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-rw-r--r--lib/stdlib/doc/src/rand.xml110
1 files changed, 93 insertions, 17 deletions
diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
index 21f680a0ee..8e657698c6 100644
--- a/lib/stdlib/doc/src/rand.xml
+++ b/lib/stdlib/doc/src/rand.xml
@@ -38,25 +38,71 @@
<p>
This module provides a pseudo random number generator.
The module contains a number of algorithms.
- The uniform distribution algorithms use the
+ The uniform distribution algorithms are based on the
<url href="http://xorshift.di.unimi.it">
- xoroshiro116+ and xorshift1024* algorithms by Sebastiano Vigna.
+ Xoroshiro and Xorshift algorithms
</url>
+ by Sebastiano Vigna.
The normal distribution algorithm uses the
<url href="http://www.jstatsoft.org/v05/i08">
Ziggurat Method by Marsaglia and Tsang
</url>
on top of the uniform distribution algorithm.
</p>
- <p>For some algorithms, jump functions are provided for generating
- non-overlapping sequences for parallel computations.
- The jump functions perform calculations
- equivalent to perform a large number of repeated calls
- for calculating new states. </p>
+ <p>
+ For most algorithms, jump functions are provided for generating
+ non-overlapping sequences for parallel computations.
+ The jump functions perform calculations
+ equivalent to perform a large number of repeated calls
+ for calculating new states.
+ </p>
<p>The following algorithms are provided:</p>
<taglist>
+ <tag><c>exsss</c></tag>
+ <item>
+ <p>Xorshift116**, 58 bits precision and period of 2^116-1</p>
+ <p>Jump function: equivalent to 2^64 calls</p>
+ <p>
+ This is the Xorshift116 generator combined with the StarStar scrambler
+ from the 2018 paper by David Blackman and Sebastiano Vigna:
+ <url href="http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf">
+ Scrambled Linear Pseudorandom Number Generators
+ </url>
+ </p>
+ <p>
+ The generator does not need 58-bit rotates so it is faster
+ than the Xoroshiro116 generator, and when combined with
+ the StarStar scrambler it does not have any weak low bits
+ like <c>exrop</c> (Xoroshiro116+).
+ </p>
+ <p>
+ Alas, this combination is about 10% slower than <c>exrop</c>,
+ but is despite that the default algorithm thanks to its
+ statistical qualities.
+ </p>
+ </item>
+ <tag><c>exro928ss</c></tag>
+ <item>
+ <p>Xoroshiro928**, 58 bits precision and a period of 2^928-1</p>
+ <p>Jump function: equivalent to 2^512 calls</p>
+ <p>
+ This is a 58 bit version of Xoroshiro1024**,
+ from the 2018 paper by David Blackman and Sebastiano Vigna:
+ <url href="http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf">
+ Scrambled Linear Pseudorandom Number Generators
+ </url>
+ that on a 64 bit Erlang system executes only about 40% slower than
+ the default <c>exsss</c> algorithm but with much longer period
+ and better statistical properties, and on the flip side
+ a larger state.
+ </p>
+ <p>
+ Many thanks to Sebastiano Vigna for his help with
+ the 58 bit adaption.
+ </p>
+ </item>
<tag><c>exrop</c></tag>
<item>
<p>Xoroshiro116+, 58 bits precision and period of 2^116-1</p>
@@ -83,7 +129,7 @@
</taglist>
<p>
- The default algorithm is <c>exrop</c> (Xoroshiro116+).
+ The default algorithm is <c>exsss</c> (Xorshift116**).
If a specific algorithm is
required, ensure to always use <seealso marker="#seed-1">
<c>seed/1</c></seealso> to initialize the state.
@@ -154,19 +200,19 @@ R1 = rand:uniform(),</pre>
<p>Use a specified algorithm:</p>
<pre>
-_ = rand:seed(exs1024s),
+_ = rand:seed(exs928ss),
R2 = rand:uniform(),</pre>
<p>Use a specified algorithm with a constant seed:</p>
<pre>
-_ = rand:seed(exs1024s, {123, 123534, 345345}),
+_ = rand:seed(exs928ss, {123, 123534, 345345}),
R3 = rand:uniform(),</pre>
<p>Use the functional API with a non-constant seed:</p>
<pre>
-S0 = rand:seed_s(exrop),
+S0 = rand:seed_s(exsss),
{R4, S1} = rand:uniform_s(S0),</pre>
<p>Textbook basic form Box-Muller standard normal deviate</p>
@@ -195,8 +241,9 @@ SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)</pre>
</note>
<p>
- For all these generators the lowest bit(s) has got
- a slightly less random behaviour than all other bits.
+ For all these generators except <c>exro928ss</c> and <c>exsss</c>
+ the lowest bit(s) has got a slightly less
+ random behaviour than all other bits.
1 bit for <c>exrop</c> (and <c>exsp</c>),
and 3 bits for <c>exs1024s</c>.
See for example the explanation in the
@@ -211,7 +258,7 @@ up to (and included) 16TB, with the exception of binary rank tests,
which fail due to the lowest bit being an LFSR; all other bits pass all
tests. We suggest to use a sign test to extract a random Boolean value.</pre>
<p>
- If this is a problem; to generate a boolean
+ If this is a problem; to generate a boolean with these algorithms
use something like this:
</p>
<pre>(rand:uniform(16) > 8)</pre>
@@ -254,21 +301,50 @@ tests. We suggest to use a sign test to extract a random Boolean value.</pre>
</desc>
</datatype>
<datatype>
- <name name="exs64_state"/>
- <desc><p>Algorithm specific internal state</p></desc>
+ <name name="seed"/>
+ <desc>
+ <p>
+ A seed value for the generator.
+ </p>
+ <p>
+ A list of integers sets the generator's internal state directly,
+ after algorithm-dependent checks of the value
+ and masking to the proper word size.
+ </p>
+ <p>
+ An integer is used as the initial state for a SplitMix64 generator.
+ The output values of that is then used for setting
+ the generator's internal state
+ after masking to the proper word size
+ and if needed avoiding zero values.
+ </p>
+ <p>
+ A traditional 3-tuple of integers seed is passed through
+ algorithm-dependent hashing functions to create
+ the generator's initial state.
+ </p>
+ </desc>
</datatype>
<datatype>
<name name="exsplus_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
- <name name="exs1024_state"/>
+ <name name="exro928_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
<name name="exrop_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
+ <datatype>
+ <name name="exs1024_state"/>
+ <desc><p>Algorithm specific internal state</p></desc>
+ </datatype>
+ <datatype>
+ <name name="exs64_state"/>
+ <desc><p>Algorithm specific internal state</p></desc>
+ </datatype>
</datatypes>
<funcs>