gmp
gmp is library providing Ruby bindings to GMP library. Here is the introduction paragraph at their homepage:
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 0.5.47 on:
Platform  Ruby  GMP (MPFR) 

Linux (Ubuntu NR 10.04) on x86 (32bit)  (MRI) Ruby 1.8.7  GMP 4.3.1 (2.4.2) GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 
(MRI) Ruby 1.9.1  GMP 4.3.1 (2.4.2) GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

(MRI) Ruby 1.9.2  GMP 4.3.1 (2.4.2) GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

Linux (Ubuntu 10.04) on x86_64 (64bit)  (MRI) Ruby 1.8.7  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 
(MRI) Ruby 1.9.1  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

(MRI) Ruby 1.9.2  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

(RBX) Rubinius 1.1.0  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

Mac OS X 10.6.8 on x86_64 (64bit)  (MRI) Ruby 1.8.7  GMP 4.3.2 (2.4.2) GMP 5.0.5 (3.1.1) 
(MRI) Ruby 1.9.3  GMP 4.3.2 (2.4.2) GMP 5.0.5 (3.1.1) 

(RBX) Rubinius 1.1.0  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

Windows 7 on x86_64 (64bit)  (MRI) Ruby 1.8.7  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 
(MRI) Ruby 1.9.1  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

(MRI) Ruby 1.9.2  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 

Windows XP on x86 (32bit)  (MRI) Ruby 1.9.1  GMP 4.3.2 (2.4.2) GMP 5.0.1 (3.0.0) 
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 GMPGMP::GMP_CFLAGS
 The CFLAGS used to compile GMPGMP::GMP\_BITS_PER_LIMB
The number of bits per limbGMP::GMP_NUMB_MAX
 The maximum value that can be stored in the number part of a limb
if MPFR is available:
* GMP::MPFR_VERSION
 A string like "2.4.2"
* GMP::MPFR\_PREC_MIN
 The minimum precision available
* GMP::MPFR_PREC_MAX
 The maximum precision available
* 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
/ division
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.
@ 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
2fac(n), double_fac(n) double factorial of n
mfac(n) mmultifactorial of n
primorial(n) primorial 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
to_i convert to Fixnum or Bignum
add! inplace addition
sub! inplace subtraction
addmul!(b,c) inplace += b*c
submul!(b,c) inplace = b*c
tdiv,fdiv,cdiv truncate, floor and ceil division
tmod,fmod,cmod truncate, floor and ceil modulus
>> shift right, floor
divisible?(b) true if divisible by b
congruent?(c,d) true if congruent to c modulus d
** power
powmod power modulo
\[\],\[\]= 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
even? is even
odd? is odd
<< shift left
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, gcdext, gcdext2 greatest common divisor
lcm least common multiple
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
hamdist the hamming distance between two integers
out_raw output to IO object
inp_raw input from IO object
export export to a byte array (String)
import import from a byte array (String)
sizeinbase(b) digits in base b
size_in_bin digits in binary
size number of limbs
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
mpfr_buildopt_tls_p returns whether MPFR was built as thread safe
mpfr_buildopt_decimal_p returns whether MPFR was compiled with decimal
float support
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
agm arithmeticgeometric mean
hypot
cos \
sin 
tan 
sin_cos 
sec 
csc 
cot 
acos 
asin 
atan  trigonometric functions
atan2 
cosh  of the object
sinh 
tanh 
sinh_cosh 
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
Functional Mappings
In order to align better with the GMP paradigms of using return arguments, I have started creating "functional mappings", singleton methods that map directly to functions in GMP. These methods take return arguments, so that passing an object to a functional mapping may change the object itself. For example:
a = GMP::Z(0)
b = GMP::Z(13)
c = GMP::Z(17)
GMP::Z.add(a, b, c)
a #=> 30
Here's a fun list of all of the functional mappings written so far:
GMP::Z
.abs .add .addmul .cdiv_q_2exp .cdiv_r_2exp .com
.congruent? .divexact .divisible? .fdiv_q_2exp .fdiv_r_2exp .lcm
.mul .mul_2exp
.neg .nextprime .sqrt .sub .submul .tdiv_q_2exp
.tdiv_r_2exp
Documentation
 This README
 Loren Segal and the guys at RubyGems.org are awesome: YARDoc.
 There should be a manual.pdf here. I spend waaay too much time working on this, but it looks very pretty.
 CHANGELOG
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.
You can also use the following shiny new rake tasks:
rake test
rake report
MPFR=nompfr rake report
Known Issues
Don't call GMP::RandState(:lc_2exp_size)
. Give a 2nd arg.
Don't use multiple assignment (a = b = GMP::Z(0)
) with functional mappings:
ruby
>> a = b = GMP::Z(0)
=> 0
>> GMP::Z.mul(a, GMP::Z(2), GMP::Z(3))
=> nil
>> a
=> 6
>> b
=> 6
JRuby has some interesting bugs and flickering tests. GMP::Z(GMP::GMPNUMBMAX) for example, blows up.
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
Please see performance
Todo
GMP::Z#to_d_2exp
,#rootrem
,#kronecker
,#bin
,#fib2
,#lucnum
,#lucnum2
,#combit
,#fits_x?
GMP::Q#to_s(base)
,GMP::F#to_s(base)
(test it!) benchmark pi
 a buttload of functional mappings. 47ish sets.
 investigate possible memory leaks when using
GMP::Q(22/7)
for example  beef up
r_gmpq_initialize
; I don't like to rely onmpz_set_value
.  finish compileresults.rb
 New in MPFR 3.1.0: mpfrfrexp, mpfrgrandom, mpfrzsub, dividebyzero exception (?)
 benchmark different orderings of type checks
The below 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
 takeover code that replaces all
Bignums
withGMP::Z
 better bignum parser (crawling into the Bignum extension)
 zerocopy method for strings generation
 benchmarks against Python GMP and Perl GMP
dup
methods for GMP::Q and GMP::F integrate
F
into system  should
Z.\[\]
bits be 0/1 or true/false, 0 is true, which might surprise users any2small_integer()
 powm with negative exponents
 check if different sorting of operations gives better cache usage
GMP::\*
opRubyFloat
andRubyFloat
opGMP::\*