File 2001-Add-jump-functions-to-rand-module.patch of Package erlang
From ff568b5e818d04048009926a7fa2ea537d2e656d Mon Sep 17 00:00:00 2001
From: Kenji Rikitake <kenji.rikitake@acm.org>
Date: Fri, 4 Nov 2016 14:53:10 +0900
Subject: [PATCH] Add jump functions to rand module
Jump functions returns the state after performing jump calculation to a
rand module internal state, which is equivalent to perform a large
number of calls of calculating new states for XorShift*/+ algorithms.
This commit add jump functions for exsplus and exs1024 algorithms, and a
wrapper function jump/1. The wrapper function will cause error with
reason "not_implemented" if the jump function for the algorithm is not
implemented.
This commit adds following new functionalities:
- Add new functions rand:jump/0 and rand:jump/1
- Add the member jump to type alg_handler(), a fun for performing
the jump function
- Add jump functions for exsplus, equivalent to 2^64 calls
- Add jump functions for exs1024, equivalent to 2^512 calls
- Revise seed_put/1, seed/1, seed/2
See <https://github.com/erlang/otp/pull/1235#discussion_r86950557>
- Add dummy jump function for exs64 calling
erlang:error(not_implemented)
- Add jump function test cases as follows:
* Add Common Test group reference_jump
* Add tests for jump/0 to reference_jump_procdict/1
* Rename tests for jump/1 to reference_jump_state/1
* Rename gen_jump/1 to gen_jump_1/1
* Add Common Tests reference_jump_procdict and reference_jump_state
to the group reference_jump
- Add jump function documentation
This commit also changes the Copyright year for Kenji Rikitake's code
from 2015 to 2015-2016.
---
lib/stdlib/doc/src/rand.xml | 35 ++++++++
lib/stdlib/src/rand.erl | 146 ++++++++++++++++++++++++++++++---
lib/stdlib/test/rand_SUITE.erl | 182 +++++++++++++++++++++++++++++++++++++++--
3 files changed, 345 insertions(+), 18 deletions(-)
diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
index 1dcc3de..1364a32 100644
--- a/lib/stdlib/doc/src/rand.xml
+++ b/lib/stdlib/doc/src/rand.xml
@@ -41,6 +41,11 @@
Sebastiano Vigna</url>. The normal distribution algorithm uses the
<url href="http://www.jstatsoft.org/v05/i08">Ziggurat Method by Marsaglia
and Tsang</url>.</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>The following algorithms are provided:</p>
@@ -48,14 +53,17 @@
<tag><c>exsplus</c></tag>
<item>
<p>Xorshift116+, 58 bits precision and period of 2^116-1</p>
+ <p>Jump function: equivalent to 2^64 calls</p>
</item>
<tag><c>exs64</c></tag>
<item>
<p>Xorshift64*, 64 bits precision and a period of 2^64-1</p>
+ <p>Jump function: not available</p>
</item>
<tag><c>exs1024</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>
</taglist>
@@ -156,6 +164,33 @@ S0 = rand:seed_s(exsplus),
</func>
<func>
+ <name name="jump" arity="0"/>
+ <fsummary>Return the seed after performing jump calculation
+ to the state in the process dictionary.</fsummary>
+ <desc><marker id="jump-0" />
+ <p>Returns the state
+ after performing jump calculation
+ to the state in the process dictionary.</p>
+ <p>This function generates a <c>not_implemented</c> error exception
+ when the jump function is not implemented for
+ the algorithm specified in the state
+ in the process dictionary.</p>
+ </desc>
+ </func>
+
+ <func>
+ <name name="jump" arity="1"/>
+ <fsummary>Return the seed after performing jump calculation.</fsummary>
+ <desc><marker id="jump-1" />
+ <p>Returns the state after performing jump calculation
+ to the given state. </p>
+ <p>This function generates a <c>not_implemented</c> error exception
+ when the jump function is not implemented for
+ the algorithm specified in the state.</p>
+ </desc>
+ </func>
+
+ <func>
<name name="normal" arity="0"/>
<fsummary>Return a standard normal distributed random float.</fsummary>
<desc>
diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl
index 93409d9..3b1767e 100644
--- a/lib/stdlib/src/rand.erl
+++ b/lib/stdlib/src/rand.erl
@@ -19,7 +19,7 @@
%%
%% =====================================================================
%% Multiple PRNG module for Erlang/OTP
-%% Copyright (c) 2015 Kenji Rikitake
+%% Copyright (c) 2015-2016 Kenji Rikitake
%% =====================================================================
-module(rand).
