## Pseudo random number generation.

This module provides a pseudo random number generator.
The module contains a number of algorithms.
The uniform distribution algorithms are based on the

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:

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:
`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that now has been superseded by Xoroshiro116+ (`exrop`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

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.

## 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

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.

#### Types

### builtin_alg() =

exsss | exro928ss | exrop | exs1024s | exsp | exs64 |

exsplus | exs1024

### 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() =

exsplus_state() |

exro928_state() |

exrop_state() |

exs1024_state() |

exs64_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.

### seed() =

[integer()] | integer() | {integer(), integer(), integer()}

A seed value for the generator.

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.

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.

A traditional 3-tuple of integers seed is passed through algorithm-dependent hashing functions to create the generator's initial state.

### exsplus_state()

Algorithm specific internal state

### exro928_state()

Algorithm specific internal state

### exrop_state()

Algorithm specific internal state

### exs1024_state()

Algorithm specific internal state

### exs64_state()

Algorithm specific internal state

#### Functions

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

.

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

.

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.

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.

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.

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

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.

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

Seeds random number generation with the specifed algorithm and
time-dependent data if

is an algorithm.

Otherwise recreates the exported seed in the process dictionary,
and returns the state. See also
`export_seed/0`

.

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

Seeds random number generation with the specifed algorithm and
time-dependent data if

is an algorithm.

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

.

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

Returns a random float uniformly distributed in the value
range `0.0 =< `

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 < `

, 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.

Returns a random float
uniformly distributed in the value range
`DBL_MIN =< `

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 / `

or `math:log(`

.

See
`uniform_real_s/1`

for more explanation.

Returns, for a specified integer

,
a random integer uniformly distributed in the value range
`1 =< `

and
updates the state in the process dictionary.

Returns, for a specified state, random float
uniformly distributed in the value range ```
0.0 =<
```

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 < `

, or instead use
`uniform_real_s/1`

.

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.

Returns, for a specified state, a random float
uniformly distributed in the value range
`DBL_MIN =< `

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 / `

or `math:log(`

.

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 =< `

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`

Returns, for a specified integer

and a state, a random integer uniformly distributed in the value
range `1 =< `

and a
new state.