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-rw-r--r--lib/stdlib/doc/src/rand.xml81
-rw-r--r--lib/stdlib/src/rand.erl148
-rw-r--r--lib/stdlib/test/rand_SUITE.erl71
3 files changed, 247 insertions, 53 deletions
diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
index 25eec216ef..8e657698c6 100644
--- a/lib/stdlib/doc/src/rand.xml
+++ b/lib/stdlib/doc/src/rand.xml
@@ -38,34 +38,50 @@
<p>
This module provides a pseudo random number generator.
The module contains a number of algorithms.
- The uniform distribution algorithms use the
+ The uniform distribution algorithms are based on the
<url href="http://xorshift.di.unimi.it">
- xoroshiro116+ and xorshift1024* algorithms by Sebastiano Vigna.
+ Xoroshiro and Xorshift algorithms
</url>
+ by Sebastiano Vigna.
The normal distribution algorithm uses the
<url href="http://www.jstatsoft.org/v05/i08">
Ziggurat Method by Marsaglia and Tsang
</url>
on top of the uniform distribution algorithm.
</p>
- <p>For some algorithms, jump functions are provided for generating
- non-overlapping sequences for parallel computations.
- The jump functions perform calculations
- equivalent to perform a large number of repeated calls
- for calculating new states. </p>
+ <p>
+ For most algorithms, jump functions are provided for generating
+ non-overlapping sequences for parallel computations.
+ The jump functions perform calculations
+ equivalent to perform a large number of repeated calls
+ for calculating new states.
+ </p>
<p>The following algorithms are provided:</p>
<taglist>
- <tag><c>exrop</c></tag>
+ <tag><c>exsss</c></tag>
<item>
- <p>Xoroshiro116+, 58 bits precision and period of 2^116-1</p>
+ <p>Xorshift116**, 58 bits precision and period of 2^116-1</p>
<p>Jump function: equivalent to 2^64 calls</p>
- </item>
- <tag><c>exs1024s</c></tag>
- <item>
- <p>Xorshift1024*, 64 bits precision and a period of 2^1024-1</p>
- <p>Jump function: equivalent to 2^512 calls</p>
+ <p>
+ This is the Xorshift116 generator combined with the StarStar scrambler
+ from the 2018 paper by David Blackman and Sebastiano Vigna:
+ <url href="http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf">
+ Scrambled Linear Pseudorandom Number Generators
+ </url>
+ </p>
+ <p>
+ The generator does not need 58-bit rotates so it is faster
+ than the Xoroshiro116 generator, and when combined with
+ the StarStar scrambler it does not have any weak low bits
+ like <c>exrop</c> (Xoroshiro116+).
+ </p>
+ <p>
+ Alas, this combination is about 10% slower than <c>exrop</c>,
+ but is despite that the default algorithm thanks to its
+ statistical qualities.
+ </p>
</item>
<tag><c>exro928ss</c></tag>
<item>
@@ -77,8 +93,8 @@
<url href="http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf">
Scrambled Linear Pseudorandom Number Generators
</url>
- that on a 64 bit Erlang system executes only about 30% slower than
- the default <c>exrop</c> algorithm but with much longer period
+ that on a 64 bit Erlang system executes only about 40% slower than
+ the default <c>exsss</c> algorithm but with much longer period
and better statistical properties, and on the flip side
a larger state.
</p>
@@ -87,6 +103,16 @@
the 58 bit adaption.
</p>
</item>
+ <tag><c>exrop</c></tag>
+ <item>
+ <p>Xoroshiro116+, 58 bits precision and period of 2^116-1</p>
+ <p>Jump function: equivalent to 2^64 calls</p>
+ </item>
+ <tag><c>exs1024s</c></tag>
+ <item>
+ <p>Xorshift1024*, 64 bits precision and a period of 2^1024-1</p>
+ <p>Jump function: equivalent to 2^512 calls</p>
+ </item>
<tag><c>exsp</c></tag>
<item>
<p>Xorshift116+, 58 bits precision and period of 2^116-1</p>
@@ -103,7 +129,7 @@
</taglist>
<p>
- The default algorithm is <c>exrop</c> (Xoroshiro116+).
+ The default algorithm is <c>exsss</c> (Xorshift116**).
If a specific algorithm is
required, ensure to always use <seealso marker="#seed-1">
<c>seed/1</c></seealso> to initialize the state.
