%% %% %CopyrightBegin% %% %% Copyright Ericsson AB 2015-2017. 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 and statistical distribution %% improvements 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, jump/0, jump/1, normal/0, normal_s/1 ]). -compile({inline, [exs64_next/1, exsplus_next/1, exs1024_next/1, exs1024_calc/2, exrop_next/1, exrop_next_s/2, get_52/1, normal_kiwi/1]}). -define(DEFAULT_ALG_HANDLER, exrop). -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(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() :: exs64_state() | exsplus_state() | exs1024_state() | exrop_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 53 bits is required for the floating point %% producing fallbacks. This field is only used when %% the 'uniform' or 'uniform_n' fields are not defined. %% %% The fields 'next', 'uniform' and 'uniform_n' %% implement the algorithm. If 'uniform' or 'uinform_n' %% is not present there is a fallback using 'next' and either %% 'bits' or the deprecated 'max'. %% -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() :: exs64 | exsplus | exsp | exs1024 | exs1024s | exrop. -type alg() :: builtin_alg() | atom(). -type export_state() :: {alg(), alg_state()}. -export_type( [builtin_alg/0, alg/0, alg_handler/0, alg_state/0, state/0, export_state/0]). -export_type( [exs64_state/0, exsplus_state/0, exs1024_state/0, exrop_state/0]). %% ===================================================================== %% Range macro and helper %% ===================================================================== -define( uniform_range(Range, Alg, R, V, MaxMinusRange, I), if 0 =< (MaxMinusRange) -> if %% Really work saving in odd cases; %% large ranges in particular (V) < (Range) -> {(V) + 1, {(Alg), (R)}}; true -> (I) = (V) rem (Range), if (V) - (I) =< (MaxMinusRange) -> {(I) + 1, {(Alg), (R)}}; true -> %% V in the truncated top range %% - try again ?FUNCTION_NAME((Range), {(Alg), (R)}) end end; true -> uniform_range((Range), (Alg), (R), (V)) end). %% For ranges larger than the algorithm bit size uniform_range(Range, #{next:=Next, bits:=Bits} = Alg, R, V) -> WeakLowBits = case Alg of #{weak_low_bits:=WLB} -> WLB; #{} -> 0 end, %% 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, {Alg, 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, {Alg, R1}}; true -> %% V1 drawn from the truncated top range %% - try again {V2, R2} = Next(R1), uniform_range(Range, Alg, 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}, Seed} -> {Alg, Seed}; _ -> undefined end. -spec export_seed_s(State :: state()) -> export_state(). export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}. %% seed(Alg) seeds RNG with runtime dependent values %% and return the NEW state %% seed({Alg,Seed}) setup RNG with a previously exported seed %% and return the NEW state -spec seed( AlgOrStateOrExpState :: builtin_alg() | state() | export_state()) -> state(). seed(Alg) -> seed_put(seed_s(Alg)). -spec seed_s( AlgOrStateOrExpState :: builtin_alg() | state() | export_state()) -> state(). seed_s({AlgHandler, _Seed} = State) when is_map(AlgHandler) -> State; seed_s({Alg0, Seed}) -> {Alg,_SeedFun} = mk_alg(Alg0), {Alg, Seed}; 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 :: {integer(), integer(), integer()}) -> state(). seed(Alg0, S0) -> seed_put(seed_s(Alg0, S0)). -spec