From c4408318c97be75915b3dff23a933bbc4b5c2046 Mon Sep 17 00:00:00 2001 From: Raimo Niskanen Date: Thu, 10 May 2018 01:13:03 +0200 Subject: Implement Xoroshiro928** --- lib/stdlib/doc/src/rand.xml | 36 ++- lib/stdlib/src/rand.erl | 317 ++++++++++++++++++++++++- lib/stdlib/test/rand_SUITE.erl | 148 +++++++++++- lib/stdlib/test/rand_Xoroshiro928ss_dev.txt | 343 ++++++++++++++++++++++++++++ 4 files changed, 815 insertions(+), 29 deletions(-) create mode 100644 lib/stdlib/test/rand_Xoroshiro928ss_dev.txt (limited to 'lib/stdlib') diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml index 21f680a0ee..5f1ec84786 100644 --- a/lib/stdlib/doc/src/rand.xml +++ b/lib/stdlib/doc/src/rand.xml @@ -67,6 +67,26 @@

Xorshift1024*, 64 bits precision and a period of 2^1024-1

Jump function: equivalent to 2^512 calls

+ exro928ss + +

Xoroshiro928**, 58 bits precision and a period of 2^928-1

+

Jump function: equivalent to 2^512 calls

+

+ This is a 58 bit version of Xoroshiro1024**, + from the 2018 paper by David Blackman and Sebastiano Vigna: + + Scrambled Linear Pseudorandom Number Generators + + that on a 64 bit Erlang system executes only about 30% slower than + the default exrop algorithm but with much longer period + and better statistical properties, and on the flip side + a larger state. +

+

+ Many thanks to Sebastiano Vigna for his help with + the 58 bit adaption. +

+ exsp

Xorshift116+, 58 bits precision and period of 2^116-1

@@ -195,8 +215,8 @@ SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)

- For all these generators the lowest bit(s) has got - a slightly less random behaviour than all other bits. + For all these generators except exro928ss the lowest bit(s) + has got a slightly less random behaviour than all other bits. 1 bit for exrop (and exsp), and 3 bits for exs1024s. See for example the explanation in the @@ -254,19 +274,23 @@ tests. We suggest to use a sign test to extract a random Boolean value. - +

