rand.erl

%%
%% %CopyrightBegin%
%%
%% Copyright Ericsson AB 2015-2024. All Rights Reserved.
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%% Licensed under the Apache License, Version 2.0 (the "License");
%% you may not use this file except in compliance with the License.
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%% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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%%
%% =====================================================================
%% Multiple PRNG module for Erlang/OTP
%% Copyright (c) 2015-2016 Kenji Rikitake
%%
%% exrop (xoroshiro116+) added, statistical distribution
%% improvements and uniform_real added by the Erlang/OTP team 2017
%% =====================================================================

-module(rand).
-moduledoc """
Pseudo random number generation

This module provides pseudo random number generation and implements
a number of base generator algorithms.  Most are provided through
a [plug-in framework](#plug-in-framework) that adds
features to the base generators.

At the end of this module documentation there are some
[niche algorithms](#niche-algorithms) that don't use
this module's normal [plug-in framework](#plug-in-framework).
They may be useful for special purposes like short generation time
when quality is not essential, for seeding other generators, and such.

[](){: #plug-in-framework } Plug-in framework
---------------------------------------------

The [plug-in framework](#plug-in-framework-api) implements
a common [API](#plug-in-framework-api) to, and enhancements
of the base generators:

* Operating on a generator state in the
  [process dictionary](#generator-state).
* [Automatic](#generator-state) [seeding](`seed/1`).
* Manual [seeding support](`seed/2`) to avoid common pitfalls.
* Generating [integers](`t:integer/0`) in any range, with
  [uniform distribution](`uniform/1`), without noticable bias.
* Generating [integers](`t:integer/0`) in any range, larger than
  the base generator's, with [uniform distribution](`uniform/1`).
* Generating [floating-point numbers](`t:float/0`) with
  [uniform distribution](`uniform/0`).
* Generating [floating-point numbers](`t:float/0`) with
  [normal distribution](`normal/0`).
* Generating any number of [bytes](`bytes/1`).

The base generator algorithms implements the
[Xoroshiro and Xorshift algorithms](http://xorshift.di.unimi.it)
by Sebastiano Vigna.  During an iteration they generate a large integer
(at least 58-bit) and operate on a state of several large integers.

To create numbers with normal distribution the
[Ziggurat Method by Marsaglia and Tsang](http://www.jstatsoft.org/v05/i08)
is used on the output from a base generator.

For most algorithms, jump functions are provided for generating
non-overlapping sequences. A jump function perform a calculation
equivalent to a large number of repeated state iterations,
but execute in a time roughly equivalent to one regular iteration
per generator bit.

[](){: #algorithms } The following algorithms are provided:

- **`exsss`**, the [_default algorithm_](#default-algorithm)
  *(Since OTP 22.0)*  
  Xorshift116\*\*, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

  This is the Xorshift116 generator combined with the StarStar scrambler from
  the 2018 paper by David Blackman and Sebastiano Vigna:
  [Scrambled Linear Pseudorandom Number Generators](http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf)

  The generator doesn't use 58-bit rotates so it is faster than the
  Xoroshiro116 generator, and when combined with the StarStar scrambler
  it doesn't have any weak low bits like `exrop` (Xoroshiro116+).

  Alas, this combination is about 10% slower than `exrop`, but despite that
  it is the [_default algorithm_](#default-algorithm) thanks to
  its statistical qualities.

- **`exro928ss`** *(Since OTP 22.0)*  
  Xoroshiro928\*\*, 58 bits precision and a period of 2^928-1

  Jump function: equivalent to 2^512 calls

  This is a 58 bit version of Xoroshiro1024\*\*, from the 2018 paper by
  David Blackman and Sebastiano Vigna:
  [Scrambled Linear Pseudorandom Number Generators](http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf)
  that on a 64 bit Erlang system executes only about 40% slower than the
  [*default `exsss` algorithm*](#default-algorithm)
  but with much longer period and better statistical properties,
  but on the flip side a larger state.

  Many thanks to Sebastiano Vigna for his help with the 58 bit adaption.

- **`exrop`** *(Since OTP 20.0)*  
  Xoroshiro116+, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

- **`exs1024s`** *(Since OTP 20.0)*  
  Xorshift1024\*, 64 bits precision and a period of 2^1024-1

  Jump function: equivalent to 2^512 calls

- **`exsp`** *(Since OTP 20.0)*  
  Xorshift116+, 58 bits precision and period of 2^116-1

  Jump function: equivalent to 2^64 calls

  This is a corrected version of a previous
  [_default algorithm_](#default-algorithm) (`exsplus`, _deprecated_),
  that was superseded by Xoroshiro116+ (`exrop`).  Since this algorithm
  doesn't use rotate it executes a little (say < 15%) faster than `exrop`
  (that has to do a 58 bit rotate, for which there is no native instruction).
  See the [algorithms' homepage](http://xorshift.di.unimi.it).

[](){: #default-algorithm }
#### Default Algorithm

The current _default algorithm_ is
[`exsss` (Xorshift116\*\*)](#algorithms). If a specific algorithm is
required, ensure to always use `seed/1` to initialize the state.

Which algorithm that is the default may change between Erlang/OTP releases,
and is selected to be one with high speed, small state and "good enough"
statistical properties.

#### Old Algorithms

Undocumented (old) algorithms are deprecated but still implemented so old code
relying on them will produce the same pseudo random sequences as before.

> #### Note {: .info }
>
> There were a number of problems in the implementation of
> the now undocumented algorithms, which is why they are deprecated.
> The new algorithms are a bit slower but do not have these problems:
>
> Uniform integer ranges had a skew in the probability distribution
> that was not noticable for small ranges but for large ranges
> less than the generator's precision the probability to produce
> a low number could be twice the probability for a high.
>
> Uniform integer ranges larger than or equal to the generator's precision
> used a floating point fallback that only calculated with 52 bits
> which is smaller than the requested range and therefore all numbers
> in the requested range weren't even possible to produce.
>
> Uniform floats had a non-uniform density so small values for example
> less than 0.5 had got smaller intervals decreasing as the generated value
> approached 0.0 although still uniformly distributed for sufficiently large
> subranges. The new algorithms produces uniformly distributed floats
> on the form `N * 2.0^(-53)` hence they are equally spaced.

[](){: #generator-state }
#### Generator State

Every time a random number is generated, a state is used to calculate it,
producing a new state. The state can either be implicit
or be an explicit argument and return value.

The functions with implicit state operates on a state stored
in the process dictionary under the key `rand_seed`.  If that key
doesn't exist when the function is called, `seed/1` is called automatically
with the [_default algorithm_](#default-algorithm) and creates
a reasonably unpredictable seed.

