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.

3 files changed, 345 insertions, 18 deletions

diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
index 1dcc3de000..1364a3277b 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 93409d95df..3b1767e731 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 02b7cb10c2..8e7ac223a7 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].