz_t Z)
Construct an `mpz_class' from an `mpz_t'. The value in Z is
copied into the new `mpz_class', there won't be any permanent
association between it and Z.
-- Function: void mpz_class::mpz_class (const char *S)
-- Function: void mpz_class::mpz_class (const char *S, int BASE = 0)
-- Function: void mpz_class::mpz_class (const string& S)
-- Function: void mpz_class::mpz_class (const string& S, int BASE = 0)
Construct an `mpz_class' converted from a string using
`mpz_set_str' (*note Assigning Integers::).
If the string is not a valid integer, an `std::invalid_argument'
exception is thrown. The same applies to `operator='.
-- Function: mpz_class operator/ (mpz_class A, mpz_class D)
-- Function: mpz_class operator% (mpz_class A, mpz_class D)
Divisions involving `mpz_class' round towards zero, as per the
`mpz_tdiv_q' and `mpz_tdiv_r' functions (*note Integer Division::).
This is the same as the C99 `/' and `%' operators.
The `mpz_fdiv...' or `mpz_cdiv...' functions can always be called
directly if desired. For example,
mpz_class q, a, d;
...
mpz_fdiv_q (q.get_mpz_t(), a.get_mpz_t(), d.get_mpz_t());
-- Function: mpz_class abs (mpz_class OP1)
-- Function: int cmp (mpz_class OP1, type OP2)
-- Function: int cmp (type OP1, mpz_class OP2)
-- Function: bool mpz_class::fits_sint_p (void)
-- Function: bool mpz_class::fits_slong_p (void)
-- Function: bool mpz_class::fits_sshort_p (void)
-- Function: bool mpz_class::fits_uint_p (void)
-- Function: bool mpz_class::fits_ulong_p (void)
-- Function: bool mpz_class::fits_ushort_p (void)
-- Function: double mpz_class::get_d (void)
-- Function: long mpz_class::get_si (void)
-- Function: string mpz_class::get_str (int BASE = 10)
-- Function: unsigned long mpz_class::get_ui (void)
-- Function: int mpz_class::set_str (const char *STR, int BASE)
-- Function: int mpz_class::set_str (const string& STR, int BASE)
-- Function: int sgn (mpz_class OP)
-- Function: mpz_class sqrt (mpz_class OP)
These functions provide a C++ class interface to the corresponding
MPIR C routines.
`cmp' can be used with any of the classes or the standard C++
types, except `long long' and `long double'.
Overloaded operators for combinations of `mpz_class' and `double'
are provided for completeness, but it should be noted that if the given
`double' is not an integer then the way any rounding is done is
currently unspecified. The rounding might take place at the start, in
the middle, or at the end of the operation, and it might change in the
future.
Conversions between `mpz_class' and `double', however, are defined
to follow the corresponding C functions `mpz_get_d' and `mpz_set_d'.
And comparisons are always made exactly, as per `mpz_cmp_d'.