The functions with explicit state don't use the process dictionary.

#### _Examples_

Simple use; create and seed the
[_default algorithm_](#default-algorithm) with a non-fixed seed,
if not already done, and generate two uniformly distibuted
floating point numbers.

```erlang
R0 = rand:uniform(),
R1 = rand:uniform(),
```

Use a specified algorithm:

```erlang
_ = rand:seed(exs928ss),
R2 = rand:uniform(),
```

Use a specified algorithm with a fixed seed:

```erlang
_ = rand:seed(exs928ss, {123, 123534, 345345}),
R3 = rand:uniform(),
```

Use the functional API with a non-fixed seed:

```erlang
S0 = rand:seed_s(exsss),
{R4, S1} = rand:uniform_s(S0),
```

Generate a textbook basic form Box-Muller standard normal distribution number:

```erlang
R5 = rand:uniform_real(),
R6 = rand:uniform(),
SND0 = math:sqrt(-2 * math:log(R5)) * math:cos(math:pi() * R6)
```

Generate a standard normal distribution number:

```erlang
{SND1, S2} = rand:normal_s(S1),
```

Generate a normal distribution number with with mean -3 and variance 0.5:

```erlang
{ND0, S3} = rand:normal_s(-3, 0.5, S2),
```

#### Quality of the Generated Numbers

> #### Note {: .info }
>
> The builtin random number generator algorithms are not cryptographically
> strong. If a cryptographically strong random number generator is needed,
> use something like `crypto:rand_seed/0`.

For all these generators except `exro928ss` and `exsss` the lowest bit(s)
have got a slightly less random behaviour than all other bits.
1 bit for `exrop` (and `exsp`), and 3 bits for `exs1024s`. See for example
this explanation in the
[Xoroshiro128+](http://xoroshiro.di.unimi.it/xoroshiro128plus.c)
generator source code:

> Beside passing BigCrush, this generator passes the PractRand test suite
> up to (and included) 16TB, with the exception of binary rank tests,
> which fail due to the lowest bit being an LFSR; all other bits pass all
> tests. We suggest to use a sign test to extract a random Boolean value.

If this is a problem; to generate a boolean with these algorithms,
use something like this:

```erlang
(rand:uniform(256) > 128) % -> boolean()
```

```erlang
((rand:uniform(256) - 1) bsr 7) % -> 0 | 1
```

For a general range, with `N = 1` for `exrop`, and `N = 3` for `exs1024s`:

```erlang
(((rand:uniform(Range bsl N) - 1) bsr N) + 1)
```

The floating point generating functions in this module waste the lowest bits
when converting from an integer so they avoid this snag.


[](){: #niche-algorithms } Niche algorithms
-------------------------------------------

The [niche algorithms API](#niche-algorithms-api) contains
special purpose algorithms that don't use the
[plug-in framework](#plug-in-framework), mainly for performance reasons.

Since these algorithms lack the plug-in framework support, generating numbers
in a range other than the base generator's range may become a problem.

There are at least four ways to do this, assuming the `Range` is less than
the generator's range:

[](){: #modulo-method }
- **Modulo**  
  To generate a number `V` in the range `0..Range-1`:

  > Generate a number `X`.  
  > Use `V = X rem Range` as your value.

  This method uses `rem`, that is, the remainder of an integer division,
  which is a slow operation.

  Low bits from the generator propagate straight through to
  the generated value, so if the generator has got weaknesses
  in the low bits this method propagates them too.

  If `Range` is not a divisor of the generator range, the generated numbers
  have a bias.  Example:

  Say the generator generates a byte, that is, the generator range
  is `0..255`, and the desired range is `0..99` (`Range = 100`).
  Then there are 3 generator outputs that produce the value `0`,
  these are; `0`, `100` and `200`.
  But there are only 2 generator outputs that produce the value `99`,
  which are; `99` and `199`. So the probability for a value `V` in `0..55`
  is 3/2 times the probability for the other values `56..99`.

  If `Range` is much smaller than the generator range, then this bias
  gets hard to detect. The rule of thumb is that if `Range` is smaller
  than the square root of the generator range, the bias is small enough.
  Example:

  A byte generator when `Range = 20`. There are 12 (`256 div 20`)
  possibilities to generate the highest numbers and one more to generate
  a number `V < 16` (`256 rem 20`). So the probability is 13/12
  for a low number versus a high. To detect that difference with
  some confidence you would need to generate a lot more numbers
  than the generator range, `256` in this small example.

[](){: #truncated-multiplication-method }
- **Truncated multiplication**  
  To generate a number `V` in the range `0..Range-1`, when you have
  a generator with a power of 2 range (`0..2^Bits-1`):

  > Generate a number `X`.  
  > Use `V = X * Range bsr Bits` as your value.

  If the multiplication `X * Range` creates a bignum
  this method becomes very slow.

  High bits from the generator propagate through to the generated value,
  so if the generator has got weaknesses in the high bits this method
  propagates them too.

  If `Range` is not a divisor of the generator range, the generated numbers
  have a bias, pretty much as for the [Modulo](#modulo-method) method above.

[](){: #shift-or-mask-method }
- **Shift or mask**  
  To generate a number in a power of 2 range (`0..2^RBits-1`),
  when you have a generator with a power of 2 range (`0..2^Bits`):

  > Generate a number `X`.  
  > Use `V = X band ((1 bsl RBits)-1)` or `V = X bsr (Bits-RBits)`
  > as your value.

  Masking with `band` preserves the low bits, and right shifting
  with `bsr` preserves the high, so if the generator has got weaknesses
  in high or low bits; choose the right operator.

  If the generator has got a range that is not a power of 2
  and this method is used anyway, it introduces bias in the same way
  as for the [Modulo](#modulo-method) method above.

[](){: #rejection-method }
- **Rejection**  

  > Generate a number `X`.  
  > If `X` is in the range, use it as your value,
  > otherwise reject it and repeat.

  In theory it is not certain that this method will ever complete,
  but in practice you ensure that the probability of rejection is low.
  Then the probability for yet another iteration decreases exponentially
  so the expected mean number of iterations will often be between 1 and 2.
  Also, since the base generator is a full length generator,
  a value that will break the loop must eventually be generated.

  These methods can be combined, such as using
  the [Modulo](#modulo-method) method and only if the generator value
  would create bias use [Rejection](#rejection-method).
  Or using [Shift or mask](#shift-or-mask-method) to reduce the size
  of a generator value so that
  [Truncated multiplication](#truncated-multiplication-method)
  will not create a bignum.

