# rand

```rand(3erl)                  Erlang Module Definition                  rand(3erl)

NAME
rand - Pseudo random number generation.

DESCRIPTION
This module provides a pseudo random number generator. The module
contains a number of algorithms. The uniform distribution algorithms use
the  xoroshiro116+ and xorshift1024* algorithms by Sebastiano Vigna.  The
normal distribution algorithm uses the  Ziggurat Method by Marsaglia and
Tsang  on top of the uniform distribution algorithm.

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.

The following algorithms are provided:

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now has been superseded by Xoroshiro116+ (exrop).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 exrop (Xoroshiro116+). 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.

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

R0 = rand:uniform(),
R1 = rand:uniform(),

Use a specified algorithm:

_ = rand:seed(exs1024s),
R2 = rand:uniform(),

Use a specified algorithm with a constant seed:

_ = rand:seed(exs1024s, {123, 123534, 345345}),
R3 = rand:uniform(),

Use the functional API with a non-constant seed:

S0 = rand:seed_s(exrop),
{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 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 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
builtin_alg() =
exs64 | exsplus | exsp | exs1024 | exs1024s | exrop

alg() = builtin_alg() | atom()

alg_handler() =
#{type := alg(),
bits => integer() >= 0,
weak_low_bits => integer() >= 0,
max => integer() >= 0,
next :=
fun((alg_state()) -> {integer() >= 0, alg_state()}),
uniform => fun((state()) -> {float(), state()}),
uniform_n =>
fun((integer() >= 1, state()) -> {integer() >= 1, state()}),
jump => fun((state()) -> state())}

alg_state() =
exs64_state() |
exsplus_state() |
exs1024_state() |
exrop_state() |
term()

state() = {alg_handler(), alg_state()}

Algorithm-dependent state.

export_state() = {alg(), alg_state()}

Algorithm-dependent state that can be printed or saved to file.

exs64_state()

Algorithm specific internal state

exsplus_state()

Algorithm specific internal state

exs1024_state()

Algorithm specific internal state

exrop_state()

Algorithm specific internal state

EXPORTS
export_seed() -> undefined | export_state()

Returns the random number state in an external format. To be used
with seed/1.

export_seed_s(State :: state()) -> export_state()

Returns the random number generator state in an external format.
To be used with seed/1.

jump() -> NewState :: state()

Returns the state after performing jump calculation to the state
in the process dictionary.

This function generates a not_implemented error exception when the
jump function is not implemented for the algorithm specified in
the state in the process dictionary.

jump(State :: state()) -> NewState :: state()

Returns the state after performing jump calculation to the given
state.

This function generates a not_implemented error exception when the
jump function is not implemented for the algorithm specified in
the state.

normal() -> float()

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.

normal(Mean :: number(), Variance :: number()) -> float()

Returns a normal N(Mean, Variance) deviate float and updates the
state in the process dictionary.

normal_s(State :: state()) -> {float(), NewState :: state()}

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.

normal_s(Mean :: number(),
Variance :: number(),
State0 :: state()) ->
{float(), NewS :: state()}

Returns, for a specified state, a normal N(Mean, Variance) deviate
float and a new state.

seed(AlgOrStateOrExpState ::
builtin_alg() | state() | export_state()) ->
state()

Seeds random number generation with the specifed algorithm and
time-dependent data if AlgOrStateOrExpState is an algorithm.

Otherwise recreates the exported seed in the process dictionary,

seed(Alg :: builtin_alg(),
Seed :: {integer(), integer(), integer()}) ->
state()

Seeds random number generation with the specified algorithm and
integers in the process dictionary and returns the state.

seed_s(AlgOrStateOrExpState ::
builtin_alg() | state() | export_state()) ->
state()

Seeds random number generation with the specifed algorithm and
time-dependent data if AlgOrStateOrExpState is an algorithm.

Otherwise recreates the exported seed and returns the state. See
also export_seed/0.

seed_s(Alg :: builtin_alg(),
Seed :: {integer(), integer(), integer()}) ->
state()

Seeds random number generation with the specified algorithm and
integers and returns the state.

uniform() -> X :: float()

Returns a random float uniformly distributed in the value range
0.0 =< X < 1.0 and updates the state in the process dictionary.

The generated numbers are on the form N * 2.0^(-53), that is;
equally spaced in the interval.

Warning:
This function may return exactly 0.0 which can be fatal for
certain applications. If that is undesired you can use (1.0 -
rand:uniform()) to get the interval 0.0 < X =< 1.0, or instead use
uniform_real/0.

If neither endpoint is desired you can test and re-try like this:

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

uniform_real() -> X :: float()

Returns a random float uniformly distributed in the value range
DBL_MIN =< X < 1.0 and updates the state in the process
dictionary.

Conceptually, a random real number R is generated from the
interval 0 =< R < 1 and then the closest rounded down normalized
number in the IEEE 754 Double precision format is returned.

Note:
The generated numbers from this function has got better
granularity for small numbers than the regular uniform/0 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 1.0 / X or math:log(X).

See uniform_real_s/1 for more explanation.

uniform(N :: integer() >= 1) -> X :: integer() >= 1

Returns, for a specified integer N >= 1, a random integer
uniformly distributed in the value range 1 =< X =< N and updates
the state in the process dictionary.

uniform_s(State :: state()) -> {X :: float(), NewState :: state()}

Returns, for a specified state, random float uniformly distributed
in the value range 0.0 =< X < 1.0 and a new state.

The generated numbers are on the form N * 2.0^(-53), that is;
equally spaced in the interval.

Warning:
This function may return exactly 0.0 which can be fatal for
certain applications. If that is undesired you can use (1.0 -
rand:uniform(State)) to get the interval 0.0 < X =< 1.0, or

If neither endpoint is desired you can test and re-try like this:

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

uniform_real_s(State :: state()) ->
{X :: float(), NewState :: state()}

Returns, for a specified state, a random float uniformly
distributed in the value range DBL_MIN =< X < 1.0 and updates the
state in the process dictionary.

Conceptually, a random real number R is generated from the
interval 0 =< R < 1 and then the closest rounded down normalized
number in the IEEE 754 Double precision format is returned.

Note:
The generated numbers from this function has got better
granularity for small numbers than the regular uniform_s/1 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 1.0 / X or math:log(X).

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
DBL_MIN, which is 2.0^(-1022).

The value range stated at the top of this function description is
technically correct, but 0.0 =< X < 1.0 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.

For example; for all sub ranges N*2.0^(-53) =< X < (N+1)*2.0^(-53)
where 0 =< integer(N) < 2.0^53 the probability is the same.
Compare that with the form of the numbers generated by
uniform_s/1.

Having to generate extra random bits for small numbers costs a
little performance. This function is about 20% slower than the
regular uniform_s/1

uniform_s(N :: integer() >= 1, State :: state()) ->
{X :: integer() >= 1, NewState :: state()}

Returns, for a specified integer N >= 1 and a state, a random
integer uniformly distributed in the value range 1 =< X =< N and a
new state.

Ericsson AB                       stdlib 3.7.1                        rand(3erl)```