File: mpir.info, Node: C++ Interface Rationals, Next: C++ Interface Floats, Prev: C++ Interface Integers, Up: C++ Class Interface
12.3 C++ Interface Rationals
============================
In all the following constructors, if a fraction is given then it
should be in canonical form, or if not then `mpq_class::canonicalize'
called.
-- Function: void mpq_class::mpq_class (type OP)
-- Function: void mpq_class::mpq_class (integer NUM, integer DEN)
Construct an `mpq_class'. The initial value can be a single value
of any type, or a pair of integers (`mpz_class' or standard C++
integer types) representing a fraction, except that `long long'
and `long double' are not supported. For example,
mpq_class q (99);
mpq_class q (1.75);
mpq_class q (1, 3);
-- Function: void mpq_class::mpq_class (mpq_t Q)
Construct an `mpq_class' from an `mpq_t'. The value in Q is
copied into the new `mpq_class', there won't be any permanent
association between it and Q.
-- Function: void mpq_class::mpq_class (const char *S)
-- Function: void mpq_class::mpq_class (const char *S, int BASE = 0)
-- Function: void mpq_class::mpq_class (const string& S)
-- Function: void mpq_class::mpq_class (const string& S, int BASE = 0)
Construct an `mpq_class' converted from a string using
`mpq_set_str' (*note Initializing Rationals::).
If the string is not a valid rational, an `std::invalid_argument'
exception is thrown. The same applies to `operator='.
-- Function: void mpq_class::canonicalize ()
Put an `mpq_class' into canonical form, as per *note Rational
Number Functions::. All arithmetic operators require their
operands in canonical form, and will return results in canonical
form.
-- Function: mpq_class abs (mpq_class OP)
-- Function: int cmp (mpq_class OP1, type OP2)
-- Function: int cmp (type OP1, mpq_class OP2)
-- Function: double mpq_class::get_d (void)
-- Function: string mpq_class::get_str (int BASE = 10)
-- Function: int mpq_class::set_str (const char *STR, int BASE)
-- Function: int mpq_class::set_str (const string& STR, int BASE)
-- Function: int sgn (mpq_class OP)
These functions provide a C++ class interface to the corresponding
MPIR 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: mpir.info, Node: C++ Interface Floats, Next: C++ Interface Random Numbers, Prev: C++ Interface Rationals, Up: C++ Class Interface
12.4 C++ Interface Floats
=========================
When an expression requires the use of temporary intermediate
`mpf_class' values, like `f=g*h+x*y', those temporaries will have the
same precision as the destination `f'. Explicit constructors can be
used if this doesn't suit.
-- Function: mpf_class::mpf_class (type OP)
-- Function: mpf_class::mpf_class (type OP, unsigned long PREC)
Construct an `mpf_class'. Any standard C++ type can be used,
except `long long' and `long double', and any of the MPIR C++
classes can be used.
If PREC is given, the initial precision is that value, in bits. If
PREC is not given, then the initial precision is determined by the
type of OP given. An `mpz_class', `mpq_class', or C++ builtin
type will give the default `mpf' precision (*note Initializing
Floats::). An `mpf_class' or expression will give the precision
of that value. The precision of a binary expression is the higher
of the two operands.
mpf_class f(1.5); // default precision
mpf_class f(1.5, 500); // 500 bits (at least)
mpf_class f(x); // precision of x
mpf_class f(abs(x)); // precision of x
mpf_class f(-g, 1000); // 1000 bits (at least)
mpf_class f(x+y); // greater of precisions of x and y
-- Function: void mpf_class::mpf_class (const char *S)
-- Function: void mpf_class::mpf_class (const char *S, unsigned long
PREC, int BASE = 0)
-- Function: void mpf_class::mpf_class (const string& S)
-- Function: void mpf_class::mpf_class (const string& S, unsigned long
PREC, int BASE = 0)
Construct an `mpf_class' converted from a string using
`mpf_set_str' (*note Assigning Floats::). If PREC is given, the
initial precision is that value, in bits. If not, the default
`mpf' precision (*note Initializing Floats::) is used.
If the string is not a valid float, an `std::invalid_argument'
exception is thrown. The same applies to `operator='.
-- Function: mpf_class& mpf_class::operator= (type OP)
Convert and store the given OP value to an `mpf_class' object. The
same types are accepted as for the constructors above.
Note that `operator=' only stores a new value, it doesn't copy or
change the precision of the destination, instead the value is
truncated if necessary. This is the same as `mpf_set' etc. Note
in particular this means for `mpf_class' a copy constructor is not
the same as a default constructor plus assignment.
mpf_class x (y); // x created with precision of y
mpf_class x; // x created with default precision
x = y; // value truncated to that precision
Applications using templated code may need to be careful about the
assumptions the code makes in this area, when working with
`mpf_class' values of various different or non-default precisions.
For instance implementations of the standard `complex' template
have been seen in both styles above, though of course `complex' is
normally only actually specified for use with the builtin float
types.
-- Function: mpf_class abs (mpf_class OP)
-- Function: mpf_class ceil (mpf_class OP)
-- Function: int cmp (mpf_class OP1, type OP2)
-- Function: int cmp (type OP1, mpf_class OP2)
-- Function: bool mpf_class::fits_sint_p (void)
-- Function: bool mpf_class::fits_slong_p (void)
-- Function: bool mpf_class::fits_sshort_p (void)
-- Function: bool mpf_class::fits_uint_p (void)
-- Function: bool mpf_class::fits_ulong_p (void)
-- Function: bool mpf_class::fits_ushort_p (void)
-- Function: mpf_class floor (mpf_class OP)
-- Function: mpf_class hypot (mpf_class OP1, mpf_class OP2)
-- Function: double mpf_class::get_d (void)
-- Function: long mpf_class::get_si (void)
-- Function: string mpf_class::get_str (mp_exp_t& EXP, int BASE = 10,
size_t DIGITS = 0)
-- Function: unsigned long mpf_class::get_ui (void)
-- Function: int mpf_class::set_str (const char *STR, int BASE)
-- Function: int mpf_class::set_str (const string& STR, int BASE)
-- Function: int sgn (mpf_class OP)
-- Function: mpf_class sqrt (mpf_class OP)
-- Function: mpf_class trunc (mpf_class OP)
These functions provide a C++ class interface to the corresponding
MPIR 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: unsigned long int mpf_class::get_prec ()
-- Function: void mpf_class::set_prec (unsigned long PREC)
-- Function: void mpf_class::set_prec_raw (unsigned long PREC)
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: mpir.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 MPIR 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: void gmp_randclass::seed (unsigned long int S)
-- Function: void gmp_randclass::seed (mpz_class S)
Seed a random number generator. See *note Random Number
Functions::, for how to choose a good seed.
-- Function: mpz_class gmp_randclass::get_z_bits (unsigned long BITS)
-- Function: mpz_class gmp_randclass::get_z_bits (mpz_class BITS)
Generate a random integer with a specified number of bits.
-- Function: mpz_class gmp_randclass::get_z_range (mpz_class N)
Generate a random integer in the range 0 to N-1 inclusive.
-- Function: mpf_class gmp_randclass::get_f ()
-- Function: mpf_class gmp_randclass::get_f (unsigned long PREC)
Generate a random float F in the range 0 <= F < 1. F will be to
PREC bits precision, or if PREC is not given then to the precision
of the destination. For example,
gmp_randclass r;
...
mpf_class f (0, 512); // 512 bits precision
f = r.get_f(); // random number, 512 bits

File: mpir.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 `http://groups.google.com/group/mpir-devel'.
Subclassing
Subclassing the MPIR 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 MPIR
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 `mpirxx.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
}

