This module provides a pseudo random number generator.
The module contains a number of algorithms.
The uniform distribution algorithms are based on the
Xoroshiro and Xorshift algorithms
by Sebastiano Vigna.
The normal distribution algorithm uses the
Ziggurat Method by Marsaglia and Tsang
on top of the uniform distribution algorithm.
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, but execute in a time
roughly equivalent to one regular iteration per generator bit.
At the end of this module documentation there are also some
niche algorithms
to be used without this module's normal
plug-in framework API
that may be useful for special purposes like
short generation time when quality is not essential,
for seeding other generators, and such.
The following algorithms are provided:
-
exsss
OTP 22.0
-
Xorshift116**, 58 bits precision and period of 2^116-1
Jump function: equivalent to 2^64 calls
This is the Xorshift116 generator combined with the StarStar scrambler
from the 2018 paper by David Blackman and Sebastiano Vigna:
Scrambled Linear Pseudorandom Number Generators
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 exrop (Xoroshiro116+).
Alas, this combination is about 10% slower than exrop,
but is despite that the
default algorithm
thanks to its statistical qualities.
-
exro928ss
OTP 22.0
-
Xoroshiro928**, 58 bits precision and a period of 2^928-1
Jump function: equivalent to 2^512 calls
This is a 58 bit version of Xoroshiro1024**,
from the 2018 paper by David Blackman and Sebastiano Vigna:
Scrambled Linear Pseudorandom Number Generators
that on a 64 bit Erlang system executes only
about 40% slower than the
default exsss algorithm
but with much longer period and better statistical properties,
but on the flip side a larger state.
Many thanks to Sebastiano Vigna for his help with
the 58 bit adaption.
-
exrop
OTP 20.0
-
Xoroshiro116+, 58 bits precision and period of 2^116-1
Jump function: equivalent to 2^64 calls
-
exs1024s
OTP 20.0
-
Xorshift1024*, 64 bits precision and a period of 2^1024-1
Jump function: equivalent to 2^512 calls
-
exsp
OTP 20.0
-
Xorshift116+, 58 bits precision and period of 2^116-1
Jump function: equivalent to 2^64 calls
This is a corrected version of the previous
default algorithm,
that now has been superseded by Xoroshiro116+ (exrop).
Since there is no native 58 bit rotate instruction this
algorithm executes a little (say < 15%) faster than exrop.
See the
algorithms' homepage.
The current default algorithm is
exsss (Xorshift116**).
If a specific algorithm is
required, ensure to always use
seed/1 to initialize the state.
Which algorithm that is the default may change between
Erlang/OTP releases, and is selected to be one with high
speed, small state and "good enough" statistical properties.
Undocumented (old) algorithms are deprecated but still implemented
so old code relying on them will produce
the same pseudo random sequences as before.
Note
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:
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.
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.
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.
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.
The functions with implicit state use the process dictionary
variable rand_seed to remember the current state.
If a process calls
uniform/0,
uniform/1 or
uniform_real/0 without
setting a seed first, seed/1
is called automatically with the
default algorithm
and creates a non-constant seed.
The functions with explicit state never use the process dictionary.
Examples:
Simple use; creates and seeds the
default algorithm
with a non-constant seed if not already done:
R0 = rand:uniform(),
R1 = rand:uniform(),
Use a specified algorithm:
_ = rand:seed(exs928ss),
R2 = rand:uniform(),
Use a specified algorithm with a constant seed:
_ = rand:seed(exs928ss, {123, 123534, 345345}),
R3 = rand:uniform(),
Use the functional API with a non-constant seed:
S0 = rand:seed_s(exsss),
{R4, S1} = rand:uniform_s(S0),
Textbook basic form Box-Muller standard normal deviate
R5 = rand:uniform_real(),
R6 = rand:uniform(),
SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)
Create a standard normal deviate:
{SND1, S2} = rand:normal_s(S1),
Create a normal deviate with mean -3 and variance 0.5:
{ND0, S3} = rand:normal_s(-3, 0.5, S2),
Note
The builtin random number generator algorithms are not
cryptographically strong. If a cryptographically strong
random number generator is needed, use something like
crypto:rand_seed/0.
For all these generators except exro928ss and exsss
the lowest bit(s) has got a slightly less
random behaviour than all other bits.
1 bit for exrop (and exsp),
and 3 bits for exs1024s.
See for example the explanation in the
Xoroshiro128+
generator source code:
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.
If this is a problem; to generate a boolean with these algorithms
use something like this:
(rand:uniform(256) > 128) % -> boolean()
((rand:uniform(256) - 1) bsr 7) % -> 0 | 1
For a general range, with N = 1 for exrop,
and N = 3 for exs1024s:
(((rand:uniform(Range bsl N) - 1) bsr N) + 1)
The floating point generating functions in this module
waste the lowest bits when converting from an integer
so they avoid this snag.