seed_s( Alg :: builtin_alg(), Seed :: {integer(), integer(), integer()}) -> state(). seed_s(Alg0, S0 = {_, _, _}) -> {Alg, Seed} = mk_alg(Alg0), AS = Seed(S0), {Alg, AS}. %%% 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, Seed} = uniform_s(seed_get()), _ = seed_put(Seed), 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, Seed} = uniform_s(N, seed_get()), _ = seed_put(Seed), 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} = Alg, R0}) -> {V, R1} = Next(R0), %% Produce floats on the form N * 2^(-53) {(V bsr (Bits - 53)) * ?TWO_POW_MINUS53, {Alg, R1}}; uniform_s({#{max:=Max, next:=Next} = Alg, R0}) -> {V, R1} = Next(R0), %% Old broken algorithm with non-uniform density {V / (Max + 1), {Alg, 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} = Alg, R0}) when is_integer(N), 1 =< N -> {V, R1} = Next(R0), MaxMinusN = ?BIT(Bits) - N, ?uniform_range(N, Alg, R1, V, MaxMinusN, I); uniform_s(N, {#{max:=Max, next:=Next} = Alg, R0}) when is_integer(N), 1 =< N -> %% Old broken algorithm with skewed probability %% and gap in ranges > Max {V, R1} = Next(R0), if N =< Max -> {(V rem N) + 1, {Alg, R1}}; true -> F = V / (Max + 1), {trunc(F * N) + 1, {Alg, R1}} end. %% 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_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. %% ===================================================================== %% 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 exsplus_next/1, jump=>fun exsplus_jump/1}, fun exsplus_seed/1}; mk_alg(exsp) -> {#{type=>exsp, bits=>58, weak_low_bits=>1, next=>fun exsplus_next/1, uniform=>fun exsp_uniform/1, uniform_n=>fun exsp_uniform/2, jump=>fun exsplus_jump/1}, fun exsplus_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}. %% ===================================================================== %% exs64 PRNG: Xorshift64* %% Algorithm by Sebastiano Vigna %% Reference URL: http://xorshift.di.unimi.it/ %% ===================================================================== -opaque exs64_state() :: uint64(). 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. %% ===================================================================== -opaque exsplus_state() :: nonempty_improper_list(uint58(), uint58()). -dialyzer({no_improper_lists, exsplus_seed/1}). exsplus_seed({A1, A2, A3}) -> {_, R1} = exsplus_next( [?MASK(58, (A1 * 4294967197) + 1)| ?MASK(58, (A2 * 4294967231) + 1)]), {_, R2} = exsplus_next( [?MASK(58, (A3 * 4294967279) + 1)| tl(R1)]), R2. -dialyzer({no_improper_lists, exsplus_next/1}). %% Advance xorshift116+ state for one step and generate 58bit unsigned integer -spec exsplus_next(exsplus_state()) -> {uint58(), exsplus_state()}. exsplus_next([S1|S0]) -> %% Note: members s0 and s1 are swapped here S11 = S1 bxor ?BSL(58, S1, 24), S12 = S11 bxor S0 bxor (S11 bsr 11) bxor (S0 bsr 41), {?MASK(58, S0 + S12), [S0|S12]}. exsp_uniform({Alg, R0}) -> {I, R1} = exsplus_next(R0), %% Waste the lowest bit since it is of lower %% randomness quality than the others {(I bsr (58-53)) * ?TWO_POW_MINUS53, {Alg, R1}}. exsp_uniform(Range, {Alg, R}) -> {V, R1} = exsplus_next(R), MaxMinusRange = ?BIT(58) - Range, ?uniform_range(Range, Alg, R1, V, MaxMinusRange, I). %% This is the jump function for the exsplus