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%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2015-2022. All Rights Reserved.
%%
%% Licensed under the Apache License, Version 2.0 (the "License");
%% you may not use this file except in compliance with the License.
%% You may obtain a copy of the License at
%%
%% http://www.apache.org/licenses/LICENSE-2.0
%%
%% Unless required by applicable law or agreed to in writing, software
%% distributed under the License is distributed on an "AS IS" BASIS,
%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
%% See the License for the specific language governing permissions and
%% limitations under the License.
%%
%% %CopyrightEnd%
%%
%% =====================================================================
%% Multiple PRNG module for Erlang/OTP
%% Copyright (c) 2015-2016 Kenji Rikitake
%%
%% exrop (xoroshiro116+) added, statistical distribution
%% improvements and uniform_real added by the Erlang/OTP team 2017
%% =====================================================================
-module(rand).
-export([seed_s/1, seed_s/2, seed/1, seed/2,
export_seed/0, export_seed_s/1,
uniform/0, uniform/1, uniform_s/1, uniform_s/2,
uniform_real/0, uniform_real_s/1,
bytes/1, bytes_s/2,
jump/0, jump/1,
normal/0, normal/2, normal_s/1, normal_s/3
]).
%% Utilities
-export([exsp_next/1, exsp_jump/1, splitmix64_next/1, mcg35/1, lcg35/1]).
%% Test, dev and internal
-export([exro928_jump_2pow512/1, exro928_jump_2pow20/1,
exro928_seed/1, exro928_next/1, exro928_next_state/1,
format_jumpconst58/1, seed58/2]).
%% Debug
-export([make_float/3, float2str/1, bc64/1]).
-compile({inline, [exs64_next/1, exsp_next/1, exsss_next/1,
exs1024_next/1, exs1024_calc/2,
exro928_next_state/4,
exrop_next/1, exrop_next_s/2,
get_52/1, normal_kiwi/1]}).
-define(DEFAULT_ALG_HANDLER, exsss).
-define(SEED_DICT, rand_seed).
%% =====================================================================
%% Bit fiddling macros
%% =====================================================================
-define(BIT(Bits), (1 bsl (Bits))).
-define(MASK(Bits), (?BIT(Bits) - 1)).
-define(MASK(Bits, X), ((X) band ?MASK(Bits))).
-define(
BSL(Bits, X, N),
%% N is evaluated 2 times
(?MASK((Bits)-(N), (X)) bsl (N))).
-define(
ROTL(Bits, X, N),
%% Bits is evaluated 2 times
%% X is evaluated 2 times
%% N i evaluated 3 times
(?BSL((Bits), (X), (N)) bor ((X) bsr ((Bits)-(N))))).
-define(
BC(V, N),
bc((V), ?BIT((N) - 1), N)).
%%-define(TWO_POW_MINUS53, (math:pow(2, -53))).
-define(TWO_POW_MINUS53, 1.11022302462515657e-16).
%% =====================================================================
%% Types
%% =====================================================================
-type uint64() :: 0..?MASK(64).
-type uint58() :: 0..?MASK(58).
%% This depends on the algorithm handler function
-type alg_state() ::
exsplus_state() | exro928_state() | exrop_state() | exs1024_state() |
exs64_state() | dummy_state() | term().
%% This is the algorithm handling definition within this module,
%% and the type to use for plugins.
%%
%% The 'type' field must be recognized by the module that implements
%% the algorithm, to interpret an exported state.
%%
%% The 'bits' field indicates how many bits the integer
%% returned from 'next' has got, i.e 'next' shall return
%% an random integer in the range 0..(2^Bits - 1).
%% At least 55 bits is required for the floating point
%% producing fallbacks, but 56 bits would be more future proof.
%%
%% The fields 'next', 'uniform' and 'uniform_n'
%% implement the algorithm. If 'uniform' or 'uniform_n'
%% is not present there is a fallback using 'next' and either
%% 'bits' or the deprecated 'max'. The 'next' function
%% must generate a word with at least 56 good random bits.
%%
%% The 'weak_low_bits' field indicate how many bits are of
%% lesser quality and they will not be used by the floating point
%% producing functions, nor by the range producing functions
%% when more bits are needed, to avoid weak bits in the middle
%% of the generated bits. The lowest bits from the range
%% functions still have the generator's quality.
%%
-type alg_handler() ::
#{type := alg(),
bits => non_neg_integer(),
weak_low_bits => non_neg_integer(),
max => non_neg_integer(), % Deprecated
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())}.
%% Algorithm state
-type state() :: {alg_handler(), alg_state()}.
-type builtin_alg() ::
exsss | exro928ss | exrop | exs1024s | exsp | exs64 | exsplus |
exs1024 | dummy.
-type alg() :: builtin_alg() | atom().
-type export_state() :: {alg(), alg_state()}.
-type seed() :: [integer()] | integer() | {integer(), integer(), integer()}.
-export_type(
[builtin_alg/0, alg/0, alg_handler/0, alg_state/0,
state/0, export_state/0, seed/0]).
