rand — stdlib v6.2.2.2

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# rand (stdlib v6.2.2.2)

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Pseudo random number generation

This module provides pseudo random number generation and implements a number of base generator algorithms. Most are provided through a plug-in framework that adds features to the base generators.

At the end of this module documentation there are some niche algorithms that don't use this module's normal plug-in framework. They may be useful for special purposes like short generation time when quality is not essential, for seeding other generators, and such.

## Plug-in framework

The plug-in framework implements a common API to, and enhancements of the base generators:

- Operating on a generator state in the process dictionary.
- Automatic seeding.
- Manual seeding support to avoid common pitfalls.
- Generating integers in any range, with uniform distribution, without noticable bias.
- Generating integers in any range, larger than the base generator's, with uniform distribution.
- Generating floating-point numbers with uniform distribution.
- Generating floating-point numbers with normal distribution.
- Generating any number of bytes.

The base generator algorithms implements the Xoroshiro and Xorshift algorithms by Sebastiano Vigna. During an iteration they generate a large integer (at least 58-bit) and operate on a state of several large integers.

To create numbers with normal distribution the Ziggurat Method by Marsaglia and Tsang is used on the output from a base generator.

For most algorithms, jump functions are provided for generating non-overlapping sequences. A jump function perform a calculation equivalent to a large number of repeated state iterations, but execute in a time roughly equivalent to one regular iteration per generator bit.

The following algorithms are provided:

- `exsss`, the default algorithm (Since 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 doesn't use 58-bit rotates so it is faster than the Xoroshiro116 generator, and when combined with the StarStar scrambler it doesn't have any weak low bits like `exrop` (Xoroshiro116+).

  Alas, this combination is about 10% slower than `exrop`, but despite that it is the default algorithm thanks to its statistical qualities.

- `exro928ss` (Since 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` (Since OTP 20.0)
  Xoroshiro116+, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

- `exs1024s` (Since OTP 20.0)
  Xorshift1024*, 64 bits precision and a period of 2^1024-1

  Jump function: equivalent to 2^512 calls

- `exsp` (Since 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 a previous default algorithm (`exsplus`, deprecated), that was superseded by Xoroshiro116+ (`exrop`). Since this algorithm doesn't use rotate it executes a little (say < 15%) faster than `exrop` (that has to do a 58 bit rotate, for which there is no native instruction). See the algorithms' homepage.

#### Default Algorithm

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.

#### Old Algorithms

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 all numbers in the requested range weren't even possible to produce.

Uniform floats had a non-uniform density so small values for example 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 they are equally spaced.

#### Generator State

Every time a random number is generated, a state is used to calculate it, producing a new state. The state can either be implicit or be an explicit argument and return value.

The functions with implicit state operates on a state stored in the process dictionary under the key `rand_seed`. If that key doesn't exist when the function is called, `seed/1` is called automatically with the default algorithm and creates a reasonably unpredictable seed.

The functions with explicit state don't use the process dictionary.

#### Examples

Simple use; create and seed the default algorithm with a non-fixed seed, if not already done, and generate two uniformly distibuted floating point numbers.

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

Use a specified algorithm:

    _ = rand:seed(exro928ss),
    R2 = rand:uniform(),

Use a specified algorithm with a fixed seed:

    _ = rand:seed(exro928ss, {123, 123534, 345345}),
    R3 = rand:uniform(),

Use the functional API with a non-fixed seed:

    S0 = rand:seed_s(exsss),
    {R4, S1} = rand:uniform_s(S0),

Generate a textbook basic form Box-Muller standard normal distribution number:

    R5 = rand:uniform_real(),
    R6 = rand:uniform(),
    SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)

Generate a standard normal distribution number:

    {SND1, S2} = rand:normal_s(S1),

Generate a normal distribution number with with mean -3 and variance 0.5:

    {ND0, S3} = rand:normal_s(-3, 0.5, S2),

#### Quality of the Generated Numbers

#### 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) have got a slightly less random behaviour than all other bits. 1 bit for `exrop` (and `exsp`), and 3 bits for `exs1024s`. See for example this 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.

