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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superseded by Xoroshiro116+ (exrop).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.