  The recommended way to generate a floating point number
  (IEEE 745 Double, that has got a 53-bit mantissa) in the range
  `0..1`, that is `0.0 =< V < 1.0` is to generate a 53-bit number `X`
  and then use `V = X * (1.0/((1 bsl 53)))` as your value.
  This will create a value on the form N*2^-53 with equal probability
  for every possible N for the range.
""".
-moduledoc(#{since => "OTP 18.0",
             titles =>
                 [{function,<<"Plug-in framework API">>},
                  {function,<<"Niche algorithms API">>}]}).

-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,
         uniform_real/0, uniform_real_s/1,
         bytes/1, bytes_s/2,
         jump/0, jump/1,
         normal/0, normal/2, normal_s/1, normal_s/3
	]).

%% Utilities
-export([exsp_next/1, exsp_jump/1, splitmix64_next/1,
         mwc59/1, mwc59_value32/1, mwc59_value/1, mwc59_float/1,
         mwc59_seed/0, mwc59_seed/1]).

%% Test, dev and internal
-export([exro928_jump_2pow512/1, exro928_jump_2pow20/1,
	 exro928_seed/1, exro928_next/1, exro928_next_state/1,
	 format_jumpconst58/1, seed58/2]).

%% Debug
-export([make_float/3, float2str/1, bc64/1]).

-compile({inline, [exs64_next/1, exsp_next/1, exsss_next/1,
		   exs1024_next/1, exs1024_calc/2,
                   exro928_next_state/4,
                   exrop_next/1, exrop_next_s/2,
                   mwc59_value/1,
		   get_52/1, normal_kiwi/1]}).

-define(DEFAULT_ALG_HANDLER, exsss).
-define(SEED_DICT, rand_seed).

%% =====================================================================
%% Bit fiddling macros
%% =====================================================================

-define(BIT(Bits), (1 bsl (Bits))).
-define(MASK(Bits), (?BIT(Bits) - 1)).
-define(MASK(Bits, X), ((X) band ?MASK(Bits))).
-define(
   BSL(Bits, X, N),
   %% N is evaluated 2 times
   (?MASK((Bits)-(N), (X)) bsl (N))).
-define(
   ROTL(Bits, X, N),
   %% Bits is evaluated 2 times
   %% X is evaluated 2 times
   %% N i evaluated 3 times
   (?BSL((Bits), (X), (N)) bor ((X) bsr ((Bits)-(N))))).

-define(
   BC(V, N),
   bc((V), ?BIT((N) - 1), N)).

%%-define(TWO_POW_MINUS53, (math:pow(2, -53))).
-define(TWO_POW_MINUS53, 1.11022302462515657e-16).

%% =====================================================================
%% Types
%% =====================================================================

-doc "`0 .. (2^64 - 1)`".
-type uint64() :: 0..?MASK(64).
-doc "`0 .. (2^58 - 1)`".
-type uint58() :: 0..?MASK(58).

%% This depends on the algorithm handler function
-type alg_state() ::
	exsplus_state() | exro928_state() |  exrop_state() | exs1024_state() |
	exs64_state() | dummy_state() | term().

%% This is the algorithm handling definition within this module,
%% and the type to use for plugins.
%%
%% The 'type' field must be recognized by the module that implements
%% the algorithm, to interpret an exported state.
%%
%% The 'bits' field indicates how many bits the integer
%% returned from 'next' has got, i.e 'next' shall return
%% an random integer in the range 0..(2^Bits - 1).
%% At least 55 bits is required for the floating point
%% producing fallbacks, but 56 bits would be more future proof.
%%
%% The fields 'next', 'uniform' and 'uniform_n'
%% implement the algorithm.  If 'uniform' or 'uniform_n'
%% is not present there is a fallback using 'next' and either
%% 'bits' or the deprecated 'max'.  The 'next' function
%% must generate a word with at least 56 good random bits.
%%
%% The 'weak_low_bits' field indicate how many bits are of
%% lesser quality and they will not be used by the floating point
%% producing functions, nor by the range producing functions
%% when more bits are needed, to avoid weak bits in the middle
%% of the generated bits.  The lowest bits from the range
%% functions still have the generator's quality.
%%
-type alg_handler() ::
        #{type := alg(),
          bits => non_neg_integer(),
          weak_low_bits => non_neg_integer(),
          max => non_neg_integer(), % Deprecated
          next :=
              fun ((alg_state()) -> {non_neg_integer(), alg_state()}),
          uniform =>
              fun ((state()) -> {float(), state()}),
          uniform_n =>
              fun ((pos_integer(), state()) -> {pos_integer(), state()}),
          jump =>
              fun ((state()) -> state())}.

%% Algorithm state
-doc "Algorithm-dependent state.".
-type state() :: {alg_handler(), alg_state()}.
-type builtin_alg() ::
	exsss | exro928ss | exrop | exs1024s | exsp | exs64 | exsplus |
        exs1024 | dummy.
-type alg() :: builtin_alg() | atom().
-doc "Algorithm-dependent state that can be printed or saved to file.".
-type export_state() :: {alg(), alg_state()}.
-doc """
Generator seed value.

A list of integers sets the generator's internal state directly, after
algorithm-dependent checks of the value and masking to the proper word size.
The number of integers must be equal to the number of state words
in the generator.

A single integer is used as the initial state for a SplitMix64 generator.
The sequential output values of that is then used for setting
the generator's internal state after masking to the proper word size
and if needed avoiding zero values.

A traditional 3-tuple of integers seed is passed through algorithm-dependent
hashing functions to create the generator's initial state.
""".
-type seed() :: [integer()] | integer() | {integer(), integer(), integer()}.
-export_type(
   [builtin_alg/0, alg/0, alg_handler/0, alg_state/0,
    state/0, export_state/0, seed/0]).
-export_type(
   [exsplus_state/0, exro928_state/0, exrop_state/0, exs1024_state/0,
    exs64_state/0, mwc59_state/0, dummy_state/0]).
-export_type(
   [uint58/0, uint64/0, splitmix64_state/0]).

%% =====================================================================
%% Range macro and helper
%% =====================================================================

-define(
   uniform_range(Range, AlgHandler, R, V, MaxMinusRange, I),
   if
       0 =< (MaxMinusRange) ->
           if
               %% Really work saving in odd cases;
               %% large ranges in particular
               (V) < (Range) ->
                   {(V) + 1, {(AlgHandler), (R)}};
               true ->
                   (I) = (V) rem (Range),
                   if
                       (V) - (I) =< (MaxMinusRange) ->
                           {(I) + 1, {(AlgHandler), (R)}};
                       true ->
                           %% V in the truncated top range
                           %% - try again
                           ?FUNCTION_NAME((Range), {(AlgHandler), (R)})
                   end
           end;
       true ->
           uniform_range((Range), (AlgHandler), (R), (V))
   end).