File: mpir.info, Node: Custom Allocation, Next: Language Bindings, Prev: C++ Class Interface, Up: Top
13 Custom Allocation
********************
By default MPIR uses `malloc', `realloc' and `free' for memory
allocation, and if they fail MPIR 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
MPIR, apart from temporary space from `alloca' if that function is
available and MPIR is configured to use it (*note Build Options::).
*Be sure to call `mp_set_memory_functions' only when there are no
active MPIR objects allocated using the previous memory functions!
Usually that means calling it before any other MPIR function.*
The functions supplied should fit the following declarations:
-- Function: void * allocate_function (size_t ALLOC_SIZE)
Return a pointer to newly allocated space with at least ALLOC_SIZE
bytes.
-- Function: void * reallocate_function (void *PTR, size_t OLD_SIZE,
size_t NEW_SIZE)
Resize a previously allocated block PTR of OLD_SIZE bytes to be
NEW_SIZE bytes.
The block may be moved if necessary or if desired, and in that
case the smaller of OLD_SIZE and NEW_SIZE bytes must be copied to
the new location. The return value is a pointer to the resized
block, that being the new location if moved or just PTR if not.
PTR is never `NULL', it's always a previously allocated block.
NEW_SIZE may be bigger or smaller than OLD_SIZE.
-- Function: void free_function (void *PTR, size_t SIZE)
De-allocate the space pointed to by PTR.
PTR is never `NULL', it's always a previously allocated block of
SIZE bytes.
A "byte" here means the unit used by the `sizeof' operator.
The OLD_SIZE parameters to REALLOCATE_FUNCTION and FREE_FUNCTION are
passed for convenience, but of course can be ignored if not needed.
The default functions using `malloc' and friends for instance don't use
them.
No error return is allowed from any of these functions, if they
return then they must have performed the specified operation. In
particular note that ALLOCATE_FUNCTION or REALLOCATE_FUNCTION mustn't
return `NULL'.
Getting a different fatal error action is a good use for custom
allocation functions, for example giving a graphical dialog rather than
the default print to `stderr'. How much is possible when genuinely out
of memory is another question though.
There's currently no defined way for the allocation functions to
recover from an error such as out of memory, they must terminate
program execution. A `longjmp' or throwing a C++ exception will have
undefined results. This may change in the future.
MPIR may use allocated blocks to hold pointers to other allocated
blocks. This will limit the assumptions a conservative garbage
collection scheme can make.
Any custom allocation functions must align pointers to limb
boundaries. Thus if a limb is eight bytes (e.g. on x86_64), then all
blocks must be aligned to eight byte boundaries. Check the
configuration options for the custom allocation library in use. It is
not necessary to align blocks to SSE boundaries even when SSE code is
used. All MPIR assembly routines assume limb boundary alignment only
(which is the default for most standard memory managers).
Since the default MPIR 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 MPIR 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: mpir.info, Node: Language Bindings, Next: Algorithms, Prev: Custom Allocation, Up: Top
14 Language Bindings
********************
The following packages and projects offer access to MPIR from languages
other than C, though perhaps with varying levels of functionality and
efficiency.
C++
* MPIR C++ class interface, *note C++ Class Interface::
Straightforward interface, expression templates to eliminate
temporaries.
* ALP `http://www-sop.inria.fr/saga/logiciels/ALP/'
Linear algebra and polynomials using templates.
* Arithmos `http://www.win.ua.ac.be/~cant/arithmos/'
Rationals with infinities and square roots.
* CLN `http://www.ginac.de/CLN/'
High level classes for arithmetic.
* LiDIA `http://www.informatik.tu-darmstadt.de/TI/LiDIA/'
A C++ library for computational number theory.
* Linbox `http://www.linalg.org/'
Sparse vectors and matrices.
* NTL `http://www.shoup.net/ntl/'
A C++ number theory library.
D
* gmp-d `http://home.comcast.net/~benhinkle/gmp-d'
Fortran
* Omni F77 `http://phase.hpcc.jp/Omni/home.html'
Arbitrary precision floats.
Haskell
* Glasgow Haskell Compiler `http://www.haskell.org/ghc/'
Java
* Kaffe `http://www.kaffe.org/'
* Kissme `http://kissme.sourceforge.net/'
Lisp
* GNU Common Lisp `http://www.gnu.org/software/gcl/gcl.html'
* Librep `http://librep.sourceforge.net/'
* XEmacs (21.5.18 beta and up) `http://www.xemacs.org'
Optional big integers, rationals and floats using MPIR.
M4
* GNU m4 betas `http://www.seindal.dk/rene/gnu/'
Optionally provides an arbitrary precision `mpeval'.
ML
* MLton compiler `http://mlton.org/'
Objective Caml
* MLGMP `http://www.di.ens.fr/~monniaux/programmes.html.en'
* Numerix `http://pauillac.inria.fr/~quercia/'
Optionally using GMP.
Oz
* Mozart `http://www.mozart-oz.org/'
Pascal
* GNU Pascal Compiler `http://www.gnu-pascal.de/'
GMP unit.
* Numerix `http://pauillac.inria.fr/~quercia/'
For Free Pascal, optionally using GMP.
Perl
* GMP module, see `demos/perl' in the MPIR sources (*note
Demonstration Programs::).
* Math::GMP `http://www.cpan.org/'
Compatible with Math::BigInt, but not as many functions as
the GMP module above.
* Math::BigInt::GMP `http://www.cpan.org/'
Plug Math::GMP into normal Math::BigInt operations.
Pike
* mpz module in the standard distribution,
`http://pike.ida.liu.se/'
Prolog
* SWI Prolog `http://www.swi-prolog.org/'
Arbitrary precision floats.
Python
* mpz module in the standard distribution,
`http://www.python.org/'
* GMPY `http://gmpy.sourceforge.net/'
Scheme
* GNU Guile (upcoming 1.8)
`http://www.gnu.org/software/guile/guile.html'
* RScheme `http://www.rscheme.org/'
* STklos `http://www.stklos.org/'
Smalltalk
* GNU Smalltalk
`http://www.smalltalk.org/versions/GNUSmalltalk.html'
Other
* Axiom `http://savannah.nongnu.org/projects/axiom'
Computer algebra using GCL.
* DrGenius `http://drgenius.seul.org/'
Geometry system and mathematical programming language.
* GiNaC `http://www.ginac.de/'
C++ computer algebra using CLN.
* GOO `http://www.googoogaga.org/'
Dynamic object oriented language.
* Maxima `http://www.ma.utexas.edu/users/wfs/maxima.html'
Macsyma computer algebra using GCL.
* Q `http://q-lang.sourceforge.net/'
Equational programming system.
* Regina `http://regina.sourceforge.net/'
Topological calculator.
* Yacas `http://www.xs4all.nl/~apinkus/yacas.html'
Yet another computer algebra system.

File: mpir.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 MPIR operations. The code is likely to be hard to understand
without knowing something about the algorithms.
Some MPIR internals are mentioned, but applications that expect to be
compatible with future MPIR releases should take care to use only the
documented functions.
Warning: this section is somewhat out-of-date.
* Menu:
* Multiplication Algorithms::
* Division Algorithms::
* Greatest Common Divisor Algorithms::
* Powering Algorithms::
* Root Extraction Algorithms::
* Radix Conversion Algorithms::
* Other Algorithms::
* Assembler Coding::

File: mpir.info, Node: Multiplication Algorithms, Next: Division Algorithms, Prev: Algorithms, Up: Algorithms
15.1 Multiplication
===================
NxN limb multiplications and squares are done using one of six
algorithms, as the size N increases.
Algorithm Threshold
Basecase (none)
Karatsuba `MUL_KARATSUBA_THRESHOLD'
Toom-3 `MUL_TOOM3_THRESHOLD'
Toom-4 `MUL_TOOM4_THRESHOLD'
Toom-7 `MUL_TOOM7_THRESHOLD'
FFT `MUL_FFT_THRESHOLD'
Similarly for squaring, with the `SQR' thresholds.
NxM multiplications of operands with different sizes above
`MUL_KARATSUBA_THRESHOLD' are currently done by splitting into MxM
pieces. The Karatsuba and Toom routines then operate only on equal size
operands. This is not very efficient, and is slated for improvement in
the future.
* Menu:
* Basecase Multiplication::
* Karatsuba Multiplication::
* Toom 3-Way Multiplication::
* FFT Multiplication::
* Other Multiplication::

