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-rw-r--r--lib/stdlib/doc/src/Makefile1
-rw-r--r--lib/stdlib/doc/src/rand.xml246
-rw-r--r--lib/stdlib/doc/src/random.xml3
-rw-r--r--lib/stdlib/doc/src/ref_man.xml1
-rw-r--r--lib/stdlib/doc/src/specs.xml1
-rw-r--r--lib/stdlib/src/Makefile1
-rw-r--r--lib/stdlib/src/rand.erl591
-rw-r--r--lib/stdlib/src/stdlib.app.src1
-rw-r--r--lib/stdlib/test/Makefile1
-rw-r--r--lib/stdlib/test/rand_SUITE.erl527
10 files changed, 1373 insertions, 0 deletions
diff --git a/lib/stdlib/doc/src/Makefile b/lib/stdlib/doc/src/Makefile
index ce1e19a2a4..a4a2ed9931 100644
--- a/lib/stdlib/doc/src/Makefile
+++ b/lib/stdlib/doc/src/Makefile
@@ -82,6 +82,7 @@ XML_REF3_FILES = \
proplists.xml \
qlc.xml \
queue.xml \
+ rand.xml \
random.xml \
re.xml \
sets.xml \
diff --git a/lib/stdlib/doc/src/rand.xml b/lib/stdlib/doc/src/rand.xml
new file mode 100644
index 0000000000..178afda5a0
--- /dev/null
+++ b/lib/stdlib/doc/src/rand.xml
@@ -0,0 +1,246 @@
+<?xml version="1.0" encoding="utf-8" ?>
+<!DOCTYPE erlref SYSTEM "erlref.dtd">
+
+<erlref>
+ <header>
+ <copyright>
+ <year>2015</year>
+ <holder>Ericsson AB. All Rights Reserved.</holder>
+ </copyright>
+ <legalnotice>
+ 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.
+
+ </legalnotice>
+
+ <title>rand</title>
+ <prepared></prepared>
+ <responsible></responsible>
+ <docno>1</docno>
+ <approved></approved>
+ <checked></checked>
+ <date></date>
+ <rev>A</rev>
+ <file>rand.xml</file>
+ </header>
+ <module>rand</module>
+ <modulesummary>Pseudo random number generation</modulesummary>
+ <description>
+ <p>Random number generator.</p>
+
+ <p>The module contains several different algorithms and can be
+ extended with more in the future. The current uniform
+ distribution algorithms uses the
+ <url href="http://xorshift.di.unimi.it">
+ scrambled Xorshift algorithms by Sebastiano Vigna</url> and the
+ normal distribution algorithm uses the
+ <url href="http://www.jstatsoft.org/v05/i08">
+ Ziggurat Method by Marsaglia and Tsang</url>.
+ </p>
+
+ <p>The implemented algorithms are:</p>
+ <taglist>
+ <tag><c>exsplus</c></tag> <item>Xorshift116+, 58 bits precision and period of 2^116-1.</item>
+ <tag><c>exs64</c></tag> <item>Xorshift64*, 64 bits precision and a period of 2^64-1.</item>
+ <tag><c>exs1024</c></tag> <item>Xorshift1024*, 64 bits precision and a period of 2^1024-1.</item>
+ </taglist>
+
+ <p>The current default algorithm is <c>exsplus</c>. The default
+ may change in future. If a specific algorithm is required make
+ sure to always use <seealso marker="#seed-1">seed/1</seealso>
+ to initialize the state.
+ </p>
+
+ <p>Every time a random number is requested, a state is used to
+ calculate it and a new state produced. The state can either be
+ implicit or it can be an explicit argument and return value.
+ </p>
+
+ <p>The functions with implicit state use the process dictionary
+ variable <c>rand_seed</c> to remember the current state.</p>
+
+ <p>If a process calls <seealso marker="#uniform-0">uniform/0</seealso> or
+ <seealso marker="#uniform-1">uniform/1</seealso> without
+ setting a seed first, <seealso marker="#seed-1">seed/1</seealso>
+ is called automatically with the default algorithm and creates a
+ non-constant seed.</p>
+
+ <p>The functions with explicit state never use the process
+ dictionary.</p>
+
+ <p>Examples:</p>
+ <pre>
+ %% Simple usage. Creates and seeds the default algorithm
+ %% with a non-constant seed if not already done.
+ R0 = rand:uniform(),
+ R1 = rand:uniform(),
+
+ %% Use a given algorithm.
+ _ = rand:seed(exs1024),
+ R2 = rand:uniform(),
+
+ %% Use a given algorithm with a constant seed.
+ _ = rand:seed(exs1024, {123, 123534, 345345}),
+ R3 = rand:uniform(),
+
+ %% Use the functional api with non-constant seed.
+ S0 = rand:seed_s(exsplus),
+ {R4, S1} = rand:uniform_s(S0),
+
+ %% Create a standard normal deviate.
+ {SND0, S2} = rand:normal_s(S1),
+ </pre>
+
+ <note><p>This random number generator is not cryptographically
+ strong. If a strong cryptographic random number generator is
+ needed, use one of functions in the
+ <seealso marker="crypto:crypto">crypto</seealso>
+ module, for example <c>crypto:rand_bytes/1</c>.</p></note>
+ </description>
+ <datatypes>
+ <datatype>
+ <name name="alg"/>
+ </datatype>
+
+ <datatype>
+ <name name="state"/>
+ <desc><p>Algorithm dependent state.</p></desc>
+ </datatype>
+
+ <datatype>
+ <name name="export_state"/>
+ <desc><p>Algorithm dependent state which can be printed or saved to file.</p></desc>
+ </datatype>
+ </datatypes>
+
+ <funcs>
+ <func>
+ <name name="seed" arity="1"/>
+ <fsummary>Seed random number generator</fsummary>
+ <desc>
+ <marker id="seed-1"/>
+ <p>Seeds random number generation with the given algorithm and time dependent
+ data if <anno>AlgOrExpState</anno> is an algorithm.</p>
+ <p>Otherwise recreates the exported seed in the process
+ dictionary, and returns the state.
+ <em>See also:</em> <seealso marker="#export_seed-0">export_seed/0</seealso>.</p>
+ </desc>
+ </func>
+ <func>
+ <name name="seed_s" arity="1"/>
+ <fsummary>Seed random number generator</fsummary>
+ <desc>
+ <p>Seeds random number generation with the given algorithm and time dependent
+ data if <anno>AlgOrExpState</anno> is an algorithm.</p>
+ <p>Otherwise recreates the exported seed and returns the state.