%% For ranges larger than the algorithm bit size
uniform_range(Range, #{next:=Next, bits:=Bits} = AlgHandler, R, V) ->
    WeakLowBits = maps:get(weak_low_bits, AlgHandler, 0),
    %% Maybe waste the lowest bit(s) when shifting in new bits
    Shift = Bits - WeakLowBits,
    ShiftMask = bnot ?MASK(WeakLowBits),
    RangeMinus1 = Range - 1,
    if
        (Range band RangeMinus1) =:= 0 -> % Power of 2
            %% Generate at least the number of bits for the range
            {V1, R1, _} =
                uniform_range(
                  Range bsr Bits, Next, R, V, ShiftMask, Shift, Bits),
            {(V1 band RangeMinus1) + 1, {AlgHandler, R1}};
        true ->
            %% Generate a value with at least two bits more than the range
            %% and try that for a fit, otherwise recurse
            %%
            %% Just one bit more should ensure that the generated
            %% number range is at least twice the size of the requested
            %% range, which would make the probability to draw a good
            %% number better than 0.5.  And repeating that until
            %% success i guess would take 2 times statistically amortized.
            %% But since the probability for fairly many attemtpts
            %% is not that low, use two bits more than the range which
            %% should make the probability to draw a bad number under 0.25,
            %% which decreases the bad case probability a lot.
            {V1, R1, B} =
                uniform_range(
                  Range bsr (Bits - 2), Next, R, V, ShiftMask, Shift, Bits),
            I = V1 rem Range,
            if
                (V1 - I) =< (1 bsl B) - Range ->
                    {I + 1, {AlgHandler, R1}};
                true ->
                    %% V1 drawn from the truncated top range
                    %% - try again
                    {V2, R2} = Next(R1),
                    uniform_range(Range, AlgHandler, R2, V2)
            end
    end.
%%
uniform_range(Range, Next, R, V, ShiftMask, Shift, B) ->
    if 
        Range =< 1 ->
            {V, R, B};
        true ->
            {V1, R1} = Next(R),
            %% Waste the lowest bit(s) when shifting in new bits
            uniform_range(
              Range bsr Shift, Next, R1,
              ((V band ShiftMask) bsl Shift) bor V1,
              ShiftMask, Shift, B + Shift)
    end.

%% =====================================================================
%% API
%% =====================================================================

%% Return algorithm and seed so that RNG state can be recreated with seed/1
-doc """
Export the seed value.

Returns the random number state in an external format.
To be used with `seed/1`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec export_seed() -> 'undefined' | export_state().
export_seed() ->
    case get(?SEED_DICT) of
	{#{type:=Alg}, AlgState} -> {Alg, AlgState};
	_ -> undefined
    end.

-doc """
Export the seed value.

Returns the random number generator state in an external format.
To be used with `seed/1`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec export_seed_s(State :: state()) -> export_state().
export_seed_s({#{type:=Alg}, AlgState}) -> {Alg, AlgState}.


%% seed(Alg) seeds RNG with runtime dependent values
%% and return the NEW state
%%
%% seed({Alg,AlgState}) setup RNG with a previously exported seed
%% and return the NEW state

-doc """
Seed the random number generator and select algorithm.

The same as [`seed_s(Alg_or_State)`](`seed_s/1`),
but also stores the generated state in the process dictionary.

The argument `default` is an alias for the
[_default algorithm_](#default-algorithm)
that has been implemented *(Since OTP 24.0)*.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec seed(Alg_or_State :: term()) -> state().
seed(Alg_or_State) ->
    seed_put(seed_s(Alg_or_State)).

-doc """
Seed the random number generator and select algorithm.

With the argument `Alg`, select that algorithm and seed random number
generation with reasonably unpredictable time dependent data.

`Alg = default` is an alias for the
[_default algorithm_](#default-algorithm)
*(Since OTP 24.0)*.

With the argument `State`, re-creates the state and returns it.
See also `export_seed/0`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec seed_s(Alg | State) -> state() when
      Alg   :: builtin_alg() | 'default',
      State :: state() | export_state().
seed_s({AlgHandler, _AlgState} = State) when is_map(AlgHandler) ->
    State;
seed_s({Alg, AlgState}) when is_atom(Alg) ->
    {AlgHandler,_SeedFun} = mk_alg(Alg),
    {AlgHandler,AlgState};
seed_s(Alg) ->
    seed_s(Alg, default_seed()).

default_seed() ->
    {erlang:phash2([{node(),self()}]),
     erlang:system_time(),
     erlang:unique_integer()}.

%% seed/2: seeds RNG with the algorithm and given values
%% and returns the NEW state.

-doc """
Seed the random number generator and select algorithm.

The same as [`seed_s(Alg, Seed)`](`seed_s/2`),
but also stores the generated state in the process dictionary.

`Alg = default` is an alias for the
[_default algorithm_](#default-algorithm)
that has been implemented *(Since OTP 24.0)*.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec seed(Alg :: term(), Seed :: term()) -> state().
seed(Alg, Seed) ->
    seed_put(seed_s(Alg, Seed)).

-doc """
Seed the random number generator and select algorithm.

Creates and returns a generator state for the specified algorithm
from the specified `t:seed/0` integers.

`Alg = default` is an alias for the [_default algorithm_](#default-algorithm)
that has been implemented *since OTP 24.0*.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec seed_s(Alg, Seed) -> state() when
      Alg  :: builtin_alg() | 'default',
      Seed :: seed().
seed_s(default, Seed) -> seed_s(?DEFAULT_ALG_HANDLER, Seed);
seed_s(Alg, Seed) ->
    {AlgHandler,SeedFun} = mk_alg(Alg),
    AlgState = SeedFun(Seed),
    {AlgHandler,AlgState}.

%%% 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.

-doc """
Generate a uniformly distributed random number `0.0 =< X < 1.0`,
using the state in the process dictionary.

Like `uniform_s/1` but operates on the state stored in
the process dictionary.  Returns the generated number `X`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec uniform() -> X :: float().
uniform() ->
    {X, State} = uniform_s(seed_get()),
    _ = seed_put(State),
    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.