generator, 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. %% -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(state()) -> state(). exsplus_jump({Alg, S}) -> {S1, AS1} = exsplus_jump(S, [0|0], ?JUMPCONST1, ?JUMPELEMLEN), {_, AS2} = exsplus_jump(S1, AS1, ?JUMPCONST2, ?JUMPELEMLEN), {Alg, AS2}. -dialyzer({no_improper_lists, exsplus_jump/4}). exsplus_jump(S, AS, _, 0) -> {S, AS}; exsplus_jump(S, [AS0|AS1], J, N) -> {_, NS} = exsplus_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({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()). exs1024_gen1024(R) -> exs1024_gen1024(16, R, []). exs1024_gen1024(0, _, L) -> L; exs1024_gen1024(N, R, L) -> {X, R2} = exs64_next(R), exs1024_gen1024(N - 1, R2, [X|L]). %% Calculation of xorshift1024*. %% exs1024_calc(S0, S1) -> {X, NS1}. %% X: random number output -spec exs1024_calc(uint64(), uint64()) -> {uint64(), uint64()}. exs1024_calc(S0, S1) -> S11 = S1 bxor ?BSL(64, S1, 31), S12 = S11 bxor (S11 bsr 11), S01 = S0 bxor (S0 bsr 30), NS1 = S01 bxor S12, {?MASK(64, NS1 * 1181783497276652981), NS1}. %% Advance xorshift1024* state for one step and generate 64bit unsigned integer -spec exs1024_next(exs1024_state()) -> {uint64(), exs1024_state()}. exs1024_next({[S0,S1|L3], RL}) -> {X, NS1} = exs1024_calc(S0, S1), {X, {[NS1|L3], [S0|RL]}}; exs1024_next({[H], RL}) -> NL = [H|lists:reverse(RL)], exs1024_next({NL, []}). %% This is the jump function for the exs1024 generator, equivalent %% to 2^512 calls to next(); it can be used to generate 2^512 %% non-overlapping subsequences for parallel computations. %% Note: the jump function takes ~2000 times of the execution time of %% next/1. %% Jump constant here split into 58 bits for speed -define(JUMPCONSTHEAD, 16#00242f96eca9c41d). -define(JUMPCONSTTAIL, [16#0196e1ddbe5a1561, 16#0239f070b5837a3c, 16#03f393cc68796cd2, 16#0248316f404489af, 16#039a30088bffbac2, 16#02fea70dc2d9891f, 16#032ae0d9644caec4, 16#0313aac17d8efa43, 16#02f132e055642626, 16#01ee975283d71c93, 16#00552321b06f5501, 16#00c41d10a1e6a569, 16#019158ecf8aa1e44, 16#004e9fc949d0b5fc, 16#0363da172811fdda, 16#030e38c3b99181f2, 16#0000000a118038fc]). -define(JUMPTOTALLEN, 1024). -define(RINGLEN, 16). -spec exs1024_jump(state()) -> state(). exs1024_jump({Alg, {L, RL}}) -> P = length(RL), AS = exs1024_jump({L, RL}, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], ?JUMPCONSTTAIL, ?JUMPCONSTHEAD, ?JUMPELEMLEN, ?JUMPTOTALLEN), {ASL, ASR} = lists:split(?RINGLEN - P, AS), {Alg, {ASL, lists:reverse(ASR)}}. exs1024_jump(_, AS, _, _, _, 0) -> AS; exs1024_jump(S, AS, [H|T], _, 0, TN) -> exs1024_jump(S, AS, T, H, ?JUMPELEMLEN, TN); exs1024_jump({L, RL}, AS, JL, J, N, TN) -> {_, NS} = exs1024_next({L, RL}), case ?MASK(1, J) of 1 -> AS2 = lists:zipwith(fun(X, Y) -> X bxor Y end, AS, L ++ lists:reverse(RL)), exs1024_jump(NS, AS2, JL, J bsr 1, N-1, TN-1); 0 -> exs1024_jump(NS, AS, JL, J bsr 1, N-1, TN-1) end. %% ===================================================================== %% exrop PRNG: Xoroshiro116+ %% %% Reference URL: http://xorshift.di.unimi.it/ %% %% 58 bits fits into an immediate on 64bits Erlang and is thus much faster. %% In fact, an immediate number is 60 bits signed in Erlang so you can %% add two positive 58 bit numbers and get a 59 bit number that still is %% a positive