-export_type(
[exsplus_state/0, exro928_state/0, exrop_state/0, exs1024_state/0,
exs64_state/0, dummy_state/0]).
-export_type(
[uint58/0, uint64/0, splitmix64_state/0, mcg35_state/0, lcg35_state/0]).
%% =====================================================================
%% Range macro and helper
%% =====================================================================
-define(
uniform_range(Range, AlgHandler, R, V, MaxMinusRange, I),
if
0 =< (MaxMinusRange) ->
if
%% Really work saving in odd cases;
%% large ranges in particular
(V) < (Range) ->
{(V) + 1, {(AlgHandler), (R)}};
true ->
(I) = (V) rem (Range),
if
(V) - (I) =< (MaxMinusRange) ->
{(I) + 1, {(AlgHandler), (R)}};
true ->
%% V in the truncated top range
%% - try again
?FUNCTION_NAME((Range), {(AlgHandler), (R)})
end
end;
true ->
uniform_range((Range), (AlgHandler), (R), (V))
end).
%% For ranges larger than the algorithm bit size
uniform_range(Range, #{next:=Next, bits:=Bits} = AlgHandler, R, V) ->
WeakLowBits = maps:get(weak_low_bits, AlgHandler, 0),
%% Maybe waste the lowest bit(s) when shifting in new bits
Shift = Bits - WeakLowBits,
ShiftMask = bnot ?MASK(WeakLowBits),
RangeMinus1 = Range - 1,
if
(Range band RangeMinus1) =:= 0 -> % Power of 2
%% Generate at least the number of bits for the range
{V1, R1, _} =
uniform_range(
Range bsr Bits, Next, R, V, ShiftMask, Shift, Bits),
{(V1 band RangeMinus1) + 1, {AlgHandler, R1}};
true ->
%% Generate a value with at least two bits more than the range
%% and try that for a fit, otherwise recurse
%%
%% Just one bit more should ensure that the generated
%% number range is at least twice the size of the requested
%% range, which would make the probability to draw a good
%% number better than 0.5. And repeating that until
%% success i guess would take 2 times statistically amortized.
%% But since the probability for fairly many attemtpts
%% is not that low, use two bits more than the range which
%% should make the probability to draw a bad number under 0.25,
%% which decreases the bad case probability a lot.
{V1, R1, B} =
uniform_range(
Range bsr (Bits - 2), Next, R, V, ShiftMask, Shift, Bits),
I = V1 rem Range,
if
(V1 - I) =< (1 bsl B) - Range ->
{I + 1, {AlgHandler, R1}};
true ->
%% V1 drawn from the truncated top range
%% - try again
{V2, R2} = Next(R1),
uniform_range(Range, AlgHandler, R2, V2)
end
end.
%%
uniform_range(Range, Next, R, V, ShiftMask, Shift, B) ->
if
Range =< 1 ->
{V, R, B};
true ->
{V1, R1} = Next(R),
%% Waste the lowest bit(s) when shifting in new bits
uniform_range(
Range bsr Shift, Next, R1,
((V band ShiftMask) bsl Shift) bor V1,
ShiftMask, Shift, B + Shift)
end.
%% =====================================================================
%% API
%% =====================================================================
%% Return algorithm and seed so that RNG state can be recreated with seed/1
-spec export_seed() -> 'undefined' | export_state().
export_seed() ->
case get(?SEED_DICT) of
{#{type:=Alg}, AlgState} -> {Alg, AlgState};
_ -> undefined
end.
-spec export_seed_s(State :: state()) -> export_state().
export_seed_s({#{type:=Alg}, AlgState}) -> {Alg, AlgState}.
%% seed(Alg) seeds RNG with runtime dependent values
%% and return the NEW state
%% seed({Alg,AlgState}) setup RNG with a previously exported seed
%% and return the NEW state
-spec seed(
AlgOrStateOrExpState :: builtin_alg() | state() | export_state()) ->
state();
(Alg :: 'default') ->
state().
seed(Alg) ->
seed_put(seed_s(Alg)).
-spec seed_s(
AlgOrStateOrExpState :: builtin_alg() | state() | export_state()) ->
state();
(Alg :: 'default') ->
state().
seed_s({AlgHandler, _AlgState} = State) when is_map(AlgHandler) ->
State;
seed_s({Alg, AlgState}) when is_atom(Alg) ->
{AlgHandler,_SeedFun} = mk_alg(Alg),
{AlgHandler,AlgState};
seed_s(Alg) ->
seed_s(Alg, {erlang:phash2([{node(),self()}]),
erlang:system_time(),
erlang:unique_integer()}).
%% seed/2: seeds RNG with the algorithm and given values
%% and returns the NEW state.
-spec seed(Alg :: builtin_alg(), Seed :: seed()) -> state();
(Alg :: 'default', Seed :: seed()) -> state().
seed(Alg, Seed) ->
seed_put(seed_s(Alg, Seed)).
