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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 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.

   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(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() =
           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

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, and returns the
              state. See also export_seed/0.

       seed(Alg :: builtin_alg(), Seed :: seed()) -> 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 :: seed()) -> 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 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.

       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.