This library is free; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you.
Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MP library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MP library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.
The precise conditions of the license for the GNU MP library are found in the Lesser General Public License version 2.1 that accompanies the source code, see `COPYING.LIB'. Certain demonstration programs are provided under the terms of the plain General Public License version 2, see `COPYING'.
GNU MP is a portable library written in C for arbitrary precision arithmetic on integers, rational numbers, and floating-point numbers. It aims to provide the fastest possible arithmetic for all applications that need higher precision than is directly supported by the basic C types.
Many applications use just a few hundred bits of precision; but some applications may need thousands or even millions of bits. GMP is designed to give good performance for both, by choosing algorithms based on the sizes of the operands, and by carefully keeping the overhead at a minimum.
The speed of GMP is achieved by using fullwords as the basic arithmetic type, by using sophisticated algorithms, by including carefully optimized assembly code for the most common inner loops for many different CPUs, and by a general emphasis on speed (as opposed to simplicity or elegance).
There is carefully optimized assembly code for these CPUs: ARM, DEC Alpha 21064, 21164, and 21264, AMD 29000, AMD K6, K6-2 and Athlon, Hitachi SuperH and SH-2, HPPA 1.0, 1.1 and 2.0, Intel Pentium, Pentium Pro/II/III, Pentium 4, generic x86, Intel IA-64, i960, Motorola MC68000, MC68020, MC88100, and MC88110, Motorola/IBM PowerPC 32 and 64, National NS32000, IBM POWER, MIPS R3000, R4000, SPARCv7, SuperSPARC, generic SPARCv8, UltraSPARC, DEC VAX, and Zilog Z8000. Some optimizations also for Cray vector systems, Clipper, IBM ROMP (RT), and Pyramid AP/XP.
There are two public mailing lists of interest. One for general questions and discussions about usage of the GMP library and one for discussions about development of GMP. There's more information about the mailing lists at http://swox.com/mailman/listinfo/. These lists are not for bug reports.
The proper place for bug reports is bug-gmp@gnu.org. See section Reporting Bugs for info about reporting bugs.
For up-to-date information on GMP, please see the GMP web pages at
http://swox.com/gmp/
The latest version of the library is available at
ftp://ftp.gnu.org/gnu/gmp
Many sites around the world mirror `ftp.gnu.org', please use a mirror near you, see http://www.gnu.org/order/ftp.html for a full list.
Everyone should read section GMP Basics. If you need to install the library yourself, then read section Installing GMP. If you have a system with multiple ABIs, then read section ABI and ISA, for the compiler options that must be used on applications.
The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.
GMP has an autoconf/automake/libtool based configuration system. On a Unix-like system a basic build can be done with
./configure make
Some self-tests can be run with
make check
And you can install (under `/usr/local' by default) with
make install
If you experience problems, please report them to bug-gmp@gnu.org. See section Reporting Bugs, for information on what to include in useful bug reports.
All the usual autoconf configure options are available, run `./configure --help' for a summary. The file `INSTALL.autoconf' has some generic installation information too.
http://www.cygnus.com/cygwin http://www.delorie.com/djgpp http://www.mingw.orgMicrosoft also publishes an Interix "Services for Unix" which can be used to build GMP on Windows (with a normal `./configure'), but it's not free software. The `macos' directory contains an unsupported port to MacOS 9 on Power Macintosh, see `macos/README'. Note that MacOS X "Darwin" should use the normal Unix-style `./configure'. It might be possible to build without the help of `configure', certainly all the code is there, but unfortunately you'll be on your own.
cd /my/build/dir /my/sources/gmp-4.1.2/configureNot all `make' programs have the necessary features (
VPATH
) to
support this. In particular, SunOS and Slowaris @command{make} have bugs that
make them unable to build in a separate directory. Use GNU @command{make}
instead.
./configure --build=ultrasparc-sun-solaris2.7In 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.
./configure --build=athlon-pc-freebsd3.5 --host=m68k-mac-linux-gnuCompiler tools are sought first with the host system type as a prefix. For example @command{m68k-mac-linux-gnu-ranlib} is tried, then plain @command{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 @env{PATH} with a suitable cross-compiling @command{cc} etc. Compiling for a different CPU in the same family as the build system is a form of cross-compilation, though very possibly this would merely be special options on a native compiler. In any case `./configure' avoids depending on being able to run code on the build system, which is important when creating binaries for a newer CPU since they very possibly won't run on the build system. In all cases the compiler must be able to produce an executable (of whatever format) from a standard C
main
. Although only object files will go to
make up `libgmp', `./configure' uses linking tests for various
purposes, such as determining what functions are available on the host system.
Currently a warning is given unless an explicit `--build' is used when
cross-compiling, because it may not be possible to correctly guess the build
system type if the @env{PATH} has only a cross-compiling @command{cc}.
Note that the `--target' option is not appropriate for GMP. It's for use
when building compiler tools, with `--host' being where they will run,
and `--target' what they'll produce code for. Ordinary programs or
libraries like GMP are only interested in the `--host' part, being where
they'll run. (Some past versions of GMP used `--target' incorrectly.)
./configure --host=none-unknown-freebsd3.5Note 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.
./configure --host=mips64-sgi-irix6 ABI=n32See section ABI and ISA, for the available choices on relevant CPUs, and what applications need to do.
alloca
if available, otherwise
`malloc-reentrant'. This is the default.
alloca
if available, otherwise
`malloc-notreentrant'.
alloca
is reentrant and fast, and is recommended, but when working with
large numbers it can overflow the available stack space, in which case one of
the two malloc methods will need to be used. Alternately it might be possible
to increase available stack with @command{limit}, @command{ulimit} or
setrlimit
, or under DJGPP with @command{stubedit} or
_stklen
. Note that depending on the system the only indication of
stack overflow might be a segmentation violation.
`malloc-reentrant' is, as the name suggests, reentrant and thread safe,
but `malloc-notreentrant' is faster and should be used if reentrancy is
not required.
The two malloc methods in fact use the memory allocation functions selected by
mp_set_memory_functions
, these being malloc
and friends by
default. See section Custom Allocation.
An additional choice `--enable-alloca=debug' is available, to help when
debugging memory related problems (see section Debugging).
mcount
calls to the assembler code.
See section Profiling.
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.
ABI (Application Binary Interface) refers to the calling conventions between functions, meaning what registers are used and what sizes the various C data types are. ISA (Instruction Set Architecture) refers to the instructions and registers a CPU has available.
Some 64-bit ISA CPUs have both a 64-bit ABI and a 32-bit ABI defined, the
latter for compatibility with older CPUs in the family. GMP supports some
CPUs like this in both ABIs. In fact within GMP `ABI' means a
combination of chip ABI, plus how GMP chooses to use it. For example in some
32-bit ABIs, GMP may support a limb as either a 32-bit long
or a 64-bit
long long
.
By default GMP chooses the best ABI available for a given system, and this generally gives significantly greater speed. But an ABI can be chosen explicitly to make GMP compatible with other libraries, or particular application requirements. For example,
./configure ABI=32
In all cases it's vital that all object code used in a given program is compiled for the same ABI.
Usually a limb is implemented as a long
. When a long long
limb
is used this is encoded in the generated `gmp.h'. This is convenient for
applications, but it does mean that `gmp.h' will vary, and can't be just
copied around. `gmp.h' remains compiler independent though, since all
compilers for a particular ABI will be expected to use the same limb type.
Currently no attempt is made to follow whatever conventions a system has for installing library or header files built for a particular ABI. This will probably only matter when installing multiple builds of GMP, and it might be as simple as configuring with a special `libdir', or it might require more than that. Note that builds for different ABIs need to done separately, with a fresh @command{./configure} and @command{make} each.
cc +DD64
long long
. This is available on HP-UX 10 or up when using
@command{cc}. No @command{gcc} support is planned for this. Applications
must be compiled with
cc +DA2.0 +e
long long
. Applications must be compiled with
gcc -mabi=n32 cc -n32
gcc -mabi=64 cc -64
gcc -maix64 xlc -q64
gcc -m64 -mptr64 -Wa,-xarch=v9 -mcpu=v9 cc -xarch=v9
gcc -mv8plus cc -xarch=v8plus@command{gcc} 2.8 and earlier only supports `-mv8' though.
./configure --build=none --host=sparcv9-sun-solaris2.7 ABI=64
GMP should present no great difficulties for packaging in a binary distribution.
Libtool is used to build the library and `-version-info' is set appropriately, having started from `3:0:0' in GMP 3.0. The GMP 4 series will be upwardly binary compatible in each release and will be upwardly binary compatible with all of the GMP 3 series. Additional function interfaces may be added in each release, so on systems where libtool versioning is not fully checked by the loader an auxiliary mechanism may be needed to express that a dynamic linked application depends on a new enough GMP.
An auxiliary mechanism may also be needed to express that `libgmpxx.la' (from @option{--enable-cxx}, see section Build Options) requires `libgmp.la' from the same GMP version, since this is not done by the libtool versioning, nor otherwise. A mismatch will result in unresolved symbols from the linker, or perhaps the loader.
Using `DESTDIR' or a `prefix' override with `make install' and a shared `libgmpxx' may run into a libtool relinking problem, see section Known Build Problems.
When building a package for a CPU family, care should be taken to use `--host' (or `--build') to choose the least common denominator among the CPUs which might use the package. For example this might necessitate `i386' for x86s, or plain `sparc' (meaning V7) for SPARCs.
Users who care about speed will want GMP built for their exact CPU type, to make use of the available optimizations. Providing a way to suitably rebuild a package may be useful. This could be as simple as making it possible for a user to omit `--build' (and `--host') so `./config.guess' will detect the CPU. But a way to manually specify a `--build' will be wanted for systems where `./config.guess' is inexact.
Note that `gmp.h' is a generated file, and will be architecture and ABI dependent.
./configure --enable-shared --disable-staticNote that the `--disable-static' is necessary because in a shared build libtool makes `libgmp.a' a symlink to `libgmp.so', apparently for the benefit of old versions of @command{ld} which only recognise `.a', but unfortunately this is done even if a fully functional @command{ld} is available.
iostream
, a standard one and
an old pre-standard one (see `man iostream_intro'). GMP can only use the
standard one, which unfortunately is not the default but must be selected by
defining __USE_STD_IOSTREAM
. Configure with for instance
./configure --enable-cxx CPPFLAGS=-D__USE_STD_IOSTREAM
./configure --disable-static --enable-sharedStatic and DLL libraries can't both be built, since certain export directives in `gmp.h' must be different. `--enable-cxx' cannot be used when building a DLL, since libtool doesn't currently support C++ DLLs. This might change in the future.
lib /machine:IX86 /def:_libs/libgmp-3.dll-def cp libgmp-3.lib /my/inst/dir/lib cp _libs/libgmp-3.dll-exp /my/inst/dir/lib/libgmp-3.expMINGW uses `msvcrt.dll' for I/O, so applications wanting to use the GMP I/O routines must be compiled with `cl /MD' to do the same. If one of the other I/O choices provided by MS C is desired then the suggestion is to use the GMP string functions and confine I/O to the application.
eval
that makes it
unsuitable for `.asm' file processing. `./configure' will detect
the problem and either abort or choose another m4 in the @env{PATH}. The bug
is fixed in OpenBSD 2.7, so either upgrade or use GNU m4.
g2
, g3
and g4
, the same way
that the GCC default `-mapp-regs' does (see section `SPARC Options' in Using the GNU Compiler Collection (GCC)).
This makes that code unsuitable for use with the special V9
`-mcmodel=embmedany' (which uses g4
as a data segment pointer),
and for applications wanting to use those registers for special purposes. In
these cases the only suggestion currently is to build GMP with CPU `none'
to avoid the assembler code.
movq
instructions, and so can't be used for MMX code.
Install a recent @command{gas} if MMX code is wanted on these systems.
You might find more up-to-date information at http://swox.com/gmp/.
./configure CC=gcc-with-my-options
LD_LIBRARY_PATH
. For example with `--prefix=/usr' but
installing under `/my/staging/area',
LD_LIBRARY_PATH=/my/staging/area/usr/lib \ make install DESTDIR=/my/staging/area
libgmp.la: $(libgmp_la_OBJECTS) $(libgmp_la_DEPENDENCIES)Either use GNU Make, or as a workaround remove
$(libgmp_la_DEPENDENCIES)
from that line (which will make the initial
build work, but if any recompiling is done `libgmp.la' might not be
rebuilt).
gmp_randinit_lc_2exp_size
. The exact cause is unknown,
`--disable-shared' is recommended.
Using functions, macros, data types, etc. not documented in this manual is strongly discouraged. If you do so your application is guaranteed to be incompatible with future versions of GMP.
All declarations needed to use GMP are collected in the include file `gmp.h'. It is designed to work with both C and C++ compilers.
#include <gmp.h>
Note however that prototypes for GMP functions with FILE *
parameters
are only provided if <stdio.h>
is included too.
#include <stdio.h> #include <gmp.h>
Likewise <stdarg.h>
(or <varargs.h>
) is required for prototypes
with va_list
parameters, such as gmp_vprintf
. And
<obstack.h>
for prototypes with struct obstack
parameters, such
as gmp_obstack_printf
, when available.
All programs using GMP must link against the `libgmp' library. On a typical Unix-like system this can be done with `-lgmp', for example
gcc myprogram.c -lgmp
GMP C++ functions are in a separate `libgmpxx' library. This is built and installed if C++ support has been enabled (see section Build Options). For example,
g++ mycxxprog.cc -lgmpxx -lgmp
GMP is built using Libtool and an application can use that to link if desired, see section `Introduction' in GNU Libtool
If GMP has been installed to a non-standard location then it may be necessary to use `-I' and `-L' compiler options to point to the right directories, and some sort of run-time path for a shared library. Consult your compiler documentation, for instance section `Introduction' in Using and Porting the GNU Compiler Collection.
In this manual, integer usually means a multiple precision integer, as
defined by the GMP library. The C data type for such integers is mpz_t
.
Here are some examples of how to declare such integers:
mpz_t sum; struct foo { mpz_t x, y; }; mpz_t vec[20];
Rational number means a multiple precision fraction. The C data type
for these fractions is mpq_t
. For example:
mpq_t quotient;
Floating point number or Float for short, is an arbitrary precision
mantissa with a limited precision exponent. The C data type for such objects
is mpf_t
.
A limb means the part of a multi-precision number that fits in a single
machine word. (We chose this word because a limb of the human body is
analogous to a digit, only larger, and containing several digits.) Normally a
limb is 32 or 64 bits. The C data type for a limb is mp_limb_t
.
There are six classes of functions in the GMP library:
mpz_
. The associated type is mpz_t
. There are about 150
functions in this class.
mpq_
. The associated type is mpq_t
. There are about 40
functions in this class, but the integer functions can be used for arithmetic
on the numerator and denominator separately.
mpf_
. The associated type is mpf_t
. There are about 60
functions is this class.
itom
, madd
, and
mult
. The associated type is MINT
.
mpn_
. The associated type is array of mp_limb_t
. There are
about 30 (hard-to-use) functions in this class.
GMP functions generally have output arguments before input arguments. This notation is by analogy with the assignment operator. The BSD MP compatibility functions are exceptions, having the output arguments last.
GMP lets you use the same variable for both input and output in one call. For
example, the main function for integer multiplication, mpz_mul
, can be
used to square x
and put the result back in x
with
mpz_mul (x, x, x);
Before you can assign to a GMP variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. Which function to use depends on the type of variable. See the chapters on integer functions, rational number functions, and floating-point functions for details.
A variable should only be initialized once, or at least cleared between each initialization. After a variable has been initialized, it may be assigned to any number of times.
For efficiency reasons, avoid excessive initializing and clearing. In general, initialize near the start of a function and clear near the end. For example,
void foo (void) { mpz_t n; int i; mpz_init (n); for (i = 1; i < 100; i++) { mpz_mul (n, ...); mpz_fdiv_q (n, ...); ... } mpz_clear (n); }
When a GMP variable is used as a function parameter, it's effectively a
call-by-reference, meaning if the function stores a value there it will change
the original in the caller. Parameters which are input-only can be designated
const
to provoke a compiler error or warning on attempting to modify
them.
When a function is going to return a GMP result, it should designate a
parameter that it sets, like the library functions do. More than one value
can be returned by having more than one output parameter, again like the
library functions. A return
of an mpz_t
etc doesn't return the
object, only a pointer, and this is almost certainly not what's wanted.
Here's an example accepting an mpz_t
parameter, doing a calculation,
and storing the result to the indicated parameter.
void foo (mpz_t result, const mpz_t param, unsigned long n) { unsigned long i; mpz_mul_ui (result, param, n); for (i = 1; i < n; i++) mpz_add_ui (result, result, i*7); } int main (void) { mpz_t r, n; mpz_init (r); mpz_init_set_str (n, "123456", 0); foo (r, n, 20L); gmp_printf ("%Zd\n", r); return 0; }
foo
works even if the mainline passes the same variable for
param
and result
, just like the library functions. But
sometimes it's tricky to make that work, and an application might not want to
bother supporting that sort of thing.
For interest, the GMP types mpz_t
etc are implemented as one-element
arrays of certain structures. This is why declaring a variable creates an
object with the fields GMP needs, but then using it as a parameter passes a
pointer to the object. Note that the actual fields in each mpz_t
etc
are for internal use only and should not be accessed directly by code that
expects to be compatible with future GMP releases.
The GMP types like mpz_t
are small, containing only a couple of sizes,
and pointers to allocated data. Once a variable is initialized, GMP takes
care of all space allocation. Additional space is allocated whenever a
variable doesn't have enough.
mpz_t
and mpq_t
variables never reduce their allocated space.
Normally this is the best policy, since it avoids frequent reallocation.
Applications that need to return memory to the heap at some particular point
can use mpz_realloc2
, or clear variables no longer needed.
mpf_t
variables, in the current implementation, use a fixed amount of
space, determined by the chosen precision and allocated at initialization, so
their size doesn't change.
All memory is allocated using malloc
and friends by default, but this
can be changed, see section Custom Allocation. Temporary memory on the stack is
also used (via alloca
), but this can be changed at build-time if
desired, see section Build Options.
GMP is reentrant and thread-safe, with some exceptions:
alloca
is not available),
then naturally GMP is not reentrant.
mpf_set_default_prec
and mpf_init
use a global variable for the
selected precision. mpf_init2
can be used instead.
mpz_random
and the other old random number functions use a global
random state and are hence not reentrant. The newer random number functions
that accept a gmp_randstate_t
parameter can be used instead.
mp_set_memory_functions
uses global variables to store the selected
memory allocation functions.
mp_set_memory_functions
(or malloc
and friends by default) are
not reentrant, then GMP will not be reentrant either.
fwrite
are not reentrant then the
GMP I/O functions using them will not be reentrant either.
gmp_randstate_t
simultaneously,
since this involves an update of that variable.
<ctype.h>
macros use per-file static
variables and may not be reentrant, depending whether the compiler optimizes
away fetches from them. The GMP text-based input functions are affected.
This version of GMP is upwardly binary compatible with all 4.x and 3.x versions, and upwardly compatible at the source level with all 2.x versions, with the following exceptions.
mpn_gcd
had its source arguments swapped as of GMP 3.0, for consistency
with other mpn
functions.
mpf_get_prec
counted precision slightly differently in GMP 3.0 and
3.0.1, but in 3.1 reverted to the 2.x style.
There are a number of compatibility issues between GMP 1 and GMP 2 that of course also apply when porting applications from GMP 1 to GMP 4. Please see the GMP 2 manual for details.
The Berkeley MP compatibility library (see section Berkeley MP Compatible Functions) is source and binary compatible with the standard `libmp'.
The `demos' subdirectory has some sample programs using GMP. These aren't built or installed, but there's a `Makefile' with rules for them. For instance,
make pexpr ./pexpr 68^975+10
The following programs are provided
mpz_probab_prime_p
function.
mpz_kronecker_ui
to estimate quadratic
class numbers.