-spec seed_s(Alg :: builtin_alg(), Seed :: seed()) -> state();
(Alg :: 'default', Seed :: seed()) -> state().
seed_s(default, Seed) -> seed_s(?DEFAULT_ALG_HANDLER, Seed);
seed_s(Alg, Seed) ->
{AlgHandler,SeedFun} = mk_alg(Alg),
AlgState = SeedFun(Seed),
{AlgHandler,AlgState}.
%%% uniform/0, uniform/1, uniform_s/1, uniform_s/2 are all
%%% uniformly distributed random numbers.
%% uniform/0: returns a random float X where 0.0 =< X < 1.0,
%% updating the state in the process dictionary.
-spec uniform() -> X :: float().
uniform() ->
{X, State} = uniform_s(seed_get()),
_ = seed_put(State),
X.
%% uniform/1: given an integer N >= 1,
%% uniform/1 returns a random integer X where 1 =< X =< N,
%% updating the state in the process dictionary.
-spec uniform(N :: pos_integer()) -> X :: pos_integer().
uniform(N) ->
{X, State} = uniform_s(N, seed_get()),
_ = seed_put(State),
X.
%% uniform_s/1: given a state, uniform_s/1
%% returns a random float X where 0.0 =< X < 1.0,
%% and a new state.
-spec uniform_s(State :: state()) -> {X :: float(), NewState :: state()}.
uniform_s(State = {#{uniform:=Uniform}, _}) ->
Uniform(State);
uniform_s({#{bits:=Bits, next:=Next} = AlgHandler, R0}) ->
{V, R1} = Next(R0),
%% Produce floats on the form N * 2^(-53)
{(V bsr (Bits - 53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}};
uniform_s({#{max:=Max, next:=Next} = AlgHandler, R0}) ->
{V, R1} = Next(R0),
%% Old algorithm with non-uniform density
{V / (Max + 1), {AlgHandler, R1}}.
%% uniform_s/2: given an integer N >= 1 and a state, uniform_s/2
%% uniform_s/2 returns a random integer X where 1 =< X =< N,
%% and a new state.
-spec uniform_s(N :: pos_integer(), State :: state()) ->
{X :: pos_integer(), NewState :: state()}.
uniform_s(N, State = {#{uniform_n:=UniformN}, _})
when is_integer(N), 1 =< N ->
UniformN(N, State);
uniform_s(N, {#{bits:=Bits, next:=Next} = AlgHandler, R0})
when is_integer(N), 1 =< N ->
{V, R1} = Next(R0),
MaxMinusN = ?BIT(Bits) - N,
?uniform_range(N, AlgHandler, R1, V, MaxMinusN, I);
uniform_s(N, {#{max:=Max, next:=Next} = AlgHandler, R0})
when is_integer(N), 1 =< N ->
%% Old algorithm with skewed probability
%% and gap in ranges > Max
{V, R1} = Next(R0),
if
N =< Max ->
{(V rem N) + 1, {AlgHandler, R1}};
true ->
F = V / (Max + 1),
{trunc(F * N) + 1, {AlgHandler, R1}}
end.
%% uniform_real/0: returns a random float X where 0.0 < X =< 1.0,
%% updating the state in the process dictionary.
-spec uniform_real() -> X :: float().
uniform_real() ->
{X, Seed} = uniform_real_s(seed_get()),
_ = seed_put(Seed),
X.
%% uniform_real_s/1: given a state, uniform_s/1
%% returns a random float X where 0.0 < X =< 1.0,
%% and a new state.
%%
%% This function does not use the same form of uniformity
%% as the uniform_s/1 function.
%%
%% Instead, this function does not generate numbers with equal
%% distance in the interval, but rather tries to keep all mantissa
%% bits random also for small numbers, meaning that the distance
%% between possible numbers decreases when the numbers
%% approaches 0.0, as does the possibility for a particular
%% number. Hence uniformity is preserved.
%%
%% To generate 56 bits at the time instead of 53 is actually
%% a speed optimization since the probability to have to
%% generate a second word decreases by 1/2 for every extra bit.
%%
%% This function generates normalized numbers, so the smallest number
%% that can be generated is 2^-1022 with the distance 2^-1074
%% to the next to smallest number, compared to 2^-53 for uniform_s/1.
%%
%% This concept of uniformity should work better for applications
%% where you need to calculate 1.0/X or math:log(X) since those
%% operations benefits from larger precision approaching 0.0,
%% and that this function does not return 0.0 nor denormalized
%% numbers very close to 0.0. The log() operation in The Box-Muller
%% transformation for normal distribution is an example of this.
%%
%%-define(TWO_POW_MINUS55, (math:pow(2, -55))).
%%-define(TWO_POW_MINUS110, (math:pow(2, -110))).
%%-define(TWO_POW_MINUS55, 2.7755575615628914e-17).
%%-define(TWO_POW_MINUS110, 7.7037197775489436e-34).
