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<h1>Uniform random number generators</h1>
<p align="center">by Agner Fog</p>
<p><a href="randomc.zip">randomc.zip</a> contains a C++ class library of uniform random number generators of good
quality.</p>
<p>The random number generators found in standard libraries are often of a poor
quality, insufficient for large Monte Carlo calculations. This C++
implementation provides
random number generators of a much better quality: Better randomness, higher
resolution, and longer cycle lengths.</p>
<p>The same random number generators are available as libraries coded in assembly language
for higher speed. These libraries can be linked into projects coded in other
programming languages under Windows, Linux, BSD, etc. The library files are
available in the
archive <a href="randoma.zip">randoma.zip</a>.</p>
<p>Non-uniform random number generators are provided in <a href="stocc.zip">stocc.zip</a>.</p>
<h2>File list</h2>
<p>The archive <a href="randomc.zip">randomc.zip</a> contains the following
files:</p>
<dl>
<dt>randomc.htm</dt>
<dd>This file. Instructions.</dd>
<dt>randomc.h</dt>
<dd>C++ header file containing class definitions.<br>
You must <code>#include</code> this in all C++ files that use this library.</dd>
<dt>mersenne.cpp</dt>
<dd>Random number generator of type Mersenne twister.</dd>
<dt>ranrotb.cpp</dt>
<dd>Random number generator of type RANROT-B.</dd>
<dt>ranrotw.cpp</dt>
<dd>Random number generator of type RANROT-W.</dd>
<dt>mother.cpp</dt>
<dd>Random number generator of type Mother-of-all (multiply with carry).</dd>
<dt>rancombi.cpp</dt>
<dd>Template class for combining any two of these random number generators.</dd>
<dt>ex-ran.cpp</dt>
<dd>Example showing how to use these random number generators.</dd>
</dl>
<h2>Quality of randomness</h2>
<p>All these random number generators provide excellent randomness and extremely
long cycle lengths.</p>
<p>For all but the most demanding applications it doesn't matter
which of the random number generators you use. The Mersenne twister is the one
that is best understood theoretically. For this reason it is recommended by
leading experts. The RANROT generators have a more chaotic behavior which is
difficult to analyze theoretically, and hence maybe also more random in the
sense that it has no mathematical regularity that might interfere with the
structure of your application. The Mother-of-all generator has the highest
bifurcation of these generators.</p>
<p>For the most demanding scientific applications you may combine any two of these
generators.</p>
<h2>Technical quality</h2>
<p>The technical qualities of these generators are compared in the table below.
If you want to simulate very low probabilities with high accuracy then it is
recommended to use a generator with a high resolution. If speed is important then
use a generator that uses a low number of clock cycles. The amount of data cache
used influences execution speed if the total amount of data handled by the
critical part of your program exceeds the size of the level-1 cache in your
microprocessor. A low cache use means faster
execution. The same applies to the code cache use if the critical part of your
program uses more than the available cache size.</p>
<table border="1" cellpadding="0" cellspacing="0" width="100%">
<tr>
<td width="12%" rowspan="2">
<p align="center">Generator type</td>
<td width="12%" rowspan="2" align="center">coding language</td>
<td width="12%" rowspan="2" align="center">resolution<br>
bits</td>
<td width="76%" colspan="4">
<p align="center">time consumption, clock cycles</td>
<td width="13%" rowspan="2" align="center">data cache use, bytes</td>
<td width="13%" rowspan="2" align="center">code cache use, bytes</td>
</tr>
<tr>
<td width="12%" align="center">bits</td>
<td width="13%" align="center">integer</td>
<td width="13%" align="center">float</td>
<td width="13%" align="center">hi.res.<BR>
float</td>
</tr>
<tr>
<td width="12%">RANROT B</td>
<td width="12%" align="center">C++</td>
<td width="12%" align="center">32</td>
<td width="12%" align="center"> </td>
<td width="13%" align="center">282</td>
<td width="13%" align="center">96</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">164</td>
<td width="13%" align="center">350</td>
</tr>
<tr>
<td width="12%">RANROT W</td>
<td width="12%" align="center">C++</td>
<td width="12%" align="center">52 or 63</td>
<td width="12%" align="center">61</td>
<td width="13%" align="center">306</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">118</td>
<td width="13%" align="center">164</td>
<td width="13%" align="center">400</td>
</tr>
<tr>
<td width="12%">RANROT W</td>
<td width="12%" align="center">ASM</td>
<td width="12%" align="center">63</td>
<td width="12%" align="center">22</td>
<td width="13%" align="center">67</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">77</td>
<td width="13%" align="center">160</td>
<td width="13%" align="center">200</td>
</tr>
<tr>
<td width="12%">Mother-of-all</td>
<td width="12%" align="center">C++</td>
<td width="12%" align="center">32</td>
<td width="12%" align="center"> </td>
<td width="13%" align="center">306</td>
<td width="13%" align="center">189</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">48</td>
<td width="13%" align="center">220</td>