File: mpir.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.
Assembler implementations of `mpn_mul_basecase' are essentially the
same as the generic C code, but have all the usual assembler tricks and
obscurities introduced for speed.
A square can be done in roughly half the time of a multiply, by
using the fact that the cross products above and below the diagonal are
the same. A triangle of products below the diagonal is formed, doubled
(left shift by one bit), and then the products on the diagonal added.
This can be seen in `mpn/generic/sqr_basecase.c'. Again the assembler
implementations take essentially the same approach.
u0 u1 u2 u3 u4
+---+---+---+---+---+
u0 | d | | | | |
+---+---+---+---+---+
u1 | | d | | | |
+---+---+---+---+---+
u2 | | | d | | |
+---+---+---+---+---+
u3 | | | | d | |
+---+---+---+---+---+
u4 | | | | | d |
+---+---+---+---+---+
In practice squaring isn't a full 2x faster than multiplying, it's
usually around 1.5x. Less than 1.5x probably indicates
`mpn_sqr_basecase' wants improving on that CPU.
On some CPUs `mpn_mul_basecase' can be faster than the generic C
`mpn_sqr_basecase' 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: mpir.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, ie. if x0 is k limbs
(y0 the same) then b=2^(k*mp_bits_per_limb). With that x=x1*b+x0 and
y=y1*b+y0, and the following holds,
x*y = (b^2+b)*x1*y1 - b*(x1-x0)*(y1-y0) + (b+1)*x0*y0
This formula means doing only three multiplies of (N/2)x(N/2) limbs,
whereas a basecase multiply of NxN limbs is equivalent to four
multiplies of (N/2)x(N/2). The factors (b^2+b) etc represent the
positions where the three products must be added.
high low
+--------+--------+ +--------+--------+
| x1*y1 | | x0*y0 |
+--------+--------+ +--------+--------+
+--------+--------+
add | x1*y1 |
+--------+--------+
+--------+--------+
add | x0*y0 |
+--------+--------+
+--------+--------+
sub | (x1-x0)*(y1-y0) |
+--------+--------+
The term (x1-x0)*(y1-y0) is best calculated as an absolute value,
and the sign used to choose to add or subtract. Notice the sum
high(x0*y0)+low(x1*y1) occurs twice, so it's possible to do 5*k limb
additions, rather than 6*k, but in MPIR extra function call overheads
outweigh the saving.
Squaring is similar to multiplying, but with x=y the formula reduces
to an equivalent with three squares,
x^2 = (b^2+b)*x1^2 - b*(x1-x0)^2 + (b+1)*x0^2
The final result is accumulated from those three squares the same
way as for the three multiplies above. The middle term (x1-x0)^2 is now
always positive.
A similar formula for both multiplying and squaring can be
constructed with a middle term (x1+x0)*(y1+y0). But those sums can
exceed k limbs, leading to more carry handling and additions than the
form above.
Karatsuba multiplication is asymptotically an O(N^1.585) algorithm,
the exponent being log(3)/log(2), representing 3 multiplies each 1/2
the size of the inputs. This is a big improvement over the basecase
multiply at O(N^2) and the advantage soon overcomes the extra additions
Karatsuba performs. `MUL_KARATSUBA_THRESHOLD' can be as little as 10
limbs. The `SQR' threshold is usually about twice the `MUL'.
The basecase algorithm will take a time of the form M(N) = a*N^2 +
b*N + c and the Karatsuba algorithm K(N) = 3*M(N/2) + d*N + e, 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: mpir.info, Node: Toom 3-Way Multiplication, Next: FFT 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 MPIR
is described here.
The operands are each considered split into 3 pieces of equal length
(or the most significant part 1 or 2 limbs shorter than the other two).
high low
+----------+----------+----------+
| x2 | x1 | x0 |
+----------+----------+----------+
+----------+----------+----------+
| y2 | y1 | y0 |
+----------+----------+----------+
These parts are treated as the coefficients of two polynomials
X(t) = x2*t^2 + x1*t + x0
Y(t) = y2*t^2 + y1*t + y0
Let b equal the power of 2 which is the size of the x0, x1, y0 and
y1 pieces, ie. if they're k limbs each then b=2^(k*mp_bits_per_limb).
With this x=X(b) and y=Y(b).
Let a polynomial W(t)=X(t)*Y(t) and suppose its coefficients are
W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0
The w[i] are going to be determined, and when they are they'll give
the final result using w=W(b), since x*y=X(b)*Y(b)=W(b). The
coefficients will be roughly b^2 each, and the final W(b) will be an
addition like,
high low
+-------+-------+
| w4 |
+-------+-------+
+--------+-------+
| w3 |
+--------+-------+
+--------+-------+
| w2 |
+--------+-------+
+--------+-------+
| w1 |
+--------+-------+
+-------+-------+
| w0 |
+-------+-------+
The w[i] coefficients could be formed by a simple set of cross
products, like w4=x2*y2, w3=x2*y1+x1*y2, w2=x2*y0+x1*y1+x0*y2 etc, but
this would need all nine x[i]*y[j] for i,j=0,1,2, and would be
equivalent merely to a basecase multiply. Instead the following
approach is used.
X(t) and Y(t) are evaluated and multiplied at 5 points, giving
values of W(t) at those points. In MPIR the following points are used,
Point Value
t=0 x0 * y0, which gives w0 immediately
t=1 (x2+x1+x0) * (y2+y1+y0)
t=-1 (x2-x1+x0) * (y2-y1+y0)
t=2 (4*x2+2*x1+x0) * (4*y2+2*y1+y0)
t=inf x2 * y2, which gives w4 immediately
At t=-1 the values can be negative and that's handled using the
absolute values and tracking the sign separately. At t=inf the value
is actually X(t)*Y(t)/t^4 in the limit as t approaches infinity, but
it's much easier to think of as simply x2*y2 giving w4 immediately
(much like x0*y0 at t=0 gives w0 immediately).
Each of the points substituted into W(t)=w4*t^4+...+w0 gives a
linear combination of the w[i] coefficients, and the value of those
combinations has just been calculated.
W(0) = w0
W(1) = w4 + w3 + w2 + w1 + w0
W(-1) = w4 - w3 + w2 - w1 + w0
W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0
W(inf) = w4
This is a set of five equations in five unknowns, and some
elementary linear algebra quickly isolates each w[i]. This involves
adding or subtracting one W(t) value from another, and a couple of
divisions by powers of 2 and one division by 3, the latter using the
special `mpn_divexact_by3' (*note Exact Division::).
The conversion of W(t) values to the coefficients is interpolation.
A polynomial of degree 4 like W(t) is uniquely determined by values
known at 5 different points. The points are arbitrary and can be
chosen to make the linear equations come out with a convenient set of
steps for quickly isolating the w[i].
Squaring follows the same procedure as multiplication, but there's
only one X(t) and it's evaluated at the 5 points, and those values
squared to give values of W(t). The interpolation is then identical,
and in fact the same `toom3_interpolate' subroutine is used for both
squaring and multiplying.
Toom-3 is asymptotically O(N^1.465), the exponent being
log(5)/log(3), representing 5 recursive multiplies of 1/3 the original
size each. This is an improvement over Karatsuba at O(N^1.585), though
Toom does more work in the evaluation and interpolation and so it only
realizes its advantage above a certain size.
Near the crossover between Toom-3 and Karatsuba there's generally a
range of sizes where the difference between the two is small.
`MUL_TOOM3_THRESHOLD' is a somewhat arbitrary point in that range and
successive runs of the tune program can give different values due to
small variations in measuring. A graph of time versus size for the two
shows the effect, see `tune/README'.
At the fairly small sizes where the Toom-3 thresholds occur it's
worth remembering that the asymptotic behaviour for Karatsuba and
Toom-3 can't be expected to make accurate predictions, due of course to
the big influence of all sorts of overheads, and the fact that only a
few recursions of each are being performed. Even at large sizes
there's a good chance machine dependent effects like cache architecture
will mean actual performance deviates from what might be predicted.
The formula given 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: mpir.info, Node: FFT Multiplication, Next: Other Multiplication, Prev: Toom 3-Way Multiplication, Up: Multiplication Algorithms
15.1.4 FFT Multiplication
-------------------------
At large to very large sizes a Fermat style FFT multiplication is used,
following Scho"nhage and Strassen (*note References::). Descriptions
of FFTs in various forms can be found in many textbooks, for instance
Knuth section 4.3.3 part C or Lipson chapter IX. A brief description
of the form used in MPIR 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: mpir.info, Node: Other Multiplication, Prev: FFT Multiplication, Up: Multiplication Algorithms
15.1.5 Other Multiplication
---------------------------
The 3-way Toom algorithm described above (*note Toom 3-Way
Multiplication::) generalizes to split into an arbitrary number of
pieces, as per Knuth section 4.3.3 algorithm C. MPIR currently
implements Toom 4 and 7 routines.
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 MPIR 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 MPIR.