## Niche algorithms

The niche algorithms API contains special purpose algorithms that don't use the plug-in framework, mainly for performance reasons.

Since these algorithms lack the plug-in framework support, generating numbers in a range other than the base generator's range may become a problem.

There are at least four ways to do this, assuming the `Range` is less than the generator's range:

- Modulo
  To generate a number `V` in the range `0..Range-1`:

  > Generate a number `X`.
  > Use `V = X rem Range` as your value.

  This method uses `rem`, that is, the remainder of an integer division, which is a slow operation.

  Low bits from the generator propagate straight through to the generated value, so if the generator has got weaknesses in the low bits this method propagates them too.

  If `Range` is not a divisor of the generator range, the generated numbers have a bias. Example:

  Say the generator generates a byte, that is, the generator range is `0..255`, and the desired range is `0..99` (`Range = 100`). Then there are 3 generator outputs that produce the value `0`, these are; `0`, `100` and `200`. But there are only 2 generator outputs that produce the value `99`, which are; `99` and `199`. So the probability for a value `V` in `0..55` is 3/2 times the probability for the other values `56..99`.

  If `Range` is much smaller than the generator range, then this bias gets hard to detect. The rule of thumb is that if `Range` is smaller than the square root of the generator range, the bias is small enough. Example:

  A byte generator when `Range = 20`. There are 12 (`256 div 20`) possibilities to generate the highest numbers and one more to generate a number `V < 16` (`256 rem 20`). So the probability is 13/12 for a low number versus a high. To detect that difference with some confidence you would need to generate a lot more numbers than the generator range, `256` in this small example.

- Truncated multiplication
  To generate a number `V` in the range `0..Range-1`, when you have a generator with a power of 2 range (`0..2^Bits-1`):

  > Generate a number `X`.
  > Use `V = X * Range bsr Bits` as your value.

  If the multiplication `X * Range` creates a bignum this method becomes very slow.

  High bits from the generator propagate through to the generated value, so if the generator has got weaknesses in the high bits this method propagates them too.

  If `Range` is not a divisor of the generator range, the generated numbers have a bias, pretty much as for the Modulo method above.

- Shift or mask
  To generate a number in a power of 2 range (`0..2^RBits-1`), when you have a generator with a power of 2 range (`0..2^Bits`):

  > Generate a number `X`.
  > Use `V = X band ((1 bsl RBits)-1)` or `V = X bsr (Bits-RBits)` as your value.

  Masking with `band` preserves the low bits, and right shifting with `bsr` preserves the high, so if the generator has got weaknesses in high or low bits; choose the right operator.

  If the generator has got a range that is not a power of 2 and this method is used anyway, it introduces bias in the same way as for the Modulo method above.

- Rejection

  > Generate a number `X`.
  > If `X` is in the range, use it as your value, otherwise reject it and repeat.

  In theory it is not certain that this method will ever complete, but in practice you ensure that the probability of rejection is low. Then the probability for yet another iteration decreases exponentially so the expected mean number of iterations will often be between 1 and 2. Also, since the base generator is a full length generator, a value that will break the loop must eventually be generated.

  These methods can be combined, such as using the Modulo method and only if the generator value would create bias use Rejection. Or using Shift or mask to reduce the size of a generator value so that Truncated multiplication will not create a bignum.

  The recommended way to generate a floating point number (IEEE 745 Double, that has got a 53-bit mantissa) in the range `0..1`, that is `0.0 =< V < 1.0` is to generate a 53-bit number `X` and then use `V = X * (1.0/((1 bsl 53)))` as your value. This will create a value on the form N*2^-53 with equal probability for every possible N for the range.