Algorithm specific internal state

- +

Algorithm specific internal state

 - +

Algorithm specific internal state

 - + +

Algorithm specific internal state

 + + +

Algorithm specific internal state

 diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl index 4951dc727b..3a32f40fcd 100644 --- a/lib/stdlib/src/rand.erl +++ b/lib/stdlib/src/rand.erl @@ -32,14 +32,19 @@ 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 + normal/0, normal/2, normal_s/1, normal_s/3 ]). +%% Test, dev and internal +-export([exro928_jump_2pow512/1, exro928_jump_2pow20/1, + format_jumpconst58/1, seed58/2]). + %% Debug -export([make_float/3, float2str/1, bc64/1]). -compile({inline, [exs64_next/1, exsplus_next/1, exs1024_next/1, exs1024_calc/2, + exro928_next_state/4, exrop_next/1, exrop_next_s/2, get_52/1, normal_kiwi/1]}). @@ -80,8 +85,8 @@ %% This depends on the algorithm handler function -type alg_state() :: - exs64_state() | exsplus_state() | exs1024_state() | - exrop_state() | term(). + exrop_state() | exs1024_state() | exro928_state() | exsplus_state() | + exs64_state() | term(). %% This is the algorithm handling definition within this module, %% and the type to use for plugins. @@ -124,14 +129,16 @@ %% Algorithm state -type state() :: {alg_handler(), alg_state()}. --type builtin_alg() :: exs64 | exsplus | exsp | exs1024 | exs1024s | exrop. +-type builtin_alg() :: + exrop | exs1024s | exro928ss | exsp | exs64 | exsplus | exs1024. -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]). + [exrop_state/0, exs1024_state/0, exro928_state/0, exsplus_state/0, + exs64_state/0]). %% ===================================================================== %% Range macro and helper @@ -260,15 +267,17 @@ seed_s(Alg) -> %% and returns the NEW state. -spec seed( - Alg :: builtin_alg(), Seed :: {integer(), integer(), integer()}) -> + 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()}) -> + Alg :: builtin_alg(), + Seed :: {integer(), integer(), integer()}) -> state(). -seed_s(Alg0, S0 = {_, _, _}) -> +seed_s(Alg0, S0) -> {Alg, Seed} = mk_alg(Alg0), AS = Seed(S0), {Alg, AS}. @@ -625,7 +634,13 @@ 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}. + fun exrop_seed/1}; +mk_alg(exro928ss) -> + {#{type=>exro928ss, bits=>58, next=>fun exro928ss_next/1, + uniform=>fun exro928ss_uniform/1, + uniform_n=>fun exro928ss_uniform/2, + jump=>fun exro928_jump/1}, + fun exro928_seed/1}. %% ===================================================================== %% exs64 PRNG: Xorshift64* @@ -708,7 +723,8 @@ exsp_uniform(Range, {Alg, R}) -> -define(JUMPELEMLEN, 58). -dialyzer({no_improper_lists, exsplus_jump/1}). --spec exsplus_jump(state()) -> state(). +-spec exsplus_jump({alg_handler(), exsplus_state()}) -> + {alg_handler(), exsplus_state()}. exsplus_jump({Alg, S}) -> {S1, AS1} = exsplus_jump(S, [0|0], ?JUMPCONST1, ?JUMPELEMLEN), {_, AS2} = exsplus_jump(S1, AS1, ?JUMPCONST2, ?JUMPELEMLEN), @@ -806,8 +822,8 @@ exs1024_next({[H], RL}) -> -define(JUMPTOTALLEN, 1024). -define(RINGLEN, 16). --spec exs1024_jump(state()) -> state(). - +-spec exs1024_jump({alg_handler(), exs1024_state()}) -> + {alg_handler(), exs1024_state()}. exs1024_jump({Alg, {L, RL}}) -> P = length(RL), AS = exs1024_jump({L, RL}, @@ -831,6 +847,255 @@ exs1024_jump({L, RL}, AS, JL, J, N, TN) -> exs1024_jump(NS, AS, JL, J bsr 1, N-1, TN-1) end. +%% ===================================================================== +%% exro928ss PRNG: Xoroshiro928** +%% +%% Reference URL: http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf +%% i.e the Xoroshiro1024 generator with ** scrambler +%% with {S, R, T} = {5, 7, 9} as recommended in the paper. +%% +%% {A, B, C} were tried out and selected as {44, 9, 45} +%% and the jump coefficients calculated. +%% +%% Standard jump function pseudocode: +%% +%% Jump constant j = 0xb10773cb...44085302f77130ca +%% Generator state: s +%% New generator state: t = 0 +%% foreach bit in j, low to high: +%% if the bit is one: +%% t ^= s +%% next s +%% s = t +%% +%% Generator used for reference value calculation: +%% +%% #include +%% #include +%% +%% int