stdlib: Add new random functionality/module

The old random module contains an old algorithm which have flaws
and the api requires the user to invoke seed and or checking if seed have
been invoked, if a non constant seed is to be used.

The api contains the following features:
- The user can invoke rand:unform/[0|1] directly and get a non constant seeding.
- The api is split in functional and non functional functions,
  i.e. usage of _s functions will not affect the process dictionary.
- The api contains several algorithms with different characteristics and
  can be extended with new algorithms in the future.
- Contains state of the art random number generators.
- Default algorithm is taylor made for erlang to be fast on 64bits machines.

1 files changed, 496 insertions, 0 deletions

diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl
new file mode 100644
index 0000000000..70d219ddaf
--- /dev/null
+++ b/lib/stdlib/test/rand_SUITE.erl
@@ -0,0 +1,496 @@
+%%
+%% %CopyrightBegin%
+%%
+%% Copyright Ericsson AB 2000-2011. All Rights Reserved.
+%%
+%% The contents of this file are subject to the Erlang Public License,
+%% Version 1.1, (the "License"); you may not use this file except in
+%% compliance with the License. You should have received a copy of the
+%% Erlang Public License along with this software. If not, it can be
+%% retrieved online at http://www.erlang.org/.
+%%
+%% Software distributed under the License is distributed on an "AS IS"
+%% basis, WITHOUT WARRANTY OF ANY KIND, either express or implied. See
+%% the License for the specific language governing rights and limitations
+%% under the License.
+%%
+%% %CopyrightEnd%
+
+-module(rand_SUITE).
+-export([all/0, suite/0,groups/0,
+	 init_per_suite/1, end_per_suite/1,
+	 init_per_group/2,end_per_group/2,
+	 init_per_testcase/2, end_per_testcase/2
+	]).
+
+-export([interval_int/1, interval_float/1, seed/1,
+         api_eq/1, reference/1, basic_stats/1,
+	 plugin/1, measure/1
+	]).
+
+-export([test/0, gen/1]).
+
+-include_lib("test_server/include/test_server.hrl").
+
+% Default timetrap timeout (set in init_per_testcase).
+-define(default_timeout, ?t:minutes(1)).
+-define(LOOP, 1000000).
+
+init_per_testcase(_Case, Config) ->
+    Dog = ?t:timetrap(?default_timeout),
+    [{watchdog, Dog} | Config].
+end_per_testcase(_Case, Config) ->
+    Dog = ?config(watchdog, Config),
+    test_server:timetrap_cancel(Dog),
+    ok.
+
+suite() -> [{ct_hooks,[ts_install_cth]}].
+
+all() ->
+    [seed, interval_int, interval_float,
+     api_eq,
+     reference,
+     basic_stats,
+     plugin, measure
+    ].
+
+groups() -> [].
+
+init_per_suite(Config) ->  Config.
+end_per_suite(_Config) -> ok.
+
+init_per_group(_GroupName, Config) -> Config.
+end_per_group(_GroupName, Config) ->  Config.
+
+%% A simple helper to test without test_server during dev
+test() ->
+    Tests = all(),
+    lists:foreach(fun(Test) ->
+			  try
+			      ok = ?MODULE:Test([]),
+			      io:format("~p: ok~n", [Test])
+			  catch _:Reason ->
+				  io:format("Failed: ~p: ~p ~p~n",
+					    [Test, Reason, erlang:get_stacktrace()])
+			  end
+		  end, Tests).
+
+algs() ->
+    [exs64, exsplus, exs1024].
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+seed(doc) ->
+    ["Test that seed and seed_s and export_seed/0 is working."];
+seed(suite) ->
+    [];
+seed(Config) when is_list(Config) ->
+    Algs = algs(),
+    Test = fun(Alg) ->
+		   try seed_1(Alg)
+		   catch _:Reason ->
+			   test_server:fail({Alg, Reason, erlang:get_stacktrace()})
+		   end
+	   end,
+    [Test(Alg) || Alg <- Algs],
+    ok.