# Summary

## Types

alg()
alg_handler()
alg_state()
builtin_alg()
dummy_state()

Algorithm specific internal state

export_state()

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

exro928_state()

Algorithm specific internal state

exrop_state()

Algorithm specific internal state

exs64_state()

Algorithm specific internal state

exs1024_state()

Algorithm specific internal state

exsplus_state()

Algorithm specific internal state

mwc59_state()

`1 .. (16#1ffb072 bsl 29) - 2`

seed()

Generator seed value.

splitmix64_state()

Algorithm specific state

state()

Algorithm-dependent state.

uint58()

`0 .. (2^58 - 1)`

uint64()

`0 .. (2^64 - 1)`

## Plug-in framework API

bytes(N)

Generate random bytes as a `t:binary()`, using the state in the process dictionary.

bytes_s(N, State)

Generate random bytes as a `t:binary()`.

export_seed()

Export the seed value.

export_seed_s(State)

Export the seed value.

jump()

Jump the generator state forward.

jump(State)

Jump the generator state forward.

normal()

Generate a random number with standard normal distribution.

normal(Mean, Variance)

Generate a random number with specified normal distribution 𝒩 (μ, σ²).

normal_s(State)

Generate a random number with standard normal distribution.

normal_s(Mean, Variance, State)

Generate a random number with specified normal distribution 𝒩 (μ, σ²).

seed(Alg_or_State)

Seed the random number generator and select algorithm.

seed(Alg, Seed)

Seed the random number generator and select algorithm.

seed_s/1

Seed the random number generator and select algorithm.

seed_s(Alg, Seed)

Seed the random number generator and select algorithm.

uniform()

Generate a uniformly distributed random number `0.0 =< X < 1.0`, using the state in the process dictionary.

uniform(N)

Generate a uniformly distributed random integer `1 =< X =< N`, using the state in the process dictionary.

uniform_real()

Generate a uniformly distributed random number `0.0 < X < 1.0`, using the state in the process dictionary.

uniform_real_s(State)

Generate a uniformly distributed random number `0.0 < X < 1.0`.

uniform_s(State)

Generate a uniformly distributed random number `0.0 =< X < 1.0`.

uniform_s(N, State)

Generate a uniformly distributed random integer `1 =< X =< N`.

## Niche algorithms API

exsp_jump(AlgState)

Jump the generator state forward.

exsp_next(AlgState)

Generate an Xorshift116+ random integer and new algorithm state.

mwc59(CX0)

Generate a new MWC59 state.

mwc59_float(CX)

Calculate a scrambled `float/0` from a MWC59 state.

mwc59_seed()

Create a MWC59 generator state.

mwc59_seed(S)

Create a MWC59 generator state.

mwc59_value32(CX)

Calculate a 32-bit scrambled value from a MWC59 state.

mwc59_value(CX)

Calculate a 59-bit scrambled value from a MWC59 state.

splitmix64_next(AlgState)

Generate a SplitMix64 random 64-bit integer and new algorithm state.

# Types

# alg()

(since OTP 18.0)

    -type alg() :: builtin_alg() | atom().

# alg_handler()

(since OTP 18.0)

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

# alg_state()

(since OTP 18.0)

    -type alg_state() ::
              exsplus_state() |
              exro928_state() |
              exrop_state() |
              exs1024_state() |
              exs64_state() |
              dummy_state() |
              term().

# builtin_alg()

(since OTP 18.0)

    -type builtin_alg() :: exsss | exro928ss | exrop | exs1024s | exsp | exs64 | exsplus | exs1024 | dummy.

# dummy_state()

(since OTP 18.0)

    -type dummy_state() :: uint58().

Algorithm specific internal state

# export_state()

(since OTP 18.0)

    -type export_state() :: {alg(), alg_state()}.

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

# exro928_state()

(since OTP 18.0)

    -opaque exro928_state()

Algorithm specific internal state

# exrop_state()

(since OTP 18.0)

    -opaque exrop_state()

Algorithm specific internal state

# exs64_state()

(since OTP 18.0)

    -opaque exs64_state()

Algorithm specific internal state

# exs1024_state()

(since OTP 18.0)

    -opaque exs1024_state()

Algorithm specific internal state

# exsplus_state()

(since OTP 18.0)

    -opaque exsplus_state()

Algorithm specific internal state

# mwc59_state()

(since OTP 18.0)

    -type mwc59_state() :: 1..133850370 bsl 32 - 1 - 1.

