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xorshift* / xorshift+ PRNGs are presented and analysed here
<http://xorshift.di.unimi.it/> and, in more mathematical as well as
computer-scientific detail, in these two papers by Sebastiano Vigna: [1]
<http://vigna.di.unimi.it/ftp/papers/xorshift.pdf> [2]
<http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf>. They are based on
scrambling (via multiplication/addition) the outputs of xorshift PRNGs
presented by George Marsaglia in his 2003 article Xorshift RNGs
<http://www.jstatsoft.org/v08/i14/>.

The reference implementations (in C) of the four algorithms of note are
provided by Sebastiano Vigna: xorshift64*
<http://xorshift.di.unimi.it/xorshift64star.c>, xorshift128+
<http://xorshift.di.unimi.it/xorshift128plus.c>, xorshift1024*
<http://xorshift.di.unimi.it/xorshift1024star.c>, xorshift4096*
<http://xorshift.di.unimi.it/xorshift4096star.c>.

There are multiple reasons why these should be considered for inclusion
into the STL alongside the already existing PRNGs:

   - All except xorshift64* pass the BigCrush test suite from TestU01
   <http://simul.iro.umontreal.ca/testu01/tu01.html>. None of the PRNG
   engines currently provided in <random> do (Mersenne Twister comes
   considerably closer to passing than others, but still fails).
   - Despite providing a measurably higher quality of pseudo-randomness
   than even Mersenne Twister, they are significantly faster than Mersenne
   Twister (up to three times) while having a much lower memory footprint.
   - As can be seen on pages 23 and 10 (respectively) of the two papers by
   Sebastiano Vigna, the escape from the "low zero state" for these PRNGs is
   considerably faster than for Mersenne Twister. Whereas Mersenne Twister
   takes ~100,000 iterations to recover, xorshift4096* takes ~1000,
   xorshift1024* ~100, xorshift128+ ~10 and xorshift64* recovers practically
   instantly.
   - Even though their period is lower than that of Mersenne Twister, the
   period of xorshift1024* (which is 2^1024 - 1) is still more than enough -
   for even if the generator outputs a new value each nanosecond, the time it
   would take to repeat the sequence while constantly generating new values is about
   4.1*10^281 times the age of the universe
   <http://www.wolframalpha.com/input/?i=%282^1024+-+1%29*%281+ns%29>.

I have included my own implementation of these PRNGs as typedefs of a
generalized class conforming with the standard PRNGs from <random> (i.e.
these can be passed to distributions from <random> etc.). For convenience,
I have also included a main.cpp which compares the outputs of my
implementation with the outputs of the reference implementations by
Sebastiano Vigna and demonstrates using a binomial_distribution with
xorshift1024*. I have also provided these source codes here
<https://sourceforge.net/projects/xorshift-cpp/files/>. Since Sebastiano
Vigna has dedicated the reference implementations to the public domain, I
have done the same.

I am interested to hear if other people think this would be a worthwhile
addition to the STL.

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<div dir=3D"ltr">xorshift* / xorshift+ PRNGs are presented and analysed <a =
href=3D"http://xorshift.di.unimi.it/">here</a> and, in more mathematical as=
 well as computer-scientific detail, in these two papers by Sebastiano Vign=
a: <a href=3D"http://vigna.di.unimi.it/ftp/papers/xorshift.pdf">[1]</a> <a =
href=3D"http://vigna.di.unimi.it/ftp/papers/xorshiftplus.pdf">[2]</a>. They=
 are based on scrambling (via multiplication/addition) the outputs of xorsh=
ift PRNGs presented by George Marsaglia in his 2003 article <a href=3D"http=
://www.jstatsoft.org/v08/i14/">Xorshift RNGs</a>.<br><br>The reference impl=
ementations (in C) of the four algorithms of note are provided by Sebastian=
o Vigna: <a href=3D"http://xorshift.di.unimi.it/xorshift64star.c">xorshift6=
4*</a>, <a href=3D"http://xorshift.di.unimi.it/xorshift128plus.c">xorshift1=
28+</a>, <a href=3D"http://xorshift.di.unimi.it/xorshift1024star.c">xorshif=
t1024*</a>, <a href=3D"http://xorshift.di.unimi.it/xorshift4096star.c">xors=
hift4096*</a>.<br><br>There are multiple reasons why these should be consid=
ered for inclusion into the STL alongside the already existing PRNGs:<br><u=
l><li>All except xorshift64* pass the BigCrush test suite from <a href=3D"h=
ttp://simul.iro.umontreal.ca/testu01/tu01.html">TestU01</a>. None of the PR=
NG engines currently provided in &lt;random&gt; do (Mersenne Twister comes =
considerably closer to passing than others, but still fails).</li><li>Despi=
te providing a measurably higher quality of pseudo-randomness than even Mer=
senne Twister, they are significantly faster than Mersenne Twister (up to t=
hree times) while having a much lower memory footprint.</li><li>As can be s=
een on pages 23 and 10 (respectively) of the two papers by Sebastiano Vigna=
, the escape from the "low zero state" for these PRNGs is considerably fast=
er than for Mersenne Twister. Whereas Mersenne Twister takes ~100,000 itera=
tions to recover, xorshift4096* takes ~1000, xorshift1024* ~100, xorshift12=
8+ ~10 and xorshift64* recovers practically instantly.</li><li>Even though =
their period is lower than that of Mersenne Twister, the period of xorshift=
1024* (which is 2^1024 - 1) is still more than enough - for even if the gen=
erator outputs a new value each nanosecond, the time it would take to repea=
t the sequence while constantly generating new values is <a href=3D"http://=
www.wolframalpha.com/input/?i=3D%282^1024+-+1%29*%281+ns%29">about 4.1*10^2=
81 times the age of the universe</a>.</li></ul>I have included my own imple=
mentation of these PRNGs as typedefs of a generalized class conforming with=
 the standard PRNGs from <span style=3D"font-family: courier new,monospace;=
">&lt;random&gt;</span> (i.e. these can be passed to distributions from <sp=
an style=3D"font-family: courier new,monospace;">&lt;random&gt;</span> etc.=
). For convenience, I have also included a <span style=3D"font-family: cour=
ier new,monospace;">main.cpp</span> which compares the outputs of my implem=
entation with the outputs of the reference implementations by Sebastiano Vi=
gna and demonstrates using a <span style=3D"font-family: courier new,monosp=
ace;">binomial_distribution</span> with xorshift1024*. I have also provided=
 these source codes <a href=3D"https://sourceforge.net/projects/xorshift-cp=
p/files/">here</a>. Since Sebastiano Vigna has dedicated the reference impl=
ementations to the public domain, I have done the same.<br><br>I am interes=
ted to hear if other people think this would be a worthwhile addition to th=
e STL.<br></div>