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
(see section Initialization Functions).
It doesn't matter if a size set with mpz_init2
or mpz_realloc2
is too small, since all functions will do a further reallocation if necessary.
Badly overestimating memory required will waste space though.
2exp
functions
mpz_mul_2exp
when
appropriate. General purpose functions like mpz_mul
make no attempt to
identify powers of two or other special forms, because such inputs will
usually be very rare and testing every time would be wasteful.
ui
and si
functions
ui
functions and the small number of si
functions exist for
convenience and should be used where applicable. But if for example an
mpz_t
contains a value that fits in an unsigned long
there's no
need extract it and call a ui
function, just use the regular mpz
function.
mpz_abs
, mpq_abs
, mpf_abs
, mpz_neg
, mpq_neg
and mpf_neg
are fast when used for in-place operations like
mpz_abs(x,x)
, since in the current implementation only a single field
of x
needs changing. On suitable compilers (GCC for instance) this is
inlined too.
mpz_add_ui
, mpz_sub_ui
, mpf_add_ui
and mpf_sub_ui
benefit from an in-place operation like mpz_add_ui(x,x,y)
, since
usually only one or two limbs of x
will need to be changed. The same
applies to the full precision mpz_add
etc if y
is small. If
y
is big then cache locality may be helped, but that's all.
mpz_mul
is currently the opposite, a separate destination is slightly
better. A call like mpz_mul(x,x,y)
will, unless y
is only one
limb, make a temporary copy of x
before forming the result. Normally
that copying will only be a tiny fraction of the time for the multiply, so
this is not a particularly important consideration.
mpz_set
, mpq_set
, mpq_set_num
, mpf_set
, etc, make
no attempt to recognise a copy of something to itself, so a call like
mpz_set(x,x)
will be wasteful. Naturally that would never be written
deliberately, but if it might arise from two pointers to the same object then
a test to avoid it might be desirable.
if (x != y) mpz_set (x, y);Note that it's never worth introducing extra
mpz_set
calls just to get
in-place operations. If a result should go to a particular variable then just
direct it there and let GMP take care of data movement.
mpz_divisible_ui_p
and mpz_congruent_ui_p
are the best functions
for testing whether an mpz_t
is divisible by an individual small
integer. They use an algorithm which is faster than mpz_tdiv_ui
, but
which gives no useful information about the actual remainder, only whether
it's zero (or a particular value).
However when testing divisibility by several small integers, it's best to take
a remainder modulo their product, to save multi-precision operations. For
instance to test whether a number is divisible by any of 23, 29 or 31 take a
remainder modulo 23@times{29@times{}31 = 20677} and then test that.
The division functions like mpz_tdiv_q_ui
which give a quotient as well
as a remainder are generally a little slower than the remainder-only functions
like mpz_tdiv_ui
. If the quotient is only rarely wanted then it's
probably best to just take a remainder and then go back and calculate the
quotient if and when it's wanted (mpz_divexact_ui
can be used if the
remainder is zero).
mpq
functions operate on mpq_t
values with no common factors
in the numerator and denominator. Common factors are checked-for and cast out
as necessary. In general, cancelling factors every time is the best approach
since it minimizes the sizes for subsequent operations.
However, applications that know something about the factorization of the
values they're working with might be able to avoid some of the GCDs used for
canonicalization, or swap them for divisions. For example when multiplying by
a prime it's enough to check for factors of it in the denominator instead of
doing a full GCD. Or when forming a big product it might be known that very
little cancellation will be possible, and so canonicalization can be left to
the end.
The mpq_numref
and mpq_denref
macros give access to the
numerator and denominator to do things outside the scope of the supplied
mpq
functions. See section Applying Integer Functions to Rationals.
The canonical form for rationals allows mixed-type mpq_t
and integer
additions or subtractions to be done directly with multiples of the
denominator. This will be somewhat faster than mpq_add
. For example,
/* mpq increment */ mpz_add (mpq_numref(q), mpq_numref(q), mpq_denref(q)); /* mpq += unsigned long */ mpz_addmul_ui (mpq_numref(q), mpq_denref(q), 123UL); /* mpq -= mpz */ mpz_submul (mpq_numref(q), mpq_denref(q), z);
mpz_fac_ui
, mpz_fib_ui
and mpz_bin_uiui
are designed for calculating isolated values. If a range of values is wanted
it's probably best to call to get a starting point and iterate from there.
init
GMP variables will have unpredictable effects, and
corruption arising elsewhere in a program may well affect GMP. Initializing
GMP variables more than once or failing to clear them will cause memory leaks.
In all such cases a malloc debugger is recommended. On a GNU or BSD system
the standard C library malloc
has some diagnostic facilities, see
section `Allocation Debugging' in The GNU C Library Reference Manual, or
`man 3 malloc'. Other possibilities, in no particular order, include
http://www.inf.ethz.ch/personal/biere/projects/ccmalloc http://quorum.tamu.edu/jon/gnu (debauch) http://dmalloc.com http://www.perens.com/FreeSoftware (electric fence) http://packages.debian.org/fda http://www.gnupdate.org/components/leakbug http://people.redhat.com/~otaylor/memprof http://www.cbmamiga.demon.co.uk/mpatrolThe GMP default allocation routines in `memory.c' also have a simple sentinel scheme which can be enabled with
#define DEBUG
in that file.
This is mainly designed for detecting buffer overruns during GMP development,
but might find other uses.
cd /my/build/dir /my/source/dir/gmp-4.1.2/configureThis works via
VPATH
, and might require GNU @command{make}.
Alternately it might be possible to change the .c.lo
rules
appropriately.
mpn
functions, however, will benefit
from @option{--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.
malloc
(or
the allocation function set with mp_set_memory_functions
).
This can help a malloc debugger detect accesses outside the intended bounds,
or detect memory not released. In a normal build, on the other hand,
temporary memory is allocated in blocks which GMP divides up for its own use,
or may be allocated with a compiler builtin alloca
which will go
nowhere near any malloc debugger hooks.
./configure --disable-shared --enable-assert \ --enable-alloca=debug --host=none CFLAGS=-gFor C++, add `--enable-cxx CXXFLAGS=-g'.
./configure --host=none-pc-linux-gnu CC=checkergcc`--host=none' must be used, since the GMP assembler code doesn't support the checking scheme. The GMP C++ features cannot be used, since current versions of checker (0.9.9.1) don't yet support the standard C++ library.
MPN_PATH
that excludes those subdirectories (see section Build Options).
Running a program under a profiler is a good way to find where it's spending most time and where improvements can be best sought.
Depending on the system, it may be possible to get a flat profile, meaning simple timer sampling of the program counter, with no special GMP build options, just a `-p' when compiling the mainline. This is a good way to ensure minimum interference with normal operation. The necessary symbol type and size information exists in most of the GMP assembler code.
The `--enable-profiling' build option can be used to add suitable
compiler flags, either for @command{prof} (`-p') or @command{gprof}
(`-pg'), see section Build Options. Which of the two is available and what
they do will depend on the system, and possibly on support available in
`libc'. For some systems appropriate corresponding mcount
calls
are added to the assembler code too.
On x86 systems @command{prof} gives call counting, so that average time spent
in a function can be determined. @command{gprof}, where supported, adds call
graph construction, so for instance calls to mpn_add_n
from
mpz_add
and from mpz_mul
can be differentiated.
On x86 and 68k systems `-pg' and `-fomit-frame-pointer' are incompatible, so the latter is not used when @command{gprof} profiling is selected, which may result in poorer code generation. If @command{prof} profiling is selected instead it should still be possible to use @command{gprof}, but only the `gprof -p' flat profile and call counts can be expected to be valid, not the `gprof -q' call graph.
Autoconf based applications can easily check whether GMP is installed. The
only thing to be noted is that GMP library symbols from version 3 onwards have
prefixes like __gmpz
. The following therefore would be a simple test,
AC_CHECK_LIB(gmp, __gmpz_init)
This just uses the default AC_CHECK_LIB
actions for found or not found,
but an application that must have GMP would want to generate an error if not
found. For example,
AC_CHECK_LIB(gmp, __gmpz_init, , [AC_MSG_ERROR( [GNU MP not found, see http://swox.com/gmp])])
If functions added in some particular version of GMP are required, then one of
those can be used when checking. For example mpz_mul_si
was added in
GMP 3.1,
AC_CHECK_LIB(gmp, __gmpz_mul_si, , [AC_MSG_ERROR( [GNU MP not found, or not 3.1 or up, see http://swox.com/gmp])])
An alternative would be to test the version number in `gmp.h' using say
AC_EGREP_CPP
. That would make it possible to test the exact version,
if some particular sub-minor release is known to be necessary.
An application that can use either GMP 2 or 3 will need to test for
__gmpz_init
(GMP 3 and up) or mpz_init
(GMP 2), and it's also
worth checking for `libgmp2' since Debian GNU/Linux systems used that
name in the past. For example,
AC_CHECK_LIB(gmp, __gmpz_init, , [AC_CHECK_LIB(gmp, mpz_init, , [AC_CHECK_LIB(gmp2, mpz_init)])])
In general it's suggested that applications should simply demand a new enough GMP rather than trying to provide supplements for features not available in past versions.
Occasionally an application will need or want to know the size of a type at
configuration or preprocessing time, not just with sizeof
in the code.
This can be done in the normal way with mp_limb_t
etc, but GMP 4.0 or
up is best for this, since prior versions needed certain `-D' defines on
systems using a long long
limb. The following would suit Autoconf 2.50
or up,
AC_CHECK_SIZEOF(mp_limb_t, , [#include <gmp.h>])
The optional mpfr
functions are provided in a separate
`libmpfr.a', and this might be from GMP with @option{--enable-mpfr} or
from MPFR installed separately. Either way `libmpfr' depends on
`libgmp', it doesn't stand alone. Currently only a static
`libmpfr.a' will be available, not a shared library, since upward binary
compatibility is not guaranteed.
AC_CHECK_LIB(mpfr, mpfr_add, , [AC_MSG_ERROR( [Need MPFR either from GNU MP 4 or separate MPFR package. See http://www.mpfr.org or http://swox.com/gmp])
C-h C-i (info-lookup-symbol
) is a good way to find documentation
on C functions while editing (see section `Info Documentation Lookup' in The Emacs Editor).
The GMP manual can be included in such lookups by putting the following in your `.emacs',
(eval-after-load "info-look" '(let ((mode-value (assoc 'c-mode (assoc 'symbol info-lookup-alist)))) (setcar (nthcdr 3 mode-value) (cons '("(gmp)Function Index" nil "^ -.* " "\\>") (nth 3 mode-value)))))
The same can be done for MPFR, with (mpfr)
in place of (gmp)
.
If you think you have found a bug in the GMP library, please investigate it and report it. We have made this library available to you, and it is not too much to ask you to report the bugs you find.
Before you report a bug, check it's not already addressed in section Known Build Problems, or perhaps section Notes for Particular Systems. You may also want to check http://swox.com/gmp/ for patches for this release.
Please include the following in any report,
Please make an effort to produce a self-contained report, with something definite that can be tested or debugged. Vague queries or piecemeal messages are difficult to act on and don't help the development effort.
It is not uncommon that an observed problem is actually due to a bug in the compiler; the GMP code tends to explore interesting corners in compilers.
If your bug report is good, we will do our best to help you get a corrected version of the library; if the bug report is poor, we won't do anything about it (except maybe ask you to send a better report).
Send your report to: bug-gmp@gnu.org.
If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.
This chapter describes the GMP functions for performing integer arithmetic.
These functions start with the prefix mpz_
.
GMP integers are stored in objects of type mpz_t
.
The functions for integer arithmetic assume that all integer objects are
initialized. You do that by calling the function mpz_init
. For
example,
{ mpz_t integ; mpz_init (integ); ... mpz_add (integ, ...); ... mpz_sub (integ, ...); /* Unless the program is about to exit, do ... */ mpz_clear (integ); }
As you can see, you can store new values any number of times, once an object is initialized.
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.
mpz_t
variables when you are done with them.
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.
The space will not be automatically increased, unlike the normal
mpz_init
, but instead an application must ensure it's sufficient for
any value stored. The following space requirements apply to various
functions,
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
.
mpz_array_init
can reduce memory usage in algorithms that need large
arrays of integers, since it avoids allocating and reallocating lots of small
memory blocks. There is no way to free the storage allocated by this
function. Don't call mpz_clear
!
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 the new size in limbs.
These functions assign new values to already initialized integers (see section Initialization Functions).
mpz_set_d
, mpz_set_q
and mpz_set_f
truncate op to
make it an integer.
This function returns 0 if the entire string is a valid number in base base. Otherwise it returns -1.
[It turns out that it is not entirely true that this function ignores white-space. It does ignore it between digits, but not after a minus sign or within or after "0x". We are considering changing the definition of this function, making it fail when there is any white-space in the input, since that makes a lot of sense. Send your opinion of this change to bug-gmp@gnu.org. Do you really want it to accept @nicode{"3 14"} as meaning 314 as it does now?]
For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there. These functions'
names have the form mpz_init_set...
Here is an example of using one:
{ mpz_t pie; mpz_init_set_str (pie, "3141592653589793238462643383279502884", 10); ... mpz_sub (pie, ...); ... mpz_clear (pie); }
Once the integer has been initialized by any of the mpz_init_set...
functions, it can be used as the source or destination operand for the ordinary
integer functions. Don't use an initialize-and-set function on a variable
already initialized!
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.)
This section describes functions for converting GMP integers to standard C types. Functions for converting to GMP integers are described in section Assignment Functions and section Input and Output Functions.
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.
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
.
double
.
If str is NULL
, the result string is allocated using the current
allocation function (see section 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.
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
.
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.
Divide n by d, forming a quotient q and/or remainder
r. For the 2exp
functions, @m{d=2^b, d=2^b}.
The rounding is in three styles, each suiting different applications.
cdiv
rounds q up towards @m{+\infty, +infinity}, and r will
have the opposite sign to d. The c
stands for "ceil".
fdiv
rounds q down towards @m{-\infty, -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 @m{n=qd+r, n=q*d+r}, and r will satisfy 0@le{@GMPabs{r}<@GMPabs{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.
The 2exp
functions are right shifts and bit masks, but of course
rounding 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.
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.
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.
mpz_divisible_2exp_p
by @m{2^b,2^b}.
mpz_congruent_2exp_p
modulo @m{2^b,2^b}.
Negative exp is supported if an inverse base^@W{-1 @bmod
mod} exists (see mpz_invert
in section Number Theoretic Functions).
If an inverse doesn't exist then a divide by zero is raised.
mpz_sqrt
. Set rop2 to the
remainder @m{(op - rop1^2),
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.
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.
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 uses a probabilistic algorithm to identify primes. For practical purposes it's adequate, the chance of a composite passing will be extremely small.
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.
If t is NULL
then that value is not computed.
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 (see section References),
or any number theory textbook. See also the example program
`demos/qcn.c' which uses mpz_kronecker_ui
.
mpz_bin_ui
, using the identity
@m{\left({-n}\atop{k}\right) = (-1)^k \left({n+k-1}\atop{k}\right),
bin(-n@C{}k) = (-1)^k * bin(n+k-1@C{}k)}, see Knuth volume 1 section 1.2.6
part G.
mpz_fib_ui
sets fn to to @m{F_n,F[n]}, the n'th Fibonacci
number. mpz_fib2_ui
sets fn to @m{F_n,F[n]}, and fnsub1 to
@m{F_{n-1},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 @m{F_{n+1} = F_n + F_{n-1}, F[n+1]=F[n]+F[n-1]} or
similar.
mpz_lucnum_ui
sets ln to to @m{L_n,L[n]}, the n'th Lucas
number. mpz_lucnum2_ui
sets ln to @m{L_n,L[n]}, and lnsub1
to @m{L_{n-1},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 @m{L_{n+1} = L_n + L_{n-1}, 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 section Lucas Numbers, the reverse is straightforward too.
Note that mpz_cmp_ui
and mpz_cmp_si
are macros and will evaluate
their arguments more than once.
Note that mpz_cmpabs_si
is a macro and will evaluate its arguments more
than once.
This function is actually implemented as a macro. It evaluates its argument multiple times.
These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0.
unsigned long
.
unsigned long
.
If the bit at starting_bit is already what's sought, then starting_bit is returned.
If there's no bit found, then MAX_ULONG is returned. This will happen
in mpz_scan0
past the end of a positive number, or mpz_scan1
past the end of a negative.
Functions that perform input from a stdio stream, and functions that output to
a stdio stream. Passing a NULL
pointer for a stream argument to any of
these functions will make them read from stdin
and write to
stdout
, respectively.
When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions.
Return the number of bytes written, or if an error occurred, return 0.
Return the number of bytes read, or if an error occurred, return 0.
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.
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.
The random number functions of GMP come in two groups; older function that rely on a global state, and newer functions that accept a state parameter that is read and modified. Please see the section Random Number Functions for more information on how to use and not to use random number functions.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization) before
invoking this function.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization)
before invoking this function.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization)
before invoking this function.
This function is obsolete. Use mpz_urandomb
or
mpz_urandomm
instead.
This function is obsolete. Use mpz_rrandomb
instead.
mpz_t
variables can be converted to and from arbitrary words of binary
data with the following functions.
The parameters specify the format of the data. count many words are read, each size bytes. order can be 1 for most significant word first or -1 for least significant first. Within each word endian can be 1 for most significant byte first, -1 for least significant first, or 0 for the native endianness of the host CPU. The most significant nails bits of each word are skipped, this can be 0 to use the full words.
There are no data alignment restrictions on op, any address is allowed.
Here's an example converting an array of unsigned long
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
.
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 *count
. rop
must have enough space for the data, or if rop is NULL
then a
result array of the necessary size is allocated using the current GMP
allocation function (see section Custom Allocation). In either case the return
value is the destination used, 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.
There are no data alignment restrictions on rop, any address is allowed. The sign of op is ignored, just the absolute value is used.
When an application is allocating space itself the required size can be
determined with a calculation like the following. Since 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.
numb = 8*size - nail; count = (mpz_sizeinbase (z, 2) + numb-1) / numb; p = malloc (count * size);
unsigned long int
,
signed long int
, unsigned int
, signed int
, unsigned
short int
, or signed short int
, respectively. Otherwise, return zero.
This function is useful in order to allocate the right amount of space before
converting op to a string. The right amount of allocation is normally
two more than the value returned by mpz_sizeinbase
(one extra for a
minus sign and one for the null-terminator).
This chapter describes the GMP functions for performing arithmetic on rational
numbers. These functions start with the prefix mpq_
.
Rational numbers are stored in objects of type mpq_t
.
All rational arithmetic functions assume operands have a canonical form, and canonicalize their result. The canonical from means that the denominator and the numerator have no common factors, and that the denominator is positive. Zero has the unique representation 0/1.
Pure assignment functions do not canonicalize the assigned variable. It is the responsibility of the user to canonicalize the assigned variable before any arithmetic operations are performed on that variable.
mpq_clear
) between each initialization.
mpq_t
variables when you are done with them.
mpq_canonicalize
before any operations are performed on rop.
The string can be an integer like "41" or a fraction like "41/152". The
fraction must be in canonical form (see section 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
(see section Assignment Functions). White space is allowed in
the string, and is simply ignored. The base can vary from 2 to 36, or
if base is 0 then the leading characters are used: 0x
for hex,
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.
double
.
If str is NULL
, the result string is allocated using the current
allocation function (see section 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.
To determine if two rationals are equal, mpq_equal
is faster than
mpq_cmp
.
num2 and den2 are allowed to have common factors.
These functions are implemented as a macros and evaluate their arguments multiple times.
This function is actually implemented as a macro. It evaluates its arguments multiple times.
mpq_cmp
can be used for the same purpose, this
function is much faster.