`1 .. (16#1ffb072 bsl 29) - 2`

# seed()

(since OTP 18.0)

    -type seed() :: [integer()] | integer() | {integer(), integer(), integer()}.

Generator seed value.

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. The number of integers must be equal to the number of state words in the generator.

A single integer is used as the initial state for a SplitMix64 generator. The sequential 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.

# splitmix64_state()

(since OTP 18.0)

    -type splitmix64_state() :: uint64().

Algorithm specific state

# state()

(since OTP 18.0)

    -type state() :: {alg_handler(), alg_state()}.

Algorithm-dependent state.

# uint58()

(since OTP 18.0)

    -type uint58() :: 0..1 bsl 58 - 1.

`0 .. (2^58 - 1)`

# uint64()

(since OTP 18.0)

    -type uint64() :: 0..1 bsl 64 - 1.

`0 .. (2^64 - 1)`

# Plug-in framework API

# bytes(N)

(since OTP 24.0)

    -spec bytes(N :: non_neg_integer()) -> Bytes :: binary().

Generate random bytes as a `t:binary()`, using the state in the process dictionary.

Like `bytes_s/2` but operates on the state stored in the process dictionary. Returns the generated `Bytes`.

# bytes_s(N, State)

(since OTP 24.0)

    -spec bytes_s(N :: non_neg_integer(), State :: state()) -> {Bytes :: binary(), NewState :: state()}.

Generate random bytes as a `t:binary()`.

For a specified integer `N >= 0`, generates a `binary/0` with that number of random bytes.

The selected algorithm is used to generate as many random numbers as required to compose the `binary/0`. Returns the generated `Bytes` and a `NewState`.

# export_seed()

(since OTP 18.0)

    -spec export_seed() -> undefined | export_state().

Export the seed value.

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

# export_seed_s(State)

(since OTP 18.0)

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

Export the seed value.

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

# jump()

(since OTP 20.0)

    -spec jump() -> NewState :: state().

Jump the generator state forward.

Like `jump/1` but operates on the state stored in the process dictionary. Returns the `NewState`.

# jump(State)

(since OTP 20.0)

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

Jump the generator state forward.

Performs an algorithm specific `State` jump calculation that is equvalent to a large number of state iterations. See this module's algorithms list.

Returns the `NewState`.

This feature can be used to create many non-overlapping random number sequences from one start state.

This function raises a `not_implemented` error exception if there is no jump function implemented for the `State`'s algorithm.

# normal()

(since OTP 18.0)

    -spec normal() -> X :: float().

Generate a random number with standard normal distribution.

Like `normal_s/1` but operates on the state stored in the process dictionary. Returns the generated number `X`.

# normal(Mean, Variance)

(since OTP 20.0)

    -spec normal(Mean :: number(), Variance :: number()) -> X :: float().

Generate a random number with specified normal distribution 𝒩 (μ, σ²).

Like `normal_s/3` but operates on the state stored in the process dictionary. Returns the generated number `X`.

# normal_s(State)

(since OTP 18.0)

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

Generate a random number with standard normal distribution.

From the specified `State`, generates a random number `X ::` `float/0`, with standard normal distribution, that is with mean value `0.0` and variance `1.0`.

Returns the generated number `X` and the `NewState`.

# normal_s(Mean, Variance, State)

(since OTP 20.0)

    -spec normal_s(Mean, Variance, State) -> {X :: float(), NewState :: state()}
                      when Mean :: number(), Variance :: number(), State :: state().

Generate a random number with specified normal distribution 𝒩 (μ, σ²).