p = 0; +%% uint64_t s[16]; +%% +%% #define MASK(x) ((x) & ((UINT64_C(1) << 58) - 1)) +%% static __inline uint64_t rotl(uint64_t x, int n) { +%% return MASK(x << n) | (x >> (58 - n)); +%% } +%% +%% uint64_t next() { +%% const int q = p; +%% const uint64_t s0 = s[p = (p + 1) & 15]; +%% uint64_t s15 = s[q]; +%% +%% const uint64_t result_starstar = MASK(rotl(MASK(s0 * 5), 7) * 9); +%% +%% s15 ^= s0; +%% s[q] = rotl(s0, 44) ^ s15 ^ MASK(s15 << 9); +%% s[p] = rotl(s15, 45); +%% +%% return result_starstar; +%% } +%% +%% static const uint64_t jump_2pow512[15] = +%% { 0x44085302f77130ca, 0xba05381fdfd14902, 0x10a1de1d7d6813d2, +%% 0xb83fe51a1eb3be19, 0xa81b0090567fd9f0, 0x5ac26d5d20f9b49f, +%% 0x4ddd98ee4be41e01, 0x0657e19f00d4b358, 0xf02f778573cf0f0a, +%% 0xb45a3a8a3cef3cc0, 0x6e62a33cc2323831, 0xbcb3b7c4cc049c53, +%% 0x83f240c6007e76ce, 0xe19f5fc1a1504acd, 0x00000000b10773cb }; +%% +%% static const uint64_t jump_2pow20[15] = +%% { 0xbdb966a3daf905e6, 0x644807a56270cf78, 0xda90f4a806c17e9e, +%% 0x4a426866bfad3c77, 0xaf699c306d8e7566, 0x8ebc73c700b8b091, +%% 0xc081a7bf148531fb, 0xdc4d3af15f8a4dfd, 0x90627c014098f4b6, +%% 0x06df2eb1feaf0fb6, 0x5bdeb1a5a90f2e6b, 0xa480c5878c3549bd, +%% 0xff45ef33c82f3d48, 0xa30bebc15fefcc78, 0x00000000cb3d181c }; +%% +%% void jump(const uint64_t *jump) { +%% uint64_t j, t[16] = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0}; +%% int m, n, k; +%% for (m = 0; m < 15; m++, jump++) { +%% for (n = 0, j = *jump; n < 64; n++, j >>= 1) { +%% if ((j & 1) != 0) { +%% for (k = 0; k < 16; k++) { +%% t[k] ^= s[(p + k) & 15]; +%% } +%% } +%% next(); +%% } +%% } +%% for (k = 0; k < 16; k++) { +%% s[(p + k) & 15] = t[k]; +%% } +%% } +%% +%% ===================================================================== + +-opaque exro928_state() :: {list(uint58()), list(uint58())}. + + +%%% %% Seed raw words +%%% exro928_seed({S0,S1,S2,S3,S4,S5,S6,S7,S8,S9,S10,S11,S12,S13,S14,S15}) +%%% when S0 bor S1 bor S2 bor S3 bor S4 bor S5 bor S6 bor S7 bor +%%% S8 bor S9 bor S10 bor S11 bor S12 bor S13 bor S14 bor S15 > 0, +%%% S0 bor S1 bor S2 bor S3 bor S4 bor S5 bor S6 bor S7 bor +%%% S8 bor S9 bor S10 bor S11 bor S12 bor S13 bor S14 bor S15 < 1 bsl 58 -> +%%% {[S0,S1,S2,S3,S4,S5,S6,S7,S8,S9,S10,S11,S12,S13,S14,S15], []}; +%%% %% +%%% %% Seed from one 64-bit integer through splitmix +%%% exro928_seed(X) when is_integer(X), 0 =< X, X =< 1 bsl 64 -> +%%% {exro928_seed(X, 16),[]}; +%% +%% Seed from traditional triple - splitmix mixed with the 3 integers +exro928_seed({A1, A2, A3}) -> + {S0, X0} = seed58(?MASK(64, A1)), + {S1, X1} = seed58(?MASK(64, A2 bxor X0)), + {S2, X2} = seed58(?MASK(64, A3 bxor X1)), + {[S0,S1,S2|exro928_seed(X2, 13)],[]}. +%% +%% Splitmix seed the rest of the state words +exro928_seed(_X, 0) -> + []; +exro928_seed(X, N) -> + {S, NewX} = seed58(X), + [S|exro928_seed(NewX, N-1)]. + + +%% Update the state and calculate output word +-spec exro928ss_next(exro928_state()) -> {uint58(), exro928_state()}. +exro928ss_next({[S15,S0|Ss], Rs}) -> + SR = exro928_next_state(Ss, Rs, S15, S0), + %% + %% {S, R, T} = {5, 7, 9} + %% const uint64_t result_starstar = rotl(s0 * S, R) * T; + %% + %% The multiply by add shifted trick avoids creating bignums + %% which improves performance significantly + %% + V0 = ?MASK(58, S0 + ?BSL(58, S0, 2)), % V0 = S0 * 5 + V1 = ?ROTL(58, V0, 7), + V = ?MASK(58, V1 + ?BSL(58, V1, 3)), % V = V1 * 9 + {V, SR}; +exro928ss_next({[S15], Rs}) -> + exro928ss_next({[S15|lists:reverse(Rs)], []}). + +%% Just update the state +exro928_next_state({[S15,S0|Ss], Rs}) -> + exro928_next_state(Ss, Rs, S15, S0); +exro928_next_state({[S15], Rs}) -> + [S0|Ss] = lists:reverse(Rs), + exro928_next_state(Ss, [], S15, S0). + +exro928_next_state(Ss, Rs, S15, S0) -> + %% {A, B, C} = {44, 9, 45}, + %% s15 ^= s0; + %% NewS15: s[q] = rotl(s0, A) ^ s15 ^ (s15 << B); + %% NewS0: s[p] = rotl(s15, C); + %% + Q = S15 bxor S0, + NewS15 = ?ROTL(58, S0, 44) bxor Q bxor ?BSL(58, Q, 9), + NewS0 = ?ROTL(58, Q, 45), + {[NewS0|Ss], [NewS15|Rs]}. + + +exro928ss_uniform({Alg, SR}) -> + {V, NewSR} = exro928ss_next(SR), + {(V