</tr>
<tr>
<td width="12%">Mother-of-all</td>
<td width="12%" align="center">ASM</td>
<td width="12%" align="center">32</td>
<td width="12%" align="center">74</td>
<td width="13%" align="center">96</td>
<td width="13%" align="center">146</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">48</td>
<td width="13%" align="center">170</td>
</tr>
<tr>
<td width="12%">Mersenne Twister</td>
<td width="12%" align="center">C++</td>
<td width="12%" align="center">32</td>
<td width="12%" align="center">38</td>
<td width="13%" align="center">258</td>
<td width="13%" align="center">91</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">1476</td>
<td width="13%" align="center">450</td>
</tr>
<tr>
<td width="12%">Mersenne Twister</td>
<td width="12%" align="center">ASM</td>
<td width="12%" align="center">32 - 63</td>
<td width="12%" align="center">24</td>
<td width="13%" align="center">44</td>
<td width="13%" align="center">65</td>
<td width="13%" align="center">81</td>
<td width="13%" align="center">1472</td>
<td width="13%" align="center">600</td>
</tr>
<tr>
<td width="12%">Combined<br>
RANROT-W+<br>
Mother-off-all</td>
<td width="12%" align="center">ASM</td>
<td width="12%" align="center">63</td>
<td width="12%" align="center">89</td>
<td width="13%" align="center">134</td>
<td width="13%" align="center"> </td>
<td width="13%" align="center">130</td>
<td width="13%" align="center">224</td>
<td width="13%" align="center">300</td>
</tr>
</table>
<p>The number of clock cycles are approximate values for a Pentium 4 microprocessor under optimal caching conditions. A count of 100 clock cycles
means that you can generate 10 million random numbers per second on a 1 GHz
computer.</p>
<h2>Instructions</h2>
<p>Choose which one of the random number generators mentioned above you want to
use.</p>
<p>Write <code>#include "randomc.h"</code> in any C++ file
that uses one of these random number generators.</p>
<p>Add the appropriate cpp file to your project, either as an <code>#include</code>
or as a separate module.</p>
<p>Make an instance (object) of the appropriate class. It needs an integer seed
as initializer.</p>
<p>The seed can be any integer, positive, negative or zero. Repeating the
calculations with the same seed will produce the same sequence of random
numbers. A different seed will give different random numbers. You may use the
time in seconds or milliseconds as seed.</p>
<p>You don't need more than one instance of the random number generator unless
you have multiple threads. If, for
any reason, you want more than one instance then make sure they don't have the
same seed. You may, for example, add 1 to the seed for each new instance.</p>
<p>The file ex-ran.cpp contains an example of how to do. Try it!</p>
<h2>Portability</h2>
<p>The C++ class library is supposed to work with all C++ compilers and all
operating systems. It has been tested on several different systems.</p>
<p>There are, however, a few system differences that you need to be aware of:</p>
<ol>
<li>Floating point representation. Most systems store floating point
numbers according to the IEEE-754 standard with 64 bits or 80 bits storage. The fast conversion to floating point relies on this standard
format. A slightly less efficient method is used when floating point numbers
are not stored in a recognized format.</li>
<li>Long double precision. The Mother-of-all generator and the RANROT-W generator
require long double precision (80 bits storage). Not all microprocessors and
compilers support this precision. A standard PC running Windows or Linux
supports the long double precision when the program is compiled with a
Borland or Gnu compiler. The Microsoft C++ compiler (Visual C++) does not support long
double precision. The Mother-of-all generator coded in C++ will not work
without long double precision. The RANROT-W generator will work, but with
lower resolution and a different random number sequence. The RANROT-W
generator coded in assembly does not require long double precision.</li>
<li>Rotate function. The RANROT-B and RANROT-W generators use a rotate
function <code>_lrotl()</code>. Some compilers have this as a
built-in function. For other compilers you need to define it. A code for
this function is given in the files ranrotb.cpp and ranrotw.cpp. If your
compiler has the built-in function then remove or comment out this function.</li>
</ol>
<h2>Theory</h2>
<p>The theory of the Mersenne twister is given in the article:<br>
M. Matsumoto & T. Nishimura: <i>"Mersenne Twister: A 623-Dimensionally
Equidistributed Uniform Pseudo-Random Number Generator".</i> ACM
Transactions on Modeling and Computer Simulation, vol. 8, no. 1, 1998, pp. 3-30.
See also <A href="http://www.math.sci.hiroshima-u.ac.jp/~m-mat/eindex.html">http://www.math.sci.hiroshima-u.ac.jp/~m-mat/eindex.html</A>.</p>
<p>The theory of Mother-of-All generators is given in George Marsaglia's DIEHARD
package, see <a href="http://stat.fsu.edu/~geo/diehard.html">stat.fsu.edu/~geo/diehard.html</a>
or <a href="http://www.cs.hku.hk/internet/randomCD.html">www.cs.hku.hk/internet/randomCD.html</a>.</p>
<p>The theory of RANROT generators is given at <a href="http://www.agner.org/random/theory/">www.agner.org/random/theory/</a>.</p>
<h2>Copyright and licence</h2>
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