@@ -27,11 +27,14 @@
-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,
+ jump/0, jump/1,
normal/0, normal_s/1
]).
-compile({inline, [exs64_next/1, exsplus_next/1,
+ exsplus_jump/1,
exs1024_next/1, exs1024_calc/2,
+ exs1024_jump/1,
get_52/1, normal_kiwi/1]}).
-define(DEFAULT_ALG_HANDLER, exsplus).
@@ -48,7 +51,8 @@
max := integer(),
next := fun(),
uniform := fun(),
- uniform_n := fun()}.
+ uniform_n := fun(),
+ jump := fun()}.
%% Internal state
-opaque state() :: {alg_handler(), alg_seed()}.
@@ -79,9 +83,7 @@ export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}.
-spec seed(AlgOrExpState::alg() | export_state()) -> state().
seed(Alg) ->
- R = seed_s(Alg),
- _ = seed_put(R),
- R.
+ seed_put(seed_s(Alg)).
-spec seed_s(AlgOrExpState::alg() | export_state()) -> state().
seed_s(Alg) when is_atom(Alg) ->
@@ -97,9 +99,7 @@ seed_s({Alg0, Seed}) ->
-spec seed(Alg :: alg(), {integer(), integer(), integer()}) -> state().
seed(Alg0, S0) ->
- State = seed_s(Alg0, S0),
- _ = seed_put(State),
- State.
+ seed_put(seed_s(Alg0, S0)).
-spec seed_s(Alg :: alg(), {integer(), integer(), integer()}) -> state().
seed_s(Alg0, S0 = {_, _, _}) ->
@@ -150,6 +150,25 @@ uniform_s(N, State0 = {#{uniform:=Uniform}, _})
{F, State} = Uniform(State0),
{trunc(F * N) + 1, State}.
+%% 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.
+%% The large number is algorithm dependent.
+
+-spec jump(state()) -> {NewS :: state()}.
+jump(State = {#{jump:=Jump}, _}) ->
+ Jump(State).
+
+%% jump/0: read the internal state and
+%% apply the jump function for the state as in jump/1
+%% and write back the new value to the internal state,
+%% then returns the new value.
+
+-spec jump() -> {NewS :: state()}.
+
+jump() ->
+ seed_put(jump(seed_get())).
+
%% normal/0: returns a random float with standard normal distribution
%% updating the state in the process dictionary.
@@ -192,9 +211,10 @@ normal_s(State0) ->
-type uint64() :: 0..16#ffffffffffffffff.
-type uint58() :: 0..16#03ffffffffffffff.
--spec seed_put(state()) -> undefined | state().
+-spec seed_put(state()) -> state().
seed_put(Seed) ->
- put(?SEED_DICT, Seed).
+ put(?SEED_DICT, Seed),
+ Seed.
seed_get() ->
case get(?SEED_DICT) of
@@ -205,15 +225,18 @@ seed_get() ->
%% Setup alg record
mk_alg(exs64) ->
{#{type=>exs64, max=>?UINT64MASK, next=>fun exs64_next/1,
- uniform=>fun exs64_uniform/1, uniform_n=>fun exs64_uniform/2},
+ uniform=>fun exs64_uniform/1, uniform_n=>fun exs64_uniform/2,
+ jump=>fun exs64_jump/1},
fun exs64_seed/1};
mk_alg(exsplus) ->
{#{type=>exsplus, max=>?UINT58MASK, next=>fun exsplus_next/1,
- uniform=>fun exsplus_uniform/1, uniform_n=>fun exsplus_uniform/2},
+ uniform=>fun exsplus_uniform/1, uniform_n=>fun exsplus_uniform/2,
+ jump=>fun exsplus_jump/1},
fun exsplus_seed/1};
mk_alg(exs1024) ->
{#{type=>exs1024, max=>?UINT64MASK, next=>fun exs1024_next/1,
- uniform=>fun exs1024_uniform/1, uniform_n=>fun exs1024_uniform/2},
+ uniform=>fun exs1024_uniform/1, uniform_n=>fun exs1024_uniform/2,
+ jump=>fun exs1024_jump/1},
fun exs1024_seed/1}.
