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authorDan Gudmundsson <dgud@erlang.org>2015-04-28 14:37:33 +0200
committerDan Gudmundsson <dgud@erlang.org>2015-04-30 13:06:58 +0200
commit401bf07f5908137cde206f2f755af83c9a7ff71e (patch)
tree2771a5b1bc341b13458209b66a16c51f5c0bdc04 /lib/stdlib/src/rand.erl
parent95aff702b5e4b21ec277b1e0125f639ce30f997a (diff)
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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.
Diffstat (limited to 'lib/stdlib/src/rand.erl')
-rw-r--r--lib/stdlib/src/rand.erl323
1 files changed, 306 insertions, 17 deletions
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},
+ {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},
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