File: mpir.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::
* Exact Division::
* Exact Remainder::
* Small Quotient Division::

File: mpir.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 section 8 of "Division by Invariant
Integers using Multiplication" by Granlund and Montgomery (*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: mpir.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: mpir.info, Node: Divide and Conquer Division, Next: Exact Division, Prev: Basecase Division, Up: Division Algorithms
15.2.3 Divide and Conquer Division
----------------------------------
For divisors larger than `DIV_DC_THRESHOLD', division is done by
dividing. Or to be precise by a recursive divide and conquer algorithm
based on work by Moenck and Borodin, Jebelean, and Burnikel and Ziegler
(*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_KARATSUBA_THRESHOLD' that divide and conquer is of use.
`DIV_DC_THRESHOLD' is based on the divisor size N, so it will be
somewhere above twice `MUL_KARATSUBA_THRESHOLD', but how much above
depends on the CPU. An optimized `mpn_mul_basecase' can lower
`DIV_DC_THRESHOLD' a little by offering a ready-made advantage over
repeated `mpn_submul_1' calls.
Divide and conquer is asymptotically O(M(N)*log(N)) where M(N) is
the time for an NxN multiplication done with FFTs. The actual time is
a sum over multiplications of the recursed sizes, as can be seen near
the end of section 2.2 of Burnikel and Ziegler. For example, within
the Toom-3 range, divide and conquer is 2.63*M(N). With higher
algorithms the M(N) term improves and the multiplier tends to log(N).
In practice, at moderate to large sizes, a 2NxN division is about 2 to
4 times slower than an NxN multiplication.
Newton's method used for division is asymptotically O(M(N)) and
should therefore be superior to divide and conquer, but it's believed
this would only be for large to very large N.

File: mpir.info, Node: Exact Division, Next: Exact Remainder, Prev: Divide and Conquer Division, Up: Division Algorithms
15.2.4 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
`modlimb_invert' in `gmp-impl.h'. This does four multiplies for a
32-bit limb, or six for a 64-bit limb. `tune/modlinv.c' has some
alternate implementations that might suit processors better at bit
twiddling than multiplying.
The sub-quadratic exact division described by Jebelean in "Exact
Division with Karatsuba Complexity" is not currently implemented. It
uses a rearrangement similar to the divide and conquer for normal
division (*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.

File: mpir.info, Node: Exact Remainder, Next: Small Quotient Division, Prev: Exact Division, Up: Division Algorithms
15.2.5 Exact Remainder
----------------------
If the exact division algorithm is done with a full subtraction at each
stage and the dividend isn't a multiple of the divisor, then low zero
limbs are produced but with a remainder in the high limbs. For
dividend a, divisor d, quotient q, and b = 2^mp_bits_per_limb, this
remainder r is of the form
a = q*d + r*b^n
n represents the number of zero limbs produced by the subtractions,
that being the number of limbs produced for q. r will be in the range
0<=r<d and can be viewed as a remainder, but one shifted up by a factor
of b^n.
Carrying out full subtractions at each stage means the same number
of cross products must be done as a normal division, but there's still
some single limb divisions saved. When d is a single limb some
simplifications arise, providing good speedups on a number of
processors.
`mpn_bdivmod', `mpn_divexact_by3', `mpn_modexact_1_odd' and the
`redc' function in `mpz_powm' differ subtly in how they return r,
leading to some negations in the above formula, but all are essentially
the same.
Clearly r is zero when a is a multiple of d, and this leads to
divisibility or congruence tests which are potentially more efficient
than a normal division.
The factor of b^n on r can be ignored in a GCD when d is odd, hence
the use of `mpn_bdivmod' in `mpn_gcd', and the use of
`mpn_modexact_1_odd' by `mpn_gcd_1' and `mpz_kronecker_ui' etc (*note
Greatest Common Divisor Algorithms::).
Montgomery's REDC method for modular multiplications uses operands
of the form of x*b^-n and y*b^-n and on calculating (x*b^-n)*(y*b^-n)
uses the factor of b^n in the exact remainder to reach a product in the
same form (x*y)*b^-n (*note Modular Powering Algorithm::).
Notice that r generally gives no useful information about the
ordinary remainder a mod d since b^n mod d could be anything. If
however b^n == 1 mod d, then r is the negative of the ordinary
remainder. This occurs whenever d is a factor of b^n-1, as for example
with 3 in `mpn_divexact_by3'. For a 32 or 64 bit limb other such
factors include 5, 17 and 257, but no particular use has been found for
this.

