rand

Module

rand

Module Summary

Pseudo random number generation.

Since

Module rand was introduced in OTP 18.0.

Description

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.

The following algorithms are provided:

exsss

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

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, and on the flip side a larger state.

Many thanks to Sebastiano Vigna for his help with the 58 bit adaption.

exrop

Xoroshiro116+, 58 bits precision and period of 2^116-1

Jump function: equivalent to 2^64 calls

exs1024s

Xorshift1024*, 64 bits precision and a period of 2^1024-1

Jump function: equivalent to 2^512 calls

exsp

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This is a corrected version of the previous default algorithm, that now has been superseded by Xoroshiro116+ (exrop).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 default algorithm is exsss (Xorshift116**). If a specific algorithm is required, ensure to always use seed/1 to initialize the state.

Undocumented (old) algorithms are deprecated but still implemented so old code relying on them will produce the same pseudo random sequences as before.

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),

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(16) > 8)

And 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.

Data Types

Exports

my_uniform() ->
    case rand:uniform() of
        0.0 -> my_uniform();
	X -> X
    end
end.

my_uniform(State) ->
    case rand:uniform(State) of
        {0.0, NewState} -> my_uniform(NewState);
	Result -> Result
    end
end.