@@ -174,19 +200,19 @@ R1 = rand:uniform(),</pre>
<p>Use a specified algorithm:</p>
<pre>
-_ = rand:seed(exs1024s),
+_ = rand:seed(exs928ss),
R2 = rand:uniform(),</pre>
<p>Use a specified algorithm with a constant seed:</p>
<pre>
-_ = rand:seed(exs1024s, {123, 123534, 345345}),
+_ = rand:seed(exs928ss, {123, 123534, 345345}),
R3 = rand:uniform(),</pre>
<p>Use the functional API with a non-constant seed:</p>
<pre>
-S0 = rand:seed_s(exrop),
+S0 = rand:seed_s(exsss),
{R4, S1} = rand:uniform_s(S0),</pre>
<p>Textbook basic form Box-Muller standard normal deviate</p>
@@ -215,8 +241,9 @@ SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)</pre>
</note>
<p>
- For all these generators except <c>exro928ss</c> the lowest bit(s)
- has got a slightly less random behaviour than all other bits.
+ For all these generators except <c>exro928ss</c> and <c>exsss</c>
+ the lowest bit(s) has got a slightly less
+ random behaviour than all other bits.
1 bit for <c>exrop</c> (and <c>exsp</c>),
and 3 bits for <c>exs1024s</c>.
See for example the explanation in the
@@ -231,7 +258,7 @@ up to (and included) 16TB, with the exception of binary rank tests,
which fail due to the lowest bit being an LFSR; all other bits pass all
tests. We suggest to use a sign test to extract a random Boolean value.</pre>
<p>
- If this is a problem; to generate a boolean
+ If this is a problem; to generate a boolean with these algorithms
use something like this:
</p>
<pre>(rand:uniform(16) > 8)</pre>
@@ -299,19 +326,19 @@ tests. We suggest to use a sign test to extract a random Boolean value.</pre>
</desc>
</datatype>
<datatype>
- <name name="exrop_state"/>
+ <name name="exsplus_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
- <name name="exs1024_state"/>
+ <name name="exro928_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
- <name name="exro928_state"/>
+ <name name="exrop_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
- <name name="exsplus_state"/>
+ <name name="exs1024_state"/>
<desc><p>Algorithm specific internal state</p></desc>
</datatype>
<datatype>
diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl
index 9854c778a1..3a9a1e007b 100644
--- a/lib/stdlib/src/rand.erl
+++ b/lib/stdlib/src/rand.erl
@@ -43,13 +43,13 @@
%% Debug
-export([make_float/3, float2str/1, bc64/1]).
--compile({inline, [exs64_next/1, exsplus_next/1,
+-compile({inline, [exs64_next/1, exsplus_next/1, exsss_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]}).
--define(DEFAULT_ALG_HANDLER, exrop).
+-define(DEFAULT_ALG_HANDLER, exsss).
-define(SEED_DICT, rand_seed).
%% =====================================================================
@@ -86,7 +86,7 @@
%% This depends on the algorithm handler function
-type alg_state() ::
- exrop_state() | exs1024_state() | exro928_state() | exsplus_state() |
+ exsplus_state() | exro928_state() | exrop_state() | exs1024_state() |
exs64_state() | term().
%% This is the algorithm handling definition within this module,
@@ -131,7 +131,7 @@
%% Algorithm state
-type state() :: {alg_handler(), alg_state()}.
-type builtin_alg() ::
- exrop | exs1024s | exro928ss | exsp | exs64 | exsplus | exs1024.
+ exsss | exro928ss | exrop | exs1024s | exsp | exs64 | exsplus | exs1024.
-type alg() :: builtin_alg() | atom().
-type export_state() :: {alg(), alg_state()}.
-type seed() :: [integer()] | integer() | {integer(), integer(), integer()}.
@@ -139,7 +139,7 @@
[builtin_alg/0, alg/0, alg_handler/0, alg_state/0,
state/0, export_state/0, seed/0]).
-export_type(
- [exrop_state/0, exs1024_state/0, exro928_state/0, exsplus_state/0,
+ [exsplus_state/0, exro928_state/0, exrop_state/0, exs1024_state/0,
exs64_state/0]).