+
+seed_1(Alg) ->
+    %% Check that uniform seeds automatically,
+    _ = rand:uniform(),
+    S00 = get(rand_seed),
+    erase(),
+    _ = rand:uniform(),
+    false = S00 =:= get(rand_seed), %% hopefully
+
+    %% Choosing algo and seed
+    S0 = rand:seed(Alg, {0, 0, 0}),
+    %% Check that (documented?) process_dict variable is correct
+    S0 = get(rand_seed),
+    S0 = rand:seed_s(Alg, {0, 0, 0}),
+    %% Check that process_dict should not be used for seed_s functionality
+    _ = rand:seed_s(Alg, {1, 0, 0}),
+    S0 = get(rand_seed),
+    %% Test export
+    ES0 = rand:export_seed(),
+    ES0 = rand:export_seed_s(S0),
+    S0 = rand:seed(ES0),
+    S0 = rand:seed_s(ES0),
+    %% seed/1 calls should be unique
+    S1 = rand:seed(Alg),
+    false = (S1 =:= rand:seed_s(Alg)),
+    %% Negative integers works
+    _ = rand:seed_s(Alg, {-1,-1,-1}),
+
+    %% Other term do not work
+    {'EXIT', _} = (catch rand:seed_s(foobar, os:timestamp())),
+    {'EXIT', _} = (catch rand:seed_s(Alg, {asd, 1, 1})),
+    {'EXIT', _} = (catch rand:seed_s(Alg, {0, 234.1234, 1})),
+    {'EXIT', _} = (catch rand:seed_s(Alg, {0, 234, [1, 123, 123]})),
+    ok.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+api_eq(doc) ->
+    ["Check that both api's are consistent with each other."];
+api_eq(suite) ->
+    [];
+api_eq(_Config) ->
+    Algs = algs(),
+    Small = fun(Alg) ->
+		    Seed = rand:seed(Alg),
+		    api_eq_1(Seed)
+	    end,
+    _ = [Small(Alg) || Alg <- Algs],
+    ok.
+
+api_eq_1(S00) ->
+    Check = fun(_, Seed) ->
+		    {V0, S0} = rand:uniform_s(Seed),
+		    V0 = rand:uniform(),
+		    {V1, S1} = rand:uniform_s(1000000, S0),
+		    V1 = rand:uniform(1000000),
+		    S1
+	    end,
+    S1 = lists:foldl(Check, S00, lists:seq(1, 200)),
+    S1 = get(rand_seed),
+    Exported = rand:export_seed(),
+    Exported = rand:export_seed_s(S1),
+    {V0, S2} = rand:uniform_s(S1),
+    V0 = rand:uniform(),
+
+    S3 = lists:foldl(Check, S2, lists:seq(1, 200)),
+    S1 = rand:seed(Exported),
+    S1 = rand:seed_s(Exported),
+
+    S4 = lists:foldl(Check, S1, lists:seq(1, 200)),
+
+    %% Verify that we do not have loops
+    false = S1 =:= S2,
+    false = S2 =:= S3,
+    false = S3 =:= S4,
+
+    S1 = rand:seed(Exported),
+    S4 = lists:foldl(Check, S1, lists:seq(1, 200)),
+    ok.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+interval_int(doc) ->
+    ["Check that uniform/1 returns values within the proper interval."];
+interval_int(suite) ->
+    [];
+interval_int(Config) when is_list(Config) ->
+    Algs = algs(),
+    Small = fun(Alg) ->
+		    _ = rand:seed(Alg),
+		    Max = interval_int_1(100000, 7, 0),
+		    Max =:= 7 orelse exit({7, Alg, Max})
+	    end,
+    _ = [Small(Alg) || Alg <- Algs],
+    %% Test large integers
+    Large = fun(Alg) ->
+		    _ = rand:seed(Alg),
+		    Max = interval_int_1(100000, 1 bsl 128, 0),
+		    Max > 1 bsl 64 orelse exit({large, Alg, Max})
+	    end,
+    [Large(Alg) || Alg <- Algs],
+    ok.