+ <em>See also:</em> <seealso marker="#export_seed-0">export_seed/0</seealso>.</p>
+ </desc>
+ </func>
+ <func>
+ <name name="seed" arity="2"/>
+ <fsummary>Seed the random number generation</fsummary>
+ <desc>
+ <p>Seeds random number generation with the given algorithm and
+ integers in the process dictionary and returns
+ the state.</p>
+ </desc>
+ </func>
+ <func>
+ <name name="seed_s" arity="2"/>
+ <fsummary>Seed the random number generation</fsummary>
+ <desc>
+ <p>Seeds random number generation with the given algorithm and
+ integers and returns the state.</p>
+ </desc>
+ </func>
+
+ <func>
+ <name name="export_seed" arity="0"/>
+ <fsummary>Export the random number generation state</fsummary>
+ <desc><marker id="export_seed-0"/>
+ <p>Returns the random number state in an external format.
+ To be used with <seealso marker="#seed-1">seed/1</seealso>.</p>
+ </desc>
+ </func>
+
+ <func>
+ <name name="export_seed_s" arity="1"/>
+ <fsummary>Export the random number generation state</fsummary>
+ <desc><marker id="export_seed_s-1"/>
+ <p>Returns the random number generator state in an external format.
+ To be used with <seealso marker="#seed-1">seed/1</seealso>.</p>
+ </desc>
+ </func>
+
+ <func>
+ <name name="uniform" arity="0"/>
+ <fsummary>Return a random float</fsummary>
+ <desc>
+ <marker id="uniform-0"/>
+ <p>Returns a random float uniformly distributed in the value
+ range <c>0.0 &lt; <anno>X</anno> &lt; 1.0 </c> and
+ updates the state in the process dictionary.</p>
+ </desc>
+ </func>
+ <func>
+ <name name="uniform_s" arity="1"/>
+ <fsummary>Return a random float</fsummary>
+ <desc>
+ <p>Given a state, <c>uniform_s/1</c> returns a random float
+ uniformly distributed in the value range <c>0.0 &lt;
+ <anno>X</anno> &lt; 1.0</c> and a new state.</p>
+ </desc>
+ </func>
+
+ <func>
+ <name name="uniform" arity="1"/>
+ <fsummary>Return a random integer</fsummary>
+ <desc>
+ <marker id="uniform-1"/>
+ <p>Given an integer <c><anno>N</anno> >= 1</c>,
+ <c>uniform/1</c> returns a random integer uniformly
+ distributed in the value range
+ <c>1 &lt;= <anno>X</anno> &lt;= <anno>N</anno></c> and
+ updates the state in the process dictionary.</p>
+ </desc>
+ </func>
+ <func>
+ <name name="uniform_s" arity="2"/>
+ <fsummary>Return a random integer</fsummary>
+ <desc>
+ <p>Given an integer <c><anno>N</anno> >= 1</c> and a state,
+ <c>uniform_s/2</c> returns a random integer uniformly
+ distributed in the value range <c>1 &lt;= <anno>X</anno> &lt;=
+ <anno>N</anno></c> and a new state.</p>
+ </desc>
+ </func>
+
+ <func>
+ <name name="normal" arity="0"/>
+ <fsummary>Return a standard normal distributed random float</fsummary>
+ <desc>
+ <p>Returns a standard normal deviate float (that is, the mean
+ is 0 and the standard deviation is 1) and updates the state in
+ the process dictionary.</p>
+ </desc>
+ </func>
+ <func>
+ <name name="normal_s" arity="1"/>
+ <fsummary>Return a standard normal distributed random float</fsummary>
+ <desc>
+ <p>Given a state, <c>normal_s/1</c> returns a standard normal
+ deviate float (that is, the mean is 0 and the standard
+ deviation is 1) and a new state.</p>
+ </desc>
+ </func>
+
+ </funcs>
+</erlref>
diff --git a/lib/stdlib/doc/src/random.xml b/lib/stdlib/doc/src/random.xml
index 2cc621ffc3..e475cda23d 100644
--- a/lib/stdlib/doc/src/random.xml
+++ b/lib/stdlib/doc/src/random.xml
@@ -48,6 +48,9 @@
<p>It should be noted that this random number generator is not cryptographically
strong. If a strong cryptographic random number generator is needed for
example <c>crypto:rand_bytes/1</c> could be used instead.</p>
+ <note><p>The new and improved <seealso
+ marker="stdlib:rand">rand</seealso> module should be used
+ instead of this module.</p></note>
</description>
<datatypes>
<datatype>
diff --git a/lib/stdlib/doc/src/ref_man.xml b/lib/stdlib/doc/src/ref_man.xml
index 94c1fb55c2..eee4a68ca1 100644
--- a/lib/stdlib/doc/src/ref_man.xml
+++ b/lib/stdlib/doc/src/ref_man.xml
@@ -79,6 +79,7 @@
<xi:include href="proplists.xml"/>
<xi:include href="qlc.xml"/>
<xi:include href="queue.xml"/>
+ <xi:include href="rand.xml"/>
<xi:include href="random.xml"/>
<xi:include href="re.xml"/>
<xi:include href="sets.xml"/>
diff --git a/lib/stdlib/doc/src/specs.xml b/lib/stdlib/doc/src/specs.xml
index 6ae0154800..0418bf7b22 100644
--- a/lib/stdlib/doc/src/specs.xml
+++ b/lib/stdlib/doc/src/specs.xml
@@ -45,6 +45,7 @@
<xi:include href="../specs/specs_proplists.xml"/>
<xi:include href="../specs/specs_qlc.xml"/>
<xi:include href="../specs/specs_queue.xml"/>
+ <xi:include href="../specs/specs_rand.xml"/>
<xi:include href="../specs/specs_random.xml"/>
<xi:include href="../specs/specs_re.xml"/>
<xi:include href="../specs/specs_sets.xml"/>
diff --git a/lib/stdlib/src/Makefile b/lib/stdlib/src/Makefile
index 85b858febd..55bda60da5 100644
--- a/lib/stdlib/src/Makefile
+++ b/lib/stdlib/src/Makefile
@@ -105,6 +105,7 @@ MODULES= \
qlc \
qlc_pt \
queue \
+ rand \
random \
sets \
shell \
diff --git a/lib/stdlib/src/rand.erl b/lib/stdlib/src/rand.erl
new file mode 100644
index 0000000000..6a805eb69e
--- /dev/null
+++ b/lib/stdlib/src/rand.erl
@@ -0,0 +1,591 @@
+%%
+%% %CopyrightBegin%
+%%
+%% Copyright Ericsson AB 2015. 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%
+%%
+%% =====================================================================
+%% Multiple PRNG module for Erlang/OTP
+%% Copyright (c) 2015 Kenji Rikitake
+%% =====================================================================
+
+-module(rand).
+
+-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,
+ normal/0, normal_s/1
+ ]).
+
+-compile({inline, [exs64_next/1, exsplus_next/1,
+ exs1024_next/1, exs1024_calc/2,
+ get_52/1, normal_kiwi/1]}).
+
+-define(DEFAULT_ALG_HANDLER, exsplus).
+-define(SEED_DICT, rand_seed).