From the specified `State`, generates a random number `X ::` `float/0`, with normal distribution 𝒩 (μ, σ²), that is 𝒩 (Mean, Variance) where `Variance >= 0.0`.

Returns `X` and the `NewState`.

# seed(Alg_or_State)

(since OTP 18.0)

    -spec seed(Alg_or_State :: term()) -> state().

Seed the random number generator and select algorithm.

The same as `seed_s(Alg_or_State)`, but also stores the generated state in the process dictionary.

The argument `default` is an alias for the default algorithm that has been implemented (Since OTP 24.0).

# seed(Alg, Seed)

(since OTP 18.0)

    -spec seed(Alg :: term(), Seed :: term()) -> state().

Seed the random number generator and select algorithm.

The same as `seed_s(Alg, Seed)`, but also stores the generated state in the process dictionary.

`Alg = default` is an alias for the default algorithm that has been implemented (Since OTP 24.0).

# seed_s/1

(since OTP 18.0)

    -spec seed_s(Alg | State) -> state()
                    when Alg :: builtin_alg() | default, State :: state() | export_state().

Seed the random number generator and select algorithm.

With the argument `Alg`, select that algorithm and seed random number generation with reasonably unpredictable time dependent data.

`Alg = default` is an alias for the default algorithm (Since OTP 24.0).

With the argument `State`, re-creates the state and returns it. See also `export_seed/0`.

# seed_s(Alg, Seed)

(since OTP 18.0)

    -spec seed_s(Alg, Seed) -> state() when Alg :: builtin_alg() | default, Seed :: seed().

Seed the random number generator and select algorithm.

Creates and returns a generator state for the specified algorithm from the specified `seed/0` integers.

`Alg = default` is an alias for the default algorithm that has been implemented since OTP 24.0.

# uniform()

(since OTP 18.0)

    -spec uniform() -> X :: float().

Generate a uniformly distributed random number `0.0 =< X < 1.0`, using the state in the process dictionary.

Like `uniform_s/1` but operates on the state stored in the process dictionary. Returns the generated number `X`.

# uniform(N)

(since OTP 18.0)

    -spec uniform(N :: pos_integer()) -> X :: pos_integer().

Generate a uniformly distributed random integer `1 =< X =< N`, using the state in the process dictionary.

Like `uniform_s/2` but operates on the state stored in the process dictionary. Returns the generated number `X`.

# uniform_real()

(since OTP 21.0)

    -spec uniform_real() -> X :: float().

Generate a uniformly distributed random number `0.0 < X < 1.0`, using the state in the process dictionary.

Like `uniform_real_s/1` but operates on the state stored in the process dictionary. Returns the generated number `X`.

See `uniform_real_s/1`.

# uniform_real_s(State)

(since OTP 21.0)

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

Generate a uniformly distributed random number `0.0 < X < 1.0`.

From the specified state, generates a random float, uniformly distributed in the value range `DBL_MIN =< X < 1.0`.

Conceptually, a random real number `R` is generated from the interval `0.0 =< R < 1.0` and then the closest rounded down nonzero 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 algorithms 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 in fact never returns `0.0`. 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, and that this function never returns exactly `0.0` is impossible to observe.

For all sub ranges `N*2.0^(-53) =< X < (N+1)*2.0^(-53)` where `0 =< integer(N) < 2.0^53`, the probability to generate a number in the range is the same. Compare with the numbers generated by `uniform_s/1`.

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

# uniform_s(State)

(since OTP 18.0)

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

Generate a uniformly distributed random number `0.0 =< X < 1.0`.

From the specified `State`, generates a random number `X ::` `float/0`, uniformly distributed in the value range `0.0 =< X < 1.0`. Returns the number `X` and the updated `NewState`.

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 achieve the range `0.0 < X < 1.0` using test and re-try like this:

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

# uniform_s(N, State)

(since OTP 18.0)

    -spec uniform_s(N :: pos_integer(), State :: state()) -> {X :: pos_integer(), NewState :: state()}.