+
+interval_int_1(0, _, Max) -> Max;
+interval_int_1(N, Top, Max) ->
+    X = rand:uniform(Top),
+    if
+	0 < X, X =< Top ->
+	    ok;
+	true ->
+	    io:format("X=~p Top=~p 0<~p<~p~n", [X,Top,X,Top]),
+	    exit({X, rand:export_seed()})
+    end,
+    interval_int_1(N-1, Top, max(X, Max)).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+interval_float(doc) ->
+    ["Check that uniform/0 returns values within the proper interval."];
+interval_float(suite) ->
+    [];
+interval_float(Config) when is_list(Config) ->
+    Algs = algs(),
+    Test = fun(Alg) ->
+		   _ = rand:seed(Alg),
+		   interval_float_1(100000)
+	   end,
+    [Test(Alg) || Alg <- Algs],
+    ok.
+
+interval_float_1(0) -> ok;
+interval_float_1(N) ->
+    X = rand:uniform(),
+    if
+	0.0 < X, X < 1.0 ->
+	    ok;
+	true ->
+	    io:format("X=~p 0<~p<1.0~n", [X,X]),
+	    exit({X, rand:export_seed()})
+    end,
+    interval_float_1(N-1).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+reference(doc) -> ["Check if exs64 algorithm generates the proper sequence."];
+reference(suite) -> [];
+reference(Config) when is_list(Config) ->
+    [reference_1(Alg) || Alg <- algs()],
+    ok.
+
+reference_1(Alg) ->
+    Refval  = reference_val(Alg),
+    Testval = gen(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)]),
+	    %% test_server:fail({Alg, Refval -- Testval}),
+	    ok
+    end.
+
+gen(Algo) ->
+    Seed = case Algo of
+	       exsplus -> %% Printed with orig 'C' code and this seed
+		   rand:seed_s({exsplus, [12345678|12345678]});
+	       exs64 -> %% Printed with orig 'C' code and this seed
+		   rand:seed_s({exs64, 12345678});
+	       exs1024 -> %% Printed with orig 'C' code and this seed
+		   rand:seed_s({exs1024, {lists:duplicate(16, 12345678), []}});
+	       _ ->
+		   rand:seed(Algo, {100, 200, 300})
+	   end,
+    gen(?LOOP, Seed, []).
+
+gen(N, State0 = {#{max:=Max}, _}, Acc) when N > 0 ->
+    {Random, State} = rand:uniform_s(Max, State0),
+    case N rem (?LOOP div 100) of
+	0 -> gen(N-1, State, [Random|Acc]);
+	_ -> gen(N-1, State, Acc)
+    end;
+gen(_, _, Acc) -> lists:reverse(Acc).
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%% This just tests the basics so we have not made any serious errors
+%% when making the conversion from the original algorithms.
+%% The algorithms must have good properties to begin with
+%%
+
+basic_stats(doc) -> ["Check that the algorithms generate sound values."];
+basic_stats(suite) -> [];
+basic_stats(Config) when is_list(Config) ->
+    [basic_stats_1(?LOOP, rand:seed_s(Alg), 0.0, array:new([{default, 0}]))
+     || Alg <- algs()],
+    [basic_stats_2(?LOOP, rand:seed_s(Alg), 0, array:new([{default, 0}]))
+     || Alg <- algs()],
+    ok.