immediate, which is a property we utilize here... %% %% Modification of the original Xororhiro128+ algorithm to 116 bits %% by Sebastiano Vigna. A lot of thanks for his help and work. %% ===================================================================== %% (a, b, c) = (24, 2, 35) %% JUMP Polynomial = 0x9863200f83fcd4a11293241fcb12a (116 bit) %% %% From http://xoroshiro.di.unimi.it/xoroshiro116plus.c: %% --------------------------------------------------------------------- %% /* Written in 2017 by Sebastiano Vigna (vigna@acm.org). %% %% To the extent possible under law, the author has dedicated all copyright %% and related and neighboring rights to this software to the public domain %% worldwide. This software is distributed without any warranty. %% %% See . */ %% %% #include %% %% #define UINT58MASK (uint64_t)((UINT64_C(1) << 58) - 1) %% %% uint64_t s[2]; %% %% static inline uint64_t rotl58(const uint64_t x, int k) { %% return (x << k) & UINT58MASK | (x >> (58 - k)); %% } %% %% uint64_t next(void) { %% uint64_t s1 = s[1]; %% const uint64_t s0 = s[0]; %% const uint64_t result = (s0 + s1) & UINT58MASK; %% %% s1 ^= s0; %% s[0] = rotl58(s0, 24) ^ s1 ^ ((s1 << 2) & UINT58MASK); // a, b %% s[1] = rotl58(s1, 35); // c %% return result; %% } %% %% void jump(void) { %% static const uint64_t JUMP[] = %% { 0x4a11293241fcb12a, 0x0009863200f83fcd }; %% %% uint64_t s0 = 0; %% uint64_t s1 = 0; %% for(int i = 0; i < sizeof JUMP / sizeof *JUMP; i++) %% for(int b = 0; b < 64; b++) { %% if (JUMP[i] & UINT64_C(1) << b) { %% s0 ^= s[0]; %% s1 ^= s[1]; %% } %% next(); %% } %% s[0] = s0; %% s[1] = s1; %% } -opaque exrop_state() :: nonempty_improper_list(uint58(), uint58()). -dialyzer({no_improper_lists, exrop_seed/1}). exrop_seed({A1, A2, A3}) -> [_|S1] = exrop_next_s( ?MASK(58, (A1 * 4294967197) + 1), ?MASK(58, (A2 * 4294967231) + 1)), exrop_next_s(?MASK(58, (A3 * 4294967279) + 1), S1). -dialyzer({no_improper_lists, exrop_next_s/2}). %% Advance xoroshiro116+ state one step %% [a, b, c] = [24, 2, 35] -define( exrop_next_s(S0, S1, S1_a), begin S1_a = S1 bxor S0, [?ROTL(58, S0, 24) bxor S1_a bxor ?BSL(58, S1_a, 2)| % a, b ?ROTL(58, S1_a, 35)] % c end). exrop_next_s(S0, S1) -> ?exrop_next_s(S0, S1, S1_a). -dialyzer({no_improper_lists, exrop_next/1}). %% Advance xoroshiro116+ state one step, generate 58 bit unsigned integer, %% and waste the lowest bit since it is of lower randomness quality exrop_next([S0|S1]) -> {?MASK(58, S0 + S1), ?exrop_next_s(S0, S1, S1_a)}. exrop_uniform({Alg, R}) -> {V, R1} = exrop_next(R), %% Waste the lowest bit since it is of lower %% randomness quality than the others {(V bsr (58-53)) * ?TWO_POW_MINUS53, {Alg, R1}}. exrop_uniform(Range, {Alg, R}) -> {V, R1} = exrop_next(R), MaxMinusRange = ?BIT(58) - Range, ?uniform_range(Range, Alg, R1, V, MaxMinusRange, I). %% Split a 116 bit constant into two '1'++58 bit words, %% the top '1' marks the top of the word -define( JUMP_116(Jump), [?BIT(58) bor ?MASK(58, (Jump)),?BIT(58) bor ((Jump) bsr 58)]). %% exrop_jump({Alg,S}) -> [J|Js] = ?JUMP_116(16#9863200f83fcd4a11293241fcb12a), {Alg, exrop_jump(S, 0, 0, J, Js)}. %% -dialyzer({no_improper_lists, exrop_jump/5}). exrop_jump(_S, S0, S1, 1, []) -> % End of jump constant [S0|S1]; exrop_jump(S, S0, S1, 1, [J|Js]) -> % End of