-doc """
Generate a uniformly distributed random integer `1 =< X =< N`,
using the state in the process dictionary.

Like `uniform_s/2` but operates on the state stored in
the process dictionary.  Returns the generated number `X`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec uniform(N :: pos_integer()) -> X :: pos_integer().
uniform(N) ->
    {X, State} = uniform_s(N, seed_get()),
    _ = seed_put(State),
    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.

-doc """
Generate a uniformly distributed random number `0.0 =< X < 1.0`.

From the specified `State`, generates a random number `X ::` `t:float/0`,
uniformly distributed in the value range `0.0 =< X < 1.0`.
Returns the number `X` and the updated `NewState`.

The generated numbers are on the form `N * 2.0^(-53)`, that is;
equally spaced in the interval.

> #### Warning {: .warning }
>
> This function may return exactly `0.0` which can be fatal for certain
> applications. If that is undesired you can use `(1.0 - rand:uniform())`
> to get the interval `0.0 < X =< 1.0`, or instead use `uniform_real/0`.
>
> If neither endpoint is desired you can achieve the range
> `0.0 < X < 1.0` using test and re-try like this:
>
> ```erlang
> my_uniform() ->
>     case rand:uniform() of
>         X when 0.0 < X -> X;
>         _ -> my_uniform()
>     end.
> ```
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec uniform_s(State :: state()) -> {X :: float(), NewState :: state()}.
uniform_s(State = {#{uniform:=Uniform}, _}) ->
    Uniform(State);
uniform_s({#{bits:=Bits, next:=Next} = AlgHandler, R0}) ->
    {V, R1} = Next(R0),
    %% Produce floats on the form N * 2^(-53)
    {(V bsr (Bits - 53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}};
uniform_s({#{max:=Max, next:=Next} = AlgHandler, R0}) ->
    {V, R1} = Next(R0),
    %% Old algorithm with non-uniform density
    {V / (Max + 1), {AlgHandler, R1}}.


%% 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.

-doc """
Generate a uniformly distributed random integer `1 =< X =< N`.

From the specified `State`, generates a random number `X ::` `t:integer/0`,
uniformly distributed in the specified range `1 =< X =< N`.
Returns the number `X` and the updated `NewState`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec uniform_s(N :: pos_integer(), State :: state()) ->
                       {X :: pos_integer(), NewState :: state()}.
uniform_s(N, State = {#{uniform_n:=UniformN}, _})
  when is_integer(N), 1 =< N ->
    UniformN(N, State);
uniform_s(N, {#{bits:=Bits, next:=Next} = AlgHandler, R0})
  when is_integer(N), 1 =< N ->
    {V, R1} = Next(R0),
    MaxMinusN = ?BIT(Bits) - N,
    ?uniform_range(N, AlgHandler, R1, V, MaxMinusN, I);
uniform_s(N, {#{max:=Max, next:=Next} = AlgHandler, R0})
  when is_integer(N), 1 =< N ->
    %% Old algorithm with skewed probability
    %% and gap in ranges > Max
    {V, R1} = Next(R0),
    if
        N =< Max ->
            {(V rem N) + 1, {AlgHandler, R1}};
        true ->
            F = V / (Max + 1),
            {trunc(F * N) + 1, {AlgHandler, R1}}
    end.

%% uniform_real/0: returns a random float X where 0.0 < X =< 1.0,
%% updating the state in the process dictionary.

-doc """
Generate a uniformly distributed random number `0.0 < X < 1.0`,
using the state in the process dictionary.

Like `uniform_real_s/1` but operates on the state stored in
the process dictionary.  Returns the generated number `X`.

See `uniform_real_s/1`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 21.0">>}).
-spec uniform_real() -> X :: float().
uniform_real() ->
    {X, Seed} = uniform_real_s(seed_get()),
    _ = seed_put(Seed),
    X.

%% uniform_real_s/1: given a state, uniform_s/1
%% returns a random float X where 0.0 < X =< 1.0,
%% and a new state.
%%
%% This function doesn't use the same form of uniformity
%% as the uniform_s/1 function.
%%
%% Instead, this function doesn't generate numbers with equal
%% distance in the interval, but rather tries to keep all mantissa
%% bits random also for small numbers, meaning that the distance
%% between possible numbers decreases when the numbers
%% approaches 0.0, as does the possibility for a particular
%% number.  Hence uniformity is preserved.
%%
%% To generate 56 bits at the time instead of 53 is actually
%% a speed optimization since the probability to have to
%% generate a second word decreases by 1/2 for every extra bit.
%%
%% This function generates normalized numbers, so the smallest number
%% that can be generated is 2^-1022 with the distance 2^-1074
%% to the next to smallest number, compared to 2^-53 for uniform_s/1.
%%
%% This concept of uniformity should work better for applications
%% where you need to calculate 1.0/X or math:log(X) since those
%% operations benefits from larger precision approaching 0.0,
%% and that this function doesn't return 0.0 nor denormalized
%% numbers very close to 0.0.  The log() operation in The Box-Muller
%% transformation for normal distribution is an example of this.
%%
%%-define(TWO_POW_MINUS55, (math:pow(2, -55))).
%%-define(TWO_POW_MINUS110, (math:pow(2, -110))).
%%-define(TWO_POW_MINUS55, 2.7755575615628914e-17).
%%-define(TWO_POW_MINUS110, 7.7037197775489436e-34).
%%
-doc """
Generate a uniformly distributed random number `0.0 < X < 1.0`.

From the specified state, generates a random float, uniformly distributed
in the value range `DBL_MIN =< X < 1.0`.

Conceptually, a random real number `R` is generated from the interval
`0.0 =< R < 1.0` and then the closest rounded down nonzero
normalized number in the IEEE 754 Double Precision Format is returned.

> #### Note {: .info }
>
> The generated numbers from this function has got better granularity
> for small numbers than the regular `uniform_s/1` because all bits
> in the mantissa are random. This property, in combination with the fact
> that exactly zero is never returned is useful for algorithms doing
> for example `1.0 / X` or `math:log(X)`.

The concept implicates that the probability to get exactly zero is extremely
low; so low that this function in fact never returns `0.0`.
The smallest number that it might return is `DBL_MIN`,
which is `2.0^(-1022)`.

The value range stated at the top of this function description is
technically correct, but `0.0 =< X < 1.0` is a better description
of the generated numbers' statistical distribution, and that
this function never returns exactly `0.0` is impossible to observe.

For all sub ranges `N*2.0^(-53) =< X < (N+1)*2.0^(-53)` where
`0 =< integer(N) < 2.0^53`, the probability to generate a number
in the range is the same.  Compare with the numbers
generated by `uniform_s/1`.

