<?xml version="1.0" encoding="utf-8" ?> <!DOCTYPE erlref SYSTEM "erlref.dtd"> <erlref> <header> <copyright> <year>2015</year><year>2016</year> <holder>Ericsson AB. All Rights Reserved.</holder> </copyright> <legalnotice> Licensed under the Apache License, Version 2.0 (the "License"); you may not use this file except in compliance with the License. You may obtain a copy of the License at http://www.apache.org/licenses/LICENSE-2.0 Unless required by applicable law or agreed to in writing, software distributed under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. See the License for the specific language governing permissions and limitations under the License. </legalnotice> <title>rand</title> <prepared></prepared> <responsible></responsible> <docno>1</docno> <approved></approved> <checked></checked> <date></date> <rev>A</rev> <file>rand.xml</file> </header> <module>rand</module> <modulesummary>Pseudo random number generation.</modulesummary> <description> <p>This module provides a random number generator. The module contains a number of algorithms. The uniform distribution algorithms use the <url href="http://xorshift.di.unimi.it">scrambled Xorshift algorithms by Sebastiano Vigna</url>. The normal distribution algorithm uses the <url href="http://www.jstatsoft.org/v05/i08">Ziggurat Method by Marsaglia and Tsang</url>.</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>The following algorithms are provided:</p> <taglist> <tag><c>exrop</c></tag> <item> <p>Xoroshiro116+, 58 bits precision and period of 2^116-1</p> <p>Jump function: equivalent to 2^64 calls</p> </item> <tag><c>exs1024s</c></tag> <item> <p>Xorshift1024*, 64 bits precision and a period of 2^1024-1</p> <p>Jump function: equivalent to 2^512 calls</p> </item> <tag><c>exsp</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 a corrected version of the previous default algorithm, that now has been superseeded by Xoroshiro116+ (<c>exrop</c>). Since there is no native 58 bit rotate instruction this algorithm executes a little (say < 15%) faster than <c>exrop</c>. See the <url href="http://xorshift.di.unimi.it">algorithms' homepage</url>. </p> </item> </taglist> <p> The default algorithm is <c>exrop</c> (Xoroshiro116+). If a specific algorithm is required, ensure to always use <seealso marker="#seed-1"> <c>seed/1</c></seealso> to initialize the state. </p> <p> Undocumented (old) algorithms are deprecated but still implemented so old code relying on them will produce the same pseudo random sequences as before. </p> <note> <p> There were a number of problems in the implementation of the now undocumented algorithms, which is why they are deprecated. The new algorithms are a bit slower but do not have these problems: </p> <p> Uniform integer ranges had a skew in the probability distribution that was not noticable for small ranges but for large ranges less than the generator's precision the probability to produce a low number could be twice the probability for a high. </p> <p> Uniform integer ranges larger than or equal to the generator's precision used a floating point fallback that only calculated with 52 bits which is smaller than the requested range and therefore were not all numbers in the requested range even possible to produce. </p> <p> Uniform floats had a non-uniform density so small values i.e less than 0.5 had got smaller intervals decreasing as the generated value approached 0.0 although still uniformly distributed for sufficiently large subranges. The new algorithms produces uniformly distributed floats on the form N * 2.0^(-53) hence equally spaced. </p> </note> <p>Every time a random number is requested, a state is used to calculate it and a new state is produced. The state can either be implicit or be an explicit argument and return value.</p> <p>The functions with implicit state use the process dictionary variable <c>rand_seed</c> to remember the current state.</p> <p>If a process calls <seealso marker="#uniform-0"><c>uniform/0</c></seealso> or <seealso marker="#uniform-1"><c>uniform/1</c></seealso> without setting a seed first, <seealso marker="#seed-1"><c>seed/1</c></seealso> is called automatically with the default algorithm and creates a non-constant seed.</p> <p>The functions with explicit state never use the process dictionary.</p> <p><em>Examples:</em></p> <p>Simple use; creates and seeds the default algorithm with a non-constant seed if not already done:</p> <pre> R0 = rand:uniform(), R1 = rand:uniform(),</pre> <p>Use a specified algorithm:</p> <pre> _ = rand:seed(exs1024s), R2 = rand:uniform(),</pre> <p>Use a specified algorithm with a constant seed:</p> <pre> _ = rand:seed(exs1024s, {123, 123534, 345345}), R3 = rand:uniform(),</pre> <p>Use the functional API with a non-constant seed:</p> <pre> S0 = rand:seed_s(exrop), {R4, S1} = rand:uniform_s(S0),</pre> <p>Create a standard normal deviate:</p> <pre> {SND0, S2} = rand:normal_s(S1),</pre> <note> <p>The builtin random number generator algorithms are not cryptographically strong. If a cryptographically strong random number generator is needed, use something like <seealso marker="crypto:crypto#rand_seed-0"><c>crypto:rand_seed/0</c></seealso>. </p> </note> <p> For all these generators 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 <url href="http://xoroshiro.di.unimi.it/xoroshiro128plus.c"> Xoroshiro128+ </url> generator source code: </p> <pre> Beside passing BigCrush, this generator passes the PractRand test suite 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 use something like this: </p> <pre>(rand:uniform(16) > 8)</pre> <p> And for a general range, with <c>N = 1</c> for <c>exrop</c>, and <c>N = 3</c> for <c>exs1024s</c>: </p> <pre>(((rand:uniform(Range bsl N) - 1) bsr N) + 1)</pre> <p> The floating point generating functions in this module waste the lowest bits when converting from an integer so they avoid this snag. </p> </description> <datatypes> <datatype> <name name="builtin_alg"/> </datatype> <datatype> <name name="alg"/> </datatype> <datatype> <name name="alg_handler"/> </datatype> <datatype> <name name="alg_state"/> </datatype> <datatype> <name name="state"/> <desc><p>Algorithm-dependent state.