<p></p>

-- <br />
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I have discovered a mistake in a comment in the file xorshift.h (in the
implementation I have provided). Instead of

* @tparam n  The state size (== n * 8 bytes).

It should be

* @tparam n  The state size (== n * sizeof(UIntType) bytes).

Since all the reference generators use only 64-bit unsigned integers, I
have mistakenly generalised.

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Content-Transfer-Encoding: quoted-printable

<div dir=3D"ltr">I have discovered a mistake in a comment in the file xorsh=
ift.h (in the implementation I have provided). Instead of<br><br><div class=
=3D"prettyprint" style=3D"background-color: rgb(250, 250, 250); border-colo=
r: rgb(187, 187, 187); border-style: solid; border-width: 1px; word-wrap: b=
reak-word;"><code class=3D"prettyprint"><div class=3D"subprettyprint"><span=
 style=3D"color: #660;" class=3D"styled-by-prettify">*</span><span style=3D=
"color: #000;" class=3D"styled-by-prettify"> </span><span style=3D"color: #=
066;" class=3D"styled-by-prettify">@tparam</span><span style=3D"color: #000=
;" class=3D"styled-by-prettify"> n &nbsp;</span><span style=3D"color: #606;=
" class=3D"styled-by-prettify">The</span><span style=3D"color: #000;" class=
=3D"styled-by-prettify"> state size </span><span style=3D"color: #660;" cla=
ss=3D"styled-by-prettify">(=3D=3D</span><span style=3D"color: #000;" class=
=3D"styled-by-prettify"> n </span><span style=3D"color: #660;" class=3D"sty=
led-by-prettify">*</span><span style=3D"color: #000;" class=3D"styled-by-pr=
ettify"> </span><span style=3D"color: #066;" class=3D"styled-by-prettify">8=
</span><span style=3D"color: #000;" class=3D"styled-by-prettify"> bytes</sp=
an><span style=3D"color: #660;" class=3D"styled-by-prettify">).</span></div=
></code></div><br>It should be<br><br><div class=3D"prettyprint" style=3D"b=
ackground-color: rgb(250, 250, 250); border-color: rgb(187, 187, 187); bord=
er-style: solid; border-width: 1px; word-wrap: break-word;"><code class=3D"=
prettyprint"><div class=3D"subprettyprint"><span style=3D"color: #660;" cla=
ss=3D"styled-by-prettify">*</span><span style=3D"color: #000;" class=3D"sty=
led-by-prettify"> </span><span style=3D"color: #066;" class=3D"styled-by-pr=
ettify">@tparam</span><span style=3D"color: #000;" class=3D"styled-by-prett=
ify"> n &nbsp;</span><span style=3D"color: #606;" class=3D"styled-by-pretti=
fy">The</span><span style=3D"color: #000;" class=3D"styled-by-prettify"> st=
ate size </span><span style=3D"color: #660;" class=3D"styled-by-prettify">(=
=3D=3D</span><span style=3D"color: #000;" class=3D"styled-by-prettify"> n <=
/span><span style=3D"color: #660;" class=3D"styled-by-prettify">*</span><sp=
an style=3D"color: #000;" class=3D"styled-by-prettify"> </span><span style=
=3D"color: #008;" class=3D"styled-by-prettify">sizeof</span><span style=3D"=
color: #660;" class=3D"styled-by-prettify">(</span><span style=3D"color: #6=
06;" class=3D"styled-by-prettify">UIntType</span><span style=3D"color: #660=
;" class=3D"styled-by-prettify">)</span><span style=3D"color: #000;" class=
=3D"styled-by-prettify"> bytes</span><span style=3D"color: #660;" class=3D"=
styled-by-prettify">).</span></div></code></div><br>Since all the reference=
 generators use only 64-bit unsigned integers, I have mistakenly generalise=
d.<br></div>

<p></p>

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/a>.<br />

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