+
+%% =====================================================================
+%% Types
+%% =====================================================================
+
+%% This depends on the algorithm handler function
+-type alg_seed() :: exs64_state() | exsplus_state() | exs1024_state().
+%% This is the algorithm handler function within this module
+-type alg_handler() :: #{type => alg(),
+ max => integer(),
+ next => fun(),
+ uniform => fun(),
+ uniform_n => fun()}.
+
+%% Internal state
+-opaque state() :: {alg_handler(), alg_seed()}.
+-type alg() :: exs64 | exsplus | exs1024.
+-opaque export_state() :: {alg(), alg_seed()}.
+-export_type([alg/0, state/0, export_state/0]).
+
+%% =====================================================================
+%% API
+%% =====================================================================
+
+%% Return algorithm and seed so that RNG state can be recreated with seed/1
+-spec export_seed() -> undefined | export_state().
+export_seed() ->
+ case seed_get() of
+ {#{type:=Alg}, Seed} -> {Alg, Seed};
+ _ -> undefined
+ end.
+
+-spec export_seed_s(state()) -> export_state().
+export_seed_s({#{type:=Alg}, Seed}) -> {Alg, Seed}.
+
+%% seed(Alg) seeds RNG with runtime dependent values
+%% and return the NEW state
+
+%% seed({Alg,Seed}) setup RNG with a previously exported seed
+%% and return the NEW state
+
+-spec seed(AlgOrExpState::alg() | export_state()) -> state().
+seed(Alg) ->
+ R = seed_s(Alg),
+ _ = seed_put(R),
+ R.
+
+-spec seed_s(AlgOrExpState::alg() | export_state()) -> state().
+seed_s(Alg) when is_atom(Alg) ->
+ seed_s(Alg, {erlang:phash2([{node(),self()}]),
+ erlang:system_time(),
+ erlang:unique_integer()});
+seed_s({Alg0, Seed}) ->
+ {Alg,_SeedFun} = mk_alg(Alg0),
+ {Alg, Seed}.
+
+%% seed/2: seeds RNG with the algorithm and given values
+%% and returns the NEW state.
+
+-spec seed(Alg :: alg(), {integer(), integer(), integer()}) -> state().
+seed(Alg0, S0) ->
+ State = seed_s(Alg0, S0),
+ _ = seed_put(State),
+ State.
+
+-spec seed_s(Alg :: alg(), {integer(), integer(), integer()}) -> state().
+seed_s(Alg0, S0 = {_, _, _}) ->
+ {Alg, Seed} = mk_alg(Alg0),
+ AS = Seed(S0),
+ {Alg, AS}.
+
+%%% uniform/0, uniform/1, uniform_s/1, uniform_s/2 are all
+%%% uniformly distributed random numbers.
+
+%% uniform/0: returns a random float X where 0.0 < X < 1.0,
+%% updating the state in the process dictionary.
+
+-spec uniform() -> X::float().
+uniform() ->
+ {X, Seed} = uniform_s(seed_get()),
+ _ = seed_put(Seed),
+ X.
+
+%% uniform/1: given an integer N >= 1,
+%% uniform/1 returns a random integer X where 1 =< X =< N,
+%% updating the state in the process dictionary.
+
+-spec uniform(N :: pos_integer()) -> X::pos_integer().
+uniform(N) ->
+ {X, Seed} = uniform_s(N, seed_get()),
+ _ = seed_put(Seed),
+ X.
+
+%% uniform_s/1: given a state, uniform_s/1
+%% returns a random float X where 0.0 < X < 1.0,
+%% and a new state.
+
+-spec uniform_s(state()) -> {X::float(), NewS :: state()}.
+uniform_s(State = {#{uniform:=Uniform}, _}) ->
+ Uniform(State).
+
+%% uniform_s/2: given an integer N >= 1 and a state, uniform_s/2
+%% uniform_s/2 returns a random integer X where 1 =< X =< N,
+%% and a new state.
+
+-spec uniform_s(N::pos_integer(), state()) -> {X::pos_integer(), NewS::state()}.
+uniform_s(N, State = {#{uniform_n:=Uniform, max:=Max}, _})
+ when 0 < N, N =< Max ->
+ Uniform(N, State);
+uniform_s(N, State0 = {#{uniform:=Uniform}, _})
+ when is_integer(N), 0 < N ->
+ {F, State} = Uniform(State0),
+ {trunc(F * N) + 1, State}.
+
+%% normal/0: returns a random float with standard normal distribution
+%% updating the state in the process dictionary.
+
+-spec normal() -> float().
+normal() ->
+ {X, Seed} = normal_s(seed_get()),
+ _ = seed_put(Seed),
+ X.
+
+%% normal_s/1: returns a random float with standard normal distribution
+%% The Ziggurat Method for generating random variables - Marsaglia and Tsang
+%% Paper and reference code: http://www.jstatsoft.org/v05/i08/
+
+-spec normal_s(state()) -> {float(), NewS :: state()}.
+normal_s(State0) ->
+ {Sign, R, State} = get_52(State0),
+ Idx = R band 16#FF,
+ Idx1 = Idx+1,
+ {Ki, Wi} = normal_kiwi(Idx1),
+ X = R * Wi,
+ case R < Ki of
+ %% Fast path 95% of the time
+ true when Sign =:= 0 -> {X, State};
+ true -> {-X, State};
+ %% Slow path
+ false when Sign =:= 0 -> normal_s(Idx, Sign, X, State);
+ false -> normal_s(Idx, Sign, -X, State)
+ end.
+
+%% =====================================================================
+%% Internal functions
+
+-define(UINT21MASK, 16#00000000001fffff).
+-define(UINT32MASK, 16#00000000ffffffff).
+-define(UINT33MASK, 16#00000001ffffffff).
+-define(UINT39MASK, 16#0000007fffffffff).
+-define(UINT58MASK, 16#03ffffffffffffff).
+-define(UINT64MASK, 16#ffffffffffffffff).
+
+-type uint64() :: 0..16#ffffffffffffffff.
+-type uint58() :: 0..16#03ffffffffffffff.
+
+-spec seed_put(state()) -> undefined | state().
+seed_put(Seed) ->
+ put(?SEED_DICT, Seed).
+
+seed_get() ->
+ case get(?SEED_DICT) of
+ undefined -> seed(?DEFAULT_ALG_HANDLER);
+ Old -> Old % no type checking here
+ end.
+
+%% 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},
+ 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},
+ 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},
+ fun exs1024_seed/1}.
+
+%% =====================================================================
+%% exs64 PRNG: Xorshift64*
+%% Algorithm by Sebastiano Vigna
+%% Reference URL: http://xorshift.di.unimi.it/
+%% =====================================================================
+
+-type exs64_state() :: uint64().