Having to generate extra random bits for occasional small numbers
costs a little performance. This function is about 20% slower
than the regular `uniform_s/1`
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 21.0">>}).
-spec uniform_real_s(State :: state()) -> {X :: float(), NewState :: state()}.
uniform_real_s({#{bits:=Bits, next:=Next} = AlgHandler, R0}) ->
    %% Generate a 56 bit number without using the weak low bits.
    %%
    %% Be sure to use only 53 bits when multiplying with
    %% math:pow(2.0, -N) to avoid rounding which would make
    %% "even" floats more probable than "odd".
    %%
    {V1, R1} = Next(R0),
    M1 = V1 bsr (Bits - 56),
    if
        ?BIT(55) =< M1 ->
            %% We have 56 bits - waste 3
            {(M1 bsr 3) * math:pow(2.0, -53), {AlgHandler, R1}};
        ?BIT(54) =< M1 ->
            %% We have 55 bits - waste 2
            {(M1 bsr 2) * math:pow(2.0, -54), {AlgHandler, R1}};
        ?BIT(53) =< M1 ->
            %% We have 54 bits - waste 1
            {(M1 bsr 1) * math:pow(2.0, -55), {AlgHandler, R1}};
        ?BIT(52) =< M1 ->
            %% We have 53 bits - use all
            {M1 * math:pow(2.0, -56), {AlgHandler, R1}};
        true ->
            %% Need more bits
            {V2, R2} = Next(R1),
            uniform_real_s(AlgHandler, Next, M1, -56, R2, V2, Bits)
    end;
uniform_real_s({#{max:=_, next:=Next} = AlgHandler, R0}) ->
    %% Generate a 56 bit number.
    %% Ignore the weak low bits for these old algorithms,
    %% just produce something reasonable.
    %%
    %% Be sure to use only 53 bits when multiplying with
    %% math:pow(2.0, -N) to avoid rounding which would make
    %% "even" floats more probable than "odd".
    %%
    {V1, R1} = Next(R0),
    M1 = ?MASK(56, V1),
    if
        ?BIT(55) =< M1 ->
            %% We have 56 bits - waste 3
            {(M1 bsr 3) * math:pow(2.0, -53), {AlgHandler, R1}};
        ?BIT(54) =< M1 ->
            %% We have 55 bits - waste 2
            {(M1 bsr 2) * math:pow(2.0, -54), {AlgHandler, R1}};
        ?BIT(53) =< M1 ->
            %% We have 54 bits - waste 1
            {(M1 bsr 1) * math:pow(2.0, -55), {AlgHandler, R1}};
        ?BIT(52) =< M1 ->
            %% We have 53 bits - use all
            {M1 * math:pow(2.0, -56), {AlgHandler, R1}};
        true ->
            %% Need more bits
            {V2, R2} = Next(R1),
            uniform_real_s(AlgHandler, Next, M1, -56, R2, V2, 56)
    end.

uniform_real_s(AlgHandler, _Next, M0, -1064, R1, V1, Bits) -> % 19*56
    %% This is a very theoretical bottom case.
    %% The odds of getting here is about 2^-1008,
    %% through a white box test case, or thanks to
    %% a malfunctioning PRNG producing 18 56-bit zeros in a row.
    %%
    %% Fill up to 53 bits, we have at most 52
    B0 = (53 - ?BC(M0, 52)), % Missing bits
    {(((M0 bsl B0) bor (V1 bsr (Bits - B0))) * math:pow(2.0, -1064 - B0)),
     {AlgHandler, R1}};
uniform_real_s(AlgHandler, Next, M0, BitNo, R1, V1, Bits) ->
    if
        %% Optimize the most probable.
        %% Fill up to 53 bits.
        ?BIT(51) =< M0 ->
            %% We have 52 bits in M0 - need 1
            {(((M0 bsl 1) bor (V1 bsr (Bits - 1)))
              * math:pow(2.0, BitNo - 1)),
             {AlgHandler, R1}};
        ?BIT(50) =< M0 ->
            %% We have 51 bits in M0 - need 2
            {(((M0 bsl 2) bor (V1 bsr (Bits - 2)))
              * math:pow(2.0, BitNo - 2)),
             {AlgHandler, R1}};
        ?BIT(49) =< M0 ->
            %% We have 50 bits in M0 - need 3
            {(((M0 bsl 3) bor (V1 bsr (Bits - 3)))
              * math:pow(2.0, BitNo - 3)),
             {AlgHandler, R1}};
        M0 == 0 ->
            M1 = V1 bsr (Bits - 56),
            if
                ?BIT(55) =< M1 ->
                    %% We have 56 bits - waste 3
                    {(M1 bsr 3) * math:pow(2.0, BitNo - 53),
                     {AlgHandler, R1}};
                ?BIT(54) =< M1 ->
                    %% We have 55 bits - waste 2
                    {(M1 bsr 2) * math:pow(2.0, BitNo - 54),
                     {AlgHandler, R1}};
                ?BIT(53) =< M1 ->
                    %% We have 54 bits - waste 1
                    {(M1 bsr 1) * math:pow(2.0, BitNo - 55),
                     {AlgHandler, R1}};
                ?BIT(52) =< M1 ->
                    %% We have 53 bits - use all
                    {M1 * math:pow(2.0, BitNo - 56),
                     {AlgHandler, R1}};
                BitNo =:= -1008 ->
                    %% Endgame
                    %% For the last round we cannot have 14 zeros or more
                    %% at the top of M1 because then we will underflow,
                    %% so we need at least 43 bits
                    if
                        ?BIT(42) =< M1 ->
                            %% We have 43 bits - get the last bits
                            uniform_real_s(
                              AlgHandler, Next, M1, BitNo - 56, R1);
                        true ->
                            %% Would underflow 2^-1022 - start all over
                            %%
                            %% We could just crash here since the odds for
                            %% the PRNG being broken is much higher than
                            %% for a good PRNG generating this many zeros
                            %% in a row.  Maybe we should write an error
                            %% report or call this a system limit...?
                            uniform_real_s({AlgHandler, R1})
                    end;
                true ->
                    %% Need more bits
                    uniform_real_s(AlgHandler, Next, M1, BitNo - 56, R1)
            end;
        true ->
            %% Fill up to 53 bits
            B0 = 53 - ?BC(M0, 49), % Number of bits we need to append
            {(((M0 bsl B0) bor (V1 bsr (Bits - B0)))
              * math:pow(2.0, BitNo - B0)),
             {AlgHandler, R1}}
    end.
%%
uniform_real_s(#{bits:=Bits} = AlgHandler, Next, M0, BitNo, R0) ->
    {V1, R1} = Next(R0),
    uniform_real_s(AlgHandler, Next, M0, BitNo, R1, V1, Bits);
uniform_real_s(#{max:=_} = AlgHandler, Next, M0, BitNo, R0) ->
    {V1, R1} = Next(R0),
    uniform_real_s(AlgHandler, Next, M0, BitNo, R1, ?MASK(56, V1), 56).