+
+basic_stats_1(N, S0, Sum, A0) when N > 0 ->
+    {X,S} = rand:uniform_s(S0),
+    I = trunc(X*100),
+    A = array:set(I, 1+array:get(I,A0), A0),
+    basic_stats_1(N-1, S, Sum+X, A);
+basic_stats_1(0, {#{type:=Alg}, _}, Sum, A) ->
+    AverN = Sum / ?LOOP,
+    io:format("~.10w: Average: ~.4f~n", [Alg, AverN]),
+    Counters = array:to_list(A),
+    Min = lists:min(Counters),
+    Max = lists:max(Counters),
+    io:format("~.10w: Min: ~p Max: ~p~n", [Alg, Min, Max]),
+
+    %% Verify that the basic statistics are ok
+    %% be gentle we don't want to see to many failing tests
+    abs(0.5 - AverN) < 0.005 orelse test_server:fail({average, Alg, AverN}),
+    abs(?LOOP div 100 - Min) < 1000 orelse test_server:fail({min, Alg, Min}),
+    abs(?LOOP div 100 - Max) < 1000 orelse test_server:fail({max, Alg, Max}),
+    ok.
+
+basic_stats_2(N, S0, Sum, A0) when N > 0 ->
+    {X,S} = rand:uniform_s(100, S0),
+    A = array:set(X-1, 1+array:get(X-1,A0), A0),
+    basic_stats_2(N-1, S, Sum+X, A);
+basic_stats_2(0, {#{type:=Alg}, _}, Sum, A) ->
+    AverN = Sum / ?LOOP,
+    io:format("~.10w: Average: ~.4f~n", [Alg, AverN]),
+    Counters = tl(array:to_list(A)),
+    Min = lists:min(Counters),
+    Max = lists:max(Counters),
+    io:format("~.10w: Min: ~p Max: ~p~n", [Alg, Min, Max]),
+
+    %% Verify that the basic statistics are ok
+    %% be gentle we don't want to see to many failing tests
+    abs(50.5 - AverN) < 0.5 orelse test_server:fail({average, Alg, AverN}),
+    abs(?LOOP div 100 - Min) < 1000 orelse test_server:fail({min, Alg, Min}),
+    abs(?LOOP div 100 - Max) < 1000 orelse test_server:fail({max, Alg, Max}),
+    ok.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+plugin(doc) ->   ["Test that the user can write algorithms"];
+plugin(suite) -> [];
+plugin(Config) when is_list(Config) ->
+    _ = lists:foldl(fun(_, S0) ->
+			    {V1, S1} = rand:uniform_s(10000, S0),
+			    true = is_integer(V1),
+			    {V2, S2} = rand:uniform_s(S1),
+			    true = is_float(V2),
+			    S2
+		    end, crypto_seed(), lists:seq(1, 200)),
+    ok.
+
+%% Test implementation
+crypto_seed() ->
+    {#{type=>crypto,
+       uniform=>fun crypto_uniform/1,
+       uniform_n=>fun crypto_uniform_n/2},
+     <<>>}.
+
+
+%% Be fair and create bignums i.e. 64bits otherwise use 58bits
+crypto_next(<<Num:64, Bin/binary>>) ->
+    {Num, Bin};
+crypto_next(_) ->
+    crypto_next(crypto:rand_bytes((64 div 8)*100)).
+
+crypto_uniform({Api, Data0}) ->
+    {Int, Data} = crypto_next(Data0),
+    {Int / (1 bsl 64), {Api, Data}}.
+
+crypto_uniform_n(N, {Api, Data0}) when N < (1 bsl 64) ->
+    {Int, Data} = crypto_next(Data0),
+    {(Int rem N)+1, {Api, Data}};
+crypto_uniform_n(N, State0) ->
+    {F,State} = crypto_uniform(State0),
+    {trunc(F * N) + 1, State}.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+
+%% Not a test but measures the time characteristics of the different algorithms
+measure(Suite) when is_atom(Suite) -> [];
+measure(_Config) ->
+    Algos = [crypto64|algs()],
+    io:format("RNG integer performance~n",[]),
+    _ = [measure_1(Algo, fun(State) -> rand:uniform_s(10000, State) end) || Algo <- Algos],
+    io:format("RNG float performance~n",[]),
+    _ = [measure_1(Algo, fun(State) -> rand:uniform_s(State) end) || Algo <- Algos],
+    ok.