Generate a uniformly distributed random integer `1 =< X =< N`.

From the specified `State`, generates a random number `X ::` `integer/0`, uniformly distributed in the specified range `1 =< X =< N`. Returns the number `X` and the updated `NewState`.

# Niche algorithms API

# exsp_jump(AlgState)

(since OTP 25.0)

    -spec exsp_jump(AlgState :: exsplus_state()) -> NewAlgState :: exsplus_state().

Jump the generator state forward.

Performs a `State` jump calculation that is equvalent to a 2^64 state iterations.

Returns the `NewState`.

This feature can be used to create many non-overlapping random number sequences from one start state.

See the description of jump functions at the top of this module description.

See `exsp_next/1` about why this internal implementation function has been exposed.

# exsp_next(AlgState)

(since OTP 25.0)

    -spec exsp_next(AlgState :: exsplus_state()) -> {X :: uint58(), NewAlgState :: exsplus_state()}.

Generate an Xorshift116+ random integer and new algorithm state.

From the specified `AlgState`, generates a random 58-bit integer `X` and a new algorithm state `NewAlgState`, according to the Xorshift116+ algorithm.

This is an API function exposing the internal implementation of the `exsp` algorithm that enables using it without the overhead of the plug-in framework, which might be useful for time critial applications. On a typical 64 bit Erlang VM this approach executes in just above 30% (1/3) of the time for the default algorithm through this module's normal plug-in framework.

To seed this generator use `{_, AlgState} = rand:seed_s(exsp)` or `{_, AlgState} = rand:seed_s(exsp, Seed)` with a specific `Seed`.

#### Note

This function offers no help in generating a number on a selected range, nor in generating floating point numbers. It is easy to accidentally mess up the statistical properties of this generator or to loose the performance advantage when doing either. See the recepies at the start of this Niche algorithms API description.

Note also the caveat about weak low bits that this generator suffers from.

The generator is exported in this form primarily for performance reasons.

# mwc59(CX0)

(since OTP 25.0)

    -spec mwc59(CX0 :: mwc59_state()) -> CX1 :: mwc59_state().

Generate a new MWC59 state.

From the specified generator state `CX0` generate a new state `CX1`, according to a Multiply With Carry generator, which is an efficient implementation of a Multiplicative Congruential Generator with a power of 2 multiplier and a prime modulus.

This generator uses the multiplier `2^32` and the modulus `16#7fa6502 * 2^32 - 1`, which have been selected, in collaboration with Sebastiano Vigna, to avoid bignum operations and still get good statistical quality. It has been named "MWC59" and can be written as:

    C = CX0 bsr 32
    X = CX0 band ((1 bsl 32)-1))
    CX1 = 16#7fa6502 * X + C

Because the generator uses a multiplier that is a power of 2 it gets statistical flaws for collision tests and birthday spacings tests in 2 and 3 dimensions, and these caveats apply even when looking only at the MWC "digit", that is the low 32 bits (the multiplier) of the generator state. The higher bits of the state are worse.

The quality of the output value improves much by using a scrambler, instead of just taking the low bits. Function `mwc59_value32` is a fast scrambler that returns a decent 32-bit number. The slightly slower `mwc59_value` scrambler returns 59 bits of very good quality, and `mwc59_float` returns a `float/0` of very good quality.

The low bits of the base generator are surprisingly good, so the lowest 16 bits actually pass fairly strict PRNG tests, despite the generator's weaknesses that lie in the high bits of the 32-bit MWC "digit". It is recommended to use `rem` on the the generator state, or bit mask extracting the lowest bits to produce numbers in a range 16 bits or less. See the recepies at the start of this Niche algorithms API description.

On a typical 64 bit Erlang VM this generator executes in below 8% (1/13) of the time for the default algorithm in the plug-in framework API of this module. With the `mwc59_value32` scrambler the total time becomes 16% (1/6), and with `mwc59_value` it becomes 20% (1/5) of the time for the default algorithm. With `mwc59_float` the total time is 60% of the time for the default algorithm generating a `float/0`.