+
+exs64_seed({A1, A2, A3}) ->
+ {V1, _} = exs64_next(((A1 band ?UINT32MASK) * 4294967197 + 1)),
+ {V2, _} = exs64_next(((A2 band ?UINT32MASK) * 4294967231 + 1)),
+ {V3, _} = exs64_next(((A3 band ?UINT32MASK) * 4294967279 + 1)),
+ ((V1 * V2 * V3) rem (?UINT64MASK - 1)) + 1.
+
+%% Advance xorshift64* state for one step and generate 64bit unsigned integer
+-spec exs64_next(exs64_state()) -> {uint64(), exs64_state()}.
+exs64_next(R) ->
+ R1 = R bxor (R bsr 12),
+ R2 = R1 bxor ((R1 band ?UINT39MASK) bsl 25),
+ R3 = R2 bxor (R2 bsr 27),
+ {(R3 * 2685821657736338717) band ?UINT64MASK, R3}.
+
+exs64_uniform({Alg, R0}) ->
+ {V, R1} = exs64_next(R0),
+ {V / 18446744073709551616, {Alg, R1}}.
+
+exs64_uniform(Max, {Alg, R}) ->
+ {V, R1} = exs64_next(R),
+ {(V rem Max) + 1, {Alg, R1}}.
+
+%% =====================================================================
+%% exsplus PRNG: Xorshift116+
+%% Algorithm by Sebastiano Vigna
+%% Reference URL: http://xorshift.di.unimi.it/
+%% 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.
+%% =====================================================================
+-type exsplus_state() :: nonempty_improper_list(uint58(), uint58()).
+
+exsplus_seed({A1, A2, A3}) ->
+ {_, R1} = exsplus_next([(((A1 * 4294967197) + 1) band ?UINT58MASK)|
+ (((A2 * 4294967231) + 1) band ?UINT58MASK)]),
+ {_, R2} = exsplus_next([(((A3 * 4294967279) + 1) band ?UINT58MASK)|
+ tl(R1)]),
+ R2.
+
+%% Advance xorshift116+ state for one step 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 (S1 bsl 24)) band ?UINT58MASK,
+ S12 = S11 bxor S0 bxor (S11 bsr 11) bxor (S0 bsr 41),
+ {(S0 + S12) band ?UINT58MASK, [S0|S12]}.
+
+exsplus_uniform({Alg, R0}) ->
+ {I, R1} = exsplus_next(R0),
+ {I / (?UINT58MASK+1), {Alg, R1}}.
+
+exsplus_uniform(Max, {Alg, R}) ->
+ {V, R1} = exsplus_next(R),
+ {(V rem Max) + 1, {Alg, R1}}.
+
+%% =====================================================================
+%% exs1024 PRNG: Xorshift1024*
+%% Algorithm by Sebastiano Vigna
+%% Reference URL: http://xorshift.di.unimi.it/
+%% =====================================================================
+
+-type exs1024_state() :: {list(uint64()), list(uint64())}.
+
+exs1024_seed({A1, A2, A3}) ->
+ B1 = (((A1 band ?UINT21MASK) + 1) * 2097131) band ?UINT21MASK,
+ B2 = (((A2 band ?UINT21MASK) + 1) * 2097133) band ?UINT21MASK,
+ B3 = (((A3 band ?UINT21MASK) + 1) * 2097143) band ?UINT21MASK,
+ {exs1024_gen1024((B1 bsl 43) bor (B2 bsl 22) bor (B3 bsl 1) bor 1),
+ []}.
+
+%% Generate a list of 16 64-bit element list
+%% of the xorshift64* random sequence
+%% from a given 64-bit seed.
+%% Note: dependent on exs64_next/1
+-spec exs1024_gen1024(uint64()) -> list(uint64()).
+exs1024_gen1024(R) ->
+ exs1024_gen1024(16, R, []).
+
+exs1024_gen1024(0, _, L) ->
+ L;
+exs1024_gen1024(N, R, L) ->
+ {X, R2} = exs64_next(R),
+ exs1024_gen1024(N - 1, R2, [X|L]).
+
+%% Calculation of xorshift1024*.
+%% exs1024_calc(S0, S1) -> {X, NS1}.
+%% X: random number output
+-spec exs1024_calc(uint64(), uint64()) -> {uint64(), uint64()}.
+exs1024_calc(S0, S1) ->
+ S11 = S1 bxor ((S1 band ?UINT33MASK) bsl 31),
+ S12 = S11 bxor (S11 bsr 11),
+ S01 = S0 bxor (S0 bsr 30),
+ NS1 = S01 bxor S12,
+ {(NS1 * 1181783497276652981) band ?UINT64MASK, NS1}.
+
+%% Advance xorshift1024* state for one step and generate 64bit unsigned integer
+-spec exs1024_next(exs1024_state()) -> {uint64(), exs1024_state()}.
+exs1024_next({[S0,S1|L3], RL}) ->
+ {X, NS1} = exs1024_calc(S0, S1),
+ {X, {[NS1|L3], [S0|RL]}};
+exs1024_next({[H], RL}) ->
+ NL = [H|lists:reverse(RL)],
+ exs1024_next({NL, []}).
+
+exs1024_uniform({Alg, R0}) ->
+ {V, R1} = exs1024_next(R0),
+ {V / 18446744073709551616, {Alg, R1}}.
+
+exs1024_uniform(Max, {Alg, R}) ->
+ {V, R1} = exs1024_next(R),
+ {(V rem Max) + 1, {Alg, R1}}.
+
+%% =====================================================================
+%% Ziggurat cont
+%% =====================================================================
+-define(NOR_R, 3.6541528853610087963519472518).
+-define(NOR_INV_R, 1/?NOR_R).
+
+%% return a {sign, Random51bits, State}
+get_52({Alg=#{next:=Next}, S0}) ->
+ {Int,S1} = Next(S0),
+ {((1 bsl 51) band Int), Int band ((1 bsl 51)-1), {Alg, S1}}.
+
+%% Slow path
+normal_s(0, Sign, X0, State0) ->
+ {U0, S1} = uniform_s(State0),
+ X = -?NOR_INV_R*math:log(U0),
+ {U1, S2} = uniform_s(S1),
+ Y = -math:log(U1),
+ case Y+Y > X*X of
+ false ->
+ normal_s(0, Sign, X0, S2);
+ true when Sign =:= 0 ->
+ {?NOR_R + X, S2};
+ true ->
+ {-?NOR_R - X, S2}
+ end;
+normal_s(Idx, _Sign, X, State0) ->
+ Fi2 = normal_fi(Idx+1),
+ {U0, S1} = uniform_s(State0),
+ case ((normal_fi(Idx) - Fi2)*U0 + Fi2) < math:exp(-0.5*X*X) of
+ true -> {X, S1};
+ false -> normal_s(S1)
+ end.