the word exrop_jump(S, S0, S1, J, Js); exrop_jump([S__0|S__1] = _S, S0, S1, J, Js) -> case ?MASK(1, J) of 1 -> NewS = exrop_next_s(S__0, S__1), exrop_jump(NewS, S0 bxor S__0, S1 bxor S__1, J bsr 1, Js); 0 -> NewS = exrop_next_s(S__0, S__1), exrop_jump(NewS, S0, S1, J bsr 1, Js) end. %% ===================================================================== %% Ziggurat cont %% ===================================================================== -define(NOR_R, 3.6541528853610087963519472518). -define(NOR_INV_R, 1/?NOR_R). %% return a {sign, Random51bits, State} get_52({Alg=#{bits:=Bits, next:=Next}, S0}) -> %% Use the high bits {Int,S1} = Next(S0), {?BIT(Bits - 51 - 1) band Int, Int bsr (Bits - 51), {Alg, S1}}; get_52({Alg=#{next:=Next}, S0}) -> {Int,S1} = Next(S0), {?BIT(51) band Int, ?MASK(51, Int), {Alg, S1}}. %% Slow path normal_s(0, Sign, X0, State0) -> {U0, S1} = uniform_s(State0), X = -?NOR_INV_R*math:log(U0), {U1, S2} = uniform_s(S1), Y = -math:log(U1), case Y+Y > X*X of false -> normal_s(0, Sign, X0, S2); true when Sign =:= 0 -> {?NOR_R + X, S2}; true -> {-?NOR_R - X, S2} end; normal_s(Idx, _Sign, X, State0) -> Fi2 = normal_fi(Idx+1), {U0, S1} = uniform_s(State0), case ((normal_fi(Idx) - Fi2)*U0 + Fi2) < math:exp(-0.5*X*X) of true -> {X, S1}; false -> normal_s(S1) end. %% Tables for generating normal_s %% ki is zipped with wi (slightly faster) normal_kiwi(Indx) -> element(Indx, {{2104047571236786,1.736725412160263e-15}, {0,9.558660351455634e-17}, {1693657211986787,1.2708704834810623e-16},{1919380038271141,1.4909740962495474e-16}, {2015384402196343,1.6658733631586268e-16},{2068365869448128,1.8136120810119029e-16}, {2101878624052573,1.9429720153135588e-16},{2124958784102998,2.0589500628482093e-16}, {2141808670795147,2.1646860576895422e-16},{2154644611568301,2.2622940392218116e-16}, {2164744887587275,2.353271891404589e-16},{2172897953696594,2.438723455742877e-16}, {2179616279372365,2.5194879829274225e-16},{2185247251868649,2.5962199772528103e-16}, {2190034623107822,2.6694407473648285e-16},{2194154434521197,2.7395729685142446e-16}, {2197736978774660,2.8069646002484804e-16},{2200880740891961,2.871905890411393e-16}, {2203661538010620,2.9346417484728883e-16},{2206138681109102,2.9953809336782113e-16}, {2208359231806599,3.054303000719244e-16},{2210361007258210,3.111563633892157e-16}, {2212174742388539,3.1672988018581815e-16},{2213825672704646,3.2216280350549905e-16}, {2215334711002614,3.274657040793975e-16},{2216719334487595,3.326479811684171e-16}, {2217994262139172,3.377180341735323e-16},{2219171977965032,3.4268340353119356e-16}, {2220263139538712,3.475508873172976e-16},{2221276900117330,3.523266384600203e-16}, {2222221164932930,3.5701624633953494e-16},{2223102796829069,3.616248057159834e-16}, {2223927782546658,3.661569752965354e-16},{2224701368170060,3.7061702777236077e-16}, {2225428170204312,3.75008892787478e-16},{2226112267248242,3.7933619401549554e-16}, {2226757276105256,3.836022812967728e-16},{2227366415328399,3.8781025861250247e-16}, {2227942558554684,3.919630085325768e-16},{2228488279492521,3.9606321366256378e-16}, {2229005890047222,4.001133755254669e-16},{2229497472775193,4.041158312414333e-16}, {2229964908627060,4.080727683096045e-16},{2230409900758597,4.119862377480744e-16}, {2230833995044585,4.1585816580828064e-16