%% =====================================================================
@@ -618,6 +618,11 @@ mk_alg(exsp) ->
uniform=>fun exsp_uniform/1, uniform_n=>fun exsp_uniform/2,
jump=>fun exsplus_jump/1},
fun exsplus_seed/1};
+mk_alg(exsss) ->
+ {#{type=>exsss, bits=>58, next=>fun exsss_next/1,
+ uniform=>fun exsss_uniform/1, uniform_n=>fun exsss_uniform/2,
+ jump=>fun exsplus_jump/1},
+ fun exsss_seed/1};
mk_alg(exs1024) ->
{#{type=>exs1024, max=>?MASK(64), next=>fun exs1024_next/1,
jump=>fun exs1024_jump/1},
@@ -675,6 +680,36 @@ exs64_next(R) ->
%% 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.
+%%
+%% Reference C code for Xorshift116+ and Xorshift116**
+%%
+%% #include <stdint.h>
+%%
+%% #define MASK(b, v) (((UINT64_C(1) << (b)) - 1) & (v))
+%% #define BSL(b, v, n) (MASK((b)-(n), (v)) << (n))
+%% #define ROTL(b, v, n) (BSL((b), (v), (n)) | ((v) >> ((b)-(n))))
+%%
+%% uint64_t s[2];
+%%
+%% uint64_t next(void) {
+%% uint64_t s1 = s[0];
+%% const uint64_t s0 = s[1];
+%%
+%% s1 ^= BSL(58, s1, 24); // a
+%% s1 ^= s0 ^ (s1 >> 11) ^ (s0 >> 41); // b, c
+%% s[0] = s0;
+%% s[1] = s1;
+%%
+%% const uint64_t result_plus = MASK(58, s0 + s1);
+%% uint64_t result_starstar = s0;
+%% result_starstar = MASK(58, result_starstar * 5);
+%% result_starstar = ROTL(58, result_starstar, 7);
+%% result_starstar = MASK(58, result_starstar * 9);
+%%
+%% return result_plus;
+%% return result_starstar;
+%% }
+%%
%% =====================================================================
-opaque exsplus_state() :: nonempty_improper_list(uint58(), uint58()).
@@ -697,16 +732,62 @@ exsplus_seed({A1, A2, A3}) ->
tl(R1)]),
R2.
+-dialyzer({no_improper_lists, exsss_seed/1}).
+
+exsss_seed(L) when is_list(L) ->
+ [S0,S1] = seed58_nz(2, L),
+ [S0|S1];
+exsss_seed(X) when is_integer(X) ->
+ [S0,S1] = seed58(2, ?MASK(64, X)),
+ [S0|S1];
+%%
+%% Seed from traditional integer triple - mix into splitmix
+exsss_seed({A1, A2, A3}) ->
+ {_, X0} = seed58(?MASK(64, A1)),
+ {S0, X1} = seed58(?MASK(64, A2) bxor X0),
+ {S1, _} = seed58(?MASK(64, A3) bxor X1),
+ [S0|S1].
+
+%% Advance Xorshift116 state one step
+-define(
+ exs_next(S0, S1, S1_b),
+ begin
+ S1_b = S1 bxor ?BSL(58, S1, 24),
+ S1_b bxor S0 bxor (S1_b bsr 11) bxor (S0 bsr 41)
+ end).
+
+-define(
+ scramble_starstar(S, V_a, V_b),
+ begin
+ %% The multiply by add shifted trick avoids creating bignums
+ %% which improves performance significantly
+ %%
+ V_a = ?MASK(58, S + ?BSL(58, S, 2)), % V_a = S * 5
+ V_b = ?ROTL(58, V_a, 7),
+ ?MASK(58, V_b + ?BSL(58, V_b, 3)) % V_b * 9
+ end).
+
-dialyzer({no_improper_lists, exsplus_next/1}).
-%% Advance xorshift116+ state for one step and generate 58bit unsigned integer
+%% Advance state 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 ?BSL(58, S1, 24),
- S12 = S11 bxor S0 bxor (S11 bsr 11) bxor (S0 bsr 41),
- {?MASK(58, S0 + S12), [S0|S12]}.
+ NewS1 = ?exs_next(S0, S1, S1_1),
+ {?MASK(58, S0 + NewS1), [S0|NewS1]}.
+%% %% Note: members s0 and s1 are swapped here
+%% S11 = S1 bxor ?BSL(58, S1, 24),
+%% S12 = S11 bxor S0 bxor (S11 bsr 11) bxor (S0 bsr 41),
+%% {?MASK(58, S0 + S12), [S0|S12]}.
+
+-dialyzer({no_improper_lists, exsss_next/1}).
+-spec exsss_next(exsplus_state()) -> {uint58(), exsplus_state()}.