</p></desc> </datatype> <datatype> <name name="export_state"/> <desc> <p> Algorithm-dependent state that can be printed or saved to file. </p> </desc> </datatype> <datatype> <name name="exs64_state"/> <desc><p>Algorithm specific internal state</p></desc> </datatype> <datatype> <name name="exsplus_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="exrop_state"/> <desc><p>Algorithm specific internal state</p></desc> </datatype> </datatypes> <funcs> <func> <name name="export_seed" arity="0"/> <fsummary>Export the random number generation state.</fsummary> <desc><marker id="export_seed-0"/> <p>Returns the random number state in an external format. To be used with <seealso marker="#seed-1"><c>seed/1</c></seealso>.</p> </desc> </func> <func> <name name="export_seed_s" arity="1"/> <fsummary>Export the random number generation state.</fsummary> <desc><marker id="export_seed_s-1"/> <p>Returns the random number generator state in an external format. To be used with <seealso marker="#seed-1"><c>seed/1</c></seealso>.</p> </desc> </func> <func> <name name="jump" arity="0"/> <fsummary>Return the seed after performing jump calculation to the state in the process dictionary.</fsummary> <desc><marker id="jump-0" /> <p>Returns the state after performing jump calculation to the state in the process dictionary.</p> <p>This function generates a <c>not_implemented</c> error exception when the jump function is not implemented for the algorithm specified in the state in the process dictionary.</p> </desc> </func> <func> <name name="jump" arity="1"/> <fsummary>Return the seed after performing jump calculation.</fsummary> <desc><marker id="jump-1" /> <p>Returns the state after performing jump calculation to the given state. </p> <p>This function generates a <c>not_implemented</c> error exception when the jump function is not implemented for the algorithm specified in the state.</p> </desc> </func> <func> <name name="normal" arity="0"/> <fsummary>Return a standard normal distributed random float.</fsummary> <desc> <p>Returns a standard normal deviate float (that is, the mean is 0 and the standard deviation is 1) and updates the state in the process dictionary.</p> </desc> </func> <func> <name name="normal_s" arity="1"/> <fsummary>Return a standard normal distributed random float.</fsummary> <desc> <p>Returns, for a specified state, a standard normal deviate float (that is, the mean is 0 and the standard deviation is 1) and a new state.</p> </desc> </func> <func> <name name="seed" arity="1"/> <fsummary>Seed random number generator.</fsummary> <desc> <marker id="seed-1"/> <p> Seeds random number generation with the specifed algorithm and time-dependent data if <c><anno>AlgOrStateOrExpState</anno></c> is an algorithm. </p> <p>Otherwise recreates the exported seed in the process dictionary, and returns the state. See also <seealso marker="#export_seed-0"><c>export_seed/0</c></seealso>.</p> </desc> </func> <func> <name name="seed" arity="2"/> <fsummary>Seed the random number generation.</fsummary> <desc> <p>Seeds random number generation with the specified algorithm and integers in the process dictionary and returns the state.</p> </desc> </func> <func> <name name="seed_s" arity="1"/> <fsummary>Seed random number generator.</fsummary> <desc> <p> Seeds random number generation with the specifed algorithm and time-dependent data if <c><anno>AlgOrStateOrExpState</anno></c> is an algorithm. </p> <p>Otherwise recreates the exported seed and returns the state. See also <seealso marker="#export_seed-0"> <c>export_seed/0</c></seealso>.</p> </desc> </func> <func> <name name="seed_s" arity="2"/> <fsummary>Seed the random number generation.</fsummary> <desc> <p>Seeds random number generation with the specified algorithm and integers and returns the state.</p> </desc> </func> <func> <name name="uniform" arity="0"/> <fsummary>Return a random float.</fsummary> <desc><marker id="uniform-0"/> <p>Returns a random float uniformly distributed in the value range <c>0.0 =< <anno>X</anno> < 1.0</c> and updates the state in the process dictionary.</p> </desc> </func> <func> <name name="uniform" arity="1"/> <fsummary>Return a random integer.</fsummary> <desc><marker id="uniform-1"/> <p>Returns, for a specified integer <c><anno>N</anno> >= 1</c>, a random integer uniformly distributed in the value range <c>1 =< <anno>X</anno> =< <anno>N</anno></c> and updates the state in the process dictionary.</p> </desc> </func> <func> <name name="uniform_s" arity="1"/> <fsummary>Return a random float.</fsummary> <desc> <p>Returns, for a specified state, random float uniformly distributed in the value range <c>0.0 =< <anno>X</anno> < 1.0</c> and a new state.</p> </desc> </func> <func> <name name="uniform_s" arity="2"/> <fsummary>Return a random integer.</fsummary> <desc> <p>Returns, for a specified integer <c><anno>N</anno> >= 1</c> and a state, a random integer uniformly distributed in the value range <c>1 =< <anno>X</anno> =< <anno>N</anno></c> and a new state.</p> </desc> </func> </funcs> </erlref>