+
+measure_1(Algo, Gen) ->
+    Parent = self(),
+    Seed = fun(crypto64) -> crypto_seed();
+	      (Alg) -> rand:seed_s(Alg)
+	   end,
+
+    Pid = spawn_link(fun() ->
+			     Fun = fun() -> measure_2(?LOOP, Seed(Algo), Gen) end,
+			     {Time, ok} = timer:tc(Fun),
+			     io:format("~.10w: ~pµs~n", [Algo, Time]),
+			     Parent ! {self(), ok},
+			     normal
+		     end),
+    receive
+	{Pid, Msg} -> Msg
+    end.
+
+measure_2(N, State0, Fun) when N > 0 ->
+    case Fun(State0) of
+	{Random, State}
+	  when is_integer(Random), Random >= 1, Random =< 100000 ->
+	    measure_2(N-1, State, Fun);
+	{Random, State} when is_float(Random), Random > 0, Random < 1 ->
+	    measure_2(N-1, State, Fun);
+	Res ->
+	    exit({error, Res, State0})
+    end;
+measure_2(0, _, _) -> ok.
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+%%% Data
+reference_val(exs64) ->
+    [16#3737ad0c703ff6c3,16#3868a78fe71adbbd,16#1f01b62b4338b605,16#50876a917437965f,
+     16#b2edfe32a10e27fc,16#995924551d8ebae1,16#9f1e6b94e94e0b58,16#27ec029eb0e94f8e,
+     16#bf654e6df7fe5c,16#b7d5ef7b79be65e3,16#4bdba4d1c159126b,16#a9c816fdc701292c,
+     16#a377b6c89d85ac8b,16#7abb5cd0e5847a6,16#62666f1fc00a0a90,16#1edc3c3d255a8113,
+     16#dfc764073767f18e,16#381783d577ca4e34,16#49693588c085ddcb,16#da6fcb16dd5163f3,
+     16#e2357a703475b1b7,16#aaa84c4924b5985a,16#b8fe07bb2bac1e49,16#23973ac0160ff064,
+     16#1afbc7b023f5d618,16#9f510f7b7caa2a0f,16#d5b0a57f7f5f1084,16#d8c49b66c5f99a29,
+     16#e920ac3b598b5213,16#1090d7e27e7a7c76,16#81171917168ee74f,16#f08489a3eb6988e,
+     16#396260c4f0b2ed46,16#4fd0a6a6caefd5b2,16#423dff07a3b888a,16#12718773ebd99987,
+     16#e50991e540807cb,16#8cfa03bbaa6679d6,16#55bdf86dfbb92dbf,16#eb7145378cce74a8,
+     16#71856c224c846595,16#20461588dae6e24d,16#c73b3e63ced74bac,16#775b11813dda0c78,
+     16#91f358e51068ede0,16#399955ef36766bc2,16#4489ee072e8a38b1,16#ba77759d52321ca0,
+     16#14f519eab5c53db8,16#1f754bd08e4f34c4,16#99e25ca29b2fcfeb,16#da11927c0d9837f8,
+     16#1eeb0f87009f5a87,16#a7c444d3b0db1089,16#49c7fbf0714849ad,16#4f2b693e7f8265cb,
+     16#80e1493cbaa8f256,16#186f345bcac2661e,16#330065ae0c698d26,16#5235ed0432c42e93,
+     16#429792e31ddb10bb,16#8769054bb6533cff,16#1ab382483444201f,16#2216368786fc7b9,
+     16#1efea1155216da0b,16#782dc868ba595452,16#2b80f6d159617f48,16#407fc35121b2fa1b,
+     16#90e8be6e618873d1,16#40ad4ec92a8abf8e,16#34e2890f583f435,16#838c0aef0a5d8427,
