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<copyright>
<year>2015</year><year>2017</year>
<holder>Ericsson AB. All Rights Reserved.</holder>
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<legalnotice>
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
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<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 pseudo random number generator.
The module contains a number of algorithms.
The uniform distribution algorithms use the
<url href="http://xorshift.di.unimi.it">
xoroshiro116+ and xorshift1024* 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>
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>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that now has been superseded by Xoroshiro116+ (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>,
<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(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>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 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" arity="2"/>
<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"/>
<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"/>
<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"/>
<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>
<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 < <anno>X</anno> =< 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"/>
<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 =< <anno>X</anno> < 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 =< R < 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"/>
<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>
<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 < <anno>X</anno> =< 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"/>
<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 =< <anno>X</anno> < 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 =< R < 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 =< <anno>X</anno> < 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) =< X < (N+1)*2.0^(-53)</c>
where
<c>0 =< integer(N) < 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"/>
<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>