+
+%% Tables for generating normal_s
+%% ki is zipped with wi (slightly faster)
+normal_kiwi(Indx) ->
+ element(Indx,
+ {{2104047571236786,1.736725412160263e-15}, {0,9.558660351455634e-17},
+ {1693657211986787,1.2708704834810623e-16},{1919380038271141,1.4909740962495474e-16},
+ {2015384402196343,1.6658733631586268e-16},{2068365869448128,1.8136120810119029e-16},
+ {2101878624052573,1.9429720153135588e-16},{2124958784102998,2.0589500628482093e-16},
+ {2141808670795147,2.1646860576895422e-16},{2154644611568301,2.2622940392218116e-16},
+ {2164744887587275,2.353271891404589e-16},{2172897953696594,2.438723455742877e-16},
+ {2179616279372365,2.5194879829274225e-16},{2185247251868649,2.5962199772528103e-16},
+ {2190034623107822,2.6694407473648285e-16},{2194154434521197,2.7395729685142446e-16},
+ {2197736978774660,2.8069646002484804e-16},{2200880740891961,2.871905890411393e-16},
+ {2203661538010620,2.9346417484728883e-16},{2206138681109102,2.9953809336782113e-16},
+ {2208359231806599,3.054303000719244e-16},{2210361007258210,3.111563633892157e-16},
+ {2212174742388539,3.1672988018581815e-16},{2213825672704646,3.2216280350549905e-16},
+ {2215334711002614,3.274657040793975e-16},{2216719334487595,3.326479811684171e-16},
+ {2217994262139172,3.377180341735323e-16},{2219171977965032,3.4268340353119356e-16},
+ {2220263139538712,3.475508873172976e-16},{2221276900117330,3.523266384600203e-16},
+ {2222221164932930,3.5701624633953494e-16},{2223102796829069,3.616248057159834e-16},
+ {2223927782546658,3.661569752965354e-16},{2224701368170060,3.7061702777236077e-16},
+ {2225428170204312,3.75008892787478e-16},{2226112267248242,3.7933619401549554e-16},
+ {2226757276105256,3.836022812967728e-16},{2227366415328399,3.8781025861250247e-16},
+ {2227942558554684,3.919630085325768e-16},{2228488279492521,3.9606321366256378e-16},
+ {2229005890047222,4.001133755254669e-16},{2229497472775193,4.041158312414333e-16},
+ {2229964908627060,4.080727683096045e-16},{2230409900758597,4.119862377480744e-16},
+ {2230833995044585,4.1585816580828064e-16},{2231238597816133,4.1969036444740733e-16},
+ {2231624991250191,4.234845407152071e-16},{2231994346765928,4.272423051889976e-16},
+ {2232347736722750,4.309651795716294e-16},{2232686144665934,4.346546035512876e-16},
+ {2233010474325959,4.383119410085457e-16},{2233321557544881,4.4193848564470665e-16},
+ {2233620161276071,4.455354660957914e-16},{2233906993781271,4.491040505882875e-16},
+ {2234182710130335,4.52645351185714e-16},{2234447917093496,4.561604276690038e-16},
+ {2234703177503020,4.596502910884941e-16},{2234949014150181,4.631159070208165e-16},
+ {2235185913274316,4.665581985600875e-16},{2235414327692884,4.699780490694195e-16},
+ {2235634679614920,4.733763047158324e-16},{2235847363174595,4.767537768090853e-16},
+ {2236052746716837,4.8011124396270155e-16},{2236251174862869,4.834494540935008e-16},
+ {2236442970379967,4.867691262742209e-16},{2236628435876762,4.900709524522994e-16},
+ {2236807855342765,4.933555990465414e-16},{2236981495548562,4.966237084322178e-16},
+ {2237149607321147,4.998759003240909e-16},{2237312426707209,5.031127730659319e-16},
+ {2237470176035652,5.0633490483427195e-16},{2237623064889403,5.095428547633892e-16},
+ {2237771290995388,5.127371639978797e-16},{2237915041040597,5.159183566785736e-16},
+ {2238054491421305,5.190869408670343e-16},{2238189808931712,5.222434094134042e-16},
+ {2238321151397660,5.253882407719454e-16},{2238448668260432,5.285218997682382e-16},
+ {2238572501115169,5.316448383216618e-16},{2238692784207942,5.34757496126473e-16},
+ {2238809644895133,5.378603012945235e-16},{2238923204068402,5.409536709623993e-16},
+ {2239033576548190,5.440380118655467e-16},{2239140871448443,5.471137208817361e-16},
+ {2239245192514958,5.501811855460336e-16},{2239346638439541,5.532407845392784e-16},
+ {2239445303151952,5.56292888151909e-16},{2239541276091442,5.593378587248462e-16},
+ {2239634642459498,5.623760510690043e-16},{2239725483455293,5.65407812864896e-16},
+ {2239813876495186,5.684334850436814e-16},{2239899895417494,5.714534021509204e-16},
+ {2239983610673676,5.744678926941961e-16},{2240065089506935,5.774772794756965e-16},
+ {2240144396119183,5.804818799107686e-16},{2240221591827230,5.834820063333892e-16},
+ {2240296735208969,5.864779662894365e-16},{2240369882240293,5.894700628185872e-16},
+ {2240441086423386,5.924585947256134e-16},{2240510398907004,5.95443856841806e-16},
+ {2240577868599305,5.984261402772028e-16},{2240643542273726,6.014057326642664e-16},
+ {2240707464668391,6.043829183936125e-16},{2240769678579486,6.073579788423606e-16},
+ {2240830224948980,6.103311925956439e-16},{2240889142947082,6.133028356617911e-16},
+ {2240946470049769,6.162731816816596e-16},{2241002242111691,6.192425021325847e-16},
+ {2241056493434746,6.222110665273788e-16},{2241109256832602,6.251791426088e-16},
+ {2241160563691400,6.281469965398895e-16},{2241210444026879,6.311148930905604e-16},
+ {2241258926538122,6.34083095820806e-16},{2241306038658137,6.370518672608815e-16},
+ {2241351806601435,6.400214690888025e-16},{2241396255408788,6.429921623054896e-16},
+ {2241439408989313,6.459642074078832e-16},{2241481290160038,6.489378645603397e-16},
+ {2241521920683062,6.519133937646159e-16},{2241561321300462,6.548910550287415e-16},
+ {2241599511767028,6.578711085350741e-16},{2241636510880960,6.608538148078259e-16},
+ {2241672336512612,6.638394348803506e-16},{2241707005631362,6.668282304624746e-16},
+ {2241740534330713,6.698204641081558e-16},{2241772937851689,6.728163993837531e-16},