+     16#ed4238f4bd6cbcfa,16#7feed11f7a8bb9f0,16#2b0636a93e26c89d,16#481ad4bea5180646,
+     16#673e5ad861afe1cc,16#298eeb519d69e74d,16#eb1dd06d168c856,16#4770651519ee7ef9,
+     16#7456ebf1bcf608f1,16#d6200f6fbd61ce05,16#c0695dfab11ab6aa,16#5bff449249983843,
+     16#7aba88471474c9ac,16#d7e9e4a21c989e91,16#c5e02ee67ccb7ce1,16#4ea8a3a912246153,
+     16#f2e6db7c9ce4ec43,16#39498a95d46d2470,16#c5294fcb8cce8aa9,16#a918fe444719f3dc,
+     16#98225f754762c0c0,16#f0721204f2cb43f5,16#b98e77b099d1f2d1,16#691d6f75aee3386,
+     16#860c7b2354ec24fd,16#33e007bd0fbcb609,16#7170ae9c20fb3d0,16#31d46938fe383a60];
+
+reference_val(exs1024) ->
+    [16#9c61311d0d4a01fd,16#ce963ef5803b703e,16#545dcffb7b644e1a,16#edd56576a8d778d5,
+     16#16bee799783c6b45,16#336f0b3caeb417fa,16#29291b8be26dedfa,16#1efed996d2e1b1a8,
+     16#c5c04757bd2dadf9,16#11aa6d194009c616,16#ab2b3e82bdb38a91,16#5011ee46fd2609eb,
+     16#766db7e5b701a9bb,16#d42cb2632c419f35,16#107c6a2667bf8557,16#3ffbf922cb306967,
+     16#1e71e3d024ac5131,16#6fdb368ec67a5f06,16#b0d8e72e7aa6d1c1,16#e5705a02dae89e3b,
+     16#9c24eb68c086a1d3,16#418de330f55f71f0,16#2917ddeb278bc8d2,16#aeba7fba67208f39,
+     16#10ceaf40f6af1d8d,16#47a6d06811d33132,16#603a661d6caf720a,16#a28bd0c9bcdacb3c,
+     16#f44754f006909762,16#6e25e8e67ccc43bc,16#174378ce374a549e,16#b5598ae9f57c4e50,
+     16#ca85807fbcd51dd,16#1816e58d6c3cc32a,16#1b4d630d3c8e96a6,16#c19b1e92b4efc5bd,
+     16#665597b20ddd721a,16#fdab4eb21b75c0ae,16#86a612dcfea0756c,16#8fc2da192f9a55f0,
+     16#d7c954eb1af31b5,16#6f5ee45b1b80101b,16#ebe8ea4e5a67cbf5,16#1cb952026b4c1400,
+     16#44e62caffe7452c0,16#b591d8f3e6d7cbcf,16#250303f8d77b6f81,16#8ef2199aae4c9b8d,
+     16#a16baa37a14d7b89,16#c006e4d2b2da158b,16#e6ec7abd54c93b31,16#e6b0d79ae2ab6fa7,
+     16#93e4b30e4ab7d4cd,16#42a01b6a4ef63033,16#9ab1e94fe94976e,16#426644e1de302a1f,
+     16#8e58569192200139,16#744f014a090107c1,16#15d056801d467c6c,16#51bdad3a8c30225f,
+     16#abfc61fb3104bd45,16#c610607122272df7,16#905e67c63116ebfc,16#1e4fd5f443bdc18,
+     16#1945d1745bc55a4c,16#f7cd2b18989595bb,16#f0d273b2c646a038,16#ee9a6fdc6fd5d734,
+     16#541a518bdb700518,16#6e67ab9a65361d76,16#bcfadc9bfe5b2e06,16#69fa334cf3c11496,
+     16#9657df3e0395b631,16#fc0d0442160108ec,16#2ee538da7b1f7209,16#8b20c9fae50a5a9e,
+     16#a971a4b5c2b3b6a,16#ff6241e32489438e,16#8fd6433f45255777,16#6e6c82f10818b0dc,
+     16#59a8fad3f6af616b,16#7eac34f43f12221c,16#6e429ec2951723ec,16#9a65179767a45c37,