%%
-spec uniform_real_s(State :: state()) -> {X :: float(), NewState :: state()}.
uniform_real_s({#{bits:=Bits, next:=Next} = AlgHandler, R0}) ->
%% Generate a 56 bit number without using the weak low bits.
%%
%% Be sure to use only 53 bits when multiplying with
%% math:pow(2.0, -N) to avoid rounding which would make
%% "even" floats more probable than "odd".
%%
{V1, R1} = Next(R0),
M1 = V1 bsr (Bits - 56),
if
?BIT(55) =< M1 ->
%% We have 56 bits - waste 3
{(M1 bsr 3) * math:pow(2.0, -53), {AlgHandler, R1}};
?BIT(54) =< M1 ->
%% We have 55 bits - waste 2
{(M1 bsr 2) * math:pow(2.0, -54), {AlgHandler, R1}};
?BIT(53) =< M1 ->
%% We have 54 bits - waste 1
{(M1 bsr 1) * math:pow(2.0, -55), {AlgHandler, R1}};
?BIT(52) =< M1 ->
%% We have 53 bits - use all
{M1 * math:pow(2.0, -56), {AlgHandler, R1}};
true ->
%% Need more bits
{V2, R2} = Next(R1),
uniform_real_s(AlgHandler, Next, M1, -56, R2, V2, Bits)
end;
uniform_real_s({#{max:=_, next:=Next} = AlgHandler, R0}) ->
%% Generate a 56 bit number.
%% Ignore the weak low bits for these old algorithms,
%% just produce something reasonable.
%%
%% Be sure to use only 53 bits when multiplying with
%% math:pow(2.0, -N) to avoid rounding which would make
%% "even" floats more probable than "odd".
%%
{V1, R1} = Next(R0),
M1 = ?MASK(56, V1),
if
?BIT(55) =< M1 ->
%% We have 56 bits - waste 3
{(M1 bsr 3) * math:pow(2.0, -53), {AlgHandler, R1}};
?BIT(54) =< M1 ->
%% We have 55 bits - waste 2
{(M1 bsr 2) * math:pow(2.0, -54), {AlgHandler, R1}};
?BIT(53) =< M1 ->
%% We have 54 bits - waste 1
{(M1 bsr 1) * math:pow(2.0, -55), {AlgHandler, R1}};
?BIT(52) =< M1 ->
%% We have 53 bits - use all
{M1 * math:pow(2.0, -56), {AlgHandler, R1}};
true ->
%% Need more bits
{V2, R2} = Next(R1),
uniform_real_s(AlgHandler, Next, M1, -56, R2, V2, 56)
end.
uniform_real_s(AlgHandler, _Next, M0, -1064, R1, V1, Bits) -> % 19*56
%% This is a very theoretical bottom case.
%% The odds of getting here is about 2^-1008,
%% through a white box test case, or thanks to
%% a malfunctioning PRNG producing 18 56-bit zeros in a row.
%%
%% Fill up to 53 bits, we have at most 52
B0 = (53 - ?BC(M0, 52)), % Missing bits
{(((M0 bsl B0) bor (V1 bsr (Bits - B0))) * math:pow(2.0, -1064 - B0)),
{AlgHandler, R1}};
uniform_real_s(AlgHandler, Next, M0, BitNo, R1, V1, Bits) ->
if
%% Optimize the most probable.
%% Fill up to 53 bits.
?BIT(51) =< M0 ->
%% We have 52 bits in M0 - need 1
{(((M0 bsl 1) bor (V1 bsr (Bits - 1)))
* math:pow(2.0, BitNo - 1)),
{AlgHandler, R1}};
?BIT(50) =< M0 ->
%% We have 51 bits in M0 - need 2
{(((M0 bsl 2) bor (V1 bsr (Bits - 2)))
* math:pow(2.0, BitNo - 2)),
{AlgHandler, R1}};
?BIT(49) =< M0 ->
%% We have 50 bits in M0 - need 3
{(((M0 bsl 3) bor (V1 bsr (Bits - 3)))
* math:pow(2.0, BitNo - 3)),
{AlgHandler, R1}};
M0 == 0 ->
M1 = V1 bsr (Bits - 56),
if
?BIT(55) =< M1 ->
%% We have 56 bits - waste 3
{(M1 bsr 3) * math:pow(2.0, BitNo - 53),
{AlgHandler, R1}};
?BIT(54) =< M1 ->
%% We have 55 bits - waste 2
{(M1 bsr 2) * math:pow(2.0, BitNo - 54),
{AlgHandler, R1}};
?BIT(53) =< M1 ->
%% We have 54 bits - waste 1
{(M1 bsr 1) * math:pow(2.0, BitNo - 55),
{AlgHandler, R1}};
?BIT(52) =< M1 ->
%% We have 53 bits - use all
{M1 * math:pow(2.0, BitNo - 56),
{AlgHandler, R1}};
BitNo =:= -1008 ->
%% Endgame
%% For the last round we cannot have 14 zeros or more
%% at the top of M1 because then we will underflow,
%% so we need at least 43 bits
if
?BIT(42) =< M1 ->
%% We have 43 bits - get the last bits
uniform_real_s(
AlgHandler, Next, M1, BitNo - 56, R1);
true ->
%% Would underflow 2^-1022 - start all over
%%
%% We could just crash here since the odds for
%% the PRNG being broken is much higher than
%% for a good PRNG generating this many zeros
%% in a row. Maybe we should write an error
%% report or call this a system limit...?