bsr (58-53)) * ?TWO_POW_MINUS53, {Alg, NewSR}}. + +exro928ss_uniform(Range, {Alg, SR}) -> + {V, NewSR} = exro928ss_next(SR), + MaxMinusRange = ?BIT(58) - Range, + ?uniform_range(Range, Alg, NewSR, V, MaxMinusRange, I). + + +-spec exro928_jump({alg_handler(), exro928_state()}) -> + {alg_handler(), exro928_state()}. +exro928_jump({Alg, SR}) -> + {Alg,exro928_jump_2pow512(SR)}. + +exro928_jump_2pow512(SR) -> + polyjump( + SR, fun exro928_next_state/1, + %% 2^512 + [16#4085302F77130CA, 16#54E07F7F4524091, + 16#5E1D7D6813D2BA0, 16#4687ACEF8644287, + 16#4567FD9F0B83FE5, 16#43E6D27EA06C024, + 16#641E015AC26D5D2, 16#6CD61377663B92F, + 16#70A0657E19F00D4, 16#43C0BDDE15CF3C3, + 16#745A3A8A3CEF3CC, 16#58A8CF308C8E0C6, + 16#7B7C4CC049C536E, 16#431801F9DB3AF2C, + 16#41A1504ACD83F24, 16#6C41DCF2F867D7F]). + +exro928_jump_2pow20(SR) -> + polyjump( + SR, fun exro928_next_state/1, + %% 2^20 + [16#5B966A3DAF905E6, 16#601E9589C33DE2F, + 16#74A806C17E9E644, 16#59AFEB4F1DF6A43, + 16#46D8E75664A4268, 16#42E2C246BDA670C, + 16#4531FB8EBC73C70, 16#537F702069EFC52, + 16#4B6DC4D3AF15F8A, 16#5A4189F0050263D, + 16#46DF2EB1FEAF0FB, 16#77AC696A43CB9AC, + 16#4C5878C3549BD5B, 16#7CCF20BCF522920, + 16#415FEFCC78FF45E, 16#72CF460728C2FAF]). + +%% ===================================================================== +%% Polynomial jump with a jump constant word list, +%% high bit in each word marking top of word, +%% SR is a {Forward, Reverse} queue tuple with Forward never empty +%% ===================================================================== + +polyjump({Ss, Rs} = SR, NextState, JumpConst) -> + %% Create new state accumulator T + Ts = lists:duplicate(length(Ss) + length(Rs), 0), + polyjump(SR, NextState, JumpConst, Ts). +%% +%% Foreach jump word +polyjump(_SR, _NextState, [], Ts) -> + %% Return new calculated state + {Ts, []}; +polyjump(SR, NextState, [J|Js], Ts) -> + polyjump(SR, NextState, Js, Ts, J). +%% +%% Foreach bit in jump word until top bit +polyjump(SR, NextState, Js, Ts, 1) -> + + polyjump(SR, NextState, Js, Ts); +polyjump({Ss, Rs} = SR, NextState, Js, Ts, J) when J =/= 0 -> + NewSR = NextState(SR), + NewJ = J bsr 1, + case ?MASK(1, J) of + 0 -> + polyjump(NewSR, NextState, Js, Ts, NewJ); + 1 -> + %% Xor this state onto T + polyjump(NewSR, NextState, Js, xorzip_sr(Ts, Ss, Rs), NewJ) + end. + +xorzip_sr([], [], undefined) -> + []; +xorzip_sr(Ts, [], Rs) -> + xorzip_sr(Ts, lists:reverse(Rs), undefined); +xorzip_sr([T|Ts], [S|Ss], Rs) -> + [T bxor S|xorzip_sr(Ts, Ss, Rs)]. + +%% ===================================================================== + +format_jumpconst58(String) -> + ReOpts = [{newline,any},{capture,all_but_first,binary},global], + {match,Matches} = re:run(String, "0x([a-zA-Z0-9]+)", ReOpts), + format_jumcons58_matches(lists:reverse(Matches), 0). + +format_jumcons58_matches([], J) -> + format_jumpconst58_value(J); +format_jumcons58_matches([[Bin]|Matches], J) -> + NewJ = (J bsl 64) bor binary_to_integer(Bin, 16), + format_jumcons58_matches(Matches, NewJ). + +format_jumpconst58_value(0) -> + ok; +format_jumpconst58_value(J) -> + io:format("16#~s,~n", [integer_to_list(?MASK(58, J) bor ?BIT(58), 16)]), + format_jumpconst58_value(J bsr 58). + %% ===================================================================== %% exrop PRNG: Xoroshiro116+ %% @@ -961,6 +1226,34 @@ exrop_jump([S__0|S__1] = _S, S0, S1, J, Js) -> exrop_jump(NewS, S0, S1, J bsr 1, Js) end. +%% ===================================================================== +%% 58-bit seeder; lowest 58 bits of SplitMix64, zeros skipped +%% +%% uint64_t splitmix64_next() { +%% uint64_t z = (x += 0x9e3779b97f4a7c15); +%% z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9; +%% z = (z ^ (z >> 27)) * 0x94d049bb133111eb; +%% return z ^ (z >> 31); +%% } +%% ===================================================================== + +seed58(0, _X) -> + []; +seed58(N, X) -> + {Z,NewX} = seed58(X), + [Z|seed58(N - 1, NewX)]. +%% +seed58(X_0) -> + X = ?MASK(64, X_0 + 16#9e3779b97f4a7c15), + Z_0 = ?MASK(64, (X bxor (X bsr 30)) * 16#bf58476d1ce4e5b9), + Z_1 = ?MASK(64, (Z_0 bxor (Z_0 bsr 27)) * 16#94d049bb133111eb), + case ?MASK(58, Z_1 bxor (Z_1 bsr 31)) of + 0 -> + seed58(X); + Z -> + {Z, X} + end. + %% ===================================================================== %% Ziggurat cont %% ===================================================================== diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl index a7094f1987..2a721dad0d 100644 --- a/lib/stdlib/test/rand_SUITE.erl +++ b/lib/stdlib/test/rand_SUITE.erl @@ -34,12 +34,15 @@ stats_standard_normal/1, uniform_real_conv/1, plugin/1, measure/1, - reference_jump_state/1, reference_jump_procdict/1]). + reference_jump_state/1, reference_jump_procdict/1, + short_jump/1]). -export([test/0, gen/1]). -export([uniform_real_gen/1, uniform_gen/2]). +-compile(export_all). + -include_lib("common_test/include/ct.hrl"). -define(LOOP, 1000000). @@ -56,7 +59,8 @@ all() -> {group, distr_stats}, uniform_real_conv, plugin, measure, - {group, reference_jump} + {group, reference_jump}, + short_jump ]. groups() -> @@ -95,7 +99,7 @@ test() -> end, Tests). algs() -> - [exrop, exsp, exs1024s, exs64, exsplus, exs1024]. + [exrop, exsp, exs1024s, exs64, exsplus, exs1024, exro928ss]. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -267,27 +271,38 @@ reference_1(Alg) -> Refval =:= not_implemented -> Refval; true -> case gen(Alg) of - Refval -> ok; + Refval -> + io:format("Ok: ~p~n",[Alg]), + ok; Testval -> io:format("Failed: ~p~n",[Alg]), io:format("Length ~p ~p~n",[length(Refval), length(Testval)]), io:format("Head ~p ~p~n",[hd(Refval), hd(Testval)]), + show_wrong(Refval, Testval), exit(wrong_value) end end. +show_wrong([], []) -> + ok; +show_wrong([H|T1], [H|T2]) -> + show_wrong(T1, T2); +show_wrong([H1|_], [H2|_]) -> + io:format("Wrong ~p ~p~n",[H1,H2]). + + gen(Algo) -> State = - case Algo of - exs64 -> %% Printed with orig 'C' code and this seed + if + Algo =:= exs64 -> %% Printed with orig 'C' code and this seed rand:seed_s({exs64, 12345678}); - _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop -> + Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop -> %% Printed with orig 'C' code and this seed rand:seed_s({Algo, [12345678|12345678]}); - _ when Algo =:= exs1024; Algo =:= exs1024s -> + Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss -> %% Printed with orig 'C' code and this seed rand:seed_s({Algo, {lists:duplicate(16, 12345678), []}}); - _ -> + true -> rand:seed(Algo, {100, 200, 300}) end, Max = range(State), @@ -1173,7 +1188,7 @@ gen_jump_1(Algo) -> %% Printed with orig 'C' code and this seed gen_jump_2( rand:seed_s({Algo, [12345678|12345678]})); - _ when Algo =:= exs1024; Algo =:= exs1024s -> + _ when Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss -> %% Printed with orig 'C' code and this seed gen_jump_2( rand:seed_s({Algo, {lists:duplicate(16, 12345678), []}})) @@ -1227,7 +1242,7 @@ gen_jump_p1(Algo) -> %% Printed with orig 'C' code and this seed gen_jump_p2( rand:seed({Algo, [12345678|12345678]})); - _ when Algo =:= exs1024; Algo =:= exs1024s -> + _ when Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss -> %% Printed with orig 'C' code and this seed gen_jump_p2( rand:seed({Algo, {lists:duplicate(16, 12345678), []}})) @@ -1247,6 +1262,37 @@ gen_jump_p3(N, Max, Acc) when N > 0 -> end; gen_jump_p3(_, _, Acc) -> lists:reverse(Acc). +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +short_jump(Config) when is_list(Config) -> + State_0 = {#{bits := Bits},_} = rand:seed(exro928ss, {1,2,3}), + Range = 1 bsl Bits, + %% + State_1a = repeat(1 bsl 20, Range, State_0), + State_1b = exro928_jump_2pow20(State_0), + check(17, Range, State_1a, State_1b), + %% + {_,State_2a} = rand:uniform_s(Range, State_1a), + State_3a = exro928_jump_2pow20(State_2a), + State_3b = repeat((1 bsl 20) + 1, Range, State_1b), + check(17, Range, State_3a, State_3b). + +exro928_jump_2pow20({Alg, AlgState}) -> + {Alg,rand:exro928_jump_2pow20(AlgState)}. + +repeat(0, _Range, State) -> + State; +repeat(N, Range, State) -> + {_, NewState} = rand:uniform_s(Range, State), + repeat(N - 