+     16#a5f8127d1e6fdf35,16#932c50bc633d8d5c,16#f3bbea4e7ebecb8,16#efc3a2bbf6a8674,
+     16#451644a99971cb6,16#cf70776d652c150d,16#c1fe0dcb87a25403,16#9523417132b2452e,
+     16#8f98bc30d06b980e,16#bb4b288ecb8daa9a,16#59e54beb32f78045,16#f9ab1562456b9d66,
+     16#6435f4130304a793,16#b4bb94c2002e1849,16#49a86d1e4bade982,16#457d63d60ed52b95];
+
+reference_val(exsplus) ->
+    [16#bc76c2e638db,16#15ede2ebb16c9fb,16#185ee2c27d6b88d,16#15d5ee9feafc3a5,
+     16#1862e91dfce3e6b,16#2c9744b0fb69e46,16#78b21bc01cef6b,16#2d16a2fae6c76ba,
+     16#13dfccb8ff86bce,16#1d9474c59e23f4d,16#d2f67dcd7f0dd6,16#2b6d489d51a0725,
+     16#1fa52ef484861d8,16#1ae9e2a38f966d4,16#2264ab1e193acca,16#23bbca085039a05,
+     16#2b6eea06a0af0e1,16#3ad47fa8866ea20,16#1ec2802d612d855,16#36c1982b134d50,
+     16#296b6a23f5b75e0,16#c5eeb600a9875c,16#2a3fd51d735f9d4,16#56fafa3593a070,
+     16#13e9d416ec0423e,16#28101a91b23e9dc,16#32e561eb55ce15a,16#94a7dbba66fe4a,
+     16#2e1845043bcec1f,16#235f7513a1b5146,16#e37af1bf2d63cb,16#2048033824a1639,
+     16#c255c750995f7,16#2c7542058e89ee3,16#204dfeefbdb62ba,16#f5a936ec63dd66,
+     16#33b3b7dbbbd8b90,16#c4f0f79026ffe9,16#20ffee2d37aca13,16#2274f931716be2c,
+     16#29b883902ba9df1,16#1a838cd5312717f,16#2edfc49ff3dc1d6,16#418145cbec84c2,
+     16#d2d8f1a17d49f,16#d41637bfa4cc6f,16#24437e03a0f5df8,16#3d1d87919b94a90,
+     16#20d6997b36769b6,16#16f9d7855cd87ca,16#821ef7e2a062a3,16#2c4d11dc4a2da70,
+     16#24a3b27f56ed26b,16#144b23c8b97387a,16#34a2ced56930d12,16#21cc0544113a017,
+     16#3e780771f634fb2,16#146c259c02e7e18,16#1d99e4cfad0ef1,16#fdf3dabefc6b3a,
+     16#7d0806e4d12dfb,16#3e3ae3580532eae,16#2456544200fbd86,16#f83aad4e88db85,
+     16#37c134779463b4d,16#21a20bf64b6e735,16#1c0585ac88b69f2,16#1b3fcea8dd30e56,
+     16#334bc301aefd97,16#37066eb7e80a946,16#15a19a6331b570f,16#35e67fa43c3f7d0,
+     16#152a4020145fb80,16#8d55139491dfbe,16#21d9cba585c059d,16#31475f363654635,
+     16#2567b17acb7a104,16#39201be3a7681c5,16#6bc675fd26b601,16#334b93232b1b1e3,
+     16#357c402cb732c6a,16#362e32efe4db46a,16#8edc7ae3da51e5,16#31573376785eac9,
+     16#6c6145ffa1169d,16#18ec2c393d45359,16#1f1a5f256e7130c,16#131cc2f49b8004f,
+     16#36f715a249f4ec2,16#1c27629826c50d3,16#914d9a6648726a,16#27f5bf5ce2301e8,
+     16#3dd493b8012970f,16#be13bed1e00e5c,16#ceef033b74ae10,16#3da38c6a50abe03,
+     16#15cbd1a421c7a8c,16#22794e3ec6ef3b1,16#26154d26e7ea99f,16#3a66681359a6ab6].