The set of mpq
functions is quite small. In particular, there are few
functions for either input or output. The following functions give direct
access to the numerator and denominator of an mpq_t
.
Note that if an assignment to the numerator and/or denominator could take an
mpq_t
out of the canonical form described at the start of this chapter
(see section Rational Number Functions) then mpq_canonicalize
must be
called before any other mpq
functions are applied to that mpq_t
.
mpz
functions can be used on the result of these macros.
mpz_set
with an appropriate mpq_numref
or
mpq_denref
. Direct use of mpq_numref
or mpq_denref
is
recommended instead of these functions.
When using any of these functions, it's a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions.
Passing a NULL
pointer for a stream argument to any of these
functions will make them read from stdin
and write to stdout
,
respectively.
Return the number of bytes written, or if an error occurred, return 0.
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 (see section 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.
GMP floating point numbers are stored in objects of type mpf_t
and
functions operating on them have an mpf_
prefix.
The mantissa of each float has a user-selectable precision, limited only by available memory. Each variable has its own precision, and that can be increased or decreased at any time.
The exponent of each float is a fixed precision, one machine word on most
systems. In the current implementation the exponent is a count of limbs, so
for example on a 32-bit system this means a range of roughly
2^@W{-68719476768} to @math{2^@W{68719476736}}, or on a 64-bit system
this will be greater. Note however mpf_get_str
can only return an
exponent which fits an mp_exp_t
and currently mpf_set_str
doesn't accept exponents bigger than a long
.
Each variable keeps a size for the mantissa data actually in use. This means that if a float is exactly represented in only a few bits then only those bits will be used in a calculation, even if the selected precision is high.
All calculations are performed to the precision of the destination variable. Each function is defined to calculate with "infinite precision" followed by a truncation to the destination precision, but of course the work done is only what's needed to determine a result under that definition.
The precision selected for a variable is a minimum value, GMP may increase it a little to facilitate efficient calculation. Currently this means rounding up to a whole limb, and then sometimes having a further partial limb, depending on the high limb of the mantissa. But applications shouldn't be concerned by such details.
The mantissa in stored in binary, as might be imagined from the fact
precisions are expressed in bits. One consequence of this is that 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.
mpf_init
will use this precision, but previously
initialized variables are unaffected.
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.
mpf_clear
, between initializations. The
precision of x is undefined unless a default precision has already been
established by a call to mpf_set_default_prec
.
mpf_clear
, between initializations.
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.
This function requires a call to realloc
, and so should not be used in
a tight loop.
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.
These functions assign new values to already initialized floats (see section Initialization Functions).
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.
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 @nicode{"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.
For convenience, GMP provides a parallel series of initialize-and-set functions
which initialize the output and then store the value there. These functions'
names have the form mpf_init_set...
Once the float has been initialized by any of the mpf_init_set...
functions, it can be used as the source or destination operand for the ordinary
float functions. Don't use an initialize-and-set function on a variable
already initialized!
The precision of rop will be taken from the active default precision, as
set by mpf_set_default_prec
.
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
.
double
.
frexp
.
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
(see section Miscellaneous Functions).
If str is NULL
, the result string is allocated using the current
allocation function (see section 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 @nicode{"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.
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.
Caution: Currently only whole limbs are compared, and only in an exact fashion. In the future values like 1000 and 0111 may be considered the same to 3 bits (on the basis that their difference is that small).
This function is actually implemented as a macro. It evaluates its arguments multiple times.
Functions that perform input from a stdio stream, and functions that output to
a stdio stream. Passing a NULL
pointer for a stream argument to
any of these functions will make them read from stdin
and write to
stdout
, respectively.
When using any of these functions, it is a good idea to include `stdio.h' before `gmp.h', since that will allow `gmp.h' to define prototypes for these functions.
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.
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.
mpf_ceil
rounds to the
next higher integer, mpf_floor
to the next lower, and mpf_trunc
to the integer towards zero.
The variable state must be initialized by calling one of the
gmp_randinit
functions (section Random State Initialization) before
invoking this function.
This chapter describes low-level GMP functions, used to implement the high-level GMP functions, but also intended for time-critical user code.
These functions start with the prefix mpn_
.
The mpn
functions are designed to be as fast as possible, not
to provide a coherent calling interface. The different functions have somewhat
similar interfaces, but there are variations that make them hard to use. These
functions do as little as possible apart from the real multiple precision
computation, so that no time is spent on things that not all callers need.
A source operand is specified by a pointer to the least significant limb and a limb count. A destination operand is specified by just a pointer. It is the responsibility of the caller to ensure that the destination has enough space for storing the result.
With this way of specifying operands, it is possible to perform computations on subranges of an argument, and store the result into a subrange of a destination.
A common requirement for all functions is that each source area needs at least one limb. No size argument may be zero. Unless otherwise stated, in-place operations are allowed where source and destination are the same, but not where they only partly overlap.
The mpn
functions are the base for the implementation of the
mpz_
, mpf_
, and mpq_
functions.
This example adds the number beginning at s1p and the number beginning at s2p and writes the sum at destp. All areas have n limbs.
cy = mpn_add_n (destp, s1p, s2p, n)
In the notation used here, a source operand is identified by the pointer to the least significant limb, and the limb count in braces. For example, {s1p, s1n}.
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.
This function requires that s1n is greater than or equal to s2n.
This is the lowest-level function for subtraction. It is the preferred function for subtraction, since it is written in assembly for most CPUs.
This function requires that s1n is greater than or equal to s2n.
The destination has to have space for 2*n limbs, even if the product's most significant limb is zero.
This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs.
Don't call this function if s2limb is a power of 2; use mpn_lshift
with a count equal to the logarithm of s2limb instead, for optimal speed.
This is a low-level function that is a building block for general multiplication as well as other operations in GMP. It is written in assembly for most CPUs.
This is a low-level function that is a building block for general multiplication and division as well as other operations in GMP. It is written in assembly for most CPUs.
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.
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.
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.
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.
mpn_tdiv_qr
instead for best
performance.]
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 @m{cb^n+a-i=3q, c*b^n + a-i = 3*q}, where
@m{b=2\GMPraise{mp\_bits\_per\_limb
}, b=2^mp_bits_per_limb}. The
return c is always 0, 1 or 2, and the initial carry i must also
be 0, 1 or 2 (these are both borrows really). When c=0 clearly
q=(a-i)/3. When @m{c \neq 0, c!=0}, the remainder @math{(a-i) @bmod{}
3} is given by 3-c, because b == 1 @bmod{ 3} (when
mp_bits_per_limb
is even, which is always so currently).
mp_bits_per_limb
bits of
q.
{s1p, s1n} - q * {s2p, s2n} mod @m{2
\GMPraise{s1n*mp\_bits\_per\_limb
},
2^(s1n*@nicode{mp\_bits\_per\_limb})} is placed at s1p. Since the
low @GMPfloor{d/@nicode{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 @le{ s1p}.
This function requires that s1n * @nicode{mp\_bits\_per\_limb @ge{} D}, and that {s2p, s2n} is odd.
This interface is preliminary. It might change incompatibly in future revisions.
count must be in the range 1 to @nicode{mp_bits_per_limb}-1. The regions {sp, n} and {rp, n} may overlap, provided rp @ge{ sp}.
This function is written in assembly for most CPUs.
count must be in the range 1 to @nicode{mp_bits_per_limb}-1. The regions {sp, n} and {rp, n} may overlap, provided rp @le{ sp}.
This function is written in assembly for most CPUs.
{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}.
{s1p, s1n} @ge{ {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 @m{r_2 s_1 + k s_2 = r_1, r2*s1 + k*s2 = r1}. The second cofactor k is not calculated but can easily be obtained from @m{(r_1 - r_2 s_1) / s_2, (r1 - r2*s1) / s2}.
The most significant limb of {sp, n} must be non-zero. The areas {r1p, @GMPceil{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
.
The most significant limb of the input {s1p, s1n} must be non-zero. The input {s1p, s1n} is clobbered, except when base is a power of 2, in which case it's unchanged.
The area at str has to have space for the largest possible number represented by a s1n long limb array, plus one extra character.
str[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.
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.
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.
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.
Everything in this section is highly experimental and may disappear or be subject to incompatible changes in a future version of GMP.
Nails are an experimental feature whereby a few bits are left unused at the
top of each mp_limb_t
. This can significantly improve carry handling
on some processors.
All the mpn
functions accepting limb data will expect the nail bits to
be zero on entry, and will return data with the nails similarly all zero.
This applies both to limb vectors and to single limb arguments.
Nails can be enabled by configuring with `--enable-nails'. By default the number of bits will be chosen according to what suits the host processor, but a particular number can be selected with `--enable-nails=N'.
At the mpn level, a nail build is neither source nor binary compatible with a non-nail build, strictly speaking. But programs acting on limbs only through the mpn functions are likely to work equally well with either build, and judicious use of the definitions below should make any program compatible with either build, at the source level.
For the higher level routines, meaning mpz
etc, a nail build should be
fully source and binary compatible with a non-nail build.
GMP_NAIL_BITS
is the number of nail bits, or 0 when nails are not in
use. GMP_NUMB_BITS
is the number of data bits in a limb.
GMP_LIMB_BITS
is the total number of bits in an mp_limb_t
. In
all cases
GMP_LIMB_BITS == GMP_NAIL_BITS + GMP_NUMB_BITS
GMP_NAIL_MASK
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.
GMP_NUMB_MASK
, but can be used for clarity when doing
comparisons rather than bit-wise operations.
The term "nails" comes from finger or toe nails, which are at the ends of a limb (arm or leg). "numb" is short for number, but is also how the developers felt after trying for a long time to come up with sensible names for these things.
In the future (the distant future most likely) a non-zero nail might be permitted, giving non-unique representations for numbers in a limb vector. This would help vector processors since carries would only ever need to propagate one or two limbs.
Sequences of pseudo-random numbers in GMP are generated using a variable of
type gmp_randstate_t
, which holds an algorithm selection and a current
state. Such a variable must be initialized by a call to one of the
gmp_randinit
functions, and can be seeded with one of the
gmp_randseed
functions.
The functions actually generating random numbers are described in section Random Number Functions, and section Miscellaneous 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.
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.
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 @ge{ 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.
Initialize state with an algorithm selected by alg. The only
choice is GMP_RAND_ALG_LC
, which is gmp_randinit_lc_2exp_size
.
A third parameter of type unsigned long
is required, this is the
size for that function. GMP_RAND_ALG_DEFAULT
or 0 are the same
as GMP_RAND_ALG_LC
.
gmp_randinit
sets bits in gmp_errno
to indicate an error.
GMP_ERROR_UNSUPPORTED_ARGUMENT
if alg is unsupported, or
GMP_ERROR_INVALID_ARGUMENT
if the size parameter is too big.
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.
gmp_printf
and friends accept format strings similar to the standard C
printf
(see section `Formatted Output' in The GNU C Library Reference Manual). A format specification is of the form
% [flags] [width] [.[precision]] [type] conv
GMP adds types `Z', `Q' and `F' for mpz_t
, mpq_t
and mpf_t
respectively, and `N' for an mp_limb_t
array.
`Z', `Q' and `N' behave like integers. `Q' will print a
`/' and a denominator, if needed. `F' behaves like a float. For
example,
mpz_t z; gmp_printf ("%s is an mpz %Zd\n", "here", z); mpq_t q; gmp_printf ("a hex rational: %#40Qx\n", q); mpf_t f; int n; gmp_printf ("fixed point mpf %.*Ff with %d digits\n", n, f, n); const mp_limb_t *ptr; mp_size_t size; gmp_printf ("limb array %Nx\n", ptr, size);
For `N' the limbs are expected least significant first, as per the
mpn
functions (see section Low-level Functions). A negative size can be
given to print the value as a negative.
All the standard C printf
types behave the same as the C library
printf
, and can be freely intermixed with the GMP extensions. In the
current implementation the standard parts of the format string are simply
handed to printf
and only the GMP extensions handled directly.
The flags accepted are as follows. GLIBC style @nisamp{'} is only for the standard C types (not the GMP types), and only if the C library supports it.
@nicode{0} pad with zeros (rather than spaces) @nicode{#} show the base with `0x', `0X' or `0' @nicode{+} always show a sign (space) show a space or a `-' sign @nicode{'} group digits, GLIBC style (not GMP types) The optional width and precision can be given as a number within the format string, or as a `*' to take an extra parameter of type
int
, the same as the standardprintf
.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.
@nicode{h} @nicode{short} @nicode{hh} @nicode{char} @nicode{j} @nicode{intmax_t} or @nicode{uintmax_t} @nicode{l} @nicode{long} or @nicode{wchar_t} @nicode{ll} @nicode{long long} @nicode{L} @nicode{long double} @nicode{q} @nicode{quad_t} or @nicode{u_quad_t} @nicode{t} @nicode{ptrdiff_t} @nicode{z} @nicode{size_t} The GMP types are
@nicode{F} @nicode{mpf_t}, float conversions @nicode{Q} @nicode{mpq_t}, integer conversions @nicode{N} @nicode{mp_limb_t} array, integer conversions @nicode{Z} @nicode{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.@nicode{a} @nicode{A} hex floats, C99 style @nicode{c} character @nicode{d} decimal integer @nicode{e} @nicode{E} scientific format float @nicode{f} fixed point float @nicode{i} same as @nicode{d} @nicode{g} @nicode{G} fixed or scientific float @nicode{m} strerror
string, GLIBC style@nicode{n} store characters written so far @nicode{o} octal integer @nicode{p} pointer @nicode{s} string @nicode{u} unsigned integer @nicode{x} @nicode{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'.
`n' can be used with any type, even the GMP types.
Other types or conversions that might be accepted by the C library
printf
cannot be used throughgmp_printf
, this includes for instance extensions registered with GLIBCregister_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 asmpf_get_str
does. Zeros will be used if necessary to pad to the requested precision. This happens even for an `f' conversion of anmpf_t
which is an integer, for instance 2^@W{1024} in anmpf_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
(see section `Locales and Internationalization' in The GNU C Library Reference Manual). 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.Functions
Each of the following functions is similar to the corresponding C library function. The basic
printf
forms take a variable argument list. Thevprintf
forms take an argument pointer, see section `Variadic Functions' in The GNU C Library Reference Manual, or `man 3 va_start'.It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions.
The file based functions
gmp_printf
andgmp_fprintf
will return -1 to indicate a write error. All the functions can return -1 if the C libraryprintf
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 @ge{ 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 libraryvsnprintf
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 (see section Custom Allocation). The block will be the size of the string and null-terminator. Put the address of the block in *pp. Return the number of characters produced, excluding the null-terminator.
Unlike the C library
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 obstack object, in the same style as
obstack_printf
. Return the number of characters written. A null-terminator is not written.fmt cannot be within the current obstack object, since the object might move as it grows.
These functions are available only when the C library provides the obstack feature, which probably means only on GNU systems, see section `Obstacks' in The GNU C Library Reference Manual.
C++ Formatted Output
The following functions are provided in `libgmpxx', which is built if C++ support is enabled (see section Build Options). Prototypes are available from
<gmp.h>
.
- Function: ostream& operator<< (ostream& stream, mpz_t op)
- Print op to stream, using its
ios
formatting settings.ios::width
is reset to 0 after output, the same as the standardostream 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 onint
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 standardostream 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 standardostream operator<<
routines do. The decimal point follows the current locale, on systems providinglocaleconv
.Hex and octal are supported, unlike the standard
operator<<
ondouble
. The mantissa will be in hex or octal, the exponent will be in decimal. For hex the exponent delimiter is an `@'. This is as permpf_out_str
.
ios::showbase
is supported, and will put a base on the mantissa, for example hex `0x1.8' or `0x0.8', or octal `01.4' or `00.4'. This last form is slightly strange, but at least differentiates itself from decimal.These operators mean that GMP types can be printed in the usual C++ way, for example,
mpz_t z; int n; ... cout << "iteration " << n << " value " << z << "\n";But note that
ostream
output (andistream
input, see section C++ Formatted Input) is the only overloading available and using for instance+
with anmpz_t
will have unpredictable results.Formatted Input
Formatted Input Strings
gmp_scanf
and friends accept format strings similar to the standard Cscanf
(see section `Formatted Input' in The GNU C Library Reference Manual). A format specification is of the form% [flags] [width] [type] convGMP adds types `Z', `Q' and `F' for
mpz_t
,mpq_t
andmpf_t
respectively. `Z' and `Q' behave like integers. `Q' will read a `/' and a denominator, if present. `F' behaves like a float.GMP variables don't require an
&
when passed togmp_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 libraryscanf
, and can be freely intermixed with the GMP extensions. In the current implementation the standard parts of the format string are simply handed toscanf
and only the GMP extensions handled directly.The flags accepted are as follows. `a' and `'' will depend on support from the C library, and `'' cannot be used with GMP types.
@nicode{*} read but don't store @nicode{a} allocate a buffer (string conversions) @nicode{'} group digits, GLIBC style (not GMP types) The standard types accepted are as follows. `h' and `l' are portable, the rest will depend on the compiler (or include files) for the type and the C library for the input.
@nicode{h} @nicode{short} @nicode{hh} @nicode{char} @nicode{j} @nicode{intmax_t} or @nicode{uintmax_t} @nicode{l} @nicode{long int}, @nicode{double} or @nicode{wchar_t} @nicode{ll} @nicode{long long} @nicode{L} @nicode{long double} @nicode{q} @nicode{quad_t} or @nicode{u_quad_t} @nicode{t} @nicode{ptrdiff_t} @nicode{z} @nicode{size_t} The GMP types are
@nicode{F} @nicode{mpf_t}, float conversions @nicode{Q} @nicode{mpq_t}, integer conversions @nicode{Z} @nicode{mpz_t}, integer conversions The conversions accepted are as follows. `p' and `[' will depend on support from the C library, the rest are standard.
@nicode{c} character or characters @nicode{d} decimal integer @nicode{e} @nicode{E} @nicode{f} @nicode{g} @nicode{G} float @nicode{i} integer with base indicator @nicode{n} characters read so far @nicode{o} octal integer @nicode{p} pointer @nicode{s} string of non-whitespace characters @nicode{u} decimal integer @nicode{x} @nicode{X} hex integer @nicode{[} string of characters in a set `e', `E', `f', `g' and `G' are identical, they all read either fixed point or scientific format, and either `e' or `E' for the exponent in scientific format.
`x' and `X' are identical, both accept both upper and lower case hexadecimal.
`o', `u', `x' and `X' all read positive or negative values. For the standard C types these are described as "unsigned" conversions, but that merely affects certain overflow handling, negatives are still allowed (see
strtoul
, section `Parsing of Integers' in The GNU C Library Reference Manual). For GMP types there are no overflows, and `d' and `u' are identical.`Q' type reads the numerator and (optional) denominator as given. If the value might not be in canonical form then
mpq_canonicalize
must be called before using it in any calculations (see section Rational Number Functions).`Qi' will read a base specification separately for the numerator and denominator. For example `0x10/11' would be 16/11, whereas `0x10/0x11' would be 16/17.
`n' can be used with any of the types above, even the GMP types. `*' to suppress assignment is allowed, though the field would then do nothing at all.
Other conversions or types that might be accepted by the C library
scanf
cannot be used throughgmp_scanf
.Whitespace is read and discarded before a field, except for `c' and `[' conversions.
For float conversions, the decimal point character (or string) expected is taken from the current locale settings on systems which provide
localeconv
(see section `Locales and Internationalization' in The GNU C Library Reference Manual). 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.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. Thevscanf
forms take an argument pointer, see section `Variadic Functions' in The GNU C Library Reference Manual, or `man 3 va_start'.It should be emphasised that if a format string is invalid, or the arguments don't match what the format specifies, then the behaviour of any of these functions will be unpredictable. GCC format string checking is not available, since it doesn't recognise the GMP extensions.
No overlap is permitted between the fmt string and any of the results produced.
- Function: int gmp_scanf (const char *fmt, ...)
