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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 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superseded by Xoroshiro116+ (exrop).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 or uniform/1 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(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),
Create a standard normal deviate:
{SND0, 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, and returns the
state. See also export_seed/0.
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.
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(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.
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_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.