%% =====================================================================
@@ -246,6 +269,9 @@ exs64_uniform(Max, {Alg, R}) ->
{V, R1} = exs64_next(R),
{(V rem Max) + 1, {Alg, R1}}.
+exs64_jump(_) ->
+ erlang:error(not_implemented).
+
%% =====================================================================
%% exsplus PRNG: Xorshift116+
%% Algorithm by Sebastiano Vigna
@@ -283,6 +309,42 @@ exsplus_uniform(Max, {Alg, R}) ->
{V, R1} = exsplus_next(R),
{(V rem Max) + 1, {Alg, R1}}.
+%% This is the jump function for the exsplus generator, 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.
+
+%% -define(JUMPCONST, 16#000d174a83e17de2302f8ea6bc32c797).
+%% split into 58-bit chunks
+%% and two iterative executions
+
+-define(JUMPCONST1, 16#02f8ea6bc32c797).
+-define(JUMPCONST2, 16#345d2a0f85f788c).
+-define(JUMPELEMLEN, 58).
+
+-spec exsplus_jump(exsplus_state()) -> exsplus_state().
+
+exsplus_jump({Alg, S}) ->
+ {S1, AS1} = exsplus_jump(S, [0|0], ?JUMPCONST1, ?JUMPELEMLEN),
+ {_, AS2} = exsplus_jump(S1, AS1, ?JUMPCONST2, ?JUMPELEMLEN),
+ {Alg, AS2}.
+
+-spec exsplus_jump(state(), state(), pos_integer(), pos_integer()) ->
+ {state(), state()}.
+
+exsplus_jump(S, AS, _, 0) ->
+ {S, AS};
+exsplus_jump(S, [AS0|AS1], J, N) ->
+ {_, NS} = exsplus_next(S),
+ case (J band 1) of
+ 1 ->
+ [S0|S1] = S,
+ exsplus_jump(NS, [(AS0 bxor S0)|(AS1 bxor S1)], J bsr 1, N-1);
+ 0 ->
+ exsplus_jump(NS, [AS0|AS1], J bsr 1, N-1)
+ end.
+
%% =====================================================================
%% exs1024 PRNG: Xorshift1024*
%% Algorithm by Sebastiano Vigna
@@ -340,6 +402,64 @@ exs1024_uniform(Max, {Alg, R}) ->
{V, R1} = exs1024_next(R),
{(V rem Max) + 1, {Alg, R1}}.
+%% This is the jump function for the exs1024 generator, equivalent
+%% to 2^512 calls to next(); it can be used to generate 2^512
+%% non-overlapping subsequences for parallel computations.
+%% Note: the jump function takes ~2000 times of the execution time of
+%% next/1.
+
+%% Jump constant here split into 58 bits for speed
+-define(JUMPCONSTHEAD, 16#00242f96eca9c41d).
+-define(JUMPCONSTTAIL,
+ [16#0196e1ddbe5a1561,
+ 16#0239f070b5837a3c,
+ 16#03f393cc68796cd2,
+ 16#0248316f404489af,
+ 16#039a30088bffbac2,
+ 16#02fea70dc2d9891f,
+ 16#032ae0d9644caec4,
+ 16#0313aac17d8efa43,
+ 16#02f132e055642626,
+ 16#01ee975283d71c93,
+ 16#00552321b06f5501,
+ 16#00c41d10a1e6a569,
+ 16#019158ecf8aa1e44,
+ 16#004e9fc949d0b5fc,
+ 16#0363da172811fdda,
+ 16#030e38c3b99181f2,
+ 16#0000000a118038fc]).
+-define(JUMPTOTALLEN, 1024).
+-define(RINGLEN, 16).
+
+-spec exs1024_jump(state()) -> state().