1, Range, NewState). + +check(0, _Range, _StateA, _StateB) -> + ok; +check(N, Range, StateA, StateB) -> + {V,NewStateA} = rand:uniform_s(Range, StateA), + {V,NewStateB} = rand:uniform_s(Range, StateB), + check(N - 1, Range, NewStateA, NewStateB). + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%% Data reference_val(exs64) -> @@ -1397,6 +1443,46 @@ reference_val(exrop) -> 98223942961505519,191144652663779840, 102425686803727694,89058927716079076,80721467542933080, 8462479817391645,2774921106204163]; + +reference_val(exro928ss) -> +%% Same as for exrop, but this state init: +%% for (n = 0; n < 16; n++) { +%% s[n] = 12345678; + [16#000000108e8d5b01,16#03604028f2769dff,16#007f92f60bc7170c, + 16#035ea81a9898a5e2,16#0104c90c5a0c8178,16#0313514025cca717, + 16#03c5506b2a2e98cf,16#0098a5405961552e,16#004ad29eabb785a0, + 16#033ea8ec4efb8058,16#00b21545e62bef1c,16#0333fc5320703482, + 16#02c3c650e51a8d47,16#03a3b7fc848c9cda,16#03775adea6cddff5, + 16#01ae5499c9049973,16#03d3c90e5504e16b,16#0383cd6b6cb852e6, + 16#009c8d0996ef543a,16#0059cf671371af60,16#03dfd68ed980b719, + 16#0290f2a0acf2c5b0,16#029061df18d63b55,16#02e702ea4b45137b, + 16#029a0ccca604d848,16#01664c7cd31f0fa6,16#00dced83e60ccddc, + 16#008764d2c9a05f3e,16#02b9ca5f6a80c4ba,16#02daf93d2c566750, + 16#0147d326ead18ace,16#014b452efc19297f,16#0242d3f7a7237eca, + 16#0141bb68c2abce39,16#02d798e1230baf45,16#0216bf8f25c1ec2d, + 16#003a43ea733f1e1f,16#036c75390db736f3,16#028cca5f5f48c6f9, + 16#0186e4a17174d6cf,16#02152679dfa4c25c,16#01429b9f15e3b9d6, + 16#0134a61411d22bb0,16#01593f7d970d1c94,16#0205a7d8a305490f, + 16#01dd092272595a9c,16#0028c95208aad2d4,16#016347c25cc24162, + 16#025306acfb891309,16#0207a07e2bebef2f,16#024ee78d86ff5288, + 16#030b53192db97613,16#03f765cb9e98e611,16#025ec35a1e237377, + 16#03d81fd73102ef6f,16#0242dc8fea9a68b2,16#00abb876c1d4ea1b, + 16#00871ffd2b7e45fb,16#03593ff73c9be08d,16#00b96b2b8aca3688, + 16#0174aba957b7cf7b,16#012b7a5d4cf4a5b7,16#032a5260f2123db8, + 16#00f9374d88ee0080,16#030df39bec2ad657,16#00dce0cb81d006c4, + 16#038213b806303c76,16#03940aafdbfabf84,16#0398dbb26aeba037, + 16#01eb28d61951587f,16#00fed3d2aacfeef4,16#03499587547d6e40, + 16#01b192fe6e979e3c,16#00e974bf5f0a26d0,16#012ed94f76459c83, + 16#02d76859e7a82587,16#00d1d2c7b791f51b,16#03988058017a031b, + 16#00bbcf4b59d8e86d,16#015ed8b73a1b767c,16#0277283ea6a5ee74, + 16#002211460dd6d422,16#001ad62761ee9fbd,16#037311b44518b067, + 16#02b5ed61bf70904e,16#011862a05c1929fa,16#014be68683c3bab4, + 16#025c29aa5c508b07,16#00895c6106f97378,16#026ce91a3d671c7f, + 16#02591f4c74784293,16#02f0ed2a70bc1853,16#00a2762ff614bfbc, + 16#008f4e354f0c20d4,16#038b66fb587ed430,16#00636296e188de89, + 16#0278fadd143e74f5,16#029697ccf1b3a4c2,16#011eccb273404458, + 16#03f204064a9fe0c0]; + reference_val(_) -> not_implemented. @@ -1527,6 +1613,46 @@ reference_jump_val(exrop) -> 241227318715885854,77323084015890802, 1663590009695191,234064400749487599,222983191707424780, 254956809144783896,203898972156838252]; + +reference_jump_val(exro928ss) -> +%% Same as for exrop, but this state init: +%% for (n = 0; n < 16; n++) { +%% s[n] = 12345678; + [16#031ee449e53b6689,16#001afeee12813137,16#005e2172711df36b, + 16#02850aea3a595d36,16#0029705187e891c7,16#001794badd489667, + 16#00ab621be15be56c,16#024b663a6924786b,16#03cab70b8ab854bf, + 16#01daa37601285320,16#02db955a53c40e89,16#01fbef51d5c65891, + 16#02fecf4116ed5f77,16#0349c2057246ac5d,16#01217f257c4fa148, + 16#0367ee84d020697d,16#01d5cf647fe23335,16#020941838adfb750, + 16#02c2da26b1d7b3e5,16#00d1583d34cea6c0,16#038be9cb5b527f50, + 16#00bfa93c1d7f4864,16#03778912a4f56b14,16#037fcabc483fa5c5, + 16#00a3c9de6aaf5fc7,16#03600b883b2f2b42,16#03797a99ffddfdfb, + 16#0189fead429945b7,16#0103ac90cd912508,16#03e3d872fd950d64, + 16#0214fc3e77dc2f02,16#02a084f4f0e580ca,16#035d2fe72266a7f3, + 16#02887c49ae7e41a4,16#0011dc026af83c51,16#02d28bfd32c2c517, + 16#022e4165c33ad4f3,16#01f053cf0687b052,16#035315e6e53c8918, + 16#01255312da07b572,16#0237f1da11ec9221,16#02faf2e282fb1fb1, + 16#0227423ec1787ebc,16#011fa5eb1505571c,16#0275ff9eaaa1abdd, + 16#03e2d032c3981cb4,16#0181bb32d51d3072,16#01b1d3939b9f16ec, + 