gmp is library providing Ruby bindings to GMP library. Here is the introduction paragraph at :

  • “GMP is a free library for arbitrary precision arithmetic, operating on signed integers, rational numbers, and floating point numbers. There is no practical limit to the precision except the ones implied by the available memory in the machine GMP runs on. GMP has a rich set of functions, and the functions have a regular interface.

  • The main target applications for GMP are cryptography applications and research, Internet security applications, algebra systems, computational algebra research, etc.

  • GMP is carefully designed to be as fast as possible, both for small operands and for huge operands. The speed is achieved by using fullwords as the basic arithmetic type, by using fast algorithms, with highly optimised assembly code for the most common inner loops for a lot of CPUs, and by a general emphasis on speed.

  • GMP is faster than any other bignum library. The advantage for GMP increases with the operand sizes for many operations, since GMP uses asymptotically faster algorithms.

  • The first GMP release was made in 1991. It is continually developed and maintained, with a new release about once a year.

  • GMP is distributed under the GNU LGPL. This license makes the library free to use, share, and improve, and allows you to pass on the result. The license gives freedoms, but also sets firm restrictions on the use with non-free programs.

  • GMP is part of the GNU project. For more information about the GNU project, please see the official GNU web site.

  • GMP's main target platforms are Unix-type systems, such as GNU/Linux, Solaris, HP-UX, Mac OS X/Darwin, BSD, AIX, etc. It also is known to work on Windoze in 32-bit mode.

  • GMP is brought to you by a team listed in the manual.

  • GMP is carefully developed and maintained, both technically and legally. We of course inspect and test contributed code carefully, but equally importantly we make sure we have the legal right to distribute the contributions, meaning users can safely use GMP. To achieve this, we will ask contributors to sign paperwork where they allow us to distribute their work.“

Only GMP 4 or newer is supported. The following environments have been tested by me: gmp gem 0.4.0 on:

|    Platform                         |       Ruby        |    GMP    |
| Cygwin 1.7 on x86                   | (MRI) Ruby 1.8.7  | GMP 4.3.1 |
|                                     |                   | GMP 4.3.2 |
|                                     |                   | GMP 5.0.0 |
| Windows XP on x86                   | (MRI) Ruby 1.9.1  | GMP 5.0.1 |
| Linux (LinuxMint 7) on x86 (32-bit) | (MRI) Ruby 1.8.7  | GMP 4.3.1 |
| Mac OS X 10.5.7 on x86 (32-bit)     | (MRI) Ruby 1.8.6  | GMP 4.3.1 |
|                                     | (MRI) Ruby 1.9.1  |           |

Note: To get this running on Mac OS X (32-bit), I compiled GMP 4.3.1 with:

./configure ABI=32 --disable-dependency-tracking


  • Tomasz Wegrzanowski

  • srawlins


The GMP module includes the following constants. Mathematical constants, such as pi, are defined under class methods of GMP::F, listed below.

GMP::GMP_VERSION        #=> A string like "5.0.1"
GMP::GMP_CC             #=> The compiler used to compile GMP
GMP::GMP_CFLAGS         #=> The CFLAGS used to compile GMP
GMP::GMP_BITS_PER_LIMB  #=> The number of bits per limb
(if MPFR is available)
GMP::MPFR_VERSION       #=> A string like "2.4.2"
GMP::GMP_RNDN           #=> The constant representing "round to nearest"
GMP::GMP_RNDZ           #=> The constant representing "round toward zero"
GMP::GMP_RNDU           #=> The constant representing "round toward plus infinity"
GMP::GMP_RNDD           #=> The constant representing "round toward minus infinity"
New in MPFR 3.0.0:
GMP::MPFR_RNDA          #=> The constant representing "round away from zero"


The GMP module is provided with following classes:

  • GMP::Z - infinite precision integer numbers

  • GMP::Q - infinite precision rational numbers

  • GMP::F - arbitrary precision floating point numbers

  • GMP::RandState - states of individual random number generators

Numbers are created by using new(). Constructors can take following arguments: GMP::Z initializer) GMP::Z initializer, any GMP::Z initializer), precision=0), precision=0), precision), precision=0), precision=0), precision=0),  precision=0)[algorithm] [, algorithm_args])

You can also call them as:



GMP::Z, GMP::Q and GMP::F
  +                        addition
  -                        substraction
  *                        multiplication
  to_s                     convert to string. For GMP::Z, this method takes
                           one optional argument, a base. The base can be a
                           Fixnum in the ranges [2, 62] or [-36, -2] or a
                           Symbol: :bin, :oct, :dec, or :hex.
  [email protected]                       negation
  neg!                     in-place negation
  abs                      absolute value
  asb!                     in-place absolute value
  coerce                   promotion of arguments
  ==                       equality test
  <=>,>=,>,<=,<            comparisions
class methods of GMP::Z
  fac(n)                   factorial of n
  fib(n)                   nth fibonacci number
  pow(n,m)                 n to mth power
GMP::Z and GMP::Q
  swap                     efficiently swap contents of two objects, there
                           is no GMP::F.swap because various GMP::F objects
                           may have different precisions, which would make
                           them unswapable
  add!                     in-place addition
  sub!                     in-place subtraction
  tdiv,fdiv,cdiv           truncate, floor and ceil division
  tmod,fmod,cmod           truncate, floor and ceil modulus
  [],[]=                   testing and setting bits (as booleans)
  scan0,scan1              starting at bitnr (1st arg), scan for a 0 or 1
                           (respectively), then return the index of the
                           first instance.
  cmpabs                   comparison of absolute value
  com                      2's complement
  com!                     in-place 2's complement
  &,|,^                    logical operations: and, or, xor
  **                       power
  powmod                   power modulo
  even?                    is even
  odd?                     is odd
  <<                       shift left
  >>                       shift right, floor
  tshr                     shift right, truncate
  lastbits_pos(n)          last n bits of object, modulo if negative
  lastbits_sgn(n)          last n bits of object, preserve sign
  power?                   is perfect power
  square?                  is perfect square
  sqrt                     square root
  sqrt!                    change the object into its square root
  sqrtrem                  square root, remainder
  root(n)                  nth root
  probab_prime?            0 if composite, 1 if probably prime, 2 if
                           certainly prime
  nextprime                next *probable* prime
  nextprime!               change the object into its next *probable* prime
  gcd                      greatest common divisor
  invert(m)                invert mod m
  jacobi                   jacobi symbol
  legendre                 legendre symbol
  remove(n)                remove all occurences of factor n
  popcount                 the number of bits equal to 1
  sizeinbase(b)            digits in base b
  size_in_bin              digits in binary
  size                     number of limbs
  to_i                     convert to Fixnum or Bignum
GMP::Q and GMP::F
  /                        division
  num                      numerator
  den                      denominator
  inv                      inversion
  inv!                     in-place inversion
  floor,ceil,trunc         nearest integer
class methods of GMP::F
  default_prec             get default precision
  default_prec=            set default precision
  prec                     get precision
  floor,ceil,trunc         nearest integer, GMP::F is returned, not GMP::Z
  floor!,ceil!,trunc!      in-place nearest integer
  seed(integer)            seed the generator with a Fixnum or GMP::Z
  urandomb(fixnum)         get uniformly distributed random number between 0
                           and 2^fixnum-1, inclusive
  urandomm(integer)        get uniformly distributed random number between 0
                           and integer-1, inclusive
GMP (timing functions for GMPbench (0.2))
  cputime                  milliseconds of cpu time since Ruby start
  time                     times the execution of a block

*only if MPFR is available*
class methods of GMP::F
  const_log2               returns the natural log of 2
  const_pi                 returns pi
  const_euler              returns euler
  const_catalan            returns catalan
  sqrt                     square root of the object
  rec_sqrt                 square root of the recprical of the object
  cbrt                     cube root of the object
  **                       power
  log                      natural logarithm of object
  log2                     binary logarithm of object
  log10                    decimal logarithm of object
  exp                      e^object
  exp2                     2^object
  exp10                    10^object
  log1p                    the same as (object + 1).log, with better
  expm1                    the same as (object.exp) - 1, with better
  eint                     exponential integral of object
  li2                      real part of the dilogarithm of object
  gamma                    Gamma fucntion of object
  lngamma                  logarithm of the Gamma function of object
  digamma                  Digamma function of object (MPFR_VERSION >= "3.0.0")
  zeta                     Reimann Zeta function of object
  erf                      error function of object
  erfc                     complementary error function of object
  j0                       first kind Bessel function of order 0 of object
  j1                       first kind Bessel function of order 1 of object
  jn                       first kind Bessel function of order n of object
  y0                       second kind Bessel function of order 0 of object
  y1                       second kind Bessel function of order 1 of object
  yn                       second kind Bessel function of order n of object
  cos                      \
  sin                      |
  tan                      |
  sec                      |
  csc                      |
  cot                      |
  acos                     |
  asin                     |
  atan                     | trigonometric functions
  cosh                     | of the object
  sinh                     |
  tanh                     |
  sec                      |
  csc                      |
  cot                      |
  acosh                    |
  asinh                    |
  atanh                    /
  nan?                     \
  infinite?                | type of floating point number
  finite?                  |
  number?                  |
  regular?                 / (MPFR_VERSION >= "3.0.0")
  mpfr_urandomb(fixnum)    get uniformly distributed random floating-point
                           number within 0 <= rop < 1


Tests can be run with:

cd test
ruby unit_tests.rb

If you have the unit_test gem installed, all tests should pass. Otherwise, one test may error. I imagine there is a bug in Ruby's built-in Test::Unit package that is fixed with the unit_test gem.