%% bytes/1: given a number N,
%% returns a random binary with N bytes

-doc """
Generate random bytes as a `t:binary()`,
using the state in the process dictionary.

Like `bytes_s/2` but operates on the state stored in
the process dictionary.  Returns the generated [`Bytes`](`t:binary/0`).
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 24.0">>}).
-spec bytes(N :: non_neg_integer()) -> Bytes :: binary().
bytes(N) ->
    {Bytes, State} = bytes_s(N, seed_get()),
    _ = seed_put(State),
    Bytes.


%% bytes_s/2: given a number N and a state,
%% returns a random binary with N bytes and a new state

-doc """
Generate random bytes as a `t:binary()`.

For a specified integer `N >= 0`, generates a `t:binary/0`
with that number of random bytes.

The selected algorithm is used to generate as many random numbers
as required to compose the `t:binary/0`.  Returns the generated
[`Bytes`](`t:binary/0`) and a [`NewState`](`t:state/0`).
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 24.0">>}).
-spec bytes_s(N :: non_neg_integer(), State :: state()) ->
                     {Bytes :: binary(), NewState :: state()}.
bytes_s(N, {#{bits:=Bits, next:=Next} = AlgHandler, R})
  when is_integer(N), 0 =< N ->
    WeakLowBits = maps:get(weak_low_bits, AlgHandler, 0),
    bytes_r(N, AlgHandler, Next, R, Bits, WeakLowBits);
bytes_s(N, {#{max:=Mask, next:=Next} = AlgHandler, R})
  when is_integer(N), 0 =< N, ?MASK(58) =< Mask ->
    %% Old spec - assume 58 bits and 2 weak low bits
    %% giving 56 bits i.e precisely 7 bytes per generated number
    Bits = 58,
    WeakLowBits = 2,
    bytes_r(N, AlgHandler, Next, R, Bits, WeakLowBits).

%% N:           Number of bytes to generate
%% Bits:        Number of bits in the generated word
%% WeakLowBits: Number of low bits in the generated word
%%              to waste due to poor quality
bytes_r(N, AlgHandler, Next, R, Bits, WeakLowBits) ->
    %% We use whole bytes from each generator word,
    %% GoodBytes: that number of bytes
    GoodBytes = (Bits - WeakLowBits) bsr 3,
    GoodBits = GoodBytes bsl 3,
    %% Shift: how many bits of each generator word to waste
    %% by shifting right - we use the bits from the big end
    Shift = Bits - GoodBits,
    bytes_r(N, AlgHandler, Next, R, <<>>, GoodBytes, GoodBits, Shift).
%%
bytes_r(N0, AlgHandler, Next, R0, Bytes0, GoodBytes, GoodBits, Shift)
  when (GoodBytes bsl 2) < N0 ->
    %% Loop unroll 4 iterations
    %% - gives about 25% shorter time for large binaries
    {V1, R1} = Next(R0),
    {V2, R2} = Next(R1),
    {V3, R3} = Next(R2),
    {V4, R4} = Next(R3),
    Bytes1 =
        <<Bytes0/binary,
          (V1 bsr Shift):GoodBits,
          (V2 bsr Shift):GoodBits,
          (V3 bsr Shift):GoodBits,
          (V4 bsr Shift):GoodBits>>,
    N1 = N0 - (GoodBytes bsl 2),
    bytes_r(N1, AlgHandler, Next, R4, Bytes1, GoodBytes, GoodBits, Shift);
bytes_r(N0, AlgHandler, Next, R0, Bytes0, GoodBytes, GoodBits, Shift)
  when GoodBytes < N0 ->
    {V, R1} = Next(R0),
    Bytes1 = <<Bytes0/binary, (V bsr Shift):GoodBits>>,
    N1 = N0 - GoodBytes,
    bytes_r(N1, AlgHandler, Next, R1, Bytes1, GoodBytes, GoodBits, Shift);
bytes_r(N, AlgHandler, Next, R0, Bytes, _GoodBytes, GoodBits, _Shift) ->
    {V, R1} = Next(R0),
    Bits = N bsl 3,
    %% Use the big end bits
    Shift = GoodBits - Bits,
    {<<Bytes/binary, (V bsr Shift):Bits>>, {AlgHandler, R1}}.


%% 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.

-doc """
Jump the generator state forward.

Performs an algorithm specific [`State`](`t:state/0`) jump calculation
that is equvalent to a large number of state iterations.
See this module's [algorithms list](#algorithms).

Returns the [`NewState`](`t:state/0`).

This feature can be used to create many non-overlapping
random number sequences from one start state.

This function raises a `not_implemented` error exception if there is
no jump function implemented for the [`State`](`t:state/0`)'s algorithm.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 20.0">>}).
-spec jump(State :: state()) -> NewState :: state().
jump(State = {#{jump:=Jump}, _}) ->
    Jump(State);
jump({#{}, _}) ->
    erlang:error(not_implemented).


%% 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.

-doc """
Jump the generator state forward.

Like `jump/1` but operates on the state stored in
the process dictionary.  Returns the [`NewState`](`t:state/0`).
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 20.0">>}).
-spec jump() -> NewState :: state().
jump() ->
    seed_put(jump(seed_get())).

%% normal/0: returns a random float with standard normal distribution
%% updating the state in the process dictionary.

-doc """
Generate a random number with standard normal distribution.

Like `normal_s/1` but operates on the state stored in
the process dictionary.  Returns the generated number `X`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec normal() -> X :: float().
normal() ->
    {X, Seed} = normal_s(seed_get()),
    _ = seed_put(Seed),
    X.

%% normal/2: returns a random float with N(μ, σ²) normal distribution
%% updating the state in the process dictionary.

-doc """
Generate a random number with specified normal distribution 𝒩 *(μ, σ²)*.

Like `normal_s/3` but operates on the state stored in
the process dictionary.  Returns the generated number `X`.
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 20.0">>}).
-spec normal(Mean :: number(), Variance :: number()) -> X :: float().
normal(Mean, Variance) ->
    Mean + (math:sqrt(Variance) * normal()).

