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<erlref>
  <header>
    <copyright>
      <year>2015</year><year>2017</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 since="OTP 18.0">rand</module>
  <modulesummary>Pseudo random number generation.</modulesummary>
  <description>
    <p>
      This module provides a pseudo random number generator.
      The module contains a number of algorithms.
      The uniform distribution algorithms are based on the
      <url href="http://xorshift.di.unimi.it">
	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 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>
        <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 superseded by Xoroshiro116+ (<c>exrop</c>).
	  Since there is no native 58 bit rotate instruction this
	  algorithm executes a little (say &lt; 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>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.
    </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>,
      <seealso marker="#uniform-1"><c>uniform/1</c></seealso> or
      <seealso marker="#uniform_real-0"><c>uniform_real/0</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(exs928ss),
R2 = rand:uniform(),</pre>

    <p>Use a specified algorithm with a constant seed:</p>

    <pre>
_ = 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(exsss),
{R4, S1} = rand:uniform_s(S0),</pre>

   <p>Textbook basic form Box-Muller standard normal deviate</p>

   <pre>
R5 = rand:uniform_real(),
R6 = rand:uniform(),
SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)</pre>

   <p>Create a standard normal deviate:</p>

   <pre>
{SND1, S2} = rand:normal_s(S1),</pre>

   <p>Create a normal deviate with mean -3 and variance 0.5:</p>

   <pre>
{ND0, S3} = rand:normal_s(-3, 0.5, S2),</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 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
      <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 with these algorithms
      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="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="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>
    <func>
      <name name="export_seed" arity="0" since="OTP 18.0"/>
      <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" since="OTP 18.0"/>
      <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" since="OTP 20.0"/>
      <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" since="OTP 20.0"/>
      <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" since="OTP 18.0"/>
      <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" arity="2" since="OTP 20.0"/>
      <fsummary>Return a normal distributed random float.</fsummary>
      <desc>
        <p>Returns a normal N(Mean, Variance) deviate float
          and updates the state in the process dictionary.</p>
      </desc>
    </func>

    <func>
      <name name="normal_s" arity="1" since="OTP 18.0"/>
      <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="normal_s" arity="3" since="OTP 20.0"/>
      <fsummary>Return a normal distributed random float.</fsummary>
      <desc>
        <p>Returns, for a specified state, a normal N(Mean, Variance)
          deviate float and a new state.</p>
      </desc>
    </func>

    <func>
      <name name="seed" arity="1" since="OTP 18.0"/>
      <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" since="OTP 18.0"/>
      <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" since="OTP 18.0"/>
      <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" since="OTP 18.0"/>
      <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" since="OTP 18.0"/>
      <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 =&lt; <anno>X</anno> &lt; 1.0</c> and
          updates the state in the process dictionary.
	</p>
	<p>
	  The generated numbers are on the form N * 2.0^(-53),
	  that is; equally spaced in the interval.
	</p>
	<warning>
	  <p>
	    This function may return exactly <c>0.0</c> which can be
	    fatal for certain applications.  If that is undesired
	    you can use <c>(1.0 - rand:uniform())</c> to get the
	    interval <c>0.0 &lt; <anno>X</anno> =&lt; 1.0</c>, or instead use
	    <seealso marker="#uniform_real-0"><c>uniform_real/0</c></seealso>.
	  </p>
	  <p>
	    If neither endpoint is desired you can test and re-try
	    like this:
	  </p>
	  <pre>
my_uniform() ->
    case rand:uniform() of
        0.0 -> my_uniform();
	X -> X
    end
end.</pre>
	</warning>
      </desc>
    </func>

    <func>
      <name name="uniform_real" arity="0" since="OTP 21.0"/>
      <fsummary>Return a random float.</fsummary>
      <desc><marker id="uniform_real-0"/>
        <p>
	  Returns a random float
	  uniformly distributed in the value range
	  <c>DBL_MIN =&lt; <anno>X</anno> &lt; 1.0</c>
	  and updates the state in the process dictionary.
	</p>
	<p>
	  Conceptually, a random real number <c>R</c> is generated
	  from the interval <c>0 =&lt; R &lt; 1</c> and then the
	  closest rounded down normalized number
	  in the IEEE 754 Double precision format
	  is returned.
	</p>
	<note>
	  <p>
	    The generated numbers from this function has got better
	    granularity for small numbers than the regular
	    <seealso marker="#uniform-0"><c>uniform/0</c></seealso>
	    because all bits in the mantissa are random.
	    This property, in combination with the fact that exactly zero
	    is never returned is useful for algoritms doing for example
	    <c>1.0 / <anno>X</anno></c> or <c>math:log(<anno>X</anno>)</c>.
	  </p>
	</note>
	<p>
	  See
	  <seealso marker="#uniform_real_s-1"><c>uniform_real_s/1</c></seealso>
	  for more explanation.
	</p>
      </desc>
    </func>