16#0259f09f55d1112f,16#0396464a2767e428,16#039777c0368bdb9e, + 16#0320925f35f36c5f,16#02a35289e0af1248,16#02e80bd4bc72254b, + 16#00a8b11af1674d68,16#027735036100a69e,16#03c8c268ded7f254, + 16#03de80aa57c65217,16#00f2247754d24000,16#005582a42b467f89, + 16#0031906569729477,16#00fd523f2ca4fefe,16#00ad223113d1e336, + 16#0238ddf026cbfca9,16#028b98211cfed876,16#0354353ebcc0de9a, + 16#009ee370c1e154f4,16#033131af3b8a7f88,16#032291baa45801e3, + 16#00941fc2b45eb217,16#035d6a61fa101647,16#03fdb51f736f1bbc, + 16#0232f7b98539faa0,16#0311b35319e3a61e,16#0048356b17860eb5, + 16#01a205b2554ce71e,16#03f873ea136e29d6,16#003c67d5c3df5ffd, + 16#00cd19e7a8641648,16#0149a8c54e4ba45e,16#0329498d134d2f6a, + 16#03b69421ae65ee2b,16#01a8d20b59447429,16#006b2292571032a2, + 16#00c193b17da22ba5,16#01faa7ab62181249,16#00acd401cd596a00, + 16#005b5086c3531402,16#0259113d5d3d058d,16#00bef3f3ce4a43b2, + 16#014837a4070b893c,16#00460a26ac2eeec1,16#026219a8b8c63d7e, + 16#03c7b8ed032cf5a6,16#004da912a1fff131,16#0297de3716215741, + 16#0079fb9b4c715466,16#00a73bad4ae5a356,16#0072e606c0d4ab86, + 16#02374382d5f9bd2e]; + reference_jump_val(_) -> not_implemented. diff --git a/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt b/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt new file mode 100644 index 0000000000..150f37fcfa --- /dev/null +++ b/lib/stdlib/test/rand_Xoroshiro928ss_dev.txt @@ -0,0 +1,343 @@ +%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% + + +Memorable facts from designing the Xoroshiro928** generator +=========================================================== +AKA: exro928ss in the rand module + +Author: Raimo Niskanen, for the Erlang/OTP team @ Ericsson. + +Reference URL: http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf +i.e the Xoroshiro1024 generator with ** scrambler: + + int p; + uint64_t s[16]; + + const int q = p; + const uint64_t s0 = s[p = (p + 1) & 15]; + + uint64_t s15 = s[q]; + + const uint64_t result_starstar = rotl(s0 * S, R) * T; + + s15 ^= s0; + s[q] = rotl(s0, A) ^ s15 ^ (s15 << B); + s[p] = rotl(s15, C); + +Where {S, R, T} = {5, 7, 9} as recommended in the paper. + +We want to scale down to 58 bit words (16 of them) +so we get a generator with period 2^928 - 1. + +{A, B, C} were deduced as follows +--------------------------------- + +First, to find which triplets that give a full period generator +one have to factor 2^928 - 1. + +https://www.alpertron.com.ar/ECM.HTM actually could do that +and gave the result: + + Value + 2^928 - 1 + + 2269 007733 883335 972287 082669 296112 915239 349672 942191 252221 331572 + 442536 403137 824056 312817 862695 551072 066953 619064 625508 194663 + 368599 769448 406663 254670 871573 830845 597595 897613 333042 429214 + 224697 474472 410882 236254 024057 110212 260250 671521 235807 709272 + 244389 361641 091086 035023 229622 419455 (280 digits) = 3 × 5 × 17 × 59 × + 233 × 257 × 929 × 1103 × 2089 × 5569 × 8353 × 59393 × 65537 × 3 033169 × + 39 594977 × 107 367629 × 536 903681 × 748 264961 × 2245 984577 × 239 + 686663 718401 × 15 929619 591127 520827 829953 × 82280 195167 144119 + 832390 568177 × 6033 312171 721035 031651 315652 130497 (34 digits) × 18 + 774318 450142 955120 650303 957350 521748 903233 (44 digits) × 15 694604 + 006012 505869 851221 169365 594050 637743 819041 (50 digits) + +Sebastiano Vigna from that calculated all full period triplets, at the +end of this document, and the ones with the highest degree were: + + 23-25-12 411 + 36-2-35 411 + 55-5-54 411 + 14-19-11 415 + 34-37-5 415 + 37-3-56 415 + 55-11-54 417 + 30-3-41 419 + 11-45-50 423 + 50-19-47 423 + 52-27-13 427 + 54-9-25 433 + 56-43-35 433 + 44-9-45 441 + +All these candidates were tested with TestU01-1.2.3: + + http://simul.iro.umontreal.ca/testu01/tu01.html + +A plugin was created with parameters for {A, B, C} and, since +TestU01 is a 32-bit test tool, with parameters to reverse +the generated bits or not, and to take the 32 highest or lowest +bits from the reversed or non-reversed 58 bit output. + +The generators were seeded with a SplitMix64 generator like the +one