diff options
Diffstat (limited to 'lib/stdlib/src/rand.erl')
| -rw-r--r-- | lib/stdlib/src/rand.erl | 261 |
1 files changed, 247 insertions, 14 deletions
diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl index 7a8a5e6d4a..362e98006e 100644 --- a/lib/stdlib/src/rand.erl +++ b/lib/stdlib/src/rand.erl @@ -21,8 +21,8 @@ %% 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 +%% exrop (xoroshiro116+) added, statistical distribution +%% improvements and uniform_real added by the Erlang/OTP team 2017 %% ===================================================================== -module(rand). @@ -30,10 +30,14 @@ -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, jump/0, jump/1, normal/0, normal/2, normal_s/1, normal_s/3 ]). +%% Debug +-export([make_float/3, float2str/1, bc64/1]). + -compile({inline, [exs64_next/1, exsplus_next/1, exs1024_next/1, exs1024_calc/2, exrop_next/1, exrop_next_s/2, @@ -60,6 +64,10 @@ %% 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). @@ -84,14 +92,21 @@ %% 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. +%% 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 'uinform_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'. +%% '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(), @@ -148,11 +163,7 @@ %% 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, + WeakLowBits = maps:get(weak_low_bits, Alg, 0), %% Maybe waste the lowest bit(s) when shifting in new bits Shift = Bits - WeakLowBits, ShiftMask = bnot ?MASK(WeakLowBits), @@ -297,7 +308,7 @@ uniform_s({#{bits:=Bits, next:=Next} = Alg, R0}) -> {(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 + %% Old algorithm with non-uniform density {V / (Max + 1), {Alg, R1}}. @@ -317,7 +328,7 @@ uniform_s(N, {#{bits:=Bits, next:=Next} = Alg, R0}) ?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 + %% Old algorithm with skewed probability %% and gap in ranges > Max {V, R1} = Next(R0), if @@ -328,6 +339,189 @@ uniform_s(N, {#{max:=Max, next:=Next} = Alg, R0}) {trunc(F * N) + 1, {Alg, 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} = Alg, 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), {Alg, R1}}; + ?BIT(54) =< M1 -> + %% We have 55 bits - waste 2 + {(M1 bsr 2) * math:pow(2.0, -54), {Alg, R1}}; + ?BIT(53) =< M1 -> + %% We have 54 bits - waste 1 + {(M1 bsr 1) * math:pow(2.0, -55), {Alg, R1}}; + ?BIT(52) =< M1 -> + %% We have 53 bits - use all + {M1 * math:pow(2.0, -56), {Alg, R1}}; + true -> + %% Need more bits + {V2, R2} = Next(R1), + uniform_real_s(Alg, Next, M1, -56, R2, V2, Bits) + end; +uniform_real_s({#{max:=_, next:=Next} = Alg, 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), {Alg, R1}}; + ?BIT(54) =< M1 -> + %% We have 55 bits - waste 2 + {(M1 bsr 2) * math:pow(2.0, -54), {Alg, R1}}; + ?BIT(53) =< M1 -> + %% We have 54 bits - waste 1 + {(M1 bsr 1) * math:pow(2.0, -55), {Alg, R1}}; + ?BIT(52) =< M1 -> + %% We have 53 bits - use all + {M1 * math:pow(2.0, -56), {Alg, R1}}; + true -> + %% Need more bits + {V2, R2} = Next(R1), + uniform_real_s(Alg, Next, M1, -56, R2, V2, 56) + end. + +uniform_real_s(Alg, _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)), + {Alg, R1}}; +uniform_real_s(Alg, 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)), + {Alg, 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)), + {Alg, 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)), + {Alg, 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), {Alg, R1}}; + ?BIT(54) =< M1 -> + %% We have 55 bits - waste 2 + {(M1 bsr 2) * math:pow(2.0, BitNo - 54), {Alg, R1}}; + ?BIT(53) =< M1 -> + %% We have 54 bits - waste 1 + {(M1 bsr 1) * math:pow(2.0, BitNo - 55), {Alg, R1}}; + ?BIT(52) =< M1 -> + %% We have 53 bits - use all + {M1 * math:pow(2.0, BitNo - 56), {Alg, R1}}; + BitNo =:= -1008 -> + %% Endgame + %% For the last round we can not 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(Alg, 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({Alg, R1}) + end; + true -> + %% Need more bits + uniform_real_s(Alg, 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)), + {Alg, R1}} + end. +%% +uniform_real_s(#{bits:=Bits} = Alg, Next, M0, BitNo, R0) -> + {V1, R1} = Next(R0), + uniform_real_s(Alg, Next, M0, BitNo, R1, V1, Bits); +uniform_real_s(#{max:=_} = Alg, Next, M0, BitNo, R0) -> + {V1, R1} = Next(R0), + uniform_real_s(Alg, Next, M0, BitNo, R1, ?MASK(56, V1), 56). + %% 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. @@ -1025,3 +1219,42 @@ normal_fi(Indx) -> 1.0214971439701471e-02,8.6165827693987316e-03,7.0508754713732268e-03, 5.5224032992509968e-03,4.0379725933630305e-03,2.6090727461021627e-03, 1.2602859304985975e-03}). + +%%%bitcount64(0) -> 0; +%%%bitcount64(V) -> 1 + bitcount(V, 64). +%%% +%%%-define( +%%% BITCOUNT(V, N), +%%% bitcount(V, N) -> +%%% if +%%% (1 bsl ((N) bsr 1)) =< (V) -> +%%% ((N) bsr 1) + bitcount((V) bsr ((N) bsr 1), ((N) bsr 1)); +%%% true -> +%%% bitcount((V), ((N) bsr 1)) +%%% end). +%%%?BITCOUNT(V, 64); +%%%?BITCOUNT(V, 32); +%%%?BITCOUNT(V, 16); +%%%?BITCOUNT(V, 8); +%%%?BITCOUNT(V, 4); +%%%?BITCOUNT(V, 2); +%%%bitcount(_, 1) -> 0. + +bc64(V) -> ?BC(V, 64). + +%% Linear from high bit - higher probability first gives faster execution +bc(V, B, N) when B =< V -> N; +bc(V, B, N) -> bc(V, B bsr 1, N - 1). + +make_float(S, E, M) -> + <<F/float>> = <<S:1, E:11, M:52>>, + F. + +float2str(N) -> + <<S:1, E:11, M:52>> = <<(float(N))/float>>, + lists:flatten( + io_lib:format( + "~c~c.~13.16.0bE~b", + [case S of 1 -> $-; 0 -> $+ end, + case E of 0 -> $0; _ -> $1 end, + M, E - 16#3ff])). |