+exsss_next([S1|S0]) ->
+ %% Note: members s0 and s1 are swapped here
+ NewS1 = ?exs_next(S0, S1, S1_1),
+ {?scramble_starstar(S0, V_0, V_1), [S0|NewS1]}.
+%% {?MASK(58, S0 + NewS1), [S0|NewS1]}.
exsp_uniform({Alg, R0}) ->
{I, R1} = exsplus_next(R0),
@@ -714,18 +795,48 @@ exsp_uniform({Alg, R0}) ->
%% randomness quality than the others
{(I bsr (58-53)) * ?TWO_POW_MINUS53, {Alg, R1}}.
+exsss_uniform({Alg, R0}) ->
+ {I, R1} = exsss_next(R0),
+ {(I bsr (58-53)) * ?TWO_POW_MINUS53, {Alg, R1}}.
+
exsp_uniform(Range, {Alg, R}) ->
{V, R1} = exsplus_next(R),
MaxMinusRange = ?BIT(58) - Range,
?uniform_range(Range, Alg, R1, V, MaxMinusRange, I).
+exsss_uniform(Range, {Alg, R}) ->
+ {V, R1} = exsss_next(R),
+ MaxMinusRange = ?BIT(58) - Range,
+ ?uniform_range(Range, Alg, R1, V, MaxMinusRange, I).
+
-%% This is the jump function for the exsplus generator, equivalent
+%% This is the jump function for the exs* generators,
+%% i.e the Xorshift116 generators, 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.
-
+%%
+%% #include <stdint.h>
+%%
+%% void jump(void) {
+%% static const uint64_t JUMP[] = { 0x02f8ea6bc32c797,
+%% 0x345d2a0f85f788c };
+%% int i, b;
+%% uint64_t s0 = 0;
+%% uint64_t s1 = 0;
+%% for(i = 0; i < sizeof JUMP / sizeof *JUMP; i++)
+%% for(b = 0; b < 58; b++) {
+%% if (JUMP[i] & 1ULL << b) {
+%% s0 ^= s[0];
+%% s1 ^= s[1];
+%% }
+%% next();
+%% }
+%% s[0] = s0;
+%% s[1] = s1;
+%% }
+%%
%% -define(JUMPCONST, 16#000d174a83e17de2302f8ea6bc32c797).
%% split into 58-bit chunks
%% and two iterative executions
@@ -973,13 +1084,14 @@ exro928ss_next({[S15,S0|Ss], Rs}) ->
%% {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};
+ {?scramble_starstar(S0, V_0, V_1), SR};
+%% %% 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)], []}).
diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl
index 6c9fab51dc..7685c17967 100644
--- a/lib/stdlib/test/rand_SUITE.erl
+++ b/lib/stdlib/test/rand_SUITE.erl
@@ -79,7 +79,7 @@ test() ->
end, Tests).
algs() ->
- [exrop, exsp, exs1024s, exs64, exsplus, exs1024, exro928ss].
+ [exsss, exrop, exsp, exs1024s, exs64, exsplus, exs1024, exro928ss].
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -276,7 +276,7 @@ gen(Algo) ->
if
Algo =:= exs64 -> %% Printed with orig 'C' code and this seed
rand:seed_s(exs64, [12345678]);
- Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
+ Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop; Algo =:= exsss ->
%% Printed with orig 'C' code and this seed
rand:seed_s(Algo, [12345678,12345678]);
Algo =:= exs1024; Algo =:= exs1024s; Algo =:= exro928ss ->
@@ -438,7 +438,7 @@ stats_standard_normal_box_muller(Config) when is_list(Config) ->
([S|Z]) ->
{Z, [S]}
end,
- State = [rand:seed(exrop)],
+ State = [rand:seed(exsss)],
stats_standard_normal(NormalS, State, 3)
catch error:_ ->
{skip, "math:erfc/1 not supported"}
@@ -463,7 +463,7 @@ stats_standard_normal_box_muller_2(Config) when is_list(Config) ->
([S|Z]) ->
{Z, [S]}
end,
- State = [rand:seed(exrop)],
+ State = [rand:seed(exsss)],
stats_standard_normal(NormalS, State, 3)
catch error:_ ->
{skip, "math:erfc/1 not supported"}
@@ -475,7 +475,7 @@ stats_standard_normal(Config) when is_list(Config) ->
try math:erfc(1.0) of
_ ->
stats_standard_normal(
- fun rand:normal_s/1, rand:seed_s(exrop), Retries)
+ fun rand:normal_s/1, rand:seed_s(exsss), Retries)
catch error:_ ->
{skip, "math:erfc/1 not supported"}
end.
