%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2015-2016. 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
%% =====================================================================
-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/2, normal_s/1, normal_s/3
]).
-compile({inline, [exs64_next/1, exsplus_next/1,
exsplus_jump/1,
exs1024_next/1, exs1024_calc/2,
exs1024_jump/1,
get_52/1, normal_kiwi/1]}).
-define(DEFAULT_ALG_HANDLER, exsplus).
-define(SEED_DICT, rand_seed).
%% =====================================================================
%% Types
%% =====================================================================
%% This depends on the algorithm handler function
-type alg_seed() :: exs64_state() | exsplus_state() | exs1024_state().
%% This is the algorithm handler function within this module
-type alg_handler() :: #{type := alg(),
max := integer(),
next := fun(),
uniform := fun(),
uniform_n := fun(),
jump := fun()}.
%% Internal state
-opaque state() :: {alg_handler(), alg_seed()}.
-type alg() :: exs64 | exsplus | exs1024.
-opaque export_state() :: {alg(), alg_seed()}.
-export_type([alg/0, state/0, export_state/0]).
%% =====================================================================
%% 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()) -> 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(AlgOrExpState::alg() | export_state()) -> state().
seed(Alg) ->
seed_put(seed_s(Alg)).
-spec seed_s(AlgOrExpState::alg() | export_state()) -> state().
seed_s(Alg) when is_atom(Alg) ->
seed_s(Alg, {erlang:phash2([{node(),self()}]),
erlang:system_time(),
erlang:unique_integer()});
seed_s({Alg0, Seed}) ->
{Alg,_SeedFun} = mk_alg(Alg0),
{Alg, Seed}.
%% seed/2: seeds RNG with the algorithm and given values
%% and returns the NEW state.
-spec seed(Alg :: alg(), {integer(), integer(), integer()}) -> state().
seed(Alg0, S0) ->
seed_put(seed_s(Alg0, S0)).
-spec seed_s(Alg :: alg(), {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()) -> {X::float(), NewS :: state()}.
uniform_s(State = {#{uniform:=Uniform}, _}) ->
Uniform(State).
%% 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()) -> {X::pos_integer(), NewS::state()}.
uniform_s(N, State = {#{uniform_n:=Uniform, max:=Max}, _})
when 0 < N, N =< Max ->
Uniform(N, State);
uniform_s(N, State0 = {#{uniform:=Uniform}, _})
when is_integer(N), 0 < N ->
{F, State} = Uniform(State0),
{trunc(F * N) + 1, State}.
%% 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()) -> NewS :: state().
jump(State = {#{jump:=Jump}, _}) ->
Jump(State).
%% 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() -> NewS :: 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()) -> {float(), NewS :: state()}.
normal_s(State0) ->
{Sign, R, State} = get_52(State0),
Idx = R band 16#FF,
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
-define(UINT21MASK, 16#00000000001fffff).
-define(UINT32MASK, 16#00000000ffffffff).
-define(UINT33MASK, 16#00000001ffffffff).
-define(UINT39MASK, 16#0000007fffffffff).
-define(UINT58MASK, 16#03ffffffffffffff).
-define(UINT64MASK, 16#ffffffffffffffff).
-type uint64() :: 0..16#ffffffffffffffff.
-type uint58() :: 0..16#03ffffffffffffff.
-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=>?UINT64MASK, next=>fun exs64_next/1,
uniform=>fun exs64_uniform/1, uniform_n=>fun exs64_uniform/2,
jump=>fun exs64_jump/1},
fun exs64_seed/1};
mk_alg(exsplus) ->
{#{type=>exsplus, max=>?UINT58MASK, next=>fun exsplus_next/1,
uniform=>fun exsplus_uniform/1, uniform_n=>fun exsplus_uniform/2,
jump=>fun exsplus_jump/1},
fun exsplus_seed/1};
mk_alg(exs1024) ->
{#{type=>exs1024, max=>?UINT64MASK, next=>fun exs1024_next/1,
uniform=>fun exs1024_uniform/1, uniform_n=>fun exs1024_uniform/2,
jump=>fun exs1024_jump/1},
fun exs1024_seed/1}.