File: mpir.info, Node: Small Quotient Division, Prev: Exact Remainder, Up: Division Algorithms
15.2.6 Small Quotient Division
------------------------------
An NxM division where the number of quotient limbs Q=N-M is small can
be optimized somewhat.
An ordinary basecase division normalizes the divisor by shifting it
to make the high bit set, shifting the dividend accordingly, and
shifting the remainder back down at the end of the calculation. This
is wasteful if only a few quotient limbs are to be formed. Instead a
division of just the top 2*Q limbs of the dividend by the top Q limbs
of the divisor can be used to form a trial quotient. This requires
only those limbs normalized, not the whole of the divisor and dividend.
A multiply and subtract then applies the trial quotient to the M-Q
unused limbs of the divisor and N-Q dividend limbs (which includes Q
limbs remaining from the trial quotient division). The starting trial
quotient can be 1 or 2 too big, but all cases of 2 too big and most
cases of 1 too big are detected by first comparing the most significant
limbs that will arise from the subtraction. An addback is done if the
quotient still turns out to be 1 too big.
This whole procedure is essentially the same as one step of the
basecase algorithm done in a Q limb base, though with the trial
quotient test done only with the high limbs, not an entire Q limb
"digit" product. The correctness of this weaker test can be
established by following the argument of Knuth section 4.3.1 exercise
20 but with the v2*q>b*r+u2 condition appropriately relaxed.

File: mpir.info, Node: Greatest Common Divisor Algorithms, Next: Powering Algorithms, Prev: Division Algorithms, Up: Algorithms
15.3 Greatest Common Divisor
============================
* Menu:
* Binary GCD::
* Accelerated GCD::
* Extended GCD::
* Jacobi Symbol::

File: mpir.info, Node: Binary GCD, Next: Accelerated GCD, Prev: Greatest Common Divisor Algorithms, Up: Greatest Common Divisor Algorithms
15.3.1 Binary GCD
-----------------
At small sizes MPIR uses an O(N^2) binary style GCD. This is described
in many textbooks, for example Knuth section 4.5.2 algorithm B. It
simply consists of successively reducing odd operands a and b using
a,b = abs(a-b),min(a,b)
strip factors of 2 from a
The Euclidean GCD algorithm, as per Knuth algorithms E and A,
reduces using a mod b but this has so far been found to be slower
everywhere. One reason the binary method does well is that the implied
quotient at each step is usually small, so often only one or two
subtractions are needed to get the same effect as a division.
Quotients 1, 2 and 3 for example occur 67.7% of the time, see Knuth
section 4.5.3 Theorem E.
When the implied quotient is large, meaning b is much smaller than
a, then a division is worthwhile. This is the basis for the initial a
mod b reductions in `mpn_gcd' and `mpn_gcd_1' (the latter for both Nx1
and 1x1 cases). But after that initial reduction, big quotients occur
too rarely to make it worth checking for them.
The final 1x1 GCD in `mpn_gcd_1' is done in the generic C code as
described above. For two N-bit operands, the algorithm takes about
0.68 iterations per bit. For optimum performance some attention needs
to be paid to the way the factors of 2 are stripped from a.
Firstly it may be noted that in twos complement the number of low
zero bits on a-b is the same as b-a, so counting or testing can begin on
a-b without waiting for abs(a-b) to be determined.
A loop stripping low zero bits tends not to branch predict well,
since the condition is data dependent. But on average there's only a
few low zeros, so an option is to strip one or two bits arithmetically
then loop for more (as done for AMD K6). Or use a lookup table to get
a count for several bits then loop for more (as done for AMD K7). An
alternative approach is to keep just one of a or b odd and iterate
a,b = abs(a-b), min(a,b)
a = a/2 if even
b = b/2 if even
This requires about 1.25 iterations per bit, but stripping of a
single bit at each step avoids any branching. Repeating the bit strip
reduces to about 0.9 iterations per bit, which may be a worthwhile
tradeoff.
Generally with the above approaches a speed of perhaps 6 cycles per
bit can be achieved, which is still not terribly fast with for instance
a 64-bit GCD taking nearly 400 cycles. It's this sort of time which
means it's not usually advantageous to combine a set of divisibility
tests into a GCD.

File: mpir.info, Node: Accelerated GCD, Next: Extended GCD, Prev: Binary GCD, Up: Greatest Common Divisor Algorithms
15.3.2 Accelerated GCD
----------------------
For sizes above `GCD_ACCEL_THRESHOLD', MPIR uses the Accelerated GCD
algorithm described independently by Weber and Jebelean (the latter as
the "Generalized Binary" algorithm), *note References::. This
algorithm is still O(N^2), but is much faster than the binary algorithm
since it does fewer multi-precision operations. It consists of
alternating the k-ary reduction by Sorenson, and a "dmod" exact
remainder reduction.
For operands u and v the k-ary reduction replaces u with n*v-d*u
where n and d are single limb values chosen to give two trailing zero
limbs on that value, which can be stripped. n and d are calculated
using an algorithm similar to half of a two limb GCD (see `find_a' in
`mpn/generic/gcd.c').
When u and v differ in size by more than a certain number of bits, a
dmod is performed to zero out bits at the low end of the larger. It
consists of an exact remainder style division applied to an appropriate
number of bits (*note Exact Division::, and *note Exact Remainder::).
This is faster than a k-ary reduction but useful only when the operands
differ in size. There's a dmod after each k-ary reduction, and if the
dmod leaves the operands still differing in size then it's repeated.
The k-ary reduction step can introduce spurious factors into the GCD
calculated, and these are eliminated at the end by taking GCDs with the
original inputs gcd(u,gcd(v,g)) using the binary algorithm. Since g is
almost always small this takes very little time.
At small sizes the algorithm needs a good implementation of
`find_a'. At larger sizes it's dominated by `mpn_addmul_1' applying n
and d.