uniform_real_s({AlgHandler, R1})
end;
true ->
%% Need more bits
uniform_real_s(AlgHandler, Next, M1, BitNo - 56, R1)
end;
true ->
%% Fill up to 53 bits
B0 = 53 - ?BC(M0, 49), % Number of bits we need to append
{(((M0 bsl B0) bor (V1 bsr (Bits - B0)))
* math:pow(2.0, BitNo - B0)),
{AlgHandler, R1}}
end.
%%
uniform_real_s(#{bits:=Bits} = AlgHandler, Next, M0, BitNo, R0) ->
{V1, R1} = Next(R0),
uniform_real_s(AlgHandler, Next, M0, BitNo, R1, V1, Bits);
uniform_real_s(#{max:=_} = AlgHandler, Next, M0, BitNo, R0) ->
{V1, R1} = Next(R0),
uniform_real_s(AlgHandler, Next, M0, BitNo, R1, ?MASK(56, V1), 56).
%% bytes/1: given a number N,
%% returns a random binary with N bytes
-spec bytes(N :: non_neg_integer()) -> Bytes :: binary().
bytes(N) ->
{Bytes, State} = bytes_s(N, seed_get()),
_ = seed_put(State),
Bytes.
%% bytes_s/2: given a number N and a state,
%% returns a random binary with N bytes and a new state
-spec bytes_s(N :: non_neg_integer(), State :: state()) ->
{Bytes :: binary(), NewState :: state()}.
bytes_s(N, {#{bits:=Bits, next:=Next} = AlgHandler, R})
when is_integer(N), 0 =< N ->
WeakLowBits = maps:get(weak_low_bits, AlgHandler, 0),
bytes_r(N, AlgHandler, Next, R, Bits, WeakLowBits);
bytes_s(N, {#{max:=Mask, next:=Next} = AlgHandler, R})
when is_integer(N), 0 =< N, ?MASK(58) =< Mask ->
%% Old spec - assume 58 bits and 2 weak low bits
%% giving 56 bits i.e precisely 7 bytes per generated number
Bits = 58,
WeakLowBits = 2,
bytes_r(N, AlgHandler, Next, R, Bits, WeakLowBits).
%% N: Number of bytes to generate
%% Bits: Number of bits in the generated word
%% WeakLowBits: Number of low bits in the generated word
%% to waste due to poor quality
bytes_r(N, AlgHandler, Next, R, Bits, WeakLowBits) ->
%% We use whole bytes from each generator word,
%% GoodBytes: that number of bytes
GoodBytes = (Bits - WeakLowBits) bsr 3,
GoodBits = GoodBytes bsl 3,
%% Shift: how many bits of each generator word to waste
%% by shifting right - we use the bits from the big end
Shift = Bits - GoodBits,
bytes_r(N, AlgHandler, Next, R, <<>>, GoodBytes, GoodBits, Shift).
%%
bytes_r(N0, AlgHandler, Next, R0, Bytes0, GoodBytes, GoodBits, Shift)
when (GoodBytes bsl 2) < N0 ->
%% Loop unroll 4 iterations
%% - gives about 25% shorter time for large binaries
{V1, R1} = Next(R0),
{V2, R2} = Next(R1),
{V3, R3} = Next(R2),
{V4, R4} = Next(R3),
Bytes1 =
<<Bytes0/binary,
(V1 bsr Shift):GoodBits,
(V2 bsr Shift):GoodBits,
(V3 bsr Shift):GoodBits,
(V4 bsr Shift):GoodBits>>,
N1 = N0 - (GoodBytes bsl 2),
bytes_r(N1, AlgHandler, Next, R4, Bytes1, GoodBytes, GoodBits, Shift);
bytes_r(N0, AlgHandler, Next, R0, Bytes0, GoodBytes, GoodBits, Shift)
when GoodBytes < N0 ->
{V, R1} = Next(R0),
Bytes1 = <<Bytes0/binary, (V bsr Shift):GoodBits>>,
N1 = N0 - GoodBytes,
bytes_r(N1, AlgHandler, Next, R1, Bytes1, GoodBytes, GoodBits, Shift);
bytes_r(N, AlgHandler, Next, R0, Bytes, _GoodBytes, GoodBits, _Shift) ->
{V, R1} = Next(R0),
Bits = N bsl 3,
%% Use the big end bits
Shift = GoodBits - Bits,
{<<Bytes/binary, (V bsr Shift):Bits>>, {AlgHandler, R1}}.
%% jump/1: given a state, jump/1
%% returns a new state which is equivalent to that
%% after a large number of call defined for each algorithm.