+ {2241804230604585,6.758163010371901e-16},{2241834426189161,6.78820435168298e-16},
+ {2241863537413311,6.818290694006254e-16},{2241891576310281,6.848424730550038e-16},
+ {2241918554154466,6.878609173251664e-16},{2241944481475843,6.908846754557169e-16},
+ {2241969368073071,6.939140229227569e-16},{2241993223025298,6.969492376174829e-16},
+ {2242016054702685,6.999906000330764e-16},{2242037870775710,7.030383934552151e-16},
+ {2242058678223225,7.060929041565482e-16},{2242078483339331,7.091544215954873e-16},
+ {2242097291739040,7.122232386196779e-16},{2242115108362774,7.152996516745303e-16},
+ {2242131937479672,7.183839610172063e-16},{2242147782689725,7.214764709364707e-16},
+ {2242162646924736,7.245774899788387e-16},{2242176532448092,7.276873311814693e-16},
+ {2242189440853337,7.308063123122743e-16},{2242201373061537,7.339347561177405e-16},
+ {2242212329317416,7.370729905789831e-16},{2242222309184237,7.4022134917658e-16},
+ {2242231311537397,7.433801711647648e-16},{2242239334556717,7.465498018555889e-16},
+ {2242246375717369,7.497305929136979e-16},{2242252431779415,7.529229026624058e-16},
+ {2242257498775893,7.561270964017922e-16},{2242261571999416,7.5934354673958895e-16},
+ {2242264645987196,7.625726339356756e-16},{2242266714504453,7.658147462610487e-16},
+ {2242267770526109,7.690702803721919e-16},{2242267806216711,7.723396417018299e-16},
+ {2242266812908462,7.756232448671174e-16},{2242264781077289,7.789215140963852e-16},
+ {2242261700316818,7.822348836756411e-16},{2242257559310145,7.855637984161084e-16},
+ {2242252345799276,7.889087141441755e-16},{2242246046552082,7.922700982152271e-16},
+ {2242238647326615,7.956484300529366e-16},{2242230132832625,7.99044201715713e-16},
+ {2242220486690076,8.024579184921259e-16},{2242209691384458,8.058900995272657e-16},
+ {2242197728218684,8.093412784821501e-16},{2242184577261310,8.128120042284501e-16},
+ {2242170217290819,8.163028415809877e-16},{2242154625735679,8.198143720706533e-16},
+ {2242137778609839,8.23347194760605e-16},{2242119650443327,8.26901927108847e-16},
+ {2242100214207556,8.304792058805374e-16},{2242079441234906,8.340796881136629e-16},
+ {2242057301132135,8.377040521420222e-16},{2242033761687079,8.413529986798028e-16},
+ {2242008788768107,8.450272519724097e-16},{2241982346215682,8.487275610186155e-16},
+ {2241954395725356,8.524547008695596e-16},{2241924896721443,8.562094740106233e-16},
+ {2241893806220517,8.599927118327665e-16},{2241861078683830,8.638052762005259e-16},
+ {2241826665857598,8.676480611245582e-16},{2241790516600041,8.715219945473698e-16},
+ {2241752576693881,8.754280402517175e-16},{2241712788642916,8.793671999021043e-16},
+ {2241671091451078,8.833405152308408e-16},{2241627420382235,8.873490703813135e-16},
+ {2241581706698773,8.913939944224086e-16},{2241533877376767,8.954764640495068e-16},
+ {2241483854795281,8.9959770648911e-16},{2241431556397035,9.037590026260118e-16},
+ {2241376894317345,9.079616903740068e-16},{2241319774977817,9.122071683134846e-16},
+ {2241260098640860,9.164968996219135e-16},{2241197758920538,9.208324163262308e-16},
+ {2241132642244704,9.252153239095693e-16},{2241064627262652,9.296473063086417e-16},
+ {2240993584191742,9.341301313425265e-16},{2240919374095536,9.38665656618666e-16},
+ {2240841848084890,9.432558359676707e-16},{2240760846432232,9.479027264651738e-16},
+ {2240676197587784,9.526084961066279e-16},{2240587717084782,9.57375432209745e-16},
+ {2240495206318753,9.622059506294838e-16},{2240398451183567,9.671026058823054e-16},
+ {2240297220544165,9.720681022901626e-16},{2240191264522612,9.771053062707209e-16},
+ {2240080312570155,9.822172599190541e-16},{2239964071293331,9.874071960480671e-16},
+ {2239842221996530,9.926785548807976e-16},{2239714417896699,9.980350026183645e-16},
+ {2239580280957725,1.003480452143618e-15},{2239439398282193,1.0090190861637457e-15},
+ {2239291317986196,1.0146553831467086e-15},{2239135544468203,1.0203941464683124e-15},
+ {2238971532964979,1.0262405372613567e-15},{2238798683265269,1.0322001115486456e-15},
+ {2238616332424351,1.03827886235154e-15},{2238423746288095,1.044483267600047e-15},
+ {2238220109591890,1.0508203448355195e-15},{2238004514345216,1.057297713900989e-15},
+ {2237775946143212,1.06392366906768e-15},{2237533267957822,1.0707072623632994e-15},
+ {2237275200846753,1.0776584002668106e-15},{2237000300869952,1.0847879564403425e-15},
+ {2236706931309099,1.0921079038149563e-15},{2236393229029147,1.0996314701785628e-15},
+ {2236057063479501,1.1073733224935752e-15},{2235695986373246,1.1153497865853155e-15},
+ {2235307169458859,1.1235791107110833e-15},{2234887326941578,1.1320817840164846e-15},
+ {2234432617919447,1.140880924258278e-15},{2233938522519765,1.1500027537839792e-15},
+ {2233399683022677,1.159477189144919e-15},{2232809697779198,1.169338578691096e-15},
+ {2232160850599817,1.17962663529558e-15},{2231443750584641,1.190387629928289e-15},
+ {2230646845562170,1.2016759392543819e-15},{2229755753817986,1.2135560818666897e-15},
+ {2228752329126533,1.2261054417450561e-15},{2227613325162504,1.2394179789163251e-15},
+ {2226308442121174,1.2536093926602567e-15},{2224797391720399,1.268824481425501e-15},
+ {2223025347823832,1.2852479319096109e-15},{2220915633329809,1.3031206634689985e-15},
+ {2218357446087030,1.3227655770195326e-15},{2215184158448668,1.3446300925011171e-15},
+ {2211132412537369,1.3693606835128518e-15},{2205758503851065,1.397943667277524e-15},
+ {2198248265654987,1.4319989869661328e-15},{2186916352102141,1.4744848603597596e-15},
+ {2167562552481814,1.5317872741611144e-15},{2125549880839716,1.6227698675312968e-15}}).
