rand - Pseudo random number generation.
Please see following description for synopsis
rand(3) Erlang Module Definition rand(3)
NAME
rand - Pseudo random number generation.
DESCRIPTION
This module provides a pseudo random number generator. The module con-
tains 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 scram-
bler 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 scram-
bler 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 quali-
ties.
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 defaultexsssalgorithm 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 adap-
tion.
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that now has been superseded by Xoroshiro116+ (exrop).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 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.
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 undoc-
umented algorithms, which is why they are deprecated. The new algo-
rithms 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 preci-
sion 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 set-
ting 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 cryptographi-
cally 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 explana-
tion 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. The number of integers must be equal to the
number of state words in the generator.
An integer is used as the initial state for a SplitMix64 genera-
tor. The output values of that is then used for setting the gen-
erator'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 algo-
rithm-dependent hashing functions to create the generator's ini-
tial 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
bytes(N :: integer() >= 0) -> Bytes :: binary()
Returns, for a specified integer N >= 0, a binary() with that
number of random bytes. Generates as many random numbers as
required using the selected algorithm to compose the binary, and
updates the state in the process dictionary accordingly.
bytes_s(N :: integer() >= 0, State :: state()) ->
{Bytes :: binary(), NewState :: state()}
Returns, for a specified integer N >= 0 and a state, a binary()
with that number of random bytes, and a new state. Generates as
many random numbers as required using the selected algorithm to
compose the binary, and the new state.
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) devi-
ate float and a new state.
seed(AlgOrStateOrExpState ::
builtin_alg() | state() | export_state()) ->
state()
seed(Alg :: default) -> state()
Seeds random number generation with the specifed algorithm and
time-dependent data if AlgOrStateOrExpState is an algorithm. Alg
= default is an alias for the default 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()
seed(Alg :: default, Seed :: seed()) -> state()
Seeds random number generation with the specified algorithm and
integers in the process dictionary and returns the state. Alg =
default is an alias for the default algorithm.
seed_s(AlgOrStateOrExpState ::
builtin_alg() | state() | export_state()) ->
state()
seed_s(Alg :: default) -> state()
Seeds random number generation with the specifed algorithm and
time-dependent data if AlgOrStateOrExpState is an algorithm. Alg
= default is an alias for the default algorithm.
Otherwise recreates the exported seed and returns the state. See
also export_seed/0.
seed_s(Alg :: builtin_alg(), Seed :: seed()) -> state()
seed_s(Alg :: default, Seed :: seed()) -> state()
Seeds random number generation with the specified algorithm and
integers and returns the state. Alg = default is an alias for
the default algorithm.
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 cer-
tain 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 dictio-
nary.
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 granu-
larity 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 uni-
formly 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 distrib-
uted 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 cer-
tain 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 dis-
tributed 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 granu-
larity for small numbers than the regular uniform_s/1 because
all bits in the mantissa are random. This property, in combina-
tion 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 guaran-
teed 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 descrip-
tion 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.
Ericsson AB stdlib 3.17 rand(3)