%% The large number is algorithm dependent.
-spec jump(state()) -> NewState :: state().
jump(State = {#{jump:=Jump}, _}) ->
Jump(State);
jump({#{}, _}) ->
erlang:error(not_implemented).
%% jump/0: read the internal state and
%% apply the jump function for the state as in jump/1
%% and write back the new value to the internal state,
%% then returns the new value.
-spec jump() -> NewState :: state().
jump() ->
seed_put(jump(seed_get())).
%% normal/0: returns a random float with standard normal distribution
%% updating the state in the process dictionary.
-spec normal() -> float().
normal() ->
{X, Seed} = normal_s(seed_get()),
_ = seed_put(Seed),
X.
%% normal/2: returns a random float with N(μ, σ²) normal distribution
%% updating the state in the process dictionary.
-spec normal(Mean :: number(), Variance :: number()) -> float().
normal(Mean, Variance) ->
Mean + (math:sqrt(Variance) * normal()).
%% normal_s/1: returns a random float with standard normal distribution
%% The Ziggurat Method for generating random variables - Marsaglia and Tsang
%% Paper and reference code: http://www.jstatsoft.org/v05/i08/
-spec normal_s(State :: state()) -> {float(), NewState :: state()}.
normal_s(State0) ->
{Sign, R, State} = get_52(State0),
Idx = ?MASK(8, R),
Idx1 = Idx+1,
{Ki, Wi} = normal_kiwi(Idx1),
X = R * Wi,
case R < Ki of
%% Fast path 95% of the time
true when Sign =:= 0 -> {X, State};
true -> {-X, State};
%% Slow path
false when Sign =:= 0 -> normal_s(Idx, Sign, X, State);
false -> normal_s(Idx, Sign, -X, State)
end.
%% normal_s/3: returns a random float with normal N(μ, σ²) distribution
-spec normal_s(Mean :: number(), Variance :: number(), state()) -> {float(), NewS :: state()}.
normal_s(Mean, Variance, State0) when Variance > 0 ->
{X, State} = normal_s(State0),
{Mean + (math:sqrt(Variance) * X), State}.
%% =====================================================================
%% Internal functions
-spec seed_put(state()) -> state().
seed_put(Seed) ->
put(?SEED_DICT, Seed),
Seed.
seed_get() ->
case get(?SEED_DICT) of
undefined -> seed(?DEFAULT_ALG_HANDLER);
Old -> Old % no type checking here
end.
%% Setup alg record
mk_alg(exs64) ->
{#{type=>exs64, max=>?MASK(64), next=>fun exs64_next/1},
fun exs64_seed/1};
mk_alg(exsplus) ->
{#{type=>exsplus, max=>?MASK(58), next=>fun exsp_next/1,
jump=>fun exsplus_jump/1},
fun exsplus_seed/1};
mk_alg(exsp) ->
{#{type=>exsp, bits=>58, weak_low_bits=>1, next=>fun exsp_next/1,
uniform=>fun exsp_uniform/1, uniform_n=>fun exsp_uniform/2,
jump=>fun exsplus_jump/1},
fun exsplus_seed/1};
mk_alg(exsss) ->
{#{type=>exsss, bits=>58, next=>fun exsss_next/1,
uniform=>fun exsss_uniform/1, uniform_n=>fun exsss_uniform/2,
jump=>fun exsplus_jump/1},
fun exsss_seed/1};
mk_alg(exs1024) ->
{#{type=>exs1024, max=>?MASK(64), next=>fun exs1024_next/1,
jump=>fun exs1024_jump/1},
fun exs1024_seed/1};
mk_alg(exs1024s) ->
{#{type=>exs1024s, bits=>64, weak_low_bits=>3, next=>fun exs1024_next/1,
jump=>fun exs1024_jump/1},
fun exs1024_seed/1};
mk_alg(exrop) ->
{#{type=>exrop, bits=>58, weak_low_bits=>1, next=>fun exrop_next/1,
uniform=>fun exrop_uniform/1, uniform_n=>fun exrop_uniform/2,
jump=>fun exrop_jump/1},
fun exrop_seed/1};
mk_alg(exro928ss) ->
{#{type=>exro928ss, bits=>58, next=>fun exro928ss_next/1,
uniform=>fun exro928ss_uniform/1,
uniform_n=>fun exro928ss_uniform/2,
jump=>fun exro928_jump/1},
fun exro928_seed/1};
mk_alg(dummy=Name) ->
{#{type=>Name, bits=>58, next=>fun dummy_next/1,
uniform=>fun dummy_uniform/1,
uniform_n=>fun dummy_uniform/2},
fun dummy_seed/1}.