},{2231238597816133,4.1969036444740733e-16}, {2231624991250191,4.234845407152071e-16},{2231994346765928,4.272423051889976e-16}, {2232347736722750,4.309651795716294e-16},{2232686144665934,4.346546035512876e-16}, {2233010474325959,4.383119410085457e-16},{2233321557544881,4.4193848564470665e-16}, {2233620161276071,4.455354660957914e-16},{2233906993781271,4.491040505882875e-16}, {2234182710130335,4.52645351185714e-16},{2234447917093496,4.561604276690038e-16}, {2234703177503020,4.596502910884941e-16},{2234949014150181,4.631159070208165e-16}, {2235185913274316,4.665581985600875e-16},{2235414327692884,4.699780490694195e-16}, {2235634679614920,4.733763047158324e-16},{2235847363174595,4.767537768090853e-16}, {2236052746716837,4.8011124396270155e-16},{2236251174862869,4.834494540935008e-16}, {2236442970379967,4.867691262742209e-16},{2236628435876762,4.900709524522994e-16}, {2236807855342765,4.933555990465414e-16},{2236981495548562,4.966237084322178e-16}, {2237149607321147,4.998759003240909e-16},{2237312426707209,5.031127730659319e-16}, {2237470176035652,5.0633490483427195e-16},{2237623064889403,5.095428547633892e-16}, {2237771290995388,5.127371639978797e-16},{2237915041040597,5.159183566785736e-16}, {2238054491421305,5.190869408670343e-16},{2238189808931712,5.222434094134042e-16}, {2238321151397660,5.253882407719454e-16},{2238448668260432,5.285218997682382e-16}, {2238572501115169,5.316448383216618e-16},{2238692784207942,5.34757496126473e-16}, {2238809644895133,5.378603012945235e-16},{2238923204068402,5.409536709623993e-16}, {2239033576548190,5.440380118655467e-16},{2239140871448443,5.471137208817361e-16}, {2239245192514958,5.501811855460336e-16},{2239346638439541,5.532407845392784e-16}, {2239445303151952,5.56292888151909e-16},{2239541276091442,5.593378587248462e-16}, {2239634642459498,5.623760510690043e-16},{2239725483455293,5.65407812864896e-16}, {2239813876495186,5.684334850436814e-16},{2239899895417494,5.714534021509204e-16}, {2239983610673676,5.744678926941961e-16},{2240065089506935,5.774772794756965e-16}, {2240144396119183,5.804818799107686e-16},{2240221591827230,5.834820063333892e-16}, {2240296735208969,5.864779662894365e-16},{2240369882240293,5.894700628185872e-16}, {2240441086423386,5.924585947256134e-16},{2240510398907004,5.95443856841806e-16}, {2240577868599305,5.984261402772028e-16},{2240643542273726,6.014057326642664e-16}, {2240707464668391,6.043829183936125e-16},{2240769678579486,6.073579788423606e-16}, {2240830224948980,6.103311925956439e-16},{2240889142947082,6.133028356617911e-16}, {2240946470049769,6.162731816816596e-16},{2241002242111691,6.192425021325847e-16}, {2241056493434746,6.222110665273788e-16},{2241109256832602,6.251791426088e-16}, {2241160563691400,6.281469965398895e-16},{2241210444026879,6.311148930905604e-16}, {2241258926538122,6.34083095820806e-16},{2241306038658137,6.370518672608815e-16}, {2241351806601435,6.400214690888025e-16},{2241396255408788,6.429921623054896e-16}, {2241439408989313,6.459642074078832e-16},{2241481290160038,6.489378645603397e-16}, {2241521920683062,6.519133937646159e-16},{2241561321300462,6.548910550287415e-16}, {2241599511767028,6.578711085350741e-16},{2241636510880960,6.608538148078259e-16}, {2241672336512612,6.638394348803506e-16},{2241707005631362,6.668282304624746e-16}, {2241