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Diffstat (limited to 'lib/stdlib/doc/src/rand.xml')
| -rw-r--r-- | lib/stdlib/doc/src/rand.xml | 110 |
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> |