%% =====================================================================
%% exs64 PRNG: Xorshift64*
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================
-opaque exs64_state() :: uint64().
exs64_seed(L) when is_list(L) ->
[R] = seed64_nz(1, L),
R;
exs64_seed(A) when is_integer(A) ->
[R] = seed64(1, A),
R;
%%
%% Traditional integer triplet seed
exs64_seed({A1, A2, A3}) ->
{V1, _} = exs64_next((?MASK(32, A1) * 4294967197 + 1)),
{V2, _} = exs64_next((?MASK(32, A2) * 4294967231 + 1)),
{V3, _} = exs64_next((?MASK(32, A3) * 4294967279 + 1)),
((V1 * V2 * V3) rem (?MASK(64) - 1)) + 1.
%% Advance xorshift64* state for one step and generate 64bit unsigned integer
-spec exs64_next(exs64_state()) -> {uint64(), exs64_state()}.
exs64_next(R) ->
R1 = R bxor (R bsr 12),
R2 = R1 bxor ?BSL(64, R1, 25),
R3 = R2 bxor (R2 bsr 27),
{?MASK(64, R3 * 2685821657736338717), R3}.
%% =====================================================================
%% exsplus PRNG: Xorshift116+
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% 58 bits fits into an immediate on 64bits erlang and is thus much faster.
%% Modification of the original Xorshift128+ algorithm to 116
%% by Sebastiano Vigna, a lot of thanks for his help and work.
%%
%% Reference C code for Xorshift116+ and Xorshift116**
%%
%% #include <stdint.h>
%%
%% #define MASK(b, v) (((UINT64_C(1) << (b)) - 1
104FF
) & (v))
%% #define BSL(b, v, n) (MASK((b)-(n), (v)) << (n))
%% #define ROTL(b, v, n) (BSL((b), (v), (n)) | ((v) >> ((b)-(n))))
%%
%% uint64_t s[2];
%%
%% uint64_t next(void) {
%% uint64_t s1 = s[0];
%% const uint64_t s0 = s[1];
%%
%% s1 ^= BSL(58, s1, 24); // a
%% s1 ^= s0 ^ (s1 >> 11) ^ (s0 >> 41); // b, c
%% s[0] = s0;
%% s[1] = s1;
%%
%% const uint64_t result_plus = MASK(58, s0 + s1);
%% uint64_t result_starstar = s0;
%% result_starstar = MASK(58, result_starstar * 5);
%% result_starstar = ROTL(58, result_starstar, 7);
%% result_starstar = MASK(58, result_starstar * 9);
%%
%% return result_plus;
%% return result_starstar;
%% }
%%
%% =====================================================================
-opaque exsplus_state() :: nonempty_improper_list(uint58(), uint58()).
-dialyzer({no_improper_lists, exsplus_seed/1}).
exsplus_seed(L) when is_list(L) ->
[S0,S1] = seed58_nz(2, L),
[S0|S1];
exsplus_seed(X) when is_integer(X) ->
[S0,S1] = seed58(2, X),
[S0|S1];
%%
%% Traditional integer triplet seed
exsplus_seed({A1, A2, A3}) ->
{_, R1} = exsp_next(
[?MASK(58, (A1 * 4294967197) + 1)|
?MASK(58, (A2 * 4294967231) + 1)]),
{_, R2} = exsp_next(
[?MASK(58, (A3 * 4294967279) + 1)|
tl(R1)]),
R2.
-dialyzer({no_improper_lists, exsss_seed/1}).
exsss_seed(L) when is_list(L) ->
[S0,S1] = seed58_nz(2, L),
[S0|S1];
exsss_seed(X) when is_integer(X) ->
[S0,S1] = seed58(2, X),
[S0|S1];
%%
%% Seed from traditional integer triple - mix into splitmix
exsss_seed({A1, A2, A3}) ->
{_, X0} = seed58(A1),
{S0, X1} = seed58(A2 bxor X0),
{S1, _} = seed58(A3 bxor X1),
[S0|S1].
%% Advance Xorshift116 state one step
-define(
exs_next(S0, S1, S1_b),
begin
S1_b = ?MASK(58, S1) bxor ?BSL(58, S1, 24),
S1_b bxor S0 bxor (S1_b bsr 11) bxor (S0 bsr 41)
end).
-define(
scramble_starstar(S, V_a, V_b),
begin
%% The multiply by add shifted trick avoids creating bignums
%% which improves performance significantly
%%
%% Scramble ** (all operations modulo word size)
%% ((S * 5) rotl 7) * 9
%%
V_a = S + ?BSL(58, S, 2), % * 5
V_b = ?BSL(58, V_a, 7) bor ?MASK(7, V_a bsr (58-7)), % rotl 7
?MASK(58, V_b + ?BSL(58, V_b, 3)) % * 9
end).
%% Advance state and generate 58bit unsigned integer
%%
-dialyzer({no_improper_lists, exsp_next/1}).
-spec exsp_next(AlgState :: exsplus_state()) ->
{X :: uint58(), NewAlgState :: exsplus_state()}.
exsp_next([S1|S0]) ->
%% Note: members s0 and s1 are swapped here
S0_1 = ?MASK(58, S0),
NewS1 = ?exs_next(S0_1, S1, S1_b),
%% Scramble + (all operations modulo word size)
%% S0 + NewS1
{?MASK(58, S0_1 + NewS1), [S0_1|NewS1]}.
