From 401bf07f5908137cde206f2f755af83c9a7ff71e Mon Sep 17 00:00:00 2001 From: Dan Gudmundsson Date: Tue, 28 Apr 2015 14:37:33 +0200 Subject: stdlib: Document and add normal distributed random value function It is needed in various tests. It uses the Ziggurat algorithm, which is the fastest that I know. --- lib/stdlib/src/rand.erl | 323 +++++++++++++++++++++++++++++++++++++++++++++--- 1 file changed, 306 insertions(+), 17 deletions(-) (limited to 'lib/stdlib/src') diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl index 0cafb35dd8..6a805eb69e 100644 --- a/lib/stdlib/src/rand.erl +++ b/lib/stdlib/src/rand.erl @@ -25,10 +25,13 @@ -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/0, uniform/1, uniform_s/1, uniform_s/2, + normal/0, normal_s/1 + ]). -compile({inline, [exs64_next/1, exsplus_next/1, - exs1024_next/1, exs1024_calc/2]}). + exs1024_next/1, exs1024_calc/2, + get_52/1, normal_kiwi/1]}). -define(DEFAULT_ALG_HANDLER, exsplus). -define(SEED_DICT, rand_seed). @@ -38,31 +41,33 @@ %% ===================================================================== %% This depends on the algorithm handler function --opaque alg_seed() :: exs64_state() | exsplus_state() | exs1024_state(). +-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()}. %% Internal state --type state() :: {alg_handler(), alg_seed()}. +-opaque state() :: {alg_handler(), alg_seed()}. -type alg() :: exs64 | exsplus | exs1024. --export_type([alg/0, alg_handler/0, state/0, alg_seed/0]). +-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 | {alg(), alg_seed()}. +-spec export_seed() -> undefined | export_state(). export_seed() -> case seed_get() of {#{type:=Alg}, Seed} -> {Alg, Seed}; _ -> undefined end. --spec export_seed_s(state()) -> {alg(), alg_seed()}. +-spec export_seed_s(state()) -> export_state(). export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}. %% seed(Alg) seeds RNG with runtime dependent values @@ -71,13 +76,13 @@ export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}. %% seed({Alg,Seed}) setup RNG with a previously exported seed %% and return the NEW state --spec seed(alg() | {alg(), alg_seed()}) -> state(). +-spec seed(AlgOrExpState::alg() | export_state()) -> state(). seed(Alg) -> R = seed_s(Alg), _ = seed_put(R), R. --spec seed_s(alg() | {alg(), alg_seed()}) -> state(). +-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(), @@ -107,7 +112,7 @@ seed_s(Alg0, S0 = {_, _, _}) -> %% uniform/0: returns a random float X where 0.0 < X < 1.0, %% updating the state in the process dictionary. --spec uniform() -> float(). +-spec uniform() -> X::float(). uniform() -> {X, Seed} = uniform_s(seed_get()), _ = seed_put(Seed), @@ -117,7 +122,7 @@ uniform() -> %% uniform/1 returns a random integer X where 1 =< X =< N, %% updating the state in the process dictionary. --spec uniform(N :: pos_integer()) -> pos_integer(). +-spec uniform(N :: pos_integer()) -> X::pos_integer(). uniform(N) -> {X, Seed} = uniform_s(N, seed_get()), _ = seed_put(Seed), @@ -127,7 +132,7 @@ uniform(N) -> %% returns a random float X where 0.0 < X < 1.0, %% and a new state. --spec uniform_s(state()) -> {float(), NewS :: state()}. +-spec uniform_s(state()) -> {X::float(), NewS :: state()}. uniform_s(State = {#{uniform:=Uniform}, _}) -> Uniform(State). @@ -135,7 +140,7 @@ uniform_s(State = {#{uniform:=Uniform}, _}) -> %% uniform_s/2 returns a random integer X where 1 =< X =< N, %% and a new state. --spec uniform_s(N::pos_integer(), state()) -> {pos_integer(), NewS::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); @@ -144,6 +149,35 @@ uniform_s(N, State0 = {#{uniform:=Uniform}, _}) {F, State} = Uniform(State0), {trunc(F * N) + 1, State}. +%% 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()) -> {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. + %% ===================================================================== %% Internal functions @@ -169,15 +203,15 @@ seed_get() -> %% Setup alg record mk_alg(exs64) -> - {#{type=>exs64, max=>?UINT64MASK, + {#{type=>exs64, max=>?UINT64MASK, next=>fun exs64_next/1, uniform=>fun exs64_uniform/1, uniform_n=>fun exs64_uniform/2}, fun exs64_seed/1}; mk_alg(exsplus) -> - {#{type=>exsplus, max=>?UINT58MASK, + {#{type=>exsplus, max=>?UINT58MASK, next=>fun exsplus_next/1, uniform=>fun exsplus_uniform/1, uniform_n=>fun exsplus_uniform/2}, fun exsplus_seed/1}; mk_alg(exs1024) -> - {#{type=>exs1024, max=>?UINT64MASK, + {#{type=>exs1024, max=>?UINT64MASK, next=>fun exs1024_next/1, uniform=>fun exs1024_uniform/1, uniform_n=>fun exs1024_uniform/2}, fun exs1024_seed/1}. @@ -219,7 +253,7 @@ exs64_uniform(Max, {Alg, R}) -> %% Modification of the original Xorshift128+ algorithm to 116 %% by Sebastiano Vigna, a lot of thanks for his help and work. %% ===================================================================== --type exsplus_state() :: [uint58()|uint58()]. +-type exsplus_state() :: nonempty_improper_list(uint58(), uint58()). exsplus_seed({A1, A2, A3}) -> {_, R1} = exsplus_next([(((A1 * 4294967197) + 1) band ?UINT58MASK)| @@ -300,3 +334,258 @@ exs1024_uniform({Alg, R0}) -> exs1024_uniform(Max, {Alg, R}) -> {V, R1} = exs1024_next(R), {(V rem Max) + 1, {Alg, R1}}. + +%% ===================================================================== +%% 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}, + 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