@@ -1071,7 +1071,7 @@ do_measure(_Config) ->
end,
State)
end,
- exrop, TMarkNormalFloat),
+ exsss, TMarkNormalFloat),
ok.
-define(LOOP_MEASURE, (?LOOP div 5)).
@@ -1170,7 +1170,7 @@ gen_jump_1(Algo) ->
catch
error:not_implemented -> [error_not_implemented]
end;
- _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
+ _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop; Algo =:= exsss ->
%% Printed with orig 'C' code and this seed
gen_jump_2(
rand:seed_s(Algo, [12345678,12345678]));
@@ -1224,7 +1224,7 @@ gen_jump_p1(Algo) ->
catch
error:not_implemented -> [error_not_implemented]
end;
- _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop ->
+ _ when Algo =:= exsplus; Algo =:= exsp; Algo =:= exrop; Algo =:= exsss ->
%% Printed with orig 'C' code and this seed
gen_jump_p2(
rand:seed(Algo, [12345678,12345678]));
@@ -1377,6 +1377,34 @@ reference_val(exsplus) ->
16#3dd493b8012970f,16#be13bed1e00e5c,16#ceef033b74ae10,16#3da38c6a50abe03,
16#15cbd1a421c7a8c,16#22794e3ec6ef3b1,16#26154d26e7ea99f,16#3a66681359a6ab6];
+reference_val(exsss) ->
+ [16#108e8d5b01,16#33b72092117209a,16#224d4d2961a2d0a,16#2c4c81aac3da48d,
+ 16#2f4bc39bfc36f3a,16#41826d4c4d243a,16#19871b8bb4e23ee,16#3e2112cdf9384b1,
+ 16#69801943bf91ab,16#2de1a603c31ec45,16#a90ca1991b831e,16#51ca29571a69a7,
+ 16#93ce3e511906cf,16#93ebc5768aef75,16#2412f284b902ae7,16#1ac10e758410c52,
+ 16#3f32494560368f6,16#39a5e82dcf0de95,16#3f4b14d59cc6a21,16#3174668db0b36ae,
+ 16#1449812fb8bd54e,16#eaca1f8ece51e1,16#2564b2545fd23c1,16#3cf3a2d2217e0d7,
+ 16#226f4164ba1d054,16#10dac9ae207ceef,16#17f2c4b2d40fcb9,16#1c1b282d386fdcb,
+ 16#a264f450ba2912,16#2a0a1dd67e52666,16#2be84eb835cb1e1,16#2a1cd9aa16ccc37,
+ 16#7dd5e8c2b3f490,16#254a3db4976c05b,16#2a0a67971ec1e63,16#13a0cbf7c0eed8a,
+ 16#3192d7edc0a20bc,16#2705ad756292e84,16#3ec429a18119c81,16#25944b38baa975b,
+ 16#291dcc43e3256f4,16#30d10b759237db,16#c1522a652058a,16#8ef1e9378381e6,
+ 16#1f442f33c2439f4,16#186087710a73818,16#12887f94b2b8387,16#3e42e8b1f3c9b4b,
+ 16#e462859d55f9d8,16#2356ae85be908de,16#15e96a927b3bc52,16#35c6dc52511ce46,
+ 16#7bc0624ce66e01,16#33ab7d95b738322,16#26f01effc182aa0,16#1b66ae7eaafea88,
+ 16#278f3dc14943b90,16#22178bc8d8faf28,16#396c37d53c11985,16#5e0d79d0b10f18,
+ 16#1be3de3b5675ec,16#d4db298f1f4b50,16#2da6cb99bb5c7b1,16#130b2dc17d03be8,
+ 16#f1847e7e059e9f,16#2da6591788326e7,16#222e4a18c24211c,16#949213ca49baab,
+ 16#b5129fec56f6a2,16#30f25f1e926f43e,16#1ddd8d04445fb4d,16#15995b542514150,
+ 16#1595fe879296296,16#e2f237a488453b,16#23e5cd2d6047890,16#3a5dc88fc954666,
+ 16#89bca9969b103,16#5e6893cd35dc63,16#1fed534feeeef5a,16#26f40e2147ee558,
+ 16#30c131a00625837,16#2618a7e617422e9,16#23630b297e45e7,16#1143b17502f3219,
+ 16#15607dac41168da,16#2886bdc314b3fb8,16#465d1cc1536546,16#30b09123e3a02e4,
