gmp
gmp is library providing Ruby bindings to GMP library. Here is the introduction paragraph at gmplib.org/#WHAT :

“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 nonfree 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 Unixtype systems, such as GNU/Linux, Solaris, HPUX, Mac OS X/Darwin, BSD, AIX, etc. It also is known to work on Windoze in 32bit 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 (32bit)  (MRI) Ruby 1.8.7  GMP 4.3.1 
++
 Mac OS X 10.5.7 on x86 (32bit)  (MRI) Ruby 1.8.6  GMP 4.3.1 
  (MRI) Ruby 1.9.1  
++++
Note: To get this running on Mac OS X (32bit), I compiled GMP 4.3.1 with:
./configure ABI=32 disabledependencytracking
Authors

Tomasz Wegrzanowski

srawlins
Constants
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_RNDN
GMP::MPFR_RNDZ
GMP::MPFR_RNDU
GMP::MPFR_RNDD
GMP::MPFR_RNDA #=> The constant representing "round away from zero"
Classes
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.new()
GMP::Z.new(GMP::Z)
GMP::Z.new(Fixnum)
GMP::Z.new(Bignum)
GMP::Z.new(String)
GMP::Q.new()
GMP::Q.new(GMP::Q)
GMP::Q.new(String)
GMP::Q.new(any GMP::Z initializer)
GMP::Q.new(any GMP::Z initializer, any GMP::Z initializer)
GMP::F.new()
GMP::F.new(GMP::Z, precision=0)
GMP::F.new(GMP::Q, precision=0)
GMP::F.new(GMP::F)
GMP::F.new(GMP::F, precision)
GMP::F.new(String, precision=0)
GMP::F.new(Fixnum, precision=0)
GMP::F.new(Bignum, precision=0)
GMP::F.new(Float, precision=0)
GMP::RandState.new([algorithm] [, algorithm_args])
You can also call them as:
GMP.Z(args)
GMP.Q(args)
GMP.F(args)
GMP.RandState()
Methods
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! inplace negation
abs absolute value
asb! inplace 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
GMP::Z
add! inplace addition
sub! inplace 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! inplace 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
GMP::Q
num numerator
den denominator
inv inversion
inv! inplace inversion
floor,ceil,trunc nearest integer
class methods of GMP::F
default_prec get default precision
default_prec= set default precision
GMP::F
prec get precision
floor,ceil,trunc nearest integer, GMP::F is returned, not GMP::Z
floor!,ceil!,trunc! inplace nearest integer
GMP::RandState
seed(integer) seed the generator with a Fixnum or GMP::Z
urandomb(fixnum) get uniformly distributed random number between 0
and 2^fixnum1, inclusive
urandomm(integer) get uniformly distributed random number between 0
and integer1, 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
GMP::F
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
precision
expm1 the same as (object.exp)  1, with better
precision
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")
GMP::RandState
mpfr_urandomb(fixnum) get uniformly distributed random floatingpoint
number within 0 <= rop < 1
Testing
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 builtin 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
Precision can be explicitely set as second argument for GMP::F.new().
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 GMP::F.new(). In particular, you can set precision of copy of GMP::F object by:
new_obj = GMP::F.new(old_obj, 0)
Precision argument, and default_precision will be rounded up to whatever GMP thinks is appropriate.
Benchmarking
“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 www.gmplib.org/gmpbench.html 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 plugin 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 plugin 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 coprime 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.
Todo
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

zerocopy 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