- Function: int gmp_vscanf (const char *fmt, va_list ap)
- Read from the standard input
stdin
.
- Function: int gmp_fscanf (FILE *fp, const char *fmt, ...)
- Function: int gmp_vfscanf (FILE *fp, const char *fmt, va_list ap)
- Read from the stream fp.
- Function: int gmp_sscanf (const char *s, const char *fmt, ...)
- Function: int gmp_vsscanf (const char *s, const char *fmt, va_list ap)
- Read from a null-terminated string s.
The return value from each of these functions is the same as the standard C99
scanf
, namely the number of fields successfully parsed and stored. `%n' fields and fields read but suppressed by `*' don't count towards the return value.If end of file or file error, or end of string, is reached when a match is required, and when no previous non-suppressed fields have matched, then the return value is EOF instead of 0. A match is required for a literal character in the format string or a field other than `%n'. Whitespace in the format string is only an optional match and won't induce an EOF in this fashion. Leading whitespace read and discarded for a field doesn't count as a match.
C++ Formatted Input
The following functions are provided in `libgmpxx', which is built only if C++ support is enabled (see section Build Options). Prototypes are available from
<gmp.h>
.
- Function: istream& operator>> (istream& stream, mpz_t rop)
- Read rop from stream, using its
ios
formatting settings.
- Function: istream& operator>> (istream& stream, mpq_t rop)
- Read rop from stream, using its
ios
formatting settings.An integer like `123' will be read, or a fraction like `5/9'. If the fraction is not in canonical form then
mpq_canonicalize
must be called (see section Rational Number Functions).
- 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.
These operators mean that GMP types can be read in the usual C++ way, for example,
mpz_t z; ... cin >> z;But note that
istream
input (andostream
output, see section C++ Formatted Output) is the only overloading available and using for instance+
with anmpz_t
will have unpredictable results.C++ Class Interface
This chapter describes the C++ class based interface to GMP.
All GMP C language types and functions can be used in C++ programs, since `gmp.h' has
extern "C"
qualifiers, but the class interface offers overloaded functions and operators which may be more convenient.Due to the implementation of this interface, a reasonably recent C++ compiler is required, one supporting namespaces, partial specialization of templates and member templates. 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.
C++ Interface General
All the C++ classes and functions are available with
#include <gmpxx.h>Programs should be linked with the `libgmpxx' and `libgmp' libraries. For example,
g++ mycxxprog.cc -lgmpxx -lgmpThe classes defined are
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 correspondingmpz_add
, without using a temporary for theb+c
part. Expressions which by their nature imply intermediate values, likea=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
anddouble
. Smaller types likeint
orfloat
can also be intermixed, since C++ will promote them.Note that
bool
is not accepted directly, but must be explicitly cast to anint
first. This is because C++ will automatically convert any pointer to abool
, so if GMP acceptedbool
it would make all sorts of invalid class and pointer combinations compile but almost certainly not do anything sensible.Conversions back from the classes to standard C++ types aren't done automatically, instead member functions like
get_si
are provided (see the following sections for details).Also there are no automatic conversions from the classes to the corresponding GMP C types, instead a reference to the underlying C object can be obtained with the following functions,
- Function: mpz_t mpz_class::get_mpz_t ()
- Function: mpq_t mpq_class::get_mpq_t ()
- Function: mpf_t mpf_class::get_mpf_t ()
These can be used to call a C function which doesn't have a C++ class interface. For example to set
a
to the GCD ofb
andc
,mpz_class a, b, c; ... mpz_gcd (a.get_mpz_t(), b.get_mpz_t(), c.get_mpz_t());In the other direction, a class can be initialized from the corresponding GMP C type, or assigned to if an explicit constructor is used. In both cases this makes a copy of the value, it doesn't create any sort of association. For example,
mpz_t z; // ... init and calculate z ... mpz_class x(z); mpz_class y; y = mpz_class (z);There are no namespace setups in `gmpxx.h', all types and functions are simply put into the global namespace. This is what `gmp.h' has done in the past, and continues to do for compatibility. The extras provided by `gmpxx.h' follow GMP naming conventions and are unlikely to clash with anything.
C++ Interface Integers
- Function: void mpz_class::mpz_class (type n)
- Construct an
mpz_class
. All the standard C++ types may be used, exceptlong long
andlong double
, and all the GMP C++ classes can be used. Any necessary conversion follows the corresponding C function, for exampledouble
followsmpz_set_d
(see section Assignment Functions).
- Function: void mpz_class::mpz_class (mpz_t z)
- Construct an
mpz_class
from anmpz_t
. The value in z is copied into the newmpz_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)
- Function: void mpz_class::mpz_class (const string& s)
- Function: void mpz_class::mpz_class (const string& s, int base)
- Construct an
mpz_class
converted from a string usingmpz_set_str
, (see section Assignment Functions). If the base is not given then 0 is used.
- 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 thempz_tdiv_q
andmpz_tdiv_r
functions (see section Division Functions). This corresponds to the rounding used for plainint
calculations on most machines.The
mpz_fdiv...
ormpz_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: double mpz_class::get_d (void)
- Function: long mpz_class::get_si (void)
- Function: unsigned long mpz_class::get_ui (void)
- @maybepagebreak
- Function: bool mpz_class::fits_sint_p (void)
- Function: bool mpz_class::fits_slong_p (void)
- Function: bool mpz_class::fits_sshort_p (void)
- @maybepagebreak
- Function: bool mpz_class::fits_uint_p (void)
- Function: bool mpz_class::fits_ulong_p (void)
- Function: bool mpz_class::fits_ushort_p (void)
- @maybepagebreak
- Function: int sgn (mpz_class op)
- Function: mpz_class sqrt (mpz_class op)
- These functions provide a C++ class interface to the corresponding GMP C routines.
cmp
can be used with any of the classes or the standard C++ types, exceptlong long
andlong double
.Overloaded operators for combinations of
mpz_class
anddouble
are provided for completeness, but it should be noted that if the givendouble
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
anddouble
, however, are defined to follow the corresponding C functionsmpz_get_d
andmpz_set_d
. And comparisons are always made exactly, as permpz_cmp_d
.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 thatlong long
andlong 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 anmpq_t
. The value in q is copied into the newmpq_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)
- Function: void mpq_class::mpq_class (const string& s)
- Function: void mpq_class::mpq_class (const string& s, int base)
- Construct an
mpq_class
converted from a string usingmpq_set_str
, (see section Initialization and Assignment Functions). If the base is not given then 0 is used.
- Function: void mpq_class::canonicalize ()
- Put an
mpq_class
into canonical form, as per section 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)
- @maybepagebreak
- Function: double mpq_class::get_d (void)
- Function: int sgn (mpq_class op)
- These functions provide a C++ class interface to the corresponding GMP C routines.
cmp
can be used with any of the classes or the standard C++ types, exceptlong long
andlong 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 anmpq_class
. This can be used both for read and write access. If the object returned is modified, it modifies the originalmpq_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 anmpq_class
. This can be passed to C functions expecting anmpz_t
. Any modifications made to thempz_t
will modify the originalmpq_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 asmpq_t operator>>
(see section C++ Formatted Input).If the rop read might not be in canonical form then
mpq_class::canonicalize
must be called.C++ Interface Floats
When an expression requires the use of temporary intermediate
mpf_class
values, likef=g*h+x*y
, those temporaries will have the same precision as the destinationf
. 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, exceptlong long
andlong double
, and any of the GMP C++ classes can be used.If prec is given, the initial precision is that value, in bits. If prec is not given, then the initial precision is determined by the type of op given. An
mpz_class
,mpq_class
, string, or C++ builtin type will give the defaultmpf
precision (see section Initialization Functions). Anmpf_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: 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)
- @maybepagebreak
- Function: mpf_class floor (mpf_class op)
- Function: mpf_class hypot (mpf_class op1, mpf_class op2)
- Function: double mpf_class::get_d (void)
- Function: long mpf_class::get_si (void)
- Function: unsigned long mpf_class::get_ui (void)
- @maybepagebreak
- Function: bool mpf_class::fits_sint_p (void)
- Function: bool mpf_class::fits_slong_p (void)
- Function: bool mpf_class::fits_sshort_p (void)
- @maybepagebreak
- Function: bool mpf_class::fits_uint_p (void)
- Function: bool mpf_class::fits_ulong_p (void)
- Function: bool mpf_class::fits_ushort_p (void)
- @maybepagebreak
- 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 GMP C routines.
cmp
can be used with any of the classes or the standard C++ types, exceptlong long
andlong 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
(see section Initialization Functions) apply tompf_class::set_prec_raw
. Note in particular that thempf_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.C++ Interface MPFR
The C++ class interface to MPFR is provided if MPFR is enabled (see section Build Options). This interface must be regarded as preliminary and possibly subject to incompatible changes in the future, since MPFR itself is preliminary. All definitions can be obtained with
#include <mpfrxx.h>This defines
which behaves similarly to
mpf_class
(see section C++ Interface Floats).C++ Interface Random Numbers
- Class: gmp_randclass
- The C++ class interface to the GMP random number functions uses
gmp_randclass
to hold an algorithm selection and current state, as pergmp_randstate_t
.
- Function: gmp_randclass::gmp_randclass (void (*randinit) (gmp_randstate_t, ...), ...)
- Construct a
gmp_randclass
, using a call to the given randinit function (see section Random State Initialization). The arguments expected are the same as randinit, but withmpz_class
instead ofmpz_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_randinit_lc_2exp_size
can fail if the size requested is too big, the behaviour ofgmp_randclass::gmp_randclass
is undefined in this case (perhaps this will change in the future).
- Function: gmp_randclass::gmp_randclass (gmp_randalg_t alg, ...)
- Construct a
gmp_randclass
using the same parameters asgmp_randinit
(see section Random State Initialization). This function is obsolete and the above randinit style should be preferred.
- Function: void gmp_randclass::seed (unsigned long int s)
- Function: void gmp_randclass::seed (mpz_class s)
- Seed a random number generator. See see section 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 bitsC++ Interface Limitations
mpq_class
and Templated Reading- A generic piece of template code probably won't know that
mpq_class
requires acanonicalize
call if inputs read withoperator>>
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 ofmpq_class
. Perhaps a mechanism to telloperator>>
what to do will be adopted in the future, maybe a preprocessor define, a global flag, or anios
flag pressed into service. Or maybe, at the risk of inconsistency, thempq_class
operator>>
could canonicalize and leavempq_t
operator>>
not doing so, for use on those occasions when that's acceptable. Send feedback or alternate ideas to bug-gmp@gnu.org.- Subclassing
- Subclassing the GMP C++ classes works, but is not currently recommended. Expressions involving subclasses resolve correctly (or seem to), but in normal C++ fashion the subclass doesn't inherit constructors and assignments. There's many of those in the GMP classes, and a good way to reestablish them in a subclass is not yet provided.
- Templated Expressions
- A subtle difficulty exists when using expressions together with application-defined template functions. Consider the following, with
T
intended to be some numeric type,template <class T> T fun (const T &, const T &);When used with, say, plainmpz_class
variables, it works fine:T
is resolved asmpz_class
.mpz_class f(1), g(2); fun (f, g); // GoodBut when one of the arguments is an expression, it doesn't work.mpz_class f(1), g(2), h(3); fun (f, g+h); // BadThis is becauseg+h
ends up being a certain expression template type internal togmpxx.h
, which the C++ template resolution rules are unable to automatically convert tompz_class
. The workaround is simply to add an explicit cast.mpz_class f(1), g(2), h(3); fun (f, mpz_class(g+h)); // GoodSimilarly, withinfun
it may be necessary to cast an expression to typeT
when calling a templatedfun2
.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 }Berkeley MP Compatible Functions
These functions are intended to be fully compatible with the Berkeley MP library which is available on many BSD derived U*ix systems. The `--enable-mpbsd' option must be used when building GNU MP to make these available (see section Installing GMP).
The original Berkeley MP library has a usage restriction: you cannot use the same variable as both source and destination in a single function call. The compatible functions in GNU MP do not share this restriction--inputs and outputs may overlap.
It is not recommended that new programs are written using these functions. Apart from the incomplete set of functions, the interface for initializing
MINT
objects is more error prone, and thepow
function collides withpow
in `libm.a'.Include the header `mp.h' to get the definition of the necessary types and functions. If you are on a BSD derived system, make sure to include GNU `mp.h' if you are going to link the GNU `libmp.a' to your program. This means that you probably need to give the `-I<dir>' option to the compiler, where `<dir>' is the directory where you have GNU `mp.h'.
- Function: MINT * itom (signed short int initial_value)
- Allocate an integer consisting of a
MINT
object and dynamic limb space. Initialize the integer to initial_value. Return a pointer to theMINT
object.
- Function: MINT * xtom (char *initial_value)
- Allocate an integer consisting of a
MINT
object and dynamic limb space. Initialize the integer from initial_value, a hexadecimal, null-terminated C string. Return a pointer to theMINT
object.
- Function: void move (MINT *src, MINT *dest)
- Set dest to src by copying. Both variables must be previously initialized.
- Function: void madd (MINT *src_1, MINT *src_2, MINT *destination)
- Add src_1 and src_2 and put the sum in destination.
- Function: void msub (MINT *src_1, MINT *src_2, MINT *destination)
- Subtract src_2 from src_1 and put the difference in destination.
- Function: void mult (MINT *src_1, MINT *src_2, MINT *destination)
- Multiply src_1 and src_2 and put the product in destination.
- Function: void mdiv (MINT *dividend, MINT *divisor, MINT *quotient, MINT *remainder)
- Function: void sdiv (MINT *dividend, signed short int divisor, MINT *quotient, signed short int *remainder)
- Set quotient to dividend/divisor, and remainder to dividend mod divisor. The quotient is rounded towards zero; the remainder has the same sign as the dividend unless it is zero.
Some implementations of these functions work differently--or not at all--for negative arguments.
- Function: void msqrt (MINT *op, MINT *root, MINT *remainder)
- Set root to @m{\lfloor\sqrt{op}\rfloor, the truncated integer part of the square root of op}, like
mpz_sqrt
. Set remainder to @m{(op - root^2), op-root*root}, i.e. zero if op is a perfect square.If root and remainder are the same variable, the results are undefined.
- Function: void pow (MINT *base, MINT *exp, MINT *mod, MINT *dest)
- Set dest to (base raised to exp) modulo mod.
- Function: void gcd (MINT *op1, MINT *op2, MINT *res)
- Set res to the greatest common divisor of op1 and op2.
- Function: int mcmp (MINT *op1, MINT *op2)
- Compare op1 and op2. Return a positive value if op1 > op2, zero if op1 = op2, and a negative value if op1 < op2.
- Function: void min (MINT *dest)
- Input a decimal string from
stdin
, and put the read integer in dest. SPC and TAB are allowed in the number string, and are ignored.
- Function: char * mtox (MINT *op)
- Convert op to a hexadecimal string, and return a pointer to the string. The returned string is allocated using the default memory allocation function,
malloc
by default. It will bestrlen(str)+1
bytes, that being exactly enough for the string and null-terminator.
- Function: void mfree (MINT *op)
- De-allocate, the space used by op. This function should only be passed a value returned by
itom
orxtom
.Custom Allocation
By default GMP uses
malloc
,realloc
andfree
for memory allocation, and if they fail GMP prints a message to the standard error output and terminates the program.Alternate functions can be specified to allocate memory in a different way or to have a different error action on running out of memory.
This feature is available in the Berkeley compatibility library (see section Berkeley MP Compatible Functions) as well as the main GMP library.
- Function: void mp_set_memory_functions (
void *(*alloc_func_ptr) (size_t),
void *(*realloc_func_ptr) (void *, size_t, size_t),
void (*free_func_ptr) (void *, size_t))- Replace the current allocation functions from the arguments. If an argument is
NULL
, the corresponding default function is used.These functions will be used for all memory allocation done by GMP, apart from temporary space from
alloca
if that function is available and GMP is configured to use it (see section Build Options).Be sure to call
mp_set_memory_functions
only when there are no active GMP objects allocated using the previous memory functions! Usually that means calling it before any other GMP function.The functions supplied should fit the following declarations:
- Function: void * allocate_function (size_t alloc_size)
- Return a pointer to newly allocated space with at least alloc_size bytes.
- Function: void * reallocate_function (void *ptr, size_t old_size, size_t new_size)
- Resize a previously allocated block ptr of old_size bytes to be new_size bytes.
The block may be moved if necessary or if desired, and in that case the smaller of old_size and new_size bytes must be copied to the new location. The return value is a pointer to the resized block, that being the new location if moved or just ptr if not.
ptr is never
NULL
, it's always a previously allocated block. new_size may be bigger or smaller than old_size.
- Function: void deallocate_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 deallocate_function are passed for convenience, but of course can be ignored if not needed. The default functions using
malloc
and friends for instance don't use them.No error return is allowed from any of these functions, if they return then they must have performed the specified operation. In particular note that allocate_function or reallocate_function mustn't return
NULL
.Getting a different fatal error action is a good use for custom allocation functions, for example giving a graphical dialog rather than the default print to
stderr
. How much is possible when genuinely out of memory is another question though.There's currently no defined way for the allocation functions to recover from an error such as out of memory, they must terminate program execution. A
longjmp
or throwing a C++ exception will have undefined results. This may change in the future.GMP may use allocated blocks to hold pointers to other allocated blocks. This will limit the assumptions a conservative garbage collection scheme can make.
Since the default GMP allocation uses
malloc
and friends, those functions will be linked in even if the first thing a program does is anmp_set_memory_functions
. It's necessary to change the GMP sources if this is a problem.Language Bindings
The following packages and projects offer access to GMP from languages other than C, though perhaps with varying levels of functionality and efficiency.
@macro spaceuref {U} \U\ @ifnottex @macro spaceuref {U} \U\
- C++
- GMP C++ class interface, see section C++ Class Interface
Straightforward interface, expression templates to eliminate temporaries.- ALP @spaceuref{http://www.inria.fr/saga/logiciels/ALP}
Linear algebra and polynomials using templates.- Arithmos @spaceuref{http://win-www.uia.ac.be/u/cant/arithmos}
Rationals with infinities and square roots.- CLN @spaceuref{http://clisp.cons.org/~haible/packages-cln.html}
High level classes for arithmetic.- LiDIA @spaceuref{http://www.informatik.tu-darmstadt.de/TI/LiDIA}
A C++ library for computational number theory.- Linbox @spaceuref{http://www.linalg.org}
Sparse vectors and matrices.- NTL @spaceuref{http://www.shoup.net/ntl}
A C++ number theory library.- Fortran
- Omni F77 @spaceuref{http://pdplab.trc.rwcp.or.jp/pdperf/Omni/home.html}
Arbitrary precision floats.- Haskell
- Glasgow Haskell Compiler @spaceuref{http://www.haskell.org/ghc}
- Java
- Kaffe @spaceuref{http://www.kaffe.org}
- Kissme @spaceuref{http://kissme.sourceforge.net}
- Lisp
- GNU Common Lisp @spaceuref{http://www.gnu.org/software/gcl/gcl.html}
In the process of switching to GMP for bignums.- Librep @spaceuref{http://librep.sourceforge.net}
- M4
- GNU m4 betas @spaceuref{http://www.seindal.dk/rene/gnu}
Optionally provides an arbitrary precisionmpeval
.- ML
- MLton compiler @spaceuref{http://www.mlton.org}
- Oz
- Mozart @spaceuref{http://www.mozart-oz.org}
- Pascal
- GNU Pascal Compiler @spaceuref{http://www.gnu-pascal.de}
GMP unit.- Perl
- GMP module, see `demos/perl' in the GMP sources.