%% =====================================================================
%% exs64 PRNG: Xorshift64*
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================
-type exs64_state() :: uint64().
exs64_seed({A1, A2, A3}) ->
{V1, _} = exs64_next(((A1 band ?UINT32MASK) * 4294967197 + 1)),
{V2, _} = exs64_next(((A2 band ?UINT32MASK) * 4294967231 + 1)),
{V3, _} = exs64_next(((A3 band ?UINT32MASK) * 4294967279 + 1)),
((V1 * V2 * V3) rem (?UINT64MASK - 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 ((R1 band ?UINT39MASK) bsl 25),
R3 = R2 bxor (R2 bsr 27),
{(R3 * 2685821657736338717) band ?UINT64MASK, R3}.
exs64_uniform({Alg, R0}) ->
{V, R1} = exs64_next(R0),
{V / 18446744073709551616, {Alg, R1}}.
exs64_uniform(Max, {Alg, R}) ->
{V, R1} = exs64_next(R),
{(V rem Max) + 1, {Alg, R1}}.
exs64_jump(_) ->
erlang:error(not_implemented).
%% =====================================================================
%% 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.
%% =====================================================================
-type exsplus_state() :: nonempty_improper_list(uint58(), uint58()).
-dialyzer({no_improper_lists, exsplus_seed/1}).
exsplus_seed({A1, A2, A3}) ->
{_, R1} = exsplus_next([(((A1 * 4294967197) + 1) band ?UINT58MASK)|
(((A2 * 4294967231) + 1) band ?UINT58MASK)]),
{_, R2} = exsplus_next([(((A3 * 4294967279) + 1) band ?UINT58MASK)|
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 (S1 bsl 24)) band ?UINT58MASK,
S12 = S11 bxor S0 bxor (S11 bsr 11) bxor (S0 bsr 41),
{(S0 + S12) band ?UINT58MASK, [S0|S12]}.
exsplus_uniform({Alg, R0}) ->
{I, R1} = exsplus_next(R0),
{I / (?UINT58MASK+1), {Alg, R1}}.
exsplus_uniform(Max, {Alg, R}) ->
{V, R1} = exsplus_next(R),
{(V rem Max) + 1, {Alg, R1}}.
%% 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 (J band 1) 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/
%% =====================================================================
-type exs1024_state() :: {list(uint64()), list(uint64())}.
exs1024_seed({A1, A2, A3}) ->
B1 = (((A1 band ?UINT21MASK) + 1) * 2097131) band ?UINT21MASK,
B2 = (((A2 band ?UINT21MASK) + 1) * 2097133) band ?UINT21MASK,
B3 = (((A3 band ?UINT21MASK) + 1) * 2097143) band ?UINT21MASK,
{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 ((S1 band ?UINT33MASK) bsl 31),
S12 = S11 bxor (S11 bsr 11),
S01 = S0 bxor (S0 bsr 30),
NS1 = S01 bxor S12,
{(NS1 * 1181783497276652981) band ?UINT64MASK, 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, []}).
exs1024_uniform({Alg, R0}) ->
{V, R1} = exs1024_next(R0),
{V / 18446744073709551616, {Alg, R1}}.
exs1024_uniform(Max, {Alg, R}) ->
{V, R1} = exs1024_next(R),
{(V rem Max) + 1, {Alg, R1}}.
%% 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 (J band 1) 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.
%% =====================================================================
%% Ziggurat cont
%% =====================================================================
-define(NOR_R, 3.6541528853610087963519472518).
-define(NOR_INV_R, 1/?NOR_R).
%% return a {sign, Random51bits, State}
get_52({Alg=#{next:=Next}, S0}) ->
{Int,S1} = Next(S0),
{((1 bsl 51) band Int), Int band ((1 bsl 51)-1), {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},
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{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,
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