+
+exs1024_jump({Alg, {L, RL}}) ->
+ P = length(RL),
+ AS = exs1024_jump({L, RL},
+ [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0],
+ ?JUMPCONSTTAIL, ?JUMPCONSTHEAD, ?JUMPELEMLEN, ?JUMPTOTALLEN),
+ {ASL, ASR} = lists:split(?RINGLEN - P, AS),
+ {Alg, {ASL, lists:reverse(ASR)}}.
+
+-spec exs1024_jump(state(), list(non_neg_integer()),
+ list(non_neg_integer()), non_neg_integer(),
+ non_neg_integer(), non_neg_integer()) -> list(non_neg_integer()).
+
+exs1024_jump(_, AS, _, _, _, 0) ->
+ AS;
+exs1024_jump(S, AS, [H|T], _, 0, TN) ->
+ exs1024_jump(S, AS, T, H, ?JUMPELEMLEN, TN);
+exs1024_jump({L, RL}, AS, JL, J, N, TN) ->
+ {_, NS} = exs1024_next({L, RL}),
+ case (J band 1) of
+ 1 ->
+ AS2 = lists:zipwith(fun(X, Y) -> X bxor Y end,
+ AS, L ++ lists:reverse(RL)),
+ exs1024_jump(NS, AS2, JL, J bsr 1, N-1, TN-1);
+ 0 ->
+ exs1024_jump(NS, AS, JL, J bsr 1, N-1, TN-1)
+ end.
+
%% =====================================================================
%% Ziggurat cont
%% =====================================================================
diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl
index 02b7cb1..8e7ac22 100644
--- a/lib/stdlib/test/rand_SUITE.erl
+++ b/lib/stdlib/test/rand_SUITE.erl
@@ -28,7 +28,8 @@
api_eq/1, reference/1,
basic_stats_uniform_1/1, basic_stats_uniform_2/1,
basic_stats_normal/1,
- plugin/1, measure/1]).
+ plugin/1, measure/1,
+ reference_jump_state/1, reference_jump_procdict/1]).
-export([test/0, gen/1]).
@@ -45,14 +46,21 @@ all() ->
api_eq,
reference,
{group, basic_stats},
- plugin, measure].
+ plugin, measure,
+ {group, reference_jump}
+ ].
groups() ->
[{basic_stats, [parallel],
- [basic_stats_uniform_1, basic_stats_uniform_2, basic_stats_normal]}].
+ [basic_stats_uniform_1, basic_stats_uniform_2, basic_stats_normal]},
+ {reference_jump, [parallel],
+ [reference_jump_state, reference_jump_procdict]}].
group(basic_stats) ->
%% valgrind needs a lot of time
+ [{timetrap,{minutes,10}}];
+group(reference_jump) ->
+ %% valgrind needs a lot of time
[{timetrap,{minutes,10}}].
%% A simple helper to test without test_server during dev
@@ -228,7 +236,7 @@ interval_float_1(N) ->
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
-%% Check if exs64 algorithm generates the proper sequence.
+%% Check if each algorithm generates the proper sequence.
reference(Config) when is_list(Config) ->
[reference_1(Alg) || Alg <- algs()],
ok.
@@ -242,7 +250,7 @@ reference_1(Alg) ->
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)]),
- ok
+ exit(wrong_value)
end.
gen(Algo) ->
@@ -434,6 +442,112 @@ measure_2(N, State0, Fun) when N > 0 ->
measure_2(0, _, _) -> ok.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% The jump sequence tests has two parts
+%% for those with the functional API (jump/1)
+%% and for those with the internal state
+%% in process dictionary (jump/0).
+
+-define(LOOP_JUMP, (?LOOP div 1000)).
+
+%% Check if each algorithm generates the proper jump sequence
+%% with the functional API.
+reference_jump_state(Config) when is_list(Config) ->
+ [reference_jump_1(Alg) || Alg <- algs()],
+ ok.
+
+reference_jump_1(Alg) ->
+ Refval = reference_jump_val(Alg),
+ Testval = gen_jump_1(Alg),
+ case Refval =:= Testval of
+ true -> ok;
+ false ->
+ 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)]),
+ exit(wrong_value)
+ end.
