=gmp
gmp is library providing Ruby bindings to GMP library. Here is the introduction
paragraph at http://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 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
=Authors
* Tomasz Wegrzanowski
* srawlins
=Classes
The module GMP 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.
-@ 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
GMP::Z
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.
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
to_i convert to Fixnum or Bignum
GMP::Q and GMP::F
/ division
GMP::Q
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
GMP::F
prec get precision
floor,ceil,trunc nearest integer, GMP::F is returned, not GMP::Z
floor!,ceil!,trunc! in-place nearest integer
GMP::F (only if MPFR is available)
exp e^object
expm1 the same as (object.exp) - 1, with better
precision
log natural logarithm of object
log2 binary logarithm of object
log10 decimal logarithm of object
log1p the same as (object + 1).log, with better
precision
sqrt square root of the object
cos \
sin |
tan |
acos |
asin |
atan | trigonometric functions
cosh | of the object
sinh |
tanh |
aconh |
asinh |
atanh /
nan? \
infinite? | type of floating point number
finite? |
number? /
** power
GMP::RandState
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
GMP (timing functions for GMPbench (0.2))
cputime milliseconds of cpu time since Ruby start
time times the execution of a block
=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 built-in Test::Unit package
that is fixed with the unit_test gem.
=Known Issues
* GMP::Z#pow does not appear to be working at all. Looking at the code, I don't
think it ever did.
* 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 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 http://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 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.
=Todo
These are inherited from Tomasz. I will go through these and see which are
still relevant.
* mpz_fits_* and 31 vs. 32 integer variables
* all appropriate module and class methods if there are any to add
* fix all sign issues (don't know what these are)
* floats with precision control
* 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 (gmpy? Is this still active?) 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)
* benchmark gcdext, pi