- Math::GMP @spaceuref{http://www.cpan.org}
Compatible with Math::BigInt, but not as many functions as the GMP module above.- Math::BigInt::GMP @spaceuref{http://www.cpan.org}
Plug Math::GMP into normal Math::BigInt operations.- Pike
- mpz module in the standard distribution, http://pike.idonex.com
- Prolog
- SWI Prolog @spaceuref{http://www.swi.psy.uva.nl/projects/SWI-Prolog}
Arbitrary precision floats.- Python
- mpz module in the standard distribution, http://www.python.org
- GMPY http://gmpy.sourceforge.net
- Scheme
- RScheme @spaceuref{http://www.rscheme.org}
- STklos @spaceuref{http://kaolin.unice.fr/STklos}
- Smalltalk
- GNU Smalltalk @spaceuref{http://www.smalltalk.org/versions/GNUSmalltalk.html}
- Other
- DrGenius @spaceuref{http://drgenius.seul.org}
Geometry system and mathematical programming language.- GiNaC @spaceuref{http://www.ginac.de}
C++ computer algebra using CLN.- Maxima http://www.ma.utexas.edu/users/wfs/maxima.html
Macsyma computer algebra using GCL.- Q @spaceuref{http://www.musikwissenschaft.uni-mainz.de/~ag/q}
Equational programming system.- Regina @spaceuref{http://regina.sourceforge.net}
Topological calculator.- Yacas @spaceuref{http://www.xs4all.nl/~apinkus/yacas.html}
Yet another computer algebra system.Algorithms
This chapter is an introduction to some of the algorithms used for various GMP operations. The code is likely to be hard to understand without knowing something about the algorithms.
Some GMP internals are mentioned, but applications that expect to be compatible with future GMP releases should take care to use only the documented functions.
Multiplication
N@cross{}N limb multiplications and squares are done using one of four algorithms, as the size N increases.
Algorithm Threshold Basecase (none) Karatsuba MUL_KARATSUBA_THRESHOLD
Toom-3 MUL_TOOM3_THRESHOLD
FFT MUL_FFT_THRESHOLD
Similarly for squaring, with the
SQR
thresholds. Note though that the FFT is only used if GMP is configured with `--enable-fft', see section Build Options.N@cross{}M multiplications of operands with different sizes above
MUL_KARATSUBA_THRESHOLD
are currently done by splitting into M@cross{}M pieces. The Karatsuba and Toom-3 routines then operate only on equal size operands. This is not very efficient, and is slated for improvement in the future.Basecase Multiplication
Basecase N@cross{}M 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 @m{O(NM),O(N*M)} algorithm. See Knuth section 4.3.1 algorithm M (see section 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.
@ifnottex
u0 u1 u2 u3 u4 +---+---+---+---+---+ u0 | d | | | | | +---+---+---+---+---+ u1 | | d | | | | +---+---+---+---+---+ u2 | | | d | | | +---+---+---+---+---+ u3 | | | | d | | +---+---+---+---+---+ u4 | | | | | d | +---+---+---+---+---+In practice squaring isn't a full 2@cross{} faster than multiplying, it's usually around 1.5@cross{}. Less than 1.5@cross{} probably indicates
mpn_sqr_basecase
wants improving on that CPU.On some CPUs
mpn_mul_basecase
can be faster than the generic Cmpn_sqr_basecase
.SQR_BASECASE_THRESHOLD
is the size at which to usempn_sqr_basecase
, this will be zero if that routine should be used always.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).
@ifnottex
high low +----------+----------+ | x1 | x0 | +----------+----------+ +----------+----------+ | y1 | y0 | +----------+----------+Let b be the power of 2 where the split occurs, ie. if @ms{x,0} is k limbs (@ms{y,0} the same) then @m{b=2\GMPraise{$k*$
mp\_bits\_per\_limb
}, b=2^(k*mp_bits_per_limb)}. With that @m{x=x_1b+x_0,x=x1*b+x0} and @m{y=y_1b+y_0,y=y1*b+y0}, and the following holds,@m{xy = (b^2+b)x_1y_1 - b(x_1-x_0)(y_1-y_0) + (b+1)x_0y_0, 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)@cross{}(N/2) limbs, whereas a basecase multiply of N@cross{}N limbs is equivalent to four multiplies of (N/2)@cross{}(N/2). The factors (b^2+b) etc represent the positions where the three products must be added.
@ifnottex
high low +--------+--------+ +--------+--------+ | x1*y1 | | x0*y0 | +--------+--------+ +--------+--------+ +--------+--------+ add | x1*y1 | +--------+--------+ +--------+--------+ add | x0*y0 | +--------+--------+ +--------+--------+ sub | (x1-x0)*(y1-y0) | +--------+--------+The term @m{(x_1-x_0)(y_1-y_0),(x1-x0)*(y1-y0)} is best calculated as an absolute value, and the sign used to choose to add or subtract. Notice the sum @m{\mathop{\rm high}(x_0y_0)+\mathop{\rm low}(x_1y_1), high(x0*y0)+low(x1*y1)} occurs twice, so it's possible to do @m{5k,5*k} limb additions, rather than @m{6k,6*k}, but in GMP extra function call overheads outweigh the saving.
Squaring is similar to multiplying, but with x=y the formula reduces to an equivalent with three squares,
@m{x^2 = (b^2+b)x_1^2 - b(x_1-x_0)^2 + (b+1)x_0^2, 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 @m{(x_1-x_0)^2,(x1-x0)^2} is now always positive.
A similar formula for both multiplying and squaring can be constructed with a middle term @m{(x_1+x_0)(y_1+y_0),(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^@W{1.585)} algorithm, the exponent being @m{\log3/\log2,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. TheSQR
threshold is usually about twice theMUL
. The basecase algorithm will take a time of the form @m{M(N) = aN^2 + bN + c, M(N) = a*N^2 + b*N + c} and the Karatsuba algorithm @m{K(N) = 3M(N/2) + dN + e, K(N) = 3*M(N/2) + d*N + e}. Clearly per-crossproduct speedups in the basecase code reduce a and decrease the threshold, but linear style speedups reducing b will actually increase the threshold. The latter can be seen for instance when adding an optimizedmpn_sqr_diagonal
tompn_sqr_basecase
. Of course all speedups reduce total time, and in that sense the algorithm thresholds are merely of academic interest.Toom-Cook 3-Way Multiplication
The Karatsuba formula is the simplest case of a general approach to splitting inputs that leads to both Toom-Cook and FFT algorithms. A description of Toom-Cook can be found in Knuth section 4.3.3, with an example 3-way calculation after Theorem A. The 3-way form used in GMP is described here.
The operands are each considered split into 3 pieces of equal length (or the most significant part 1 or 2 limbs shorter than the others).
@ifnottex
high low +----------+----------+----------+ | x2 | x1 | x0 | +----------+----------+----------+ +----------+----------+----------+ | y2 | y1 | y0 | +----------+----------+----------+These parts are treated as the coefficients of two polynomials
@m{X(t) = x_2t^2 + x_1t + x_0, X(t) = x2*t^2 + x1*t + x0} @m{Y(t) = y_2t^2 + y_1t + y_0, Y(t) = y2*t^2 + y1*t + y0}Again let b equal the power of 2 which is the size of the @ms{x,0}, @ms{x,1}, @ms{y,0} and @ms{y,1} pieces, ie. if they're k limbs each then @m{b=2\GMPraise{$k*$
mp\_bits\_per\_limb
}, b=2^(k*mp_bits_per_limb)}. With this x=X(b) and y=Y(b).Let a polynomial @m{W(t)=X(t)Y(t),W(t)=X(t)*Y(t)} and suppose its coefficients are
@m{W(t) = w_4t^4 + w_3t^3 + w_2t^2 + w_1t + w_0, W(t) = w4*t^4 + w3*t^3 + w2*t^2 + w1*t + w0}The @m{w_i,w[i]} are going to be determined, and when they are they'll give the final result using w=W(b), since @m{xy=X(b)Y(b),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,
@ifnottex
high low +-------+-------+ | w4 | +-------+-------+ +--------+-------+ | w3 | +--------+-------+ +--------+-------+ | w2 | +--------+-------+ +--------+-------+ | w1 | +--------+-------+ +-------+-------+ | w0 | +-------+-------+The @m{w_i,w[i]} coefficients could be formed by a simple set of cross products, like @m{w_4=x_2y_2,w4=x2*y2}, @m{w_3=x_2y_1+x_1y_2,w3=x2*y1+x1*y2}, @m{w_2=x_2y_0+x_1y_1+x_0y_2,w2=x2*y0+x1*y1+x0*y2} etc, but this would need all nine @m{x_iy_j,x[i]*y[j]} for i,j=0,1,2, and would be equivalent merely to a basecase multiply. Instead the following approach is used.
X(t) and Y(t) are evaluated and multiplied at 5 points, giving values of W(t) at those points. The points used can be chosen in various ways, but in GMP the following are used
Point Value t=0 @m{x_0y_0,x0*y0}, which gives @ms{w,0} immediately t=2 @m{(4x_2+2x_1+x_0)(4y_2+2y_1+y_0),(4*x2+2*x1+x0)*(4*y2+2*y1+y0)} t=1 @m{(x_2+x_1+x_0)(y_2+y_1+y_0),(x2+x1+x0)*(y2+y1+y0)} @m{t={1\over2},t=1/2} @m{(x_2+2x_1+4x_0)(y_2+2y_1+4y_0),(x2+2*x1+4*x0)*(y2+2*y1+4*y0)} @m{t=\infty,t=inf} @m{x_2y_2,x2*y2}, which gives @ms{w,4} immediately At @m{t={1\over2},t=1/2} the value calculated is actually @m{16X({1\over2})Y({1\over2}), 16*X(1/2)*Y(1/2)}, giving a value for @m{16W({1\over2}),16*W(1/2)}, and this is always an integer. At @m{t=\infty,t=inf} the value is actually @m{\lim_{t\to\infty} {X(t)Y(t)\over t^4}, X(t)*Y(t)/t^4 in the limit as t approaches infinity}, but it's much easier to think of as simply @m{x_2y_2,x2*y2} giving @ms{w,4} immediately (much like @m{x_0y_0,x0*y0} at t=0 gives @ms{w,0} immediately).
Now each of the points substituted into @m{W(t)=w_4t^4+\cdots+w_0,W(t)=w4*t^4+...+w0} gives a linear combination of the @m{w_i,w[i]} coefficients, and the value of those combinations has just been calculated.
@ifnottex
W(0) = w0 16*W(1/2) = w4 + 2*w3 + 4*w2 + 8*w1 + 16*w0 W(1) = w4 + w3 + w2 + w1 + w0 W(2) = 16*w4 + 8*w3 + 4*w2 + 2*w1 + w0 W(inf) = w4This is a set of five equations in five unknowns, and some elementary linear algebra quickly isolates each @m{w_i,w[i]}, by subtracting multiples of one equation from another.
In the code the set of five values W(0),...,@m{W(\infty),W(inf)} will represent those certain linear combinations. By adding or subtracting one from another as necessary, values which are each @m{w_i,w[i]} alone are arrived at. This involves only a few subtractions of small multiples (some of which are powers of 2), and so is fast. A couple of divisions remain by powers of 2 and one division by 3 (or by 6 rather), and that last uses the special
mpn_divexact_by3
(see section Exact Division).In the code the values @ms{w,4}, @ms{w,2} and @ms{w,0} are formed in the destination with pointers
E
,C
andA
, and @ms{w,3} and @ms{w,1} in temporary spaceD
andB
are added to them. There are extra limbstD
,tC
andtB
at the high end of @ms{w,3}, @ms{w,2} and @ms{w,1} which are handled separately. The final addition then is as follows.@ifnottex
high low +-------+-------+-------+-------+-------+-------+ | E | C | A | +-------+-------+-------+-------+-------+-------+ +------+-------++------+-------+ | D || B | +------+-------++------+-------+ -- -- -- |tD| |tC| |tB| -- -- --The conversion of W(t) values to the coefficients is interpolation. A polynomial of degree 4 like W(t) is uniquely determined by values known at 5 different points. The points can be chosen to make the linear equations come out with a convenient set of steps for isolating the @m{w_i,w[i]}.
In `mpn/generic/mul_n.c' the
interpolate3
routine performs the interpolation. The open-coded one-pass version may be a bit hard to understand, the steps performed can be better seen in theUSE_MORE_MPN
version.Squaring follows the same procedure as multiplication, but there's only one X(t) and it's evaluated at 5 points, and those values squared to give values of W(t). The interpolation is then identical, and in fact the same
interpolate3
subroutine is used for both squaring and multiplying.Toom-3 is asymptotically O(N^@W{1.465)}, the exponent being @m{\log5/\log3,log(5)/log(3)}, representing 5 recursive multiplies of 1/3 the original size. This is an improvement over Karatsuba at O(N^@W{1.585)}, though Toom-Cook does more work in the evaluation and interpolation and so it only realizes its advantage above a certain size.
Near the crossover between Toom-3 and Karatsuba there's generally a range of sizes where the difference between the two is small.
MUL_TOOM3_THRESHOLD
is a somewhat arbitrary point in that range and successive runs of the tune program can give different values due to small variations in measuring. A graph of time versus size for the two shows the effect, see `tune/README'.At the fairly small sizes where the Toom-3 thresholds occur it's worth remembering that the asymptotic behaviour for Karatsuba and Toom-3 can't be expected to make accurate predictions, due of course to the big influence of all sorts of overheads, and the fact that only a few recursions of each are being performed. Even at large sizes there's a good chance machine dependent effects like cache architecture will mean actual performance deviates from what might be predicted.
The formula given above for the Karatsuba algorithm has an equivalent for Toom-3 involving only five multiplies, but this would be complicated and unenlightening.
An alternate view of Toom-3 can be found in Zuras (see section 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.
FFT Multiplication
At large to very large sizes a Fermat style FFT multiplication is used, following Sch@"onhage and Strassen (see section References). Descriptions of FFTs in various forms can be found in many textbooks, for instance Knuth section 4.3.3 part C or Lipson chapter IX. A brief description of the form used in GMP is given here.
The multiplication done is @m{xy \bmod 2^N+1, x*y mod 2^N+1}, for a given N. A full product @m{xy,x*y} is obtained by choosing @m{N \ge \mathop{\rm bits}(x)+\mathop{\rm bits}(y), 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 @m{2^k\times
mp\_bits\_per\_limb
, (2^k)*@nicode{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 @m{2^{N'}+1, 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. @ifnottex--- \ b w[n] = / (-1) * x[i] * y[j] --- i+j==b*2^k+n b=0,1The points used for the evaluation are g^i for i=0 to 2^k-1 where @m{g=2^{2N'/2^k}, g=2^(2N'/2^k)}. g is a @m{2^k,2^k'}th root of unity mod @m{2^{N'}+1,2^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 @m{2^{N'}+1, 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 @m{x_iy_j, 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 @m{O(N^{k/(k-1)}), O(N^(k/(k-1)))} algorithm, the exponent representing 2^k recursed modular multiplies each @m{1/2^{k-1},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
andSQR_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 N@cross{N@rightarrow{}2N} bit multiply plus a subtraction, so an FFT and Toom-3 etc can be compared directly. A k=4 FFT at O(N^@W{1.333)} can be expected to be the first faster than Toom-3 at O(N^@W{1.465)}. In practice this is what's found, with
MUL_FFT_MODF_THRESHOLD
andSQR_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 @m{1/2^{k-2},1/2^(k-2)} the size of the inputs, making it @m{O(N^{k/(k-2)}), O(N^(k/(k-2)))}. This would mean k=7 at O(N^@W{1.4)} would be the first FFT faster than Toom-3. In practice
MUL_FFT_THRESHOLD
andSQR_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@ge{@nicode{mp\_bits\_per\_limb}} the effective N is a multiple of @m{2^{2k-1},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 @m{2^{2k-1},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
andSQR_FFT_TABLE
will be needed.Other Multiplication
The 3-way Toom-Cook algorithm described above (see section Toom-Cook 3-Way Multiplication) generalizes to split into an arbitrary number of pieces, as per Knuth section 4.3.3 algorithm C. This is not currently used, though it's possible a Toom-4 might fit in between Toom-3 and the FFTs. The notes here are merely for interest.
In general a split into r+1 pieces is made, and evaluations and pointwise multiplications done at @m{2r+1,2*r+1} points. A 4-way split does 7 pointwise multiplies, 5-way does 9, etc. Asymptotically an (r+1)-way algorithm is @m{O(N^{log(2r+1)/log(r+1)}, 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 @m{O(Nr),O(N*r)}, whereas in an r=2^k FFT they grow only as @m{O(N \log r), O(N*log(r))}.
Knuth algorithm C evaluates at points 0,1,2,...,@m{2r,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 @m{2tj-j^2,2*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 @m{i=\sqrt{-1}, i=sqrt(-1)} doesn't help, essentially because in cartesian form it takes three real multiplies to do a complex multiply. The existence of @m{2^k,2^k'}th roots of unity in a suitable ring or field lets the fast fourier transform keep splitting and get to @m{O(N \log r), O(N*log(r))}.
Floating point FFTs use complex numbers approximating Nth roots of unity. Some processors have special support for such FFTs. But these are not used in GMP since it's very difficult to guarantee an exact result (to some number of bits). An occasional difference of 1 in the last bit might not matter to a typical signal processing algorithm, but is of course of vital importance to GMP.
Division Algorithms
Single Limb Division
N@cross{}1 division is implemented using repeated 2@cross{}1 divisions from high to low, either with a hardware divide instruction or a multiplication by inverse, whichever is best on a given CPU.
The multiply by inverse follows section 8 of "Division by Invariant Integers using Multiplication" by Granlund and Montgomery (see section References) and is implemented as
udiv_qrnnd_preinv
in `gmp-impl.h'. The idea is to have a fixed-point approximation to 1/d (seeinvert_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 @m{qd,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 @m{a2^k,a*2^k} by @m{d2^k,d*2^k} where a is the dividend and k is the power necessary to have the high bit of @m{d2^k,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 @m{r = a \bmod d2^k, r = a mod d*2^k} followed by an extra final step @m{r2^k \bmod d2^k, 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 2@cross{}2 multiply, but the four 1@cross{}1s to do that are independent and can therefore be done partly or wholly in parallel. Likewise for a 2@cross{}1 calculating @m{qd,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 1@cross{}1 multiply for the inverse effectively becomes two @m{1\over2@cross{}1, (1/2)x1} for each limb, which can be a saving on CPUs with a fast half limb multiply, or in fact if the only multiply is a half limb, and especially if it's not pipelined.
Basecase Division
Basecase N@cross{}M division is like long division done by hand, but in base @m{2\GMPraise{
mp\_bits\_per\_limb
}, 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 N@cross{}1 product @m{qd,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 2@cross{}1 division and a 1@cross{}1 multiplication plus some subtractions. The 2@cross{}1 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 section 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 @m{O(QM),O(Q*M)} algorithm and will run at a speed similar to a basecase Q@cross{}M multiplication, differing in fact only in the extra multiply and divide for each of the Q quotient limbs.
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 (see section References).The algorithm consists essentially of recognising that a 2N@cross{}N division can be done with the basecase division algorithm (see section Basecase Division), but using N/2 limbs as a base, not just a single limb. This way the multiplications that arise are (N/2)@cross{}(N/2) and can take advantage of Karatsuba and higher multiplication algorithms (see section Multiplication). The "digits" of the quotient are formed by recursive N@cross{}(N/2) divisions.
If the (N/2)@cross{}(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 twiceMUL_KARATSUBA_THRESHOLD
, but how much above depends on the CPU. An optimizedmpn_mul_basecase
can lowerDIV_DC_THRESHOLD
a little by offering a ready-made advantage over repeatedmpn_submul_1
calls.Divide and conquer is asymptotically @m{O(M(N)\log N),O(M(N)*log(N))} where M(N) is the time for an N@cross{}N 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 @m{2.63M(N), 2.63*M(N)}. With higher algorithms the M(N) term improves and the multiplier tends to @m{\log N, log(N)}. In practice, at moderate to large sizes, a 2N@cross{}N division is about 2 to 4 times slower than an N@cross{}N 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.