740534330713,6.698204641081558e-16},{2241772937851689,6.728163993837531e-16}, {2241804230604585,6.758163010371901e-16},{2241834426189161,6.78820435168298e-16}, {2241863537413311,6.818290694006254e-16},{2241891576310281,6.848424730550038e-16}, {2241918554154466,6.878609173251664e-16},{2241944481475843,6.908846754557169e-16}, {2241969368073071,6.939140229227569e-16},{2241993223025298,6.969492376174829e-16}, {2242016054702685,6.999906000330764e-16},{2242037870775710,7.030383934552151e-16}, {2242058678223225,7.060929041565482e-16},{2242078483339331,7.091544215954873e-16}, {2242097291739040,7.122232386196779e-16},{2242115108362774,7.152996516745303e-16}, {2242131937479672,7.183839610172063e-16},{2242147782689725,7.214764709364707e-16}, {2242162646924736,7.245774899788387e-16},{2242176532448092,7.276873311814693e-16}, {2242189440853337,7.308063123122743e-16},{2242201373061537,7.339347561177405e-16}, {2242212329317416,7.370729905789831e-16},{2242222309184237,7.4022134917658e-16}, {2242231311537397,7.433801711647648e-16},{2242239334556717,7.465498018555889e-16}, {2242246375717369,7.497305929136979e-16},{2242252431779415,7.529229026624058e-16}, {2242257498775893,7.561270964017922e-16},{2242261571999416,7.5934354673958895e-16}, {2242264645987196,7.625726339356756e-16},{2242266714504453,7.658147462610487e-16}, {2242267770526109,7.690702803721919e-16},{2242267806216711,7.723396417018299e-16}, {2242266812908462,7.756232448671174e-16},{2242264781077289,7.789215140963852e-16}, {2242261700316818,7.822348836756411e-16},{2242257559310145,7.855637984161084e-16}, {2242252345799276,7.889087141441755e-16},{2242246046552082,7.922700982152271e-16}, {2242238647326615,7.956484300529366e-16},{2242230132832625,7.99044201715713e-16}, {2242220486690076,8.024579184921259e-16},{2242209691384458,8.058900995272657e-16}, {2242197728218684,8.093412784821501e-16},{2242184577261310,8.128120042284501e-16}, {2242170217290819,8.163028415809877e-16},{2242154625735679,8.198143720706533e-16}, {2242137778609839,8.23347194760605e-16},{2242119650443327,8.26901927108847e-16}, {2242100214207556,8.304792058805374e-16},{2242079441234906,8.340796881136629e-16}, {2242057301132135,8.377040521420222e-16},{2242033761687079,8.413529986798028e-16}, {2242008788768107,8.450272519724097e-16},{2241982346215682,8.487275610186155e-16}, {2241954395725356,8.524547008695596e-16},{2241924896721443,8.562094740106233e-16}, {2241893806220517,8.599927118327665e-16},{2241861078683830,8.638052762005259e-16}, {2241826665857598,8.676480611245582e-16},{2241790516600041,8.715219945473698e-16}, {2241752576693881,8.754280402517175e-16},{2241712788642916,8.793671999021043e-16}, {2241671091451078,8.833405152308408e-16},{2241627420382235,8.873490703813135e-16}, {2241581706698773,8.913939944224086e-16},{2241533877376767,8.954764640495068e-16}, {2241483854795281,8.9959770648911e-16},{2241431556397035,9.037590026260118e-16}, {2241376894317345,9.079616903740068e-16},{2241319774977817,9.122071683134846e-16}, {2241260098640860,9.164968996219135e-16},{2241197758920538,9.208324163262308e-16}, {2241132642244704,9.252153239095693e-16},{2241064627262652,9.296473063086417e-16}, {2240993584191742,9.341301313425265e-16},{2240919374095536,9.38665656618666e-16}, {2240841848084890,9.432558359676707e-16},{2240760846432232