-dialyzer({no_improper_lists, exsss_next/1}).
-spec exsss_next(exsplus_state()) -> {uint58(), exsplus_state()}.
exsss_next([S1|S0]) ->
%% Note: members s0 and s1 are swapped here
S0_1 = ?MASK(58, S0),
NewS1 = ?exs_next(S0_1, S1, S1_b),
{?scramble_starstar(S0_1, V_1, V_2), [S0_1|NewS1]}.
exsp_uniform({AlgHandler, R0}) ->
{I, R1} = exsp_next(R0),
%% Waste the lowest bit since it is of lower
%% randomness quality than the others
{(I bsr (58-53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}}.
exsss_uniform({AlgHandler, R0}) ->
{I, R1} = exsss_next(R0),
{(I bsr (58-53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}}.
exsp_uniform(Range, {AlgHandler, R}) ->
{V, R1} = exsp_next(R),
MaxMinusRange = ?BIT(58) - Range,
?uniform_range(Range, AlgHandler, R1, V, MaxMinusRange, I).
exsss_uniform(Range, {AlgHandler, R}) ->
{V, R1} = exsss_next(R),
MaxMinusRange = ?BIT(58) - Range,
?uniform_range(Range, AlgHandler, R1, V, MaxMinusRange, I).
%% This is the jump function for the exs... generators,
%% i.e the Xorshift116 generators, equivalent
%% to 2^64 calls to next/1; it can be used to generate 2^52
%% non-overlapping subsequences for parallel computations.
%% Note: the jump function takes 116 times of the execution time of
%% next/1.
%%
%% #include <stdint.h>
%%
%% void jump(void) {
%% static const uint64_t JUMP[] = { 0x02f8ea6bc32c797,
%% 0x345d2a0f85f788c };
%% int i, b;
%% uint64_t s0 = 0;
%% uint64_t s1 = 0;
%% for(i = 0; i < sizeof JUMP / sizeof *JUMP; i++)
%% for(b = 0; b < 58; b++) {
%% if (JUMP[i] & 1ULL << b) {
%% s0 ^= s[0];
%% s1 ^= s[1];
%% }
%% next();
%% }
%% s[0] = s0;
%% s[1] = s1;
%% }
%%
%% -define(JUMPCONST, 16#000d174a83e17de2302f8ea6bc32c797).
%% split into 58-bit chunks
%% and two iterative executions
-define(JUMPCONST1, 16#02f8ea6bc32c797).
-define(JUMPCONST2, 16#345d2a0f85f788c).
-define(JUMPELEMLEN, 58).
-dialyzer({no_improper_lists, exsplus_jump/1}).
-spec exsplus_jump({alg_handler(), exsplus_state()}) ->
{alg_handler(), exsplus_state()}.
exsplus_jump({AlgHandler, S}) ->
{AlgHandler, exsp_jump(S)}.
-dialyzer({no_improper_lists, exsp_jump/1}).
-spec exsp_jump(AlgState :: exsplus_state()) ->
NewAlgState :: exsplus_state().
exsp_jump(S) ->
{S1, AS1} = exsplus_jump(S, [0|0], ?JUMPCONST1, ?JUMPELEMLEN),
{_, AS2} = exsplus_jump(S1, AS1, ?JUMPCONST2, ?JUMPELEMLEN),
AS2.
-dialyzer({no_improper_lists, exsplus_jump/4}).
exsplus_jump(S, AS, _, 0) ->
{S, AS};
exsplus_jump(S, [AS0|AS1], J, N) ->
{_, NS} = exsp_next(S),
case ?MASK(1, J) of
1 ->
[S0|S1] = S,
exsplus_jump(NS, [(AS0 bxor S0)|(AS1 bxor S1)], J bsr 1, N-1);
0 ->
exsplus_jump(NS, [AS0|AS1], J bsr 1, N-1)
end.
%% =====================================================================
%% exs1024 PRNG: Xorshift1024*
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================
-opaque exs1024_state() :: {list(uint64()), list(uint64())}.
exs1024_seed(L) when is_list(L) ->
{seed64_nz(16, L), []};
exs1024_seed(X) when is_integer(X) ->
{seed64(16, X), []};
%%
%% Seed from traditional triple, remain backwards compatible
exs1024_seed({A1, A2, A3}) ->
B1 = ?MASK(21, (?MASK(21, A1) + 1) * 2097131),
B2 = ?MASK(21, (?MASK(21, A2) + 1) * 2097133),
B3 = ?MASK(21, (?MASK(21, A3) + 1) * 2097143),
{exs1024_gen1024((B1 bsl 43) bor (B2 bsl 22) bor (B3 bsl 1) bor 1),
[]}.
%% Generate a list of 16 64-bit element list
%% of the xorshift64* random sequence
%% from a given 64-bit seed.
%% Note: dependent on exs64_next/1
-spec exs1024_gen1024(uint64()) -> list(uint64()).