+ 16#245a375f810be52,16#6a1b0792376a03,16#221425f59f2470f,16#867ce16dfac81c,
+ 16#9c62d95fae9b58,16#380381db1394426,16#34908dedc01c324,16#1f0ff517089b561,
+ 16#1571366dd873d32,16#3ee353dc56e192,16#15a1dee8d889b11,16#41036ad76d9888
+ ];
+
reference_val(exsp) ->
reference_val(exsplus);
reference_val(exs1024s) ->
@@ -1517,6 +1545,33 @@ reference_jump_val(exsplus) ->
12504080415362731, 45083100453836317, 270968267812126657, 93505647407734103,
252852934678537969, 258758309277167202, 74250882143432077, 141629095984552833];
+reference_jump_val(exsss) ->
+ [16#304ae783d40db2b,16#1dfb196b3a5600a,16#2a24116effc6a0d,16#1f138d68c56725,
+ 16#9360a445e2f989,16#32ed8080390e242,16#294ca85a270cff6,16#1418e6296a88bf,
+ 16#114fae3dc578ba7,16#479c42c760eb72,16#334a40655df22d6,16#e7a85dd4d37d72,
+ 16#181db16c8925c77,16#1b8a5a8afd16cbd,16#329107bf9777a39,16#2fc915c08535e42,
+ 16#16696d142c6078,16#2e2a2601c919448,16#2246150d1000568,16#26109007cb3dd44,
+ 16#3761360723e3175,16#169abd352db74de,16#1c97d520983684f,16#12455f0adee8c66,
+ 16#46719cff00622d,16#1fc92792ed4e437,16#18e2edae21affb5,16#3a67fa9e3e7d46e,
+ 16#1313fdc2728aa74,16#1c1a2b577581db8,16#db49357ea196b1,16#10e219a21d93fc7,
+ 16#3c43abede083666,16#3eef5055a58bbf9,16#1975056f95d90e3,16#3916c133ab16d87,
+ 16#2bc0bea891c26f1,16#391e4b369fc6b36,16#183f83155a359f6,16#1d9f137e9d2e488,
+ 16#ef084de5f4cd3c,16#36a9cf7e29e55d3,16#19eca704e0409a7,16#1bdb99902896c69,
+ 16#21777e2ad128203,16#5d0369ec0563e4,16#36db40b863bd74a,16#33feb71b7515159,
+ 16#208d923ce26f257,16#3841b32891c082d,16#2748f224c2ba226,16#2fcd93b2daf79bb,
+ 16#2c8e6cacad58ec4,16#39850131a1a85f,16#134648d6eea624d,16#2e102e197d5725c,
+ 16#12ac280fa744758,16#1c18266c7442d16,16#22b5f91b15fe17e,16#316740ca870f7c8,
+ 16#720ed4836c426,16#1aac0f738d04f8c,16#34fcd2a647b462c,16#3d430ac755114a3,
+ 16#3692e3670fdf2a,16#265279ab0fc0a15,16#10bd883dee80945,16#10e7843413175e4,
+ 16#b291deba08cee2,16#3915a8234caf11,16#34b911b96707dbd,16#ae63fcda15fde6,
+ 16#b13b9091e82e41,16#29de1b6d70dc04f,16#23fbcbc409617e8,16#1389a0738061066,
+ 16#360f39af790f5d1,16#f436da2a7d12f5,16#2d06ba8da21e08,16#3601a6492b887d,
+ 16#2b2590b8c6cc186,16#f8d613b6904464,16#e5456786e46b78,16#201b8b1f96ed80c,
+ 16#1b75b86d9b843f2,16#2e8bfaa7243a630,16#125ff068a78c3b4,16#3875a28c48bd26e,
+ 16#f09a06941fc9d7,16#107c4de8ca77744,16#357c34144bb9ed6,16#3ccc55d3ebb3378,
+ 16#28db7cea7d3fdee,16#3197fd0b49f6370,16#11af6fedb708ea6,16#2bde0382e37469e,
+ 16#10666171abddb3f,16#1a8876c1f4e78a8,16#169c0efd4422043,16#1501c49abf0440f];
+
reference_jump_val(exs1024) ->
[2655961906500790629, 17003395417078685063, 10466831598958356428, 7603399148503548021,
1650550950190587188, 12294992315080723704, 15743995773860389219, 5492181000145247327,
generated by cgit v1.2.3 (git 2.25.1) at 2024-07-08 18:27:18 +0000