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 (see section 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 @m{4 \mathord{\times} 7 \bmod 10, 4*7 mod 10}, using the fact 7 is the modular inverse of 3 (the low digit of the divisor), since @m{3 \mathord{\times} 7 \mathop{\equiv} 1 \bmod 10, 3*7 == 1 mod 10}. So @m{8\mathord{\times}543 = 4344,8*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 (@m{1 \mathord{\times} 7 \bmod 10, 7 == 1*7 mod 10}), subtracting @m{7\mathord{\times}543 = 3801,7*543=3801}, leaving 325800. And finally at the third digit with quotient digit 6 (@m{8 \mathord{\times} 7 \bmod 10, 8*7 mod 10}), subtracting @m{6\mathord{\times}543 = 3258,6*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 2N@cross{}N division like this one, only about half the work of a normal basecase division is necessary.
For an N@cross{}M 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 2@cross{}1 divide and multiply. Secondly, the crossproducts are reduced when Q>M to @m{QM-M(M+1)/2,Q*M-M*(M+1)/2}, or when @math{Q@le{}M} to @m{Q(Q-1)/2, Q*(Q-1)/2}. Notice the savings are complementary. If Q is big then many divisions are saved, or if Q is small then the crossproducts reduce to a small number.
The modular inverse used is calculated efficiently by
modlimb_invert
in `gmp-impl.h'. This does four multiplies for a 32-bit limb, or six for a 64-bit limb. `tune/modlinv.c' has some alternate implementations that might suit processors better at bit twiddling than multiplying.The sub-quadratic exact division described by Jebelean in "Exact Division with Karatsuba Complexity" is not currently implemented. It uses a rearrangement similar to the divide and conquer for normal division (see section 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 2N@cross{}N division.
A special case exact division by 3 exists in
mpn_divexact_by3
, supporting Toom-3 multiplication andmpq
canonicalizations. It forms quotient digits with a multiply by the modular inverse of 3 (which is0xAA..AAB
) and uses two comparisons to determine a borrow for the next limb. The multiplications don't need to be on the dependent chain, as long as the effect of the borrows is applied. Only a few optimized assembler implementations currently exist.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 @m{b = 2 \GMPraise{
mp\_bits\_per\_limb
}, b = 2^mp_bits_per_limb}, then this remainder r is of the form @ifnottexa = q*d + r*b^nn 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@le{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 theredc
function inmpz_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
inmpn_gcd
, and the use ofmpn_modexact_1_odd
bympn_gcd_1
andmpz_kronecker_ui
etc (see section Greatest Common Divisor).Montgomery's REDC method for modular multiplications uses operands of the form of @m{xb^{-n}, x*b^-n} and @m{yb^{-n}, y*b^-n} and on calculating @m{(xb^{-n}) (yb^{-n}), (x*b^-n)*(y*b^-n)} uses the factor of b^n in the exact remainder to reach a product in the same form @m{(xy)b^{-n}, (x*y)*b^-n} (see section Modular Powering).
Notice that r generally gives no useful information about the ordinary remainder a @bmod d since b^n @bmod d could be anything. If however b^n == 1 @bmod d, then r is the negative of the ordinary remainder. This occurs whenever d is a factor of b^n-1, as for example with 3 in
mpn_divexact_by3
. Other such factors include 5, 17 and 257, but no particular use has been found for this.Small Quotient Division
An N@cross{}M 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 @m{\rm2Q,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 @m{v_2 \GMPhat q > b \GMPhat r + u_2, v2*q>b*r+u2} condition appropriately relaxed.
Greatest Common Divisor
Binary GCD
At small sizes GMP uses an O(N^2) binary style GCD. This is described in many textbooks, for example Knuth section 4.5.2 algorithm B. It simply consists of successively reducing operands a and b using @gcd{(a,b) = @gcd{}(@min{}(a,b),@abs{}(a-b))}, and also that if a and b are first made odd then @abs{(a-b)} is even and factors of two can be discarded.
Variants like letting a-b become negative and doing a different next step are of interest only as far as they suit particular CPUs, since on small operands it's machine dependent factors that determine performance.
The Euclidean GCD algorithm, as per Knuth algorithms E and A, reduces using a @bmod 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 @bmod b reductions in
mpn_gcd
andmpn_gcd_1
(the latter for both N@cross{}1 and 1@cross{}1 cases). But after that initial reduction, big quotients occur too rarely to make it worth checking for them.Accelerated GCD
For sizes above
GCD_ACCEL_THRESHOLD
, GMP uses the Accelerated GCD algorithm described independently by Weber and Jebelean (the latter as the "Generalized Binary" algorithm), see section 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 @m{nv-du,n*v-d*u} where n and d are single limb values chosen to give two trailing zero limbs on that value, which can be stripped. n and d are calculated using an algorithm similar to half of a two limb GCD (see
find_a
in `mpn/generic/gcd.c').When u and v differ in size by more than a certain number of bits, a dmod is performed to zero out bits at the low end of the larger. It consists of an exact remainder style division applied to an appropriate number of bits (see section Exact Division, and see section 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 bympn_addmul_1
applying n and d.Extended GCD
The extended GCD calculates @gcd{(a,b)} and also cofactors x and y satisfying @m{ax+by=\gcd(a@C{}b), a*x+b*y=gcd(a@C{}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 thempn_addmul_1
calls.When a CPU has a data-dependent multiplier, meaning one which is faster on operands with fewer bits, the extra work in the double-limb calculation might only save some looping overheads, leading to a large
GCDEXT_THRESHOLD
.Currently the single limb calculation doesn't optimize for the small quotients that often occur, and this can lead to unusually low values of
GCDEXT_THRESHOLD
, depending on the CPU.An analysis of double-limb calculations can be found in "A Double-Digit Lehmer-Euclid Algorithm" by Jebelean (see section References). The code in GMP was developed independently.
It should be noted that when a double limb calculation is used, it's used for the whole of that GCD, it doesn't fall back to single limb part way through. This is because as the algorithm proceeds, the inputs a and b are reduced, but the cofactors x and y grow, so the multipliers at each step are applied to a roughly constant total number of limbs.
Jacobi Symbol
mpz_jacobi
andmpz_kronecker
are currently implemented with a simple binary algorithm similar to that described for the GCDs (see section 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 eithermpn_mod_1
ormpn_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.
Powering Algorithms
Normal Powering
Normal
mpz
ormpf
powering uses a simple binary algorithm, successively squaring and then multiplying by the base when a 1 bit is seen in the exponent, as per Knuth section 4.6.3. The "left to right" variant described there is used rather than algorithm A, since it's just as easy and can be done with somewhat less temporary memory.Modular Powering
Modular powering is implemented using a 2^k-ary sliding window algorithm, as per "Handbook of Applied Cryptography" algorithm 14.85 (see section References). k is chosen according to the size of the exponent. Larger exponents use larger values of k, the choice being made to minimize the average number of multiplications that must supplement the squaring.
The modular multiplies and squares use either a simple division or the REDC method by Montgomery (see section References). REDC is a little faster, essentially saving N single limb divisions in a fashion similar to an exact remainder (see section 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.Root Extraction Algorithms
Square Root
Square roots are taken using the "Karatsuba Square Root" algorithm by Paul Zimmermann (see section References). This is expressed in a divide and conquer form, but as noted in the paper it can also be viewed as a discrete variant of Newton's method.
In the Karatsuba multiplication range this is an @m{O({3\over2} M(N/2)),O(1.5*M(N/2))} algorithm, where M(n) is the time to multiply two numbers of n limbs. In the FFT multiplication range this grows to a bound of @m{O(6 M(N/2)),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 bothmpz_sqrt
andmpf_sqrt
. The extended precision given bympf_sqrt_ui
is obtained by padding with zero limbs.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. @ifnottex
1 A a[i+1] = - * ( --------- + (n-1)*a[i] ) n a[i]^(n-1)The initial approximation @m{a_1,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.
Perfect Square
mpz_perfect_square_p
is able to quickly exclude most non-squares by checking whether the input is a quadratic residue modulo some small integers.The first test is modulo 256 which means simply examining the least significant byte. Only 44 different values occur as the low byte of a square, so 82.8% of non-squares can be immediately excluded. Similar tests modulo primes from 3 to 29 exclude 99.5% of those remaining, or if a limb is 64 bits then primes up to 53 are used, excluding 99.99%. A single N@cross{}1 remainder using
PP
from `gmp-impl.h' quickly gives all these remainders.A square root must still be taken for any value that passes the residue tests, to verify it's really a square and not one of the 0.086% (or 0.000156% for 64 bits) non-squares that get through. See section Square Root.
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. (See section Nth Root.)If a prime divisor p with multiplicity e can be found, then only roots which are divisors of e need to be considered, much reducing the work necessary. To this end divisibility by a set of small primes is checked.
Radix Conversion
Radix conversions are less important than other algorithms. A program dominated by conversions should probably use a different data representation.
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 @m{b^{n2^i},b^(n*2^i)} of the radix are calculated, until a power between t and @m{\sqrt{t},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 thanGET_STR_DC_THRESHOLD
limbs, the basecase algorithm described above is used.The advantage of this algorithm is that big divisions can make use of the sub-quadratic divide and conquer division (see section 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 @m{b^{n2^i},b^(n*2^i)} must first be overcome.
GET_STR_PRECOMPUTE_THRESHOLD
andGET_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, whereasGET_STR_DC_THRESHOLD
assumes that's already available, which is the case when recursing.Since the base case produces digits from least to most significant but they want to be stored from most to least, it's necessary to calculate in advance how many digits there will be, or at least be sure not to underestimate that. For GMP the number of input bits is multiplied by
chars_per_bit_exactly
frommp_bases
, rounding up. The result is either correct or one too big.Examining some of the high bits of the input could increase the chance of getting the exact number of digits, but an exact result every time would not be practical, since in general the difference between numbers 100... and 99... is only in the last few bits and the work to identify 99... might well be almost as much as a full conversion.
mpf_get_str
doesn't currently use the algorithm described here, it multiplies or divides by a power of b to move the radix point to the just above the highest non-zero digit (or at worst one above that location), then multiplies by b^n to bring out digits. This is O(N^2) and is certainly not optimal.The r/b^n scheme described above for using multiplications to bring out digits might be useful for more than a single limb. Some brief experiments with it on the base case when recursing didn't give a noticable improvement, but perhaps that was only due to the implementation. Something similar would work for the sub-quadratic divisions too, though there would be the cost of calculating a bigger radix power.
Another possible improvement for the sub-quadratic part would be to arrange for radix powers that balanced the sizes of quotient and remainder produced, ie. the highest power would be an @m{b^{nk},b^(n*k)} approximately equal to @m{\sqrt{t},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.
Radix to Binary
Conversions from a power-of-2 radix into binary use a simple and fast O(N) bitwise concatenation algorithm.
Conversions from other radices use one of two algorithms. Sizes below
SET_STR_THRESHOLD
use a basic O(N^2) method. Groups of n digits are converted to limbs, where n is the biggest power of the base b which will fit in a limb, then those groups are accumulated into the result by multiplying by b^n and adding. This saves multi-precision operations, as per Knuth section 4.4 part E (see section References). Some special case code is provided for decimal, giving the compiler a chance to optimize multiplications by 10.Above
SET_STR_THRESHOLD
a sub-quadratic algorithm is used. First groups of n digits are converted into limbs. Then adjacent limbs are combined into limb pairs with @m{xb^n+y,x*b^n+y}, where x and y are the limbs. Adjacent limb pairs are combined into quads similarly with @m{xb^{2n}+y,x*b^(2n)+y}. This continues until a single block remains, that being the result.The advantage of this method is that the multiplications for each x are big blocks, allowing Karatsuba and higher algorithms to be used. But the cost of calculating the powers @m{b^{n2^i},b^(n*2^i)} must be overcome.
SET_STR_THRESHOLD
usually ends up quite big, around 5000 digits, and on some processors much bigger still.
SET_STR_THRESHOLD
is based on the input digits (and tuned for decimal), though it might be better based on a limb count, so as to be independent of the base. But that sort of count isn't used by the base case and so would need some sort of initial calculation or estimate.The main reason
SET_STR_THRESHOLD
is so much bigger than the correspondingGET_STR_PRECOMPUTE_THRESHOLD
is thatmpn_mul_1
is much faster thanmpn_divrem_1
(often by a factor of 10, or more).Other Algorithms
Factorial
Factorials n! are calculated by a simple product from 1 to n, but arranged into certain sub-products.
First as many factors as fit in a limb are accumulated, then two of those multiplied to give a 2-limb product. When two 2-limb products are ready they're multiplied to a 4-limb product, and when two 4-limbs are ready they're multiplied to an 8-limb product, etc. A stack of outstanding products is built up, with two of the same size multiplied together when ready.
Arranging for multiplications to have operands the same (or nearly the same) size means the Karatsuba and higher multiplication algorithms can be used. And even on sizes below the Karatsuba threshold an N@cross{}N multiply will give a basecase multiply more to work on.
An obvious improvement not currently implemented would be to strip factors of 2 from the products and apply them at the end with a bit shift. Another possibility would be to determine the prime factorization of the result (which can be done easily), and use a powering method, at each stage squaring then multiplying in those primes with a 1 in their exponent at that point. The advantage would be some multiplies turned into squares.
Binomial Coefficients
Binomial coefficients @m{\left({n}\atop{k}\right), C(n@C{}k)} are calculated by first arranging k @le{ n/2} using @m{\left({n}\atop{k}\right) = \left({n}\atop{n-k}\right), C(n@C{}k) = C(n@C{}n-k)} if necessary, and then evaluating the following product simply from i=2 to i=k. @ifnottex
k (n-k+i) C(n,k) = (n-k+1) * prod ------- i=2 iIt's easy to show that each denominator i will divide the product so far, so the exact division algorithm is used (see section Exact Division).
The numerators n-k+i and denominators i are first accumulated into as many fit a limb, to save multi-precision operations, though for
mpz_bin_ui
this applies only to the divisors, since n is anmpz_t
and n-k+i in general won't fit in a limb at all.An obvious improvement would be to strip factors of 2 from each multiplier and divisor and count them separately, to be applied with a bit shift at the end. Factors of 3 and perhaps 5 could even be handled similarly. Another possibility, if n is not too big, would be to determine the prime factorization of the result based on the factorials involved, and power up those primes appropriately. This would help most when k is near n/2.
Fibonacci Numbers
The Fibonacci functions
mpz_fib_ui
andmpz_fib2_ui
are designed for calculating isolated @m{F_n,F[n]} or @m{F_n,F[n]},@m{F_{n-1},F[n-1]} values efficiently.For small n, a table of single limb values in
__gmp_fib_table
is used. On a 32-bit limb this goes up to @m{F_{47},F[47]}, or on a 64-bit limb up to @m{F_{93},F[93]}. For convenience the table starts at @m{F_{-1},F[-1]}.Beyond the table, values are generated with a binary powering algorithm, calculating a pair @m{F_n,F[n]} and @m{F_{n-1},F[n-1]} working from high to low across the bits of n. The formulas used are @ifnottex
F[2k+1] = 4*F[k]^2 - F[k-1]^2 + 2*(-1)^k F[2k-1] = F[k]^2 + F[k-1]^2 F[2k] = F[2k+1] - F[2k-1]At each step, k is the high b bits of n. If the next bit of n is 0 then @m{F_{2k},F[2k]},@m{F_{2k-1},F[2k-1]} is used, or if it's a 1 then @m{F_{2k+1},F[2k+1]},@m{F_{2k},F[2k]} is used, and the process repeated until all bits of n are incorporated. Notice these formulas require just two squares per bit of n.
It'd be possible to handle the first few n above the single limb table with simple additions, using the defining Fibonacci recurrence @m{F_{k+1} = F_k + F_{k-1}, F[k+1]=F[k]+F[k-1]}, but this is not done since it usually turns out to be faster for only about 10 or 20 values of n, and including a block of code for just those doesn't seem worthwhile. If they really mattered it'd be better to extend the data table.
Using a table avoids lots of calculations on small numbers, and makes small n go fast. A bigger table would make more small n go fast, it's just a question of balancing size against desired speed. For GMP the code is kept compact, with the emphasis primarily on a good powering algorithm.
mpz_fib2_ui
returns both @m{F_n,F[n]} and @m{F_{n-1},F[n-1]}, butmpz_fib_ui
is only interested in @m{F_n,F[n]}. In this case the last step of the algorithm can become one multiply instead of two squares. One of the following two formulas is used, according as n is odd or even. @ifnottexF[2k] = F[k]*(F[k]+2F[k-1]) F[2k+1] = (2F[k]+F[k-1])*(2F[k]-F[k-1]) + 2*(-1)^k@m{F_{2k+1},F[2k+1]} here is the same as above, just rearranged to be a multiply. For interest, the @m{2(-1)^k, 2*(-1)^k} term both here and above can be applied just to the low limb of the calculation, without a carry or borrow into further limbs, which saves some code size. See comments with
mpz_fib_ui
and the internalmpn_fib2_ui
for how this is done.Lucas Numbers
mpz_lucnum2_ui
derives a pair of Lucas numbers from a pair of Fibonacci numbers with the following simple formulas. @ifnottexL[k] = F[k] + 2*F[k-1] L[k-1] = 2*F[k] - F[k-1]
mpz_lucnum_ui
is only interested in @m{L_n,L[n]}, and some work can be saved. Trailing zero bits on n can be handled with a single square each. @ifnottexL[2k] = L[k]^2 - 2*(-1)^kAnd the lowest 1 bit can be handled with one multiply of a pair of Fibonacci numbers, similar to what
mpz_fib_ui
does. @ifnottexL[2k+1] = 5*F[k-1]*(2*F[k]+F[k-1]) - 4*(-1)^kAssembler Coding
The assembler subroutines in GMP are the most significant source of speed at small to moderate sizes. At larger sizes algorithm selection becomes more important, but of course speedups in low level routines will still speed up everything proportionally.
Carry handling and widening multiplies that are important for GMP can't be easily expressed in C. GCC
asm
blocks help a lot and are provided in `longlong.h', but hand coding low level routines invariably offers a speedup over generic C by a factor of anything from 2 to 10.Code Organisation
The various `mpn' subdirectories contain machine-dependent code, written in C or assembler. The `mpn/generic' subdirectory contains default code, used when there's no machine-specific version of a particular file.
Each `mpn' subdirectory is for an ISA family. Generally 32-bit and 64-bit variants in a family cannot share code and will have separate directories. Within a family further subdirectories may exist for CPU variants.
Assembler Basics
mpn_addmul_1
andmpn_submul_1
are the most important routines for overall GMP performance. All multiplications and divisions come down to repeated calls to these.mpn_add_n
,mpn_sub_n
,mpn_lshift
andmpn_rshift
are next most important.On some CPUs assembler versions of the internal functions
mpn_mul_basecase
andmpn_sqr_basecase
give significant speedups, mainly through avoiding function call overheads. They can also potentially make better use of a wide superscalar processor.The restrictions on overlaps between sources and destinations (see section Low-level Functions) are designed to facilitate a variety of implementations. For example, knowing
mpn_add_n
won't have partly overlapping sources and destination means reading can be done far ahead of writing on superscalar processors, and loops can be vectorized on a vector processor, depending on the carry handling.Carry Propagation
The problem that presents most challenges in GMP is propagating carries from one limb to the next. In functions like
mpn_addmul_1
andmpn_add_n
, carries are the only dependencies between limb operations.On processors with carry flags, a straightforward CISC style
adc
is generally best. AMD K6mpn_addmul_1
however is an example of an unusual set of circumstances where a branch works out better.On RISC processors generally an add and compare for overflow is used. This sort of thing can be seen in `mpn/generic/aors_n.c'. Some carry propagation schemes require 4 instructions, meaning at least 4 cycles per limb, but other schemes may use just 1 or 2. On wide superscalar processors performance may be completely determined by the number of dependent instructions between carry-in and carry-out for each limb.