used for seeding this generator in the rand module, +taking the 58 lowest bits and wasting all zero values. + +3 runs were made with all candidates and all four bit selection variants. +For these runs the seeder was initialized with 12345678, 876543212345678 +and 1234567890. + +After all these runs the candidate with the highest degree: 44-9-45 +had not gotten any suspicious p-value at all. All the other +got p-values around 5.0e-4 worst 9.8e-6 suggesting only random +failures, so they would probably have worked about as well. + +Finally 44-9-45 was run through PractRand-0.93: + + http://pracrand.sourceforge.net/ + +Again, all 4 bit selection variants of 32 bits were run. +Random failures with p-values around e-3..e-4 for the "smaller" +tests, but for 8, 16 and 32 TB tests no anomalies were found +(with the internal seeder masked to 58 bits): + + Xoroshiro928High seed: 0x3178ec5d + Xoroshiro928Low seed: 0xa9a04fb9 + Xoroshiro928ReverseHigh seed: 0xfa0bdbab + Xoroshiro928ReverseLow seed: 0xada51705 + + +Then, S. Vigna calculated the 2^512 jump coefficient as well +as a 2^20 jump coefficient (for testing purposes) for 44-9-45. + +2^512: + { 0x44085302f77130ca, 0xba05381fdfd14902, 0x10a1de1d7d6813d2, + 0xb83fe51a1eb3be19, 0xa81b0090567fd9f0, 0x5ac26d5d20f9b49f, + 0x4ddd98ee4be41e01, 0x0657e19f00d4b358, 0xf02f778573cf0f0a, + 0xb45a3a8a3cef3cc0, 0x6e62a33cc2323831, 0xbcb3b7c4cc049c53, + 0x83f240c6007e76ce, 0xe19f5fc1a1504acd, 0x00000000b10773cb } + +2^20: + { 0xbdb966a3daf905e6, 0x644807a56270cf78, 0xda90f4a806c17e9e, + 0x4a426866bfad3c77, 0xaf699c306d8e7566, 0x8ebc73c700b8b091, + 0xc081a7bf148531fb, 0xdc4d3af15f8a4dfd, 0x90627c014098f4b6, + 0x06df2eb1feaf0fb6, 0x5bdeb1a5a90f2e6b, 0xa480c5878c3549bd, + 0xff45ef33c82f3d48, 0xa30bebc15fefcc78, 0x00000000cb3d181c } + +Standard jump function pseudocode: + + Jump constant j = 0xb10773cb...44085302f77130ca + Generator state: s + New generator state: t = 0 + foreach bit in j, low to high: + if the bit is one: + t ^= s + next s + s = t + + +The complete list of full period constants +------------------------------------------ + +29-48-54 27 x^928 + x^874 + x^870 + x^840 + x^814 + x^784 + x^759 + x^750 + x^724 + x^720 + x^634 + x^630 + x^600 + x^574 + x^544 + x^519 + x^510 + x^484 + x^480 + x^390 + x^360 + x^270 + x^240 + x^150 + x^120 + x^30 + 1 + +29-18-56 65 x^928 + x^870 + x^850 + x^840 + x^832 + x^821 + x^802 + x^791 + x^761 + x^750 + x^734 + x^731 + x^720 + x^712 + x^701 + x^686 + x^674 + x^671 + x^663 + x^641 + x^630 + x^611 + x^610 + x^603 + x^600 + x^596 + x^577 + x^566 + x^547 + x^524 + x^517 + x^510 + x^487 + x^480 + x^476 + x^464 + x^457 + x^446 + x^427 + x^423 + x^397 + x^390 + x^367 + x^363 + x^360 + x^356 + x^352 + x^341 + x^326 + x^322 + x^311 + x^281 + x^270 + x^251 + x^240 + x^236 + x^232 + x^221 + x^191 + x^161 + x^150 + x^131 + x^120 + x^30 + 1 + +29-32-36 81 x^928 + x^874 + x^870 + x^840 + x^820 + x^819 + x^794 + x^790 + x^770 + x^765 + x^764 + x^760 + x^754 + x^750 + x^740 + x^739 + x^735 + x^734 + x^730 + x^709 + x^704 + x^674 + x^670 + x^665 + x^660 + x^650 + x^649 + x^635 + x^634 + x^619 + x^614 + x^605 + x^595 + x^585 + x^584 + x^579 + x^564 + x^555 + x^545 + x^540 + x^524 + x^515 + x^514 + x^495 + x^490 + x^485 + x^460 + x^455 + x^435 + x^425 + x^400 + x^395 + x^375 + x^370 + x^365 + x^340 + x^315 + x^310 + x^305 + x^300 + x^290 + x^285 + x^275 + x^260 + x^250 + x^245 + x^240 + x^215 + x^190 + x^180 + x^170 + x^150 + x^130 + x^120 + x^115 + x^105 + x^100 + x^75 + x^40 + x^30 + 1 + +39-10-20 105 x^928 + x^898 + x^870 + x^841 + x^840 + x^808 + x^783 + x^782 + x^781 + x^780 + x^778 + x^754 + x^752 + x^724 + x^721 + x^720 + x^696 + x^692 + x^688 + x^667 + x^666 + x^661 + x^660 + x^658 + x^638 + x^636 + x^632 + x^609 + x^607 + x^606 + x^605 + x^604 + x^601 + x^600 + x^580 + x^578 + x^576 + x^572 + x^568 + x^549 + x^548 + x^541 + x^540 + x^538 + x^516 + x^512 + x^489 + x^485 + x^484 + x^481 + x^480 + x^456 + x^452 + x^448 + x^429 + x^428 + x^421 + x^420 + x^418 + x^400 + x^398 + x^396 + x^392 + 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