+
+normal_fi(Indx) ->
+ element(Indx,
+ {1.0000000000000000e+00,9.7710170126767082e-01,9.5987909180010600e-01,
+ 9.4519895344229909e-01,9.3206007595922991e-01,9.1999150503934646e-01,
+ 9.0872644005213032e-01,8.9809592189834297e-01,8.8798466075583282e-01,
+ 8.7830965580891684e-01,8.6900868803685649e-01,8.6003362119633109e-01,
+ 8.5134625845867751e-01,8.4291565311220373e-01,8.3471629298688299e-01,
+ 8.2672683394622093e-01,8.1892919160370192e-01,8.1130787431265572e-01,
+ 8.0384948317096383e-01,7.9654233042295841e-01,7.8937614356602404e-01,
+ 7.8234183265480195e-01,7.7543130498118662e-01,7.6863731579848571e-01,
+ 7.6195334683679483e-01,7.5537350650709567e-01,7.4889244721915638e-01,
+ 7.4250529634015061e-01,7.3620759812686210e-01,7.2999526456147568e-01,
+ 7.2386453346862967e-01,7.1781193263072152e-01,7.1183424887824798e-01,
+ 7.0592850133275376e-01,7.0009191813651117e-01,6.9432191612611627e-01,
+ 6.8861608300467136e-01,6.8297216164499430e-01,6.7738803621877308e-01,
+ 6.7186171989708166e-01,6.6639134390874977e-01,6.6097514777666277e-01,
+ 6.5561147057969693e-01,6.5029874311081637e-01,6.4503548082082196e-01,
+ 6.3982027745305614e-01,6.3465179928762327e-01,6.2952877992483625e-01,
+ 6.2445001554702606e-01,6.1941436060583399e-01,6.1442072388891344e-01,
+ 6.0946806492577310e-01,6.0455539069746733e-01,5.9968175261912482e-01,
+ 5.9484624376798689e-01,5.9004799633282545e-01,5.8528617926337090e-01,
+ 5.8055999610079034e-01,5.7586868297235316e-01,5.7121150673525267e-01,
+ 5.6658776325616389e-01,5.6199677581452390e-01,5.5743789361876550e-01,
+ 5.5291049042583185e-01,5.4841396325526537e-01,5.4394773119002582e-01,
+ 5.3951123425695158e-01,5.3510393238045717e-01,5.3072530440366150e-01,
+ 5.2637484717168403e-01,5.2205207467232140e-01,5.1775651722975591e-01,
+ 5.1348772074732651e-01,5.0924524599574761e-01,5.0502866794346790e-01,
+ 5.0083757512614835e-01,4.9667156905248933e-01,4.9253026364386815e-01,
+ 4.8841328470545758e-01,4.8432026942668288e-01,4.8025086590904642e-01,
+ 4.7620473271950547e-01,4.7218153846772976e-01,4.6818096140569321e-01,
+ 4.6420268904817391e-01,4.6024641781284248e-01,4.5631185267871610e-01,
+ 4.5239870686184824e-01,4.4850670150720273e-01,4.4463556539573912e-01,
+ 4.4078503466580377e-01,4.3695485254798533e-01,4.3314476911265209e-01,
+ 4.2935454102944126e-01,4.2558393133802180e-01,4.2183270922949573e-01,
+ 4.1810064983784795e-01,4.1438753404089090e-01,4.1069314827018799e-01,
+ 4.0701728432947315e-01,4.0335973922111429e-01,3.9972031498019700e-01,
+ 3.9609881851583223e-01,3.9249506145931540e-01,3.8890886001878855e-01,
+ 3.8534003484007706e-01,3.8178841087339344e-01,3.7825381724561896e-01,
+ 3.7473608713789086e-01,3.7123505766823922e-01,3.6775056977903225e-01,
+ 3.6428246812900372e-01,3.6083060098964775e-01,3.5739482014578022e-01,
+ 3.5397498080007656e-01,3.5057094148140588e-01,3.4718256395679348e-01,
+ 3.4380971314685055e-01,3.4045225704452164e-01,3.3711006663700588e-01,
+ 3.3378301583071823e-01,3.3047098137916342e-01,3.2717384281360129e-01,
+ 3.2389148237639104e-01,3.2062378495690530e-01,3.1737063802991350e-01,
+ 3.1413193159633707e-01,3.1090755812628634e-01,3.0769741250429189e-01,
+ 3.0450139197664983e-01,3.0131939610080288e-01,2.9815132669668531e-01,
+ 2.9499708779996164e-01,2.9185658561709499e-01,2.8872972848218270e-01,
+ 2.8561642681550159e-01,2.8251659308370741e-01,2.7943014176163772e-01,
+ 2.7635698929566810e-01,2.7329705406857691e-01,2.7025025636587519e-01,
+ 2.6721651834356114e-01,2.6419576399726080e-01,2.6118791913272082e-01,
+ 2.5819291133761890e-01,2.5521066995466168e-01,2.5224112605594190e-01,
+ 2.4928421241852824e-01,2.4633986350126363e-01,2.4340801542275012e-01,
+ 2.4048860594050039e-01,2.3758157443123795e-01,2.3468686187232990e-01,
+ 2.3180441082433859e-01,2.2893416541468023e-01,2.2607607132238020e-01,
+ 2.2323007576391746e-01,2.2039612748015194e-01,2.1757417672433113e-01,
+ 2.1476417525117358e-01,2.1196607630703015e-01,2.0917983462112499e-01,
+ 2.0640540639788071e-01,2.0364274931033485e-01,2.0089182249465656e-01,
+ 1.9815258654577511e-01,1.9542500351413428e-01,1.9270903690358912e-01,
+ 1.9000465167046496e-01,1.8731181422380025e-01,1.8463049242679927e-01,
+ 1.8196065559952254e-01,1.7930227452284767e-01,1.7665532144373500e-01,
+ 1.7401977008183875e-01,1.7139559563750595e-01,1.6878277480121151e-01,
+ 1.6618128576448205e-01,1.6359110823236570e-01,1.6101222343751107e-01,
+ 1.5844461415592431e-01,1.5588826472447920e-01,1.5334316106026283e-01,
+ 1.5080929068184568e-01,1.4828664273257453e-01,1.4577520800599403e-01,
+ 1.4327497897351341e-01,1.4078594981444470e-01,1.3830811644855071e-01,
+ 1.3584147657125373e-01,1.3338602969166913e-01,1.3094177717364430e-01,
+ 1.2850872227999952e-01,1.2608687022018586e-01,1.2367622820159654e-01,
+ 1.2127680548479021e-01,1.1888861344290998e-01,1.1651166562561080e-01,
+ 1.1414597782783835e-01,1.1179156816383801e-01,1.0944845714681163e-01,
+ 1.0711666777468364e-01,1.0479622562248690e-01,1.0248715894193508e-01,
+ 1.0018949876880981e-01,9.7903279038862284e-02,9.5628536713008819e-02,
+ 9.3365311912690860e-02,9.1113648066373634e-02,8.8873592068275789e-02,
+ 8.6645194450557961e-02,8.4428509570353374e-02,8.2223595813202863e-02,
+ 8.0030515814663056e-02,7.7849336702096039e-02,7.5680130358927067e-02,
+ 7.3522973713981268e-02,7.1377949058890375e-02,6.9245144397006769e-02,
+ 6.7124653827788497e-02,6.5016577971242842e-02,6.2921024437758113e-02,
+ 6.0838108349539864e-02,5.8767952920933758e-02,5.6710690106202902e-02,
+ 5.4666461324888914e-02,5.2635418276792176e-02,5.0617723860947761e-02,
+ 4.8613553215868521e-02,4.6623094901930368e-02,4.4646552251294443e-02,
+ 4.2684144916474431e-02,4.0736110655940933e-02,3.8802707404526113e-02,
+ 3.6884215688567284e-02,3.4980941461716084e-02,3.3093219458578522e-02,
+ 3.1221417191920245e-02,2.9365939758133314e-02,2.7527235669603082e-02,
+ 2.5705804008548896e-02,2.3902203305795882e-02,2.2117062707308864e-02,
+ 2.0351096230044517e-02,1.8605121275724643e-02,1.6880083152543166e-02,
+ 1.5177088307935325e-02,1.3497450601739880e-02,1.1842757857907888e-02,
+ 1.0214971439701471e-02,8.6165827693987316e-03,7.0508754713732268e-03,
+ 5.5224032992509968e-03,4.0379725933630305e-03,2.6090727461021627e-03,
+ 1.2602859304985975e-03}).