,9.479027264651738e-16}, {2240676197587784,9.526084961066279e-16},{2240587717084782,9.57375432209745e-16}, {2240495206318753,9.622059506294838e-16},{2240398451183567,9.671026058823054e-16}, {2240297220544165,9.720681022901626e-16},{2240191264522612,9.771053062707209e-16}, {2240080312570155,9.822172599190541e-16},{2239964071293331,9.874071960480671e-16}, {2239842221996530,9.926785548807976e-16},{2239714417896699,9.980350026183645e-16}, {2239580280957725,1.003480452143618e-15},{2239439398282193,1.0090190861637457e-15}, {2239291317986196,1.0146553831467086e-15},{2239135544468203,1.0203941464683124e-15}, {2238971532964979,1.0262405372613567e-15},{2238798683265269,1.0322001115486456e-15}, {2238616332424351,1.03827886235154e-15},{2238423746288095,1.044483267600047e-15}, {2238220109591890,1.0508203448355195e-15},{2238004514345216,1.057297713900989e-15}, {2237775946143212,1.06392366906768e-15},{2237533267957822,1.0707072623632994e-15}, {2237275200846753,1.0776584002668106e-15},{2237000300869952,1.0847879564403425e-15}, {2236706931309099,1.0921079038149563e-15},{2236393229029147,1.0996314701785628e-15}, {2236057063479501,1.1073733224935752e-15},{2235695986373246,1.1153497865853155e-15}, {2235307169458859,1.1235791107110833e-15},{2234887326941578,1.1320817840164846e-15}, {2234432617919447,1.140880924258278e-15},{2233938522519765,1.1500027537839792e-15}, {2233399683022677,1.159477189144919e-15},{2232809697779198,1.169338578691096e-15}, {2232160850599817,1.17962663529558e-15},{2231443750584641,1.190387629928289e-15}, {2230646845562170,1.2016759392543819e-15},{2229755753817986,1.2135560818666897e-15}, {2228752329126533,1.2261054417450561e-15},{2227613325162504,1.2394179789163251e-15}, {2226308442121174,1.2536093926602567e-15},{2224797391720399,1.268824481425501e-15}, {2223025347823832,1.2852479319096109e-15},{2220915633329809,1.3031206634689985e-15}, {2218357446087030,1.3227655770195326e-15},{2215184158448668,1.3446300925011171e-15}, {2211132412537369,1.3693606835128518e-15},{2205758503851065,1.397943667277524e-15}, {2198248265654987,1.4319989869661328e-15},{2186916352102141,1.4744848603597596e-15}, {2167562552481814,1.5317872741611144e-15},{2125549880839716,1.6227698675312968e-15}}). normal_fi(Indx) -> element(Indx, {1.0000000000000000e+00,9.7710170126767082e-01,9.5987909180010600e-01, 9.4519895344229909e-01,9.3206007595922991e-01,9.1999150503934646e-01, 9.0872644005213032e-01,8.9809592189834297e-01,8.8798466075583282e-01, 8.7830965580891684e-01,8.6900868803685649e-01,8.6003362119633109e-01, 8.5134625845867751e-01,8.4291565311220373e-01,8.3471629298688299e-01, 8.2672683394622093e-01,8.1892919160370192e-01,8.1130787431265572e-01, 8.0384948317096383e-01,7.9654233042295841e-01,7.8937614356602404e-01, 7.8234183265480195e-01,7.7543130498118662e-01,7.6863731579848571e-01, 7.6195334683679483e-01,7.5537350650709567e-01,7.4889244721915638e-01, 7.4250529634015061e-01,7.3620759812686210e-01,7.2999526456147568e-01, 7.2386453346862967e-01,7.1781193263072152e-01,7.1183424887824798e-01, 7.0592850133275376e-01,7.0009191813651117e-01,6.9432191612611627e-01, 6.8861608300467136e-01,6.8297216164499430e-01,6.7738803621877308e-01, 6.7186171989708166e-01,6.6639134390874977e-01,6.6097514777666277e-01, 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