On vector processors good use can be made of the fact that a carry bit only very rarely propagates more than one limb. When adding a single bit to a limb, there's only a carry out if that limb was
0xFF...FF
which on random data will be only 1 in @m{2\GMPraise{mp\_bits\_per\_limb
}, 2^mp_bits_per_limb}. `mpn/cray/add_n.c' is an example of this, it adds all limbs in parallel, adds one set of carry bits in parallel and then only rarely needs to fall through to a loop propagating further carries.On the x86s, GCC (as of version 2.95.2) doesn't generate particularly good code for the RISC style idioms that are necessary to handle carry bits in C. Often conditional jumps are generated where
adc
orsbb
forms would be better. And so unfortunately almost any loop involving carry bits needs to be coded in assembler for best results.Cache Handling
GMP aims to perform well both on operands that fit entirely in L1 cache and those which don't.
Basic routines like
mpn_add_n
ormpn_lshift
are often used on large operands, so L2 and main memory performance is important for them.mpn_mul_1
andmpn_addmul_1
are mostly used for multiply and square basecases, so L1 performance matters most for them, unless assembler versions ofmpn_mul_basecase
andmpn_sqr_basecase
exist, in which case the remaining uses are mostly for larger operands.For L2 or main memory operands, memory access times will almost certainly be more than the calculation time. The aim therefore is to maximize memory throughput, by starting a load of the next cache line which processing the contents of the previous one. Clearly this is only possible if the chip has a lock-up free cache or some sort of prefetch instruction. Most current chips have both these features.
Prefetching sources combines well with loop unrolling, since a prefetch can be initiated once per unrolled loop (or more than once if the loop covers more than one cache line).
On CPUs without write-allocate caches, prefetching destinations will ensure individual stores don't go further down the cache hierarchy, limiting bandwidth. Of course for calculations which are slow anyway, like
mpn_divrem_1
, write-throughs might be fine.The distance ahead to prefetch will be determined by memory latency versus throughput. The aim of course is to have data arriving continuously, at peak throughput. Some CPUs have limits on the number of fetches or prefetches in progress.
If a special prefetch instruction doesn't exist then a plain load can be used, but in that case care must be taken not to attempt to read past the end of an operand, since that might produce a segmentation violation.
Some CPUs or systems have hardware that detects sequential memory accesses and initiates suitable cache movements automatically, making life easy.
Floating Point
Floating point arithmetic is used in GMP for multiplications on CPUs with poor integer multipliers. It's mostly useful for
mpn_mul_1
,mpn_addmul_1
andmpn_submul_1
on 64-bit machines, andmpn_mul_basecase
on both 32-bit and 64-bit machines.With IEEE 53-bit double precision floats, integer multiplications producing up to 53 bits will give exact results. Breaking a 64@cross{}64 multiplication into eight 16@cross{}@math{32@rightarrow{}48} bit pieces is convenient. With some care though six 21@cross{}@math{32@rightarrow{}53} bit products can be used, if one of the lower two 21-bit pieces also uses the sign bit.
For the
mpn_mul_1
family of functions on a 64-bit machine, the invariant single limb is split at the start, into 3 or 4 pieces. Inside the loop, the bignum operand is split into 32-bit pieces. Fast conversion of these unsigned 32-bit pieces to floating point is highly machine-dependent. In some cases, reading the data into the integer unit, zero-extending to 64-bits, then transferring to the floating point unit back via memory is the only option.Converting partial products back to 64-bit limbs is usually best done as a signed conversion. Since all values are smaller than @m{2^{53},2^53}, signed and unsigned are the same, but most processors lack unsigned conversions.
Here is a diagram showing 16@cross{}32 bit products for an
mpn_mul_1
ormpn_addmul_1
with a 64-bit limb. The single limb operand V is split into four 16-bit parts. The multi-limb operand U is split in the loop into two 32-bit parts.@ifnottex
+---+---+---+---+ |v48|v32|v16|v00| V operand +---+---+---+---+ +-------+---+---+ x | u32 | u00 | U operand (one limb) +---------------+ --------------------------------- +-----------+ | u00 x v00 | p00 48-bit products +-----------+ +-----------+ | u00 x v16 | p16 +-----------+ +-----------+ | u00 x v32 | p32 +-----------+ +-----------+ | u00 x v48 | p48 +-----------+ +-----------+ | u32 x v00 | r32 +-----------+ +-----------+ | u32 x v16 | r48 +-----------+ +-----------+ | u32 x v32 | r64 +-----------+ +-----------+ | u32 x v48 | r80 +-----------+p32 and r32 can be summed using floating-point addition, and likewise p48 and r48. p00 and p16 can be summed with r64 and r80 from the previous iteration.
For each loop then, four 49-bit quantities are transfered to the integer unit, aligned as follows,
@ifnottex
|-----64bits----|-----64bits----| +------------+ | p00 + r64' | i00 +------------+ +------------+ | p16 + r80' | i16 +------------+ +------------+ | p32 + r32 | i32 +------------+ +------------+ | p48 + r48 | i48 +------------+The challenge then is to sum these efficiently and add in a carry limb, generating a low 64-bit result limb and a high 33-bit carry limb (i48 extends 33 bits into the high half).
SIMD Instructions
The single-instruction multiple-data support in current microprocessors is aimed at signal processing algorithms where each data point can be treated more or less independently. There's generally not much support for propagating the sort of carries that arise in GMP.
SIMD multiplications of say four 16@cross{}16 bit multiplies only do as much work as one 32@cross{}32 from GMP's point of view, and need some shifts and adds besides. But of course if say the SIMD form is fully pipelined and uses less instruction decoding then it may still be worthwhile.
On the 80x86 chips, MMX has so far found a use in
mpn_rshift
andmpn_lshift
since it allows 64-bit operations, and is used in a special case for 16-bit multipliers in the P55mpn_mul_1
. 3DNow and SSE haven't found a use so far.Software Pipelining
Software pipelining consists of scheduling instructions around the branch point in a loop. For example a loop taking a checksum of an array of limbs might have a load and an add, but the load wouldn't be for that add, rather for the one next time around the loop. Each load then is effectively scheduled back in the previous iteration, allowing latency to be hidden.
Naturally this is wanted only when doing things like loads or multiplies that take a few cycles to complete, and only where a CPU has multiple functional units so that other work can be done while waiting.
A pipeline with several stages will have a data value in progress at each stage and each loop iteration moves them along one stage. This is like juggling.
Within the loop some moves between registers may be necessary to have the right values in the right places for each iteration. Loop unrolling can help this, with each unrolled block able to use different registers for different values, even if some shuffling is still needed just before going back to the top of the loop.
Loop Unrolling
Loop unrolling consists of replicating code so that several limbs are processed in each loop. At a minimum this reduces loop overheads by a corresponding factor, but it can also allow better register usage, for example alternately using one register combination and then another. Judicious use of @command{m4} macros can help avoid lots of duplication in the source code.
Unrolling is commonly done to a power of 2 multiple so the number of unrolled loops and the number of remaining limbs can be calculated with a shift and mask. But other multiples can be used too, just by subtracting each n limbs processed from a counter and waiting for less than n remaining (or offsetting the counter by n so it goes negative when there's less than n remaining).
The limbs not a multiple of the unrolling can be handled in various ways, for example
- A simple loop at the end (or the start) to process the excess. Care will be wanted that it isn't too much slower than the unrolled part.
- A set of binary tests, for example after an 8-limb unrolling, test for 4 more limbs to process, then a further 2 more or not, and finally 1 more or not. This will probably take more code space than a simple loop.
- A
switch
statement, providing separate code for each possible excess, for example an 8-limb unrolling would have separate code for 0 remaining, 1 remaining, etc, up to 7 remaining. This might take a lot of code, but may be the best way to optimize all cases in combination with a deep pipelined loop.- A computed jump into the middle of the loop, thus making the first iteration handle the excess. This should make times smoothly increase with size, which is attractive, but setups for the jump and adjustments for pointers can be tricky and could become quite difficult in combination with deep pipelining.
One way to write the setups and finishups for a pipelined unrolled loop is simply to duplicate the loop at the start and the end, then delete instructions at the start which have no valid antecedents, and delete instructions at the end whose results are unwanted. Sizes not a multiple of the unrolling can then be handled as desired.
Internals
This chapter is provided only for informational purposes and the various internals described here may change in future GMP releases. Applications expecting to be compatible with future releases should use only the documented interfaces described in previous chapters.
Integer Internals
mpz_t
variables represent integers using sign and magnitude, in space dynamically allocated and reallocated. The fields are as follows.
_mp_size
- The number of limbs, or the negative of that when representing a negative integer. Zero is represented by
_mp_size
set to zero, in which case the_mp_d
data is unused._mp_d
- A pointer to an array of limbs which is the magnitude. These are stored "little endian" as per the
mpn
functions, so_mp_d[0]
is the least significant limb and_mp_d[ABS(_mp_size)-1]
is the most significant. Whenever_mp_size
is non-zero, the most significant limb is non-zero. Currently there's always at least one limb allocated, so for instancempz_set_ui
never needs to reallocate, andmpz_get_ui
can fetch_mp_d[0]
unconditionally (though its value is then only wanted if_mp_size
is non-zero)._mp_alloc
_mp_alloc
is the number of limbs currently allocated at_mp_d
, and naturally_mp_alloc >= ABS(_mp_size)
. When anmpz
routine is about to (or might be about to) increase_mp_size
, it checks_mp_alloc
to see whether there's enough space, and reallocates if not.MPZ_REALLOC
is generally used for this.The various bitwise logical functions like
mpz_and
behave as if negative values were twos complement. But sign and magnitude is always used internally, and necessary adjustments are made during the calculations. Sometimes this isn't pretty, but sign and magnitude are best for other routines.Some internal temporary variables are setup with
MPZ_TMP_INIT
and these have_mp_d
space obtained fromTMP_ALLOC
rather than the memory allocation functions. Care is taken to ensure that these are big enough that no reallocation is necessary (since it would have unpredictable consequences).Rational Internals
mpq_t
variables represent rationals using anmpz_t
numerator and denominator (see section Integer Internals).The canonical form adopted is denominator positive (and non-zero), no common factors between numerator and denominator, and zero uniquely represented as 0/1.
It's believed that casting out common factors at each stage of a calculation is best in general. A GCD is an O(N^2) operation so it's better to do a few small ones immediately than to delay and have to do a big one later. Knowing the numerator and denominator have no common factors can be used for example in
mpq_mul
to make only two cross GCDs necessary, not four.This general approach to common factors is badly sub-optimal in the presence of simple factorizations or little prospect for cancellation, but GMP has no way to know when this will occur. As per section Efficiency, that's left to applications. The
mpq_t
framework might still suit, withmpq_numref
andmpq_denref
for direct access to the numerator and denominator, or of coursempz_t
variables can be used directly.Float Internals
Efficient calculation is the primary aim of GMP floats and the use of whole limbs and simple rounding facilitates this.
mpf_t
floats have a variable precision mantissa and a single machine word signed exponent. The mantissa is represented using sign and magnitude.@ifnottex
most least significant significant limb limb _mp_d |---- _mp_exp ---> | _____ _____ _____ _____ _____ |_____|_____|_____|_____|_____| . <------------ radix point <-------- _mp_size --------->The fields are as follows.
_mp_size
- The number of limbs currently in use, or the negative of that when representing a negative value. Zero is represented by
_mp_size
and_mp_exp
both set to zero, and in that case the_mp_d
data is unused. (In the future_mp_exp
might be undefined when representing zero.)_mp_prec
- The precision of the mantissa, in limbs. In any calculation the aim is to produce
_mp_prec
limbs of result (the most significant being non-zero)._mp_d
- A pointer to the array of limbs which is the absolute value of the mantissa. These are stored "little endian" as per the
mpn
functions, so_mp_d[0]
is the least significant limb and_mp_d[ABS(_mp_size)-1]
the most significant. The most significant limb is always non-zero, but there are no other restrictions on its value, in particular the highest 1 bit can be anywhere within the limb._mp_prec+1
limbs are allocated to_mp_d
, the extra limb being for convenience (see below). There are no reallocations during a calculation, only in a change of precision withmpf_set_prec
._mp_exp
- The exponent, in limbs, determining the location of the implied radix point. Zero means the radix point is just above the most significant limb. Positive values mean a radix point offset towards the lower limbs and hence a value @ge{ 1}, as for example in the diagram above. Negative exponents mean a radix point further above the highest limb. Naturally the exponent can be any value, it doesn't have to fall within the limbs as the diagram shows, it can be a long way above or a long way below. Limbs other than those included in the
{_mp_d,_mp_size}
data are treated as zero.The following various points should be noted.
- Low Zeros
- The least significant limbs
_mp_d[0]
etc can be zero, though such low zeros can always be ignored. Routines likely to produce low zeros check and avoid them to save time in subsequent calculations, but for most routines they're quite unlikely and aren't checked.- Mantissa Size Range
- The
_mp_size
count of limbs in use can be less than_mp_prec
if the value can be represented in less. This means low precision values or small integers stored in a high precisionmpf_t
can still be operated on efficiently._mp_size
can also be greater than_mp_prec
. Firstly a value is allowed to use all of the_mp_prec+1
limbs available at_mp_d
, and secondly whenmpf_set_prec_raw
lowers_mp_prec
it leaves_mp_size
unchanged and so the size can be arbitrarily bigger than_mp_prec
.- Rounding
- All rounding is done on limb boundaries. Calculating
_mp_prec
limbs with the high non-zero will ensure the application requested minimum precision is obtained. The use of simple "trunc" rounding towards zero is efficient, since there's no need to examine extra limbs and increment or decrement.- Bit Shifts
- Since the exponent is in limbs, there are no bit shifts in basic operations like
mpf_add
andmpf_mul
. When differing exponents are encountered all that's needed is to adjust pointers to line up the relevant limbs. Of coursempf_mul_2exp
andmpf_div_2exp
will require bit shifts, but the choice is between an exponent in limbs which requires shifts there, or one in bits which requires them almost everywhere else.- Use of
_mp_prec+1
Limbs- The extra limb on
_mp_d
(_mp_prec+1
rather than just_mp_prec
) helps when anmpf
routine might get a carry from its operation.mpf_add
for instance will do anmpn_add
of_mp_prec
limbs. If there's no carry then that's the result, but if there is a carry then it's stored in the extra limb of space and_mp_size
becomes_mp_prec+1
. Whenever_mp_prec+1
limbs are held in a variable, the low limb is not needed for the intended precision, only the_mp_prec
high limbs. But zeroing it out or moving the rest down is unnecessary. Subsequent routines reading the value will simply take the high limbs they need, and this will be_mp_prec
if their target has that same precision. This is no more than a pointer adjustment, and must be checked anyway since the destination precision can be different from the sources. Copy functions likempf_set
will retain a full_mp_prec+1
limbs if available. This ensures that a variable which has_mp_size
equal to_mp_prec+1
will get its full exact value copied. Strictly speaking this is unnecessary since only_mp_prec
limbs are needed for the application's requested precision, but it's considered that anmpf_set
from one variable into another of the same precision ought to produce an exact copy.- Application Precisions
__GMPF_BITS_TO_PREC
converts an application requested precision to an_mp_prec
. The value in bits is rounded up to a whole limb then an extra limb is added since the most significant limb of_mp_d
is only non-zero and therefore might contain only one bit.__GMPF_PREC_TO_BITS
does the reverse conversion, and removes the extra limb from_mp_prec
before converting to bits. The net effect of reading back withmpf_get_prec
is simply the precision rounded up to a multiple ofmp_bits_per_limb
. Note that the extra limb added here for the high only being non-zero is in addition to the extra limb allocated to_mp_d
. For example with a 32-bit limb, an application request for 250 bits will be rounded up to 8 limbs, then an extra added for the high being only non-zero, giving an_mp_prec
of 9._mp_d
then gets 10 limbs allocated. Reading back withmpf_get_prec
will take_mp_prec
subtract 1 limb and multiply by 32, giving 256 bits. Strictly speaking, the fact the high limb has at least one bit means that a float with, say, 3 limbs of 32-bits each will be holding at least 65 bits, but for the purposes ofmpf_t
it's considered simply to be 64 bits, a nice multiple of the limb size.Raw Output Internals
mpz_out_raw
uses the following format.@ifnottex
+------+------------------------+ | size | data bytes | +------+------------------------+The size is 4 bytes written most significant byte first, being the number of subsequent data bytes, or the twos complement negative of that when a negative integer is represented. The data bytes are the absolute value of the integer, written most significant byte first.
The most significant data byte is always non-zero, so the output is the same on all systems, irrespective of limb size.
In GMP 1, leading zero bytes were written to pad the data bytes to a multiple of the limb size.
mpz_inp_raw
will still accept this, for compatibility.The use of "big endian" for both the size and data fields is deliberate, it makes the data easy to read in a hex dump of a file. Unfortunately it also means that the limb data must be reversed when reading or writing, so neither a big endian nor little endian system can just read and write
_mp_d
.C++ Interface Internals
A system of expression templates is used to ensure something like
a=b+c
turns into a simple call tompz_add
etc. Formpf_class
andmpfr_class
the scheme also ensures the precision of the final destination is used for any temporaries within a statement likef=w*x+y*z
. These are important features which a naive implementation cannot provide.A simplified description of the scheme follows. The true scheme is complicated by the fact that expressions have different return types. For detailed information, refer to the source code.
To perform an operation, say, addition, we first define a "function object" evaluating it,
struct __gmp_binary_plus { static void eval(mpf_t f, mpf_t g, mpf_t h) { mpf_add(f, g, h); } };And an "additive expression" object,
__gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> > operator+(const mpf_class &f, const mpf_class &g) { return __gmp_expr <__gmp_binary_expr<mpf_class, mpf_class, __gmp_binary_plus> >(f, g); }The seemingly redundant
__gmp_expr<__gmp_binary_expr<...>>
is used to encapsulate any possible kind of expression into a single template type. In fact evenmpf_class
etc aretypedef
specializations of__gmp_expr
.Next we define assignment of
__gmp_expr
tompf_class
.template <class T> mpf_class & mpf_class::operator=(const __gmp_expr<T> &expr) { expr.eval(this->get_mpf_t(), this->precision()); return *this; } template <class Op> void __gmp_expr<__gmp_binary_expr<mpf_class, mpf_class, Op> >::eval (mpf_t f, unsigned long int precision) { Op::eval(f, expr.val1.get_mpf_t(), expr.val2.get_mpf_t()); }where
expr.val1
andexpr.val2
are references to the expression's operands (hereexpr
is the__gmp_binary_expr
stored within the__gmp_expr
).This way, the expression is actually evaluated only at the time of assignment, when the required precision (that of
f
) is known. Furthermore the targetmpf_t
is now available, thus we can callmpf_add
directly withf
as the output argument.Compound expressions are handled by defining operators taking subexpressions as their arguments, like this:
template <class T, class U> __gmp_expr <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > operator+(const __gmp_expr<T> &expr1, const __gmp_expr<U> &expr2) { return __gmp_expr <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, __gmp_binary_plus> > (expr1, expr2); }And the corresponding specializations of
__gmp_expr::eval
:template <class T, class U, class Op> void __gmp_expr <__gmp_binary_expr<__gmp_expr<T>, __gmp_expr<U>, Op> >::eval (mpf_t f, unsigned long int precision) { // declare two temporaries mpf_class temp1(expr.val1, precision), temp2(expr.val2, precision); Op::eval(f, temp1.get_mpf_t(), temp2.get_mpf_t()); }The expression is thus recursively evaluated to any level of complexity and all subexpressions are evaluated to the precision of
f
.Contributors
Torbjorn Granlund wrote the original GMP library and is still developing and maintaining it. Several other individuals and organizations have contributed to GMP in various ways. Here is a list in chronological order:
Gunnar Sjoedin and Hans Riesel helped with mathematical problems in early versions of the library.