Known Issues

  • Don't call GMP::RandState(:lc_2exp_size). Give a 2nd arg.


Precision can be explicitely set as second argument for

If there is no explicit precision, highest precision of all GMP::F arguments is used. That doesn't ensure that result will be exact. For details, consult any paper about floating point arithmetics.

Default precision can be explicitely set by passing 0 as the second argument for to In particular, you can set precision of copy of GMP::F object by:

new_obj =, 0)

Precision argument, and default_precision will be rounded up to whatever GMP thinks is appropriate.


“GMP is carefully designed to be as fast as possible.” Therefore, I believe it is very important for GMP, and its various language bindings to be benchmarked. In recent years, the GMP team developed GMPbench, an elegant, weighted benchmark. Currently, at they maintain a list of recent benchmark results, broken down by CPU, CPU freq, ABI, and compiler flags; GMPbench compares different processor's performance against eachother, rather than GMP against other bignum libraries, or comparing different versions of GMP.

I intend to build a plug-in to GMPbench that will test the ruby gmp gem. The results of this benchmark should be directly comparable with the results of GMP (on same CPU, etc.). Rather than write a benchmark from the ground up, or try to emulate what GMPbench does, a plug-in will allow for this type of comparison. And in fact, GMPbench is (perhaps intentionally) written perfectly to allow for plugging in.

Various scores are derived from GMPbench by running the runbench script. This script compiles and runs various individual programs that measure the performance of base functions, such as multiply, and app functions such as rsa.

The gmp gem benchmark uses the GMPbench framework (that is, runbench, gexpr, and the timing methods), and plugs in ruby scripts as the individual programs. Right now, there are only three such plugged in ruby scripts:

  • multiply - measures performance of multiplying (or squaring) GMP::Z objects whose size (in bits) is given by 1 or 2 operands.

  • divide - measures performance of dividing two GMP::Z objects (using tdiv) whose size (in bits) is given by 2 operands.

  • rsa - measures performance of using RSA to sign messages. The size of pq, the product of the two co-prime GMP::Z objects, p and q, is given by 1 operand.

Results: on my little Intel Core Duo T2400 @ 1.83GHz:

| GMP 4.3.1* compiled with GCC 3.4.4, I think (cygwin did |
| it)                                                     |
|  test      |  GMP      |  ruby gmp gem                  |
|  multiply  |  4660     |        2473.8  (47% overhead)  |
|  divide    |  2744     |        2253.1  (18% overhead)  |
|  gcd       |  1004.5   |         865.13 (14% overhead)  |
|  rsa       |   515.49  |         506.69 ( 2% overhead)  |
| GMP 5.0.0 compiled with GCC 3.4.4, I think (cygwin did  |
| it)                                                     |
|  test      |  GMP      |  ruby gmp gem                  |
|  multiply  |  4905     |        2572.1  (48% overhead)  |
|  divide    |  4873     |        3427.4  (30% overhead)  |
|  gcd       |  1083.5   |         931.75 (14% overhead)  |
|  rsa       |   520.20  |         506.14 ( 3% overhead)  |
| GMP 5.0.1 compiled with GCC 3.4.5 in MinGW              |
|  test      |  GMP      |  ruby gmp gem                  |
|  multiply  |  4950     |        xxxx.x  (xx% overhead)  |
|  divide    |  4809     |        xxxx.x  (xx% overhead)  |
|  gcd       |  1071.3   |         xxx.xx (xx% overhead)  |
|  rsa       |   524.96  |         xxx.xx ( x% overhead)  |

* GMP 4.3.2 evaluated to almost the same benchmarks.

My guess is that the increase in ruby gmp gem overhead is caused by increased efficiency in GMP; the inefficiencies of the gmp gem are relatively greater.


These are inherited from Tomasz. I will go through these and see which are still relevant, and which I understand.

  • mpz_fits_* and 31 vs. 32 integer variables

  • fix all sign issues (don't know what these are)

  • to_s vs. inspect

  • check if mpz_addmul_ui would optimize some statements

  • some system that allows using denref and numref as normal ruby objects (?)

  • should we allocate global temporary variables like Perl GMP does?

  • takeover code that replaces all Bignums with GMP::Z

  • better bignum parser

  • zero-copy method for strings generation

  • put rb_raise into nice macros

  • benchmarks against Python GMP and Perl GMP

  • dup methods

  • integrate F into system

  • should Z.[] bits be 0/1 or true/false, 0 is true, what might badly surprise users

  • any2small_integer()

  • check asm output, especially local memory efficiency

  • it might be better to use `register' for some local variables

  • powm with negative exponents

  • check if different sorting of operatations gives better cache usage

  • GMP::* op RubyFloat and RubyFloat op GMP::*

  • sort checks

  • GMP::Q.to_s(base), GMP::F.to_s(base) (test it!)

  • benchmark gcdext, pi