File: mpir.info, Node: Extended GCD, Next: Jacobi Symbol, Prev: Accelerated GCD, Up: Greatest Common Divisor Algorithms
15.3.3 Extended GCD
-------------------
The extended GCD calculates gcd(a,b) and also cofactors x and y
satisfying a*x+b*y=gcd(a,b). Lehmer's multi-step improvement of the
extended Euclidean algorithm is used. See Knuth section 4.5.2
algorithm L, and `mpn/generic/gcdext.c'. This is an O(N^2) algorithm.
The multipliers at each step are found using single limb
calculations for sizes up to `GCDEXT_THRESHOLD', or double limb
calculations above that. The single limb code is faster but doesn't
produce full-limb multipliers, hence not making full use of the
`mpn_addmul_1' calls.
When a CPU has a data-dependent multiplier, meaning one which is
faster on operands with fewer bits, the extra work in the double-limb
calculation might only save some looping overheads, leading to a large
`GCDEXT_THRESHOLD'.
Currently the single limb calculation doesn't optimize for the small
quotients that often occur, and this can lead to unusually low values of
`GCDEXT_THRESHOLD', depending on the CPU.
An analysis of double-limb calculations can be found in "A
Double-Digit Lehmer-Euclid Algorithm" by Jebelean (*note References::).
The code in MPIR was developed independently.
It should be noted that when a double limb calculation is used, it's
used for the whole of that GCD, it doesn't fall back to single limb
part way through. This is because as the algorithm proceeds, the
inputs a and b are reduced, but the cofactors x and y grow, so the
multipliers at each step are applied to a roughly constant total number
of limbs.

File: mpir.info, Node: Jacobi Symbol, Prev: Extended GCD, Up: Greatest Common Divisor Algorithms
15.3.4 Jacobi Symbol
--------------------
`mpz_jacobi' and `mpz_kronecker' are currently implemented with a
simple binary algorithm similar to that described for the GCDs (*note
Binary GCD::). They're not very fast when both inputs are large.
Lehmer's multi-step improvement or a binary based multi-step algorithm
is likely to be better.
When one operand fits a single limb, and that includes
`mpz_kronecker_ui' and friends, an initial reduction is done with
either `mpn_mod_1' or `mpn_modexact_1_odd', followed by the binary
algorithm on a single limb. The binary algorithm is well suited to a
single limb, and the whole calculation in this case is quite efficient.
In all the routines sign changes for the result are accumulated
using some bit twiddling, avoiding table lookups or conditional jumps.

File: mpir.info, Node: Powering Algorithms, Next: Root Extraction Algorithms, Prev: Greatest Common Divisor Algorithms, Up: Algorithms
15.4 Powering Algorithms
========================
* Menu:
* Normal Powering Algorithm::
* Modular Powering Algorithm::

File: mpir.info, Node: Normal Powering Algorithm, Next: Modular Powering Algorithm, Prev: Powering Algorithms, Up: Powering Algorithms
15.4.1 Normal Powering
----------------------
Normal `mpz' or `mpf' powering uses a simple binary algorithm,
successively squaring and then multiplying by the base when a 1 bit is
seen in the exponent, as per Knuth section 4.6.3. The "left to right"
variant described there is used rather than algorithm A, since it's
just as easy and can be done with somewhat less temporary memory.

File: mpir.info, Node: Modular Powering Algorithm, Prev: Normal Powering Algorithm, Up: Powering Algorithms
15.4.2 Modular Powering
-----------------------
Modular powering is implemented using a 2^k-ary sliding window
algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85
(*note References::). k is chosen according to the size of the
exponent. Larger exponents use larger values of k, the choice being
made to minimize the average number of multiplications that must
supplement the squaring.
The modular multiplies and squares use either a simple division or
the REDC method by Montgomery (*note References::). REDC is a little
faster, essentially saving N single limb divisions in a fashion similar
to an exact remainder (*note Exact Remainder::). The current REDC has
some limitations. It's only O(N^2) so above `POWM_THRESHOLD' division
becomes faster and is used. It doesn't attempt to detect small bases,
but rather always uses a REDC form, which is usually a full size
operand. And lastly it's only applied to odd moduli.

File: mpir.info, Node: Root Extraction Algorithms, Next: Radix Conversion Algorithms, Prev: Powering Algorithms, Up: Algorithms
15.5 Root Extraction Algorithms
===============================
* Menu:
* Square Root Algorithm::
* Nth Root Algorithm::
* Perfect Square Algorithm::
* Perfect Power Algorithm::

File: mpir.info, Node: Square Root Algorithm, Next: Nth Root Algorithm, Prev: Root Extraction Algorithms, Up: Root Extraction Algorithms
15.5.1 Square Root
------------------
Square roots are taken using the "Karatsuba Square Root" algorithm by
Paul Zimmermann (*note References::).
An input n is split into four parts of k bits each, so with b=2^k we
have n = a3*b^3 + a2*b^2 + a1*b + a0. Part a3 must be "normalized" so
that either the high or second highest bit is set. In MPIR, k is kept
on a limb boundary and the input is left shifted (by an even number of
bits) to normalize.
The square root of the high two parts is taken, by recursive
application of the algorithm (bottoming out in a one-limb Newton's
method),
s1,r1 = sqrtrem (a3*b + a2)
This is an approximation to the desired root and is extended by a
division to give s,r,
q,u = divrem (r1*b + a1, 2*s1)
s = s1*b + q
r = u*b + a0 - q^2
The normalization requirement on a3 means at this point s is either
correct or 1 too big. r is negative in the latter case, so
if r < 0 then
r = r + 2*s - 1
s = s - 1
The algorithm is expressed in a divide and conquer form, but as
noted in the paper it can also be viewed as a discrete variant of
Newton's method, or as a variation on the schoolboy method (no longer
taught) for square roots two digits at a time.
If the remainder r is not required then usually only a few high limbs
of r and u need to be calculated to determine whether an adjustment to
s is required. This optimization is not currently implemented.
In the Karatsuba multiplication range this algorithm is
O(1.5*M(N/2)), where M(n) is the time to multiply two numbers of n
limbs. In the FFT multiplication range this grows to a bound of
O(6*M(N/2)). In practice a factor of about 1.5 to 1.8 is found in the
Karatsuba and Toom-3 ranges, growing to 2 or 3 in the FFT range.
The algorithm does all its calculations in integers and the resulting
`mpn_sqrtrem' is used for both `mpz_sqrt' and `mpf_sqrt'. The extended
precision given by `mpf_sqrt_ui' is obtained by padding with zero limbs.

File: mpir.info, Node: Nth Root Algorithm, Next: Perfect Square Algorithm, Prev: Square Root Algorithm, Up: Root Extraction Algorithms
15.5.2 Nth Root
---------------
Integer Nth roots are taken using Newton's method with the following
iteration, where A is the input and n is the root to be taken.
1 A
a[i+1] = - * ( --------- + (n-1)*a[i] )
n a[i]^(n-1)
The initial approximation a[1] is generated bitwise by successively
powering a trial root with or without new 1 bits, aiming to be just
above the true root. The iteration converges quadratically when
started from a good approximation. When n is large more initial bits
are needed to get good convergence. The current implementation is not
particularly well optimized.