#### Note

This generator is a niche generator for high speed applications. It has a much shorter period than the default generator, which in itself is a quality concern, although when used with the value scramblers it passes strict PRNG tests. The generator is much faster than `exsp_next/1` but with a bit lower quality and much shorter period.

# mwc59_float(CX)

(since OTP 25.0)

    -spec mwc59_float(CX :: mwc59_state()) -> V :: float().

Calculate a scrambled `float/0` from a MWC59 state.

Returns a value `V ::` `float/0` from a generator state `CX`, in the range `0.0 =< V < 1.0` like for example `uniform_s/1`.

The generator state is scrambled as with `mwc59_value/1` before converted to a `float/0`.

# mwc59_seed()

(since OTP 25.0)

    -spec mwc59_seed() -> CX :: mwc59_state().

Create a MWC59 generator state.

Like `mwc59_seed/1` but it hashes the default seed value of `seed_s(atom())`.

# mwc59_seed(S)

(since OTP 25.0)

    -spec mwc59_seed(S :: 0..1 bsl 58 - 1) -> CX :: mwc59_state().

Create a MWC59 generator state.

Returns a generator state `CX`. The 58-bit seed value `S` is hashed to create the generator state, to avoid that similar seeds create similar sequences.

# mwc59_value32(CX)

(since OTP 25.0)

    -spec mwc59_value32(CX :: mwc59_state()) -> V :: 0..1 bsl 32 - 1.

Calculate a 32-bit scrambled value from a MWC59 state.

Returns a 32-bit value `V` from a generator state `CX`. The generator state is scrambled using an 8-bit xorshift which masks the statistical imperfecions of the base generator `mwc59` enough to produce numbers of decent quality. Still some problems in 2- and 3-dimensional birthday spacing and collision tests show through.

When using this scrambler it is in general better to use the high bits of the value than the low. The lowest 8 bits are of good quality and are passed right through from the base generator. They are combined with the next 8 in the xorshift making the low 16 good quality, but in the range 16..31 bits there are weaker bits that should not become high bits of the generated values.

Therefore it is in general safer to shift out low bits. See the recepies at the start of this Niche algorithms API description.

For a non power of 2 range less than about 16 bits (to not get too much bias and to avoid bignums) truncated multiplication can be used, that is: `(Range*V) bsr 32`, which is much faster than using `rem`.

# mwc59_value(CX)

(since OTP 25.0)

    -spec mwc59_value(CX :: mwc59_state()) -> V :: 0..1 bsl 59 - 1.

Calculate a 59-bit scrambled value from a MWC59 state.

Returns a 59-bit value `V` from a generator state `CX`. The generator state is scrambled using an 4-bit followed by a 27-bit xorshift, which masks the statistical imperfecions of the MWC59 base generator enough that all 59 bits are of very good quality.

Be careful to not accidentaly create a bignum when handling the value `V`.

It is in general general better to use the high bits from this scrambler than the low. See the recepies at the start of this Niche algorithms API description.

For a non power of 2 range less than about 29 bits (to not get too much bias and to avoid bignums) truncated multiplication can be used, which is much faster than using `rem`. Example for range 1'000'000'000; the range is 30 bits, we use 29 bits from the generator, adding up to 59 bits, which is not a bignum (on a 64-bit VM ): `(1000000000 * (V bsr (59-29))) bsr 29`.

# splitmix64_next(AlgState)

(since OTP 25.0)

    -spec splitmix64_next(AlgState :: integer()) -> {X :: uint64(), NewAlgState :: splitmix64_state()}.

Generate a SplitMix64 random 64-bit integer and new algorithm state.

From the specified `AlgState` generates a random 64-bit integer `X` and a new generator state `NewAlgState`, according to the SplitMix64 algorithm.

This generator is used internally in the `rand` module for seeding other generators since it is of a quite different breed which reduces the probability for creating an accidentally bad seed.