Richard Stallman contributed to the interface design and revised the first version of this manual.
Brian Beuning and Doug Lea helped with testing of early versions of the library and made creative suggestions.
John Amanatides of York University in Canada contributed the function
mpz_probab_prime_p
.Paul Zimmermann of Inria sparked the development of GMP 2, with his comparisons between bignum packages.
Ken Weber (Kent State University, Universidade Federal do Rio Grande do Sul) contributed
mpz_gcd
,mpz_divexact
,mpn_gcd
, andmpn_bdivmod
, partially supported by CNPq (Brazil) grant 301314194-2.Per Bothner of Cygnus Support helped to set up GMP to use Cygnus' configure. He has also made valuable suggestions and tested numerous intermediary releases.
Joachim Hollman was involved in the design of the
mpf
interface, and in thempz
design revisions for version 2.Bennet Yee contributed the initial versions of
mpz_jacobi
andmpz_legendre
.Andreas Schwab contributed the files `mpn/m68k/lshift.S' and `mpn/m68k/rshift.S' (now in `.asm' form).
The development of floating point functions of GNU MP 2, were supported in part by the ESPRIT-BRA (Basic Research Activities) 6846 project POSSO (POlynomial System SOlving).
GNU MP 2 was finished and released by SWOX AB, SWEDEN, in cooperation with the IDA Center for Computing Sciences, USA.
Robert Harley of Inria, France and David Seal of ARM, England, suggested clever improvements for population count.
Robert Harley also wrote highly optimized Karatsuba and 3-way Toom multiplication functions for GMP 3. He also contributed the ARM assembly code.
Torsten Ekedahl of the Mathematical department of Stockholm University provided significant inspiration during several phases of the GMP development. His mathematical expertise helped improve several algorithms.
Paul Zimmermann wrote the Divide and Conquer division code, the REDC code, the REDC-based mpz_powm code, the FFT multiply code, and the Karatsuba square root. The ECMNET project Paul is organizing was a driving force behind many of the optimizations in GMP 3.
Linus Nordberg wrote the new configure system based on autoconf and implemented the new random functions.
Kent Boortz made the Macintosh port.
Kevin Ryde worked on a number of things: optimized x86 code, m4 asm macros, parameter tuning, speed measuring, the configure system, function inlining, divisibility tests, bit scanning, Jacobi symbols, Fibonacci and Lucas number functions, printf and scanf functions, perl interface, demo expression parser, the algorithms chapter in the manual, `gmpasm-mode.el', and various miscellaneous improvements elsewhere.
Steve Root helped write the optimized alpha 21264 assembly code.
Gerardo Ballabio wrote the `gmpxx.h' C++ class interface and the C++
istream
input routines.GNU MP 4.0 was finished and released by Torbjorn Granlund and Kevin Ryde. Torbjorn's work was partially funded by the IDA Center for Computing Sciences, USA.
(This list is chronological, not ordered after significance. If you have contributed to GMP but are not listed above, please tell tege@swox.com about the omission!)
Thanks goes to Hans Thorsen for donating an SGI system for the GMP test system environment.
References
Books
- Jonathan M. Borwein and Peter B. Borwein, "Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity", Wiley, John & Sons, 1998.
- Henri Cohen, "A Course in Computational Algebraic Number Theory", Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. @texlinebreak{} http://www.math.u-bordeaux.fr/~cohen
- Donald E. Knuth, "The Art of Computer Programming", volume 2, "Seminumerical Algorithms", 3rd edition, Addison-Wesley, 1998. @texlinebreak{} http://www-cs-faculty.stanford.edu/~knuth/taocp.html
- John D. Lipson, "Elements of Algebra and Algebraic Computing", The Benjamin Cummings Publishing Company Inc, 1981.
- Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone, "Handbook of Applied Cryptography", http://www.cacr.math.uwaterloo.ca/hac/
- Richard M. Stallman, "Using and Porting GCC", Free Software Foundation, 1999, available online http://www.gnu.org/software/gcc/onlinedocs/, and in the GCC package ftp://ftp.gnu.org/gnu/gcc/
Papers
- Christoph Burnikel and Joachim Ziegler, "Fast Recursive Division", Max-Planck-Institut fuer Informatik Research Report MPI-I-98-1-022, @texlinebreak{} http://data.mpi-sb.mpg.de/internet/reports.nsf/NumberView/1998-1-022
- Torbjorn Granlund and Peter L. Montgomery, "Division by Invariant Integers using Multiplication", in Proceedings of the SIGPLAN PLDI'94 Conference, June 1994. Also available ftp://ftp.cwi.nl/pub/pmontgom/divcnst.psa4.gz (and .psl.gz).
- Peter L. Montgomery, "Modular Multiplication Without Trial Division", in Mathematics of Computation, volume 44, number 170, April 1985.
- Tudor Jebelean, "An algorithm for exact division", Journal of Symbolic Computation, volume 15, 1993, pp. 169-180. Research report version available @texlinebreak{} ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-35.ps.gz
- Tudor Jebelean, "Exact Division with Karatsuba Complexity - Extended Abstract", RISC-Linz technical report 96-31, @texlinebreak{} ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-31.ps.gz
- Tudor Jebelean, "Practical Integer Division with Karatsuba Complexity", ISSAC 97, pp. 339-341. Technical report available @texlinebreak{} ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1996/96-29.ps.gz
- Tudor Jebelean, "A Generalization of the Binary GCD Algorithm", ISSAC 93, pp. 111-116. Technical report version available @texlinebreak{} ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1993/93-01.ps.gz
- Tudor Jebelean, "A Double-Digit Lehmer-Euclid Algorithm for Finding the GCD of Long Integers", Journal of Symbolic Computation, volume 19, 1995, pp. 145-157. Technical report version also available @texlinebreak{} ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1992/92-69.ps.gz
- Werner Krandick and Tudor Jebelean, "Bidirectional Exact Integer Division", Journal of Symbolic Computation, volume 21, 1996, pp. 441-455. Early technical report version also available ftp://ftp.risc.uni-linz.ac.at/pub/techreports/1994/94-50.ps.gz
- R. Moenck and A. Borodin, "Fast Modular Transforms via Division", Proceedings of the 13th Annual IEEE Symposium on Switching and Automata Theory, October 1972, pp. 90-96. Reprinted as "Fast Modular Transforms", Journal of Computer and System Sciences, volume 8, number 3, June 1974, pp. 366-386.
- Arnold Sch@"onhage and Volker Strassen, "Schnelle Multiplikation grosser Zahlen", Computing 7, 1971, pp. 281-292.
- Kenneth Weber, "The accelerated integer GCD algorithm", ACM Transactions on Mathematical Software, volume 21, number 1, March 1995, pp. 111-122.
- Paul Zimmermann, "Karatsuba Square Root", INRIA Research Report 3805, November 1999, http://www.inria.fr/RRRT/RR-3805.html
- Paul Zimmermann, "A Proof of GMP Fast Division and Square Root Implementations", @texlinebreak{} http://www.loria.fr/~zimmerma/papers/proof-div-sqrt.ps.gz
- Dan Zuras, "On Squaring and Multiplying Large Integers", ARITH-11: IEEE Symposium on Computer Arithmetic, 1993, pp. 260 to 271. Reprinted as "More on Multiplying and Squaring Large Integers", IEEE Transactions on Computers, volume 43, number 8, August 1994, pp. 899-908.
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Concept Index
Jump to: a - b - c - d - e - f - g - h - i - j - k - l - m - n - o - p - r - s - t - u - v - w
a
ABI About this manual Algorithms alloca
Allocation of memory Anonymous FTP of latest version Application Binary Interface Arithmetic functions, Arithmetic functions, Arithmetic functions Assignment functions, Assignment functions Autoconf detections b
Basics Berkeley MP compatible functions Binomial coefficient functions Bit manipulation functions Bit shift left Bit shift right Bits per limb BSD MP compatible functions Bug reporting Build notes for binary packaging Build notes for particular systems Build options Build problems known Building GMP c
C++ Interface C++ istream
inputC++ ostream
outputComparison functions, Comparison functions, Comparison functions Compatibility with older versions Conditions for copying GNU MP Configuring GMP Constants Contributors Conventions for parameters Conventions for variables Conversion functions, Conversion functions, Conversion functions Copying conditions CPUs supported Custom allocation d
Debugging Demonstration programs `DESTDIR' Division algorithms Division functions, Division functions, Division functions e
Efficiency Emacs Exact division functions Example programs Exponentiation functions, Exponentiation functions Export Extended GCD f
Factorial functions FDL, GNU Free Documentation License Fibonacci sequence functions Float arithmetic functions Float assignment functions Float comparison functions Float conversion functions Float functions Float init and assign functions Float initialization functions Float input and output functions Float miscellaneous functions Float sign tests Floating-point functions Floating-point number Formatted input Formatted output FTP of latest version Function classes g
GMP version number `gmp.h' gmpxx.h GNU Free Documentation License Greatest common divisor algorithms Greatest common divisor functions h
Headers Home page i
I/O functions, I/O functions, I/O functions Import Initialization and assignment functions, Initialization and assignment functions, Initialization and assignment functions Initialization functions, Initialization functions Input functions, Input functions, Input functions Installing GMP Instruction Set Architecture Integer Integer arithmetic functions Integer assignment functions Integer bit manipulation functions Integer comparison functions Integer conversion functions Integer division functions Integer exponentiation functions Integer export Integer functions Integer import Integer init and assign Integer initialization functions Integer input and output functions Integer miscellaneous functions Integer random number functions Integer root functions Integer sign tests Introduction ISA istream
inputj
Jacobi symbol functions k
Kronecker symbol functions l
Latest version of GMP Least common multiple functions Libraries Libtool versioning License conditions Limb Limb size Linking Logical functions Low-level functions Lucas number functions m
Mailing lists Memory allocation Memory Management Miscellaneous float functions Miscellaneous integer functions Modular inverse functions `mp.h' MPFR mpfrxx.h Multi-threading Multiplication algorithms n
Nails Nomenclature Number theoretic functions Numerator and denominator o
ostream
outputOutput functions, Output functions, Output functions p
Packaged builds Parameter conventions Particular systems perl
Powering algorithms Powering functions, Powering functions Precision of floats Prime testing functions printf
formatted outputProfiling r
Radix conversion algorithms Random number functions, Random number functions Random number seeding Random number state Rational arithmetic functions Rational comparison functions Rational conversion functions Rational init and assign Rational input and output functions Rational number Rational number functions Rational numerator and denominator Rational sign tests Reentrancy References Reporting bugs Root extraction algorithms Root extraction functions, Root extraction functions s
Sample programs scanf
formatted inputShared library versioning Sign tests, Sign tests, Sign tests Sparc Stack overflow segfaults Stripped libraries Systems t
Thread safety Types u
Upward compatibility Useful macros and constants User-defined precision v
Variable conventions Version number w
Web page Function and Type Index
Jump to: * - _ - a - c - d - f - g - h - i - m - o - p - r - s - t - x
*
*mpz_export _
__GNU_MP_VERSION __GNU_MP_VERSION_MINOR __GNU_MP_VERSION_PATCHLEVEL _mpz_realloc a
abs, abs, abs allocate_function c
ceil cmp, cmp, cmp, cmp, cmp, cmp d
deallocate_function f
floor g
gcd gmp_asprintf gmp_fprintf gmp_fscanf GMP_LIMB_BITS GMP_NAIL_BITS GMP_NAIL_MASK GMP_NUMB_BITS GMP_NUMB_MASK GMP_NUMB_MAX gmp_obstack_printf gmp_obstack_vprintf gmp_printf gmp_randclass gmp_randclass::get_f, gmp_randclass::get_f gmp_randclass::get_z_bits, gmp_randclass::get_z_bits gmp_randclass::get_z_range gmp_randclass::gmp_randclass, gmp_randclass::gmp_randclass gmp_randclass::seed, gmp_randclass::seed gmp_randclear gmp_randinit gmp_randinit_default gmp_randinit_lc_2exp gmp_randinit_lc_2exp_size gmp_randseed gmp_randseed_ui gmp_scanf gmp_snprintf gmp_sprintf gmp_sscanf gmp_vasprintf gmp_version gmp_vfprintf gmp_vfscanf gmp_vprintf gmp_vscanf gmp_vsnprintf gmp_vsprintf gmp_vsscanf h
hypot i
itom m
madd mcmp mdiv mfree min mout move mp_bits_per_limb mp_limb_t
mp_set_memory_functions mpf_abs mpf_add mpf_add_ui mpf_ceil mpf_class mpf_class::fits_sint_p mpf_class::fits_slong_p mpf_class::fits_sshort_p mpf_class::fits_uint_p mpf_class::fits_ulong_p mpf_class::fits_ushort_p mpf_class::get_d mpf_class::get_mpf_t mpf_class::get_prec mpf_class::get_si mpf_class::get_ui mpf_class::mpf_class, mpf_class::mpf_class mpf_class::set_prec mpf_class::set_prec_raw mpf_clear mpf_cmp mpf_cmp_d mpf_cmp_si mpf_cmp_ui mpf_div mpf_div_2exp mpf_div_ui mpf_eq mpf_fits_sint_p mpf_fits_slong_p mpf_fits_sshort_p mpf_fits_uint_p mpf_fits_ulong_p mpf_fits_ushort_p mpf_floor mpf_get_d mpf_get_d_2exp mpf_get_default_prec mpf_get_prec mpf_get_si mpf_get_str mpf_get_ui mpf_init mpf_init2 mpf_init_set mpf_init_set_d mpf_init_set_si mpf_init_set_str mpf_init_set_ui mpf_inp_str mpf_integer_p mpf_mul mpf_mul_2exp mpf_mul_ui mpf_neg mpf_out_str mpf_pow_ui mpf_random2 mpf_reldiff mpf_set mpf_set_d mpf_set_default_prec mpf_set_prec mpf_set_prec_raw mpf_set_q mpf_set_si mpf_set_str mpf_set_ui mpf_set_z mpf_sgn mpf_sqrt mpf_sqrt_ui mpf_sub mpf_sub_ui mpf_swap mpf_t
mpf_trunc mpf_ui_div mpf_ui_sub mpf_urandomb mpfr_class mpn_add mpn_add_1 mpn_add_n mpn_addmul_1 mpn_bdivmod mpn_cmp mpn_divexact_by3 mpn_divexact_by3c mpn_divmod mpn_divmod_1 mpn_divrem mpn_divrem_1 mpn_gcd mpn_gcd_1 mpn_gcdext mpn_get_str mpn_hamdist mpn_lshift mpn_mod_1 mpn_mul mpn_mul_1 mpn_mul_n mpn_perfect_square_p mpn_popcount mpn_random mpn_random2 mpn_rshift mpn_scan0 mpn_scan1 mpn_set_str mpn_sqrtrem mpn_sub mpn_sub_1 mpn_sub_n mpn_submul_1 mpn_tdiv_qr mpq_abs mpq_add mpq_canonicalize mpq_class mpq_class::canonicalize mpq_class::get_d mpq_class::get_den mpq_class::get_den_mpz_t mpq_class::get_mpq_t mpq_class::get_num mpq_class::get_num_mpz_t mpq_class::mpq_class, mpq_class::mpq_class, mpq_class::mpq_class, mpq_class::mpq_class, mpq_class::mpq_class, mpq_class::mpq_class, mpq_class::mpq_class mpq_clear mpq_cmp mpq_cmp_si mpq_cmp_ui mpq_denref mpq_div mpq_div_2exp mpq_equal mpq_get_d mpq_get_den mpq_get_num mpq_get_str mpq_init mpq_inp_str mpq_inv mpq_mul mpq_mul_2exp mpq_neg mpq_numref mpq_out_str mpq_set mpq_set_d mpq_set_den mpq_set_f mpq_set_num mpq_set_si mpq_set_str mpq_set_ui mpq_set_z mpq_sgn mpq_sub mpq_swap mpq_t
mpz_abs mpz_add mpz_add_ui mpz_addmul mpz_addmul_ui mpz_and mpz_array_init mpz_bin_ui mpz_bin_uiui mpz_cdiv_q mpz_cdiv_q_2exp mpz_cdiv_q_ui mpz_cdiv_qr mpz_cdiv_qr_ui mpz_cdiv_r mpz_cdiv_r_2exp mpz_cdiv_r_ui mpz_cdiv_ui mpz_class mpz_class::fits_sint_p mpz_class::fits_slong_p mpz_class::fits_sshort_p mpz_class::fits_uint_p mpz_class::fits_ulong_p mpz_class::fits_ushort_p mpz_class::get_d mpz_class::get_mpz_t mpz_class::get_si mpz_class::get_ui mpz_class::mpz_class, mpz_class::mpz_class, mpz_class::mpz_class, mpz_class::mpz_class, mpz_class::mpz_class, mpz_class::mpz_class mpz_clear mpz_clrbit mpz_cmp mpz_cmp_d mpz_cmp_si mpz_cmp_ui mpz_cmpabs mpz_cmpabs_d mpz_cmpabs_ui mpz_com mpz_congruent_2exp_p mpz_congruent_p mpz_congruent_ui_p mpz_divexact mpz_divexact_ui mpz_divisible_2exp_p mpz_divisible_p mpz_divisible_ui_p mpz_even_p mpz_fac_ui mpz_fdiv_q mpz_fdiv_q_2exp mpz_fdiv_q_ui mpz_fdiv_qr mpz_fdiv_qr_ui mpz_fdiv_r mpz_fdiv_r_2exp mpz_fdiv_r_ui mpz_fdiv_ui mpz_fib2_ui mpz_fib_ui mpz_fits_sint_p mpz_fits_slong_p mpz_fits_sshort_p mpz_fits_uint_p mpz_fits_ulong_p mpz_fits_ushort_p mpz_gcd mpz_gcd_ui mpz_gcdext mpz_get_d mpz_get_d_2exp mpz_get_si mpz_get_str mpz_get_ui mpz_getlimbn mpz_hamdist mpz_import mpz_init mpz_init2 mpz_init_set mpz_init_set_d mpz_init_set_si mpz_init_set_str mpz_init_set_ui mpz_inp_raw mpz_inp_str mpz_invert mpz_ior mpz_jacobi mpz_kronecker mpz_kronecker_si mpz_kronecker_ui mpz_lcm mpz_lcm_ui mpz_legendre mpz_lucnum2_ui mpz_lucnum_ui mpz_mod mpz_mod_ui mpz_mul mpz_mul_2exp mpz_mul_si mpz_mul_ui mpz_neg mpz_nextprime mpz_odd_p mpz_out_raw mpz_out_str mpz_perfect_power_p mpz_perfect_square_p mpz_popcount mpz_pow_ui mpz_powm mpz_powm_ui mpz_probab_prime_p mpz_random mpz_random2 mpz_realloc2 mpz_remove mpz_root mpz_rrandomb mpz_scan0 mpz_scan1 mpz_set mpz_set_d mpz_set_f mpz_set_q mpz_set_si mpz_set_str mpz_set_ui mpz_setbit mpz_sgn mpz_si_kronecker mpz_size mpz_sizeinbase mpz_sqrt mpz_sqrtrem mpz_sub mpz_sub_ui mpz_submul mpz_submul_ui mpz_swap mpz_t
mpz_tdiv_q mpz_tdiv_q_2exp mpz_tdiv_q_ui mpz_tdiv_qr mpz_tdiv_qr_ui mpz_tdiv_r mpz_tdiv_r_2exp mpz_tdiv_r_ui mpz_tdiv_ui mpz_tstbit mpz_ui_kronecker mpz_ui_pow_ui mpz_ui_sub mpz_urandomb mpz_urandomm mpz_xor msqrt msub mtox mult o
operator% operator/ operator<<, operator<<, operator<< operator>>, operator>>, operator>>, operator>> p
pow r
reallocate_function rpow s
sdiv sgn, sgn, sgn sqrt, sqrt t
trunc x
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