diff --git a/lib/stdlib/src/stdlib.app.src b/lib/stdlib/src/stdlib.app.src
index d4d2237b38..a27a35dca2 100644
--- a/lib/stdlib/src/stdlib.app.src
+++ b/lib/stdlib/src/stdlib.app.src
@@ -84,6 +84,7 @@
qlc,
qlc_pt,
queue,
+ rand,
random,
re,
sets,
diff --git a/lib/stdlib/test/Makefile b/lib/stdlib/test/Makefile
index 9bf10ea494..61eb34d565 100644
--- a/lib/stdlib/test/Makefile
+++ b/lib/stdlib/test/Makefile
@@ -54,6 +54,7 @@ MODULES= \
proc_lib_SUITE \
qlc_SUITE \
queue_SUITE \
+ rand_SUITE \
random_SUITE \
re_SUITE \
run_pcre_tests \
diff --git a/lib/stdlib/test/rand_SUITE.erl b/lib/stdlib/test/rand_SUITE.erl
new file mode 100644
index 0000000000..9a1f37aa75
--- /dev/null
+++ b/lib/stdlib/test/rand_SUITE.erl
@@ -0,0 +1,527 @@
+%%
+%% %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),
+ io:format("Seed ~p~n",[rand:export_seed_s(Seed)]),
+ 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),
+ {V2, S2} = rand:normal_s(S1),
+ V2 = rand:normal(),
+ S2
+ end,
+ S1 = lists:foldl(Check, S00, lists:seq(1, 200)),
+ S1 = get(rand_seed),
+ {V0, S2} = rand:uniform_s(S1),
+ V0 = rand:uniform(),
+ S2 = get(rand_seed),
+
+ Exported = rand:export_seed(),
+ Exported = rand:export_seed_s(S2),
+
+ S3 = lists:foldl(Check, S2, lists:seq(1, 200)),
+ S3 = get(rand_seed),
+
+ S4 = lists:foldl(Check, S3, lists:seq(1, 200)),
+ S4 = get(rand_seed),
+ %% Verify that we do not have loops
+ false = S1 =:= S2,
+ false = S2 =:= S3,
+ false = S3 =:= S4,
+
+ S2 = rand:seed(Exported),
+ S3 = lists:foldl(Check, S2, 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) ->
+ Seed = rand:seed(Alg),
+ io:format("Seed ~p~n",[rand:export_seed_s(Seed)]),
+ 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) ->
+ Seed = rand:seed(Alg),
+ io:format("Seed ~p~n",[rand:export_seed_s(Seed)]),
+ 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) ->
+ io:format("Testing uniform~n",[]),
+ [basic_uniform_1(?LOOP, rand:seed_s(Alg), 0.0, array:new([{default, 0}]))
+ || Alg <- algs()],
+ [basic_uniform_2(?LOOP, rand:seed_s(Alg), 0, array:new([{default, 0}]))
+ || Alg <- algs()],
+ io:format("Testing normal~n",[]),
+ [basic_normal_1(?LOOP, rand:seed_s(Alg), 0, 0) || Alg <- algs()],
+ ok.
+
+basic_uniform_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_uniform_1(N-1, S, Sum+X, A);
+basic_uniform_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_uniform_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_uniform_2(N-1, S, Sum+X, A);
+basic_uniform_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.
+
+basic_normal_1(N, S0, Sum, Sq) when N > 0 ->
+ {X,S} = rand:normal_s(S0),
+ basic_normal_1(N-1, S, X+Sum, X*X+Sq);
+basic_normal_1(0, {#{type:=Alg}, _}, Sum, SumSq) ->
+ Mean = Sum / ?LOOP,
+ StdDev = math:sqrt((SumSq - (Sum*Sum/?LOOP))/(?LOOP - 1)),
+ io:format("~.10w: Average: ~7.4f StdDev ~6.4f~n", [Alg, Mean, StdDev]),
+ %% Verify that the basic statistics are ok
+ %% be gentle we don't want to see to many failing tests
+ abs(Mean) < 0.005 orelse test_server:fail({average, Alg, Mean}),
+ abs(StdDev - 1.0) < 0.005 orelse test_server:fail({stddev, Alg, StdDev}),
+ 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,
+ max=>(1 bsl 64)-1,
+ next=>fun crypto_next/1,
+ 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 uniform integer performance~n",[]),
+ _ = measure_1(random, fun(State) -> {int, random:uniform_s(10000, State)} end),
+ _ = [measure_1(Algo, fun(State) -> {int, rand:uniform_s(10000, State)} end) || Algo <- Algos],
+ io:format("RNG uniform float performance~n",[]),
+ _ = measure_1(random, fun(State) -> {uniform, random:uniform_s(State)} end),
+ _ = [measure_1(Algo, fun(State) -> {uniform, rand:uniform_s(State)} end) || Algo <- Algos],
+ io:format("RNG normal float performance~n",[]),
+ io:format("~.10w: not implemented (too few bits)~n", [random]),
+ _ = [measure_1(Algo, fun(State) -> {normal, rand:normal_s(State)} end) || Algo <- Algos],
+ ok.
+
+measure_1(Algo, Gen) ->
+ Parent = self(),
+ Seed = fun(crypto64) -> crypto_seed();
+ (random) -> random:seed(os:timestamp()), get(random_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
+ {int, {Random, State}}
+ when is_integer(Random), Random >= 1, Random =< 100000 ->
+ measure_2(N-1, State, Fun);
+ {uniform, {Random, State}} when is_float(Random), Random > 0, Random < 1 ->
+ measure_2(N-1, State, Fun);
+ {normal, {Random, State}} when is_float(Random) ->
+ 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].
generated by cgit v1.2.3 (git 2.25.1) at 2024-11-09 18:32:19 +0000