    <func>
      <name name="uniform" arity="1" since="OTP 18.0"/>
      <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 =&lt; <anno>X</anno> =&lt; <anno>N</anno></c> and
          updates the state in the process dictionary.</p>
      </desc>
    </func>

    <func>
      <name name="uniform_s" arity="1" since="OTP 18.0"/>
      <fsummary>Return a random float.</fsummary>
      <desc>
        <p>
	  Returns, for a specified state, random float
          uniformly distributed in the value range <c>0.0 =&lt;
          <anno>X</anno> &lt; 1.0</c> and a new state.
	</p>
	<p>
	  The generated numbers are on the form N * 2.0^(-53),
	  that is; equally spaced in the interval.
	</p>
	<warning>
	  <p>
	    This function may return exactly <c>0.0</c> which can be
	    fatal for certain applications.  If that is undesired
	    you can use <c>(1.0 - rand:uniform(State))</c> to get the
	    interval <c>0.0 &lt; <anno>X</anno> =&lt; 1.0</c>, or instead use
	    <seealso marker="#uniform_real_s-1"><c>uniform_real_s/1</c></seealso>.
	  </p>
	  <p>
	    If neither endpoint is desired you can test and re-try
	    like this:
	  </p>
	  <pre>
my_uniform(State) ->
    case rand:uniform(State) of
        {0.0, NewState} -> my_uniform(NewState);
	Result -> Result
    end
end.</pre>
	</warning>
      </desc>
    </func>

    <func>
      <name name="uniform_real_s" arity="1" since="OTP 21.0"/>
      <fsummary>Return a random float.</fsummary>
      <desc>
        <p>
	  Returns, for a specified state, a random float
	  uniformly distributed in the value range
	  <c>DBL_MIN =&lt; <anno>X</anno> &lt; 1.0</c>
	  and updates the state in the process dictionary.
	</p>
	<p>
	  Conceptually, a random real number <c>R</c> is generated
	  from the interval <c>0 =&lt; R &lt; 1</c> and then the
	  closest rounded down normalized number
	  in the IEEE 754 Double precision format
	  is returned.
	</p>
	<note>
	  <p>
	    The generated numbers from this function has got better
	    granularity for small numbers than the regular
	    <seealso marker="#uniform_s-1"><c>uniform_s/1</c></seealso>
	    because all bits in the mantissa are random.
	    This property, in combination with the fact that exactly zero
	    is never returned is useful for algoritms doing for example
	    <c>1.0 / <anno>X</anno></c> or <c>math:log(<anno>X</anno>)</c>.
	  </p>
	</note>
	<p>
	  The concept implicates that the probability to get
	  exactly zero is extremely low; so low that this function
	  is in fact guaranteed to never return zero.  The smallest
	  number that it might return is <c>DBL_MIN</c>, which is
	  2.0^(-1022).
	</p>
	<p>
	  The value range stated at the top of this function
	  description is technically correct, but
	  <c>0.0 =&lt; <anno>X</anno> &lt; 1.0</c>
	  is a better description of the generated numbers'
	  statistical distribution.  Except that exactly 0.0
	  is never returned, which is not possible to observe
	  statistically.
	</p>
	<p>
	  For example; for all sub ranges
	  <c>N*2.0^(-53) =&lt; X &lt; (N+1)*2.0^(-53)</c>
	  where
	  <c>0 =&lt; integer(N) &lt; 2.0^53</c>
	  the probability is the same.
	  Compare that with the form of the numbers generated by
	  <seealso marker="#uniform_s-1"><c>uniform_s/1</c></seealso>.
	</p>
	<p>
	  Having to generate extra random bits for
	  small numbers costs a little performance.
	  This function is about 20% slower than the regular
	  <seealso marker="#uniform_s-1"><c>uniform_s/1</c></seealso>
	</p>
      </desc>
    </func>

    <func>
      <name name="uniform_s" arity="2" since="OTP 18.0"/>
      <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 =&lt; <anno>X</anno> =&lt; <anno>N</anno></c> and a
          new state.</p>
      </desc>
    </func>
  </funcs>
</erlref>