File: mpir.info, Node: Perfect Square Algorithm, Next: Perfect Power Algorithm, Prev: Nth Root Algorithm, Up: Root Extraction Algorithms
15.5.3 Perfect Square
---------------------
A significant fraction of non-squares can be quickly identified by
checking whether the input is a quadratic residue modulo small integers.
`mpz_perfect_square_p' first tests the input mod 256, which means
just examining the low byte. Only 44 different values occur for
squares mod 256, so 82.8% of inputs can be immediately identified as
non-squares.
On a 32-bit system similar tests are done mod 9, 5, 7, 13 and 17,
for a total 99.25% of inputs identified as non-squares. On a 64-bit
system 97 is tested too, for a total 99.62%.
These moduli are chosen because they're factors of 2^24-1 (or 2^48-1
for 64-bits), and such a remainder can be quickly taken just using
additions (see `mpn_mod_34lsub1').
When nails are in use moduli are instead selected by the `gen-psqr.c'
program and applied with an `mpn_mod_1'. The same 2^24-1 or 2^48-1
could be done with nails using some extra bit shifts, but this is not
currently implemented.
In any case each modulus is applied to the `mpn_mod_34lsub1' or
`mpn_mod_1' remainder and a table lookup identifies non-squares. By
using a "modexact" style calculation, and suitably permuted tables,
just one multiply each is required, see the code for details. Moduli
are also combined to save operations, so long as the lookup tables
don't become too big. `gen-psqr.c' does all the pre-calculations.
A square root must still be taken for any value that passes these
tests, to verify it's really a square and not one of the small fraction
of non-squares that get through (ie. a pseudo-square to all the tested
bases).
Clearly more residue tests could be done, `mpz_perfect_square_p' only
uses a compact and efficient set. Big inputs would probably benefit
from more residue testing, small inputs might be better off with less.
The assumed distribution of squares versus non-squares in the input
would affect such considerations.

File: mpir.info, Node: Perfect Power Algorithm, Prev: Perfect Square Algorithm, Up: Root Extraction Algorithms
15.5.4 Perfect Power
--------------------
Detecting perfect powers is required by some factorization algorithms.
Currently `mpz_perfect_power_p' is implemented using repeated Nth root
extractions, though naturally only prime roots need to be considered.
(*Note Nth Root Algorithm::.)
If a prime divisor p with multiplicity e can be found, then only
roots which are divisors of e need to be considered, much reducing the
work necessary. To this end divisibility by a set of small primes is
checked.

File: mpir.info, Node: Radix Conversion Algorithms, Next: Other Algorithms, Prev: Root Extraction Algorithms, Up: Algorithms
15.6 Radix Conversion
=====================
Radix conversions are less important than other algorithms. A program
dominated by conversions should probably use a different data
representation.
* Menu:
* Binary to Radix::
* Radix to Binary::

File: mpir.info, Node: Binary to Radix, Next: Radix to Binary, Prev: Radix Conversion Algorithms, Up: Radix Conversion Algorithms
15.6.1 Binary to Radix
----------------------
Conversions from binary to a power-of-2 radix use a simple and fast
O(N) bit extraction algorithm.
Conversions from binary to other radices use one of two algorithms.
Sizes below `GET_STR_PRECOMPUTE_THRESHOLD' use a basic O(N^2) method.
Repeated divisions by b^n are made, where b is the radix and n is the
biggest power that fits in a limb. But instead of simply using the
remainder r from such divisions, an extra divide step is done to give a
fractional limb representing r/b^n. The digits of r can then be
extracted using multiplications by b rather than divisions. Special
case code is provided for decimal, allowing multiplications by 10 to
optimize to shifts and adds.
Above `GET_STR_PRECOMPUTE_THRESHOLD' a sub-quadratic algorithm is
used. For an input t, powers b^(n*2^i) of the radix are calculated,
until a power between t and sqrt(t) is reached. t is then divided by
that largest power, giving a quotient which is the digits above that
power, and a remainder which is those below. These two parts are in
turn divided by the second highest power, and so on recursively. When
a piece has been divided down to less than `GET_STR_DC_THRESHOLD'
limbs, the basecase algorithm described above is used.
The advantage of this algorithm is that big divisions can make use
of the sub-quadratic divide and conquer division (*note Divide and
Conquer Division::), and big divisions tend to have less overheads than
lots of separate single limb divisions anyway. But in any case the
cost of calculating the powers b^(n*2^i) must first be overcome.
`GET_STR_PRECOMPUTE_THRESHOLD' and `GET_STR_DC_THRESHOLD' represent
the same basic thing, the point where it becomes worth doing a big
division to cut the input in half. `GET_STR_PRECOMPUTE_THRESHOLD'
includes the cost of calculating the radix power required, whereas
`GET_STR_DC_THRESHOLD' assumes that's already available, which is the
case when recursing.
Since the base case produces digits from least to most significant
but they want to be stored from most to least, it's necessary to
calculate in advance how many digits there will be, or at least be sure
not to underestimate that. For MPIR the number of input bits is
multiplied by `chars_per_bit_exactly' from `mp_bases', rounding up.
The result is either correct or one too big.
Examining some of the high bits of the input could increase the
chance of getting the exact number of digits, but an exact result every
time would not be practical, since in general the difference between
numbers 100... and 99... is only in the last few bits and the work to
identify 99... might well be almost as much as a full conversion.
`mpf_get_str' doesn't currently use the algorithm described here, it
multiplies or divides by a power of b to move the radix point to the
just above the highest non-zero digit (or at worst one above that
location), then multiplies by b^n to bring out digits. This is O(N^2)
and is certainly not optimal.
The r/b^n scheme described above for using multiplications to bring
out digits might be useful for more than a single limb. Some brief
experiments with it on the base case when recursing didn't give a
noticeable improvement, but perhaps that was only due to the
implementation. Something similar would work for the sub-quadratic
divisions too, though there would be the cost of calculating a bigger
radix power.
Another possible improvement for the sub-quadratic part would be to
arrange for radix powers that balanced the sizes of quotient and
remainder produced, ie. the highest power would be an b^(n*k)
approximately equal to sqrt(t), not restricted to a 2^i factor. That
ought to smooth out a graph of times against sizes, but may or may not
be a net speedup.
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