+
+gen_jump_1(Algo) ->
+ Seed = case Algo of
+ exsplus -> %% Printed with orig 'C' code and this seed
+ rand:seed_s({exsplus, [12345678|12345678]});
+ exs1024 -> %% Printed with orig 'C' code and this seed
+ rand:seed_s({exs1024, {lists:duplicate(16, 12345678), []}});
+ exs64 -> %% Test exception of not_implemented notice
+ try rand:jump(rand:seed_s(exs64))
+ catch
+ error:not_implemented -> not_implemented
+ end;
+ _ -> % unimplemented
+ not_implemented
+ end,
+ case Seed of
+ not_implemented -> [not_implemented];
+ S -> gen_jump_1(?LOOP_JUMP, S, [])
+ end.
+
+gen_jump_1(N, State0 = {#{max:=Max}, _}, Acc) when N > 0 ->
+ {_, State1} = rand:uniform_s(Max, State0),
+ {Random, State2} = rand:uniform_s(Max, rand:jump(State1)),
+ case N rem (?LOOP_JUMP div 100) of
+ 0 -> gen_jump_1(N-1, State2, [Random|Acc]);
+ _ -> gen_jump_1(N-1, State2, Acc)
+ end;
+gen_jump_1(_, _, Acc) -> lists:reverse(Acc).
+
+%% Check if each algorithm generates the proper jump sequence
+%% with the internal state in the process dictionary.
+reference_jump_procdict(Config) when is_list(Config) ->
+ [reference_jump_0(Alg) || Alg <- algs()],
+ ok.
+
+reference_jump_0(Alg) ->
+ Refval = reference_jump_val(Alg),
+ Testval = gen_jump_0(Alg),
+ case Refval =:= Testval of
+ true -> ok;
+ false ->
+ 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)]),
+ exit(wrong_value)
+ end.
+
+gen_jump_0(Algo) ->
+ Seed = case Algo of
+ exsplus -> %% Printed with orig 'C' code and this seed
+ rand:seed({exsplus, [12345678|12345678]});
+ exs1024 -> %% Printed with orig 'C' code and this seed
+ rand:seed({exs1024, {lists:duplicate(16, 12345678), []}});
+ exs64 -> %% Test exception of not_implemented notice
+ try
+ _ = rand:seed(exs64),
+ rand:jump()
+ catch
+ error:not_implemented -> not_implemented
+ end;
+ _ -> % unimplemented
+ not_implemented
+ end,
+ case Seed of
+ not_implemented -> [not_implemented];
+ S ->
+ {Seedmap=#{}, _} = S,
+ Max = maps:get(max, Seedmap),
+ gen_jump_0(?LOOP_JUMP, Max, [])
+ end.
+
+gen_jump_0(N, Max, Acc) when N > 0 ->
+ _ = rand:uniform(Max),
+ _ = rand:jump(),
+ Random = rand:uniform(Max),
+ case N rem (?LOOP_JUMP div 100) of
+ 0 -> gen_jump_0(N-1, Max, [Random|Acc]);
+ _ -> gen_jump_0(N-1, Max, Acc)
+ end;
+gen_jump_0(_, _, Acc) -> lists:reverse(Acc).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% Data
reference_val(exs64) ->
[16#3737ad0c703ff6c3,16#3868a78fe71adbbd,16#1f01b62b4338b605,16#50876a917437965f,
@@ -515,3 +629,61 @@ reference_val(exsplus) ->
16#36f715a249f4ec2,16#1c27629826c50d3,16#914d9a6648726a,16#27f5bf5ce2301e8,
16#3dd493b8012970f,16#be13bed1e00e5c,16#ceef033b74ae10,16#3da38c6a50abe03,
16#15cbd1a421c7a8c,16#22794e3ec6ef3b1,16#26154d26e7ea99f,16#3a66681359a6ab6].
+
+%%%
+
+reference_jump_val(exsplus) ->
+ [82445318862816932, 145810727464480743, 16514517716894509, 247642377064868650,
+ 162385642339156908, 251810707075252101, 82288275771998924, 234412731596926322,
+ 49960883129071044, 200690077681656596, 213743196668671647, 131182800982967108,
+ 144200072021941728, 263557425008503277, 194858522616874272, 185869394820993172,
+ 80384502675241453, 262654144824057588, 90033295011291362, 4494510449302659,
+ 226005372746479588, 116780561309220553, 47048528594475843, 39168929349768743,
+ 139615163424415552, 55330632656603925, 237575574720486569, 102381140288455025,
+ 18452933910354323, 150248612130579752, 269358096791922740, 61313433522002187,
+ 160327361842676597, 185187983548528938, 57378981505594193, 167510799293984067,
+ 105117045862954303, 176126685946302943, 123590876906828803, 69185336947273487,
+ 9098689247665808, 49906154674145057, 131575138412788650, 161843880211677185,
+ 30743946051071186, 187578920583823612, 45008401528636978, 122454158686456658,
+ 111195992644229524, 17962783958752862, 13579507636941108, 130137843317798663,
+ 144202635170576832, 132539563255093922, 159785575703967124, 187241848364816640,
+ 183044737781926478, 12921559769912263, 83553932242922001, 96698298841984688,
+ 281664320227537824, 224233030818578263, 77812932110318774, 169729351013291728,
+ 164475402723178734, 242780633011249051, 51095111179609125, 19249189591963554,
+ 221412426221439180, 265700202856282653, 265342254311932308, 241218503498385511,
+ 255400887248486575, 212083616929812076, 227947034485840579, 268261881651571692,
+ 104846262373404908, 49690734329496661, 213259196633566308, 186966479726202436,
+ 282157378232384574, 11272948584603747, 166540426999573480, 50628164001018755,
+ 65235580992800860, 230664399047956956, 64575592354687978, 40519393736078511,
+ 108341851194332747, 115426411532008961, 120656817002338193, 234537867870809797,
+ 12504080415362731, 45083100453836317, 270968267812126657, 93505647407734103,
+ 252852934678537969, 258758309277167202, 74250882143432077, 141629095984552833];
+
+reference_jump_val(exs1024) ->
+ [2655961906500790629, 17003395417078685063, 10466831598958356428, 7603399148503548021,
+ 1650550950190587188, 12294992315080723704, 15743995773860389219, 5492181000145247327,
+ 14118165228742583601, 1024386975263610703, 10124872895886669513, 6445624517813169301,
+ 6238575554686562601, 14108646153524288915, 11804141635807832816, 8421575378006186238,
+ 6354993374304550369, 838493020029548163, 14759355804308819469, 12212491527912522022,
+ 16943204735100571602, 198964074252287588, 7325922870779721649, 15853102065526570574,
+ 16294058349151823341, 6153379962047409781, 15874031679495957261, 17299265255608442340,
+ 984658421210027171, 17408042033939375278, 3326465916992232353, 5222817718770538733,
+ 13262385796795170510, 15648751121811336061, 6718721549566546451, 7353765235619801875,
+ 16110995049882478788, 14559143407227563441, 4189805181268804683, 10938587948346538224,
+ 1635025506014383478, 12619562911869525411, 17469465615861488695, 125252234176411528,
+ 2004192558503448853, 13175467866790974840, 17712272336167363518, 1710549840100880318,
+ 17486892343528340916, 5337910082227550967, 8333082060923612691, 6284787745504163856,
+ 8072221024586708290, 6077032673910717705, 11495200863352251610, 11722792537523099594,
+ 14642059504258647996, 8595733246938141113, 17223366528010341891, 17447739753327015776,
+ 6149800490736735996, 11155866914574313276, 7123864553063709909, 15982886296520662323,
+ 5775920250955521517, 8624640108274906072, 8652974210855988961, 8715770416136907275,
+ 11841689528820039868, 10991309078149220415, 11758038663970841716, 7308750055935299261,
+ 15939068400245256963, 6920341533033919644, 8017706063646646166, 15814376391419160498,
+ 13529376573221932937, 16749061963269842448, 14639730709921425830, 3265850480169354066,
+ 4569394597532719321, 16594515239012200038, 13372824240764466517, 16892840440503406128,
+ 11260004846380394643, 2441660009097834955, 10566922722880085440, 11463315545387550692,
+ 5252492021914937692, 10404636333478845345, 11109538423683960387, 5525267334484537655,
+ 17936751184378118743, 4224632875737239207, 15888641556987476199, 9586888813112229805,
+ 9476861567287505094, 14909536929239540332, 17996844556292992842, 2699310519182298856];
+
+reference_jump_val(exs64) -> [not_implemented].
--
2.10.2