%% 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/

-doc """
Generate a random number with standard normal distribution.

From the specified `State`, generates a random number `X ::` `t:float/0`,
with standard normal distribution, that is with mean value `0.0`
and variance `1.0`.

Returns the generated number [`X`](`t:float/0`)
and the [`NewState`](`t:state/0`).
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 18.0">>}).
-spec normal_s(State :: state()) -> {X :: float(), NewState :: state()}.
normal_s(State0) ->
    {Sign, R, State} = get_52(State0),
    Idx = ?MASK(8, R),
    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.

%% normal_s/3: returns a random float with normal N(μ, σ²) distribution

-doc """
Generate a random number with specified normal distribution 𝒩 *(μ, σ²)*.

From the specified `State`, generates a random number `X ::` `t:float/0`,
with normal distribution 𝒩 *(μ, σ²)*, that is 𝒩 (Mean, Variance)
where `Variance >= 0.0`.

Returns [`X`](`t:float/0`) and the [`NewState`](`t:state/0`).
""".
-doc(#{title => <<"Plug-in framework API">>,since => <<"OTP 20.0">>}).
-spec normal_s(Mean, Variance, State) -> {X :: float(), NewState :: state()}
              when
      Mean     :: number(),
      Variance :: number(),
      State    :: state().
normal_s(Mean, Variance, State0) when 0 =< Variance ->
    {X, State} = normal_s(State0),
    {Mean + (math:sqrt(Variance) * X), State}.

%% =====================================================================
%% Internal functions

-spec seed_put(state()) -> state().
seed_put(Seed) ->
    put(?SEED_DICT, Seed),
    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=>?MASK(64), next=>fun exs64_next/1},
     fun exs64_seed/1};
mk_alg(exsplus) ->
    {#{type=>exsplus, max=>?MASK(58), next=>fun exsp_next/1,
       jump=>fun exsplus_jump/1},
     fun exsplus_seed/1};
mk_alg(exsp) ->
    {#{type=>exsp, bits=>58, weak_low_bits=>1, next=>fun exsp_next/1,
       uniform=>fun exsp_uniform/1, uniform_n=>fun exsp_uniform/2,
       jump=>fun exsplus_jump/1},
     fun exsplus_seed/1};
mk_alg(exsss) ->
    {#{type=>exsss, bits=>58, next=>fun exsss_next/1,
       uniform=>fun exsss_uniform/1, uniform_n=>fun exsss_uniform/2,
       jump=>fun exsplus_jump/1},
     fun exsss_seed/1};
mk_alg(exs1024) ->
    {#{type=>exs1024, max=>?MASK(64), next=>fun exs1024_next/1,
       jump=>fun exs1024_jump/1},
     fun exs1024_seed/1};
mk_alg(exs1024s) ->
    {#{type=>exs1024s, bits=>64, weak_low_bits=>3, next=>fun exs1024_next/1,
       jump=>fun exs1024_jump/1},
     fun exs1024_seed/1};
mk_alg(exrop) ->
    {#{type=>exrop, bits=>58, weak_low_bits=>1, next=>fun exrop_next/1,
       uniform=>fun exrop_uniform/1, uniform_n=>fun exrop_uniform/2,
       jump=>fun exrop_jump/1},
     fun exrop_seed/1};
mk_alg(exro928ss) ->
    {#{type=>exro928ss, bits=>58, next=>fun exro928ss_next/1,
       uniform=>fun exro928ss_uniform/1,
       uniform_n=>fun exro928ss_uniform/2,
       jump=>fun exro928_jump/1},
     fun exro928_seed/1};
mk_alg(dummy=Name) ->
    {#{type=>Name, bits=>58, next=>fun dummy_next/1,
       uniform=>fun dummy_uniform/1,
       uniform_n=>fun dummy_uniform/2},
     fun dummy_seed/1}.

%% =====================================================================
%% exs64 PRNG: Xorshift64*
%% Algorithm by Sebastiano Vigna
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================

-doc "Algorithm specific internal state".
-opaque exs64_state() :: uint64().

exs64_seed(L) when is_list(L) ->
    [R] = seed64_nz(1, L),
    R;
exs64_seed(A) when is_integer(A) ->
    [R] = seed64(1, A),
    R;
%%
%% Traditional integer triplet seed
exs64_seed({A1, A2, A3}) ->
    {V1, _} = exs64_next((?MASK(32, A1) * 4294967197 + 1)),
    {V2, _} = exs64_next((?MASK(32, A2) * 4294967231 + 1)),
    {V3, _} = exs64_next((?MASK(32, A3) * 4294967279 + 1)),
    ((V1 * V2 * V3) rem (?MASK(64) - 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 ?BSL(64, R1, 25),
    R3 = R2 bxor (R2 bsr 27),
    {?MASK(64, R3 * 2685821657736338717), R3}.

%% =====================================================================
%% 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.
%%
%% Reference C code for Xorshift116+ and Xorshift116**
%%
%% #include <stdint.h>
%%
%% #define MASK(b, v) (((UINT64_C(1) << (b)) - 1) & (v))
%% #define BSL(b, v, n) (MASK((b)-(n), (v)) << (n))
%% #define ROTL(b, v, n) (BSL((b), (v), (n)) | ((v) >> ((b)-(n))))
%%
%% uint64_t s[2];
%%
%% uint64_t next(void) {
%%     uint64_t s1 = s[0];
%%     const uint64_t s0 = s[1];
%%
%%     s1 ^= BSL(58, s1, 24); // a
%%     s1 ^= s0 ^ (s1 >> 11) ^ (s0 >> 41); // b, c
%%     s[0] = s0;
%%     s[1] = s1;
%%
%%     const uint64_t result_plus = MASK(58, s0 + s1);
%%     uint64_t result_starstar = s0;
%%     result_starstar = MASK(58, result_starstar * 5);
%%     result_starstar = ROTL(58, result_starstar, 7);
%%     result_starstar = MASK(58, result_starstar * 9);
%%
%%     return result_plus;
%%     return result_starstar;
%% }
%%
%% =====================================================================
-doc "Algorithm specific internal state".
-opaque exsplus_state() :: nonempty_improper_list(uint58(), uint58()).

-dialyzer({no_improper_lists, exsplus_seed/1}).

exsplus_seed(L) when is_list(L) ->
    [S0,S1] = seed58_nz(2, L),
    [S0|S1];
exsplus_seed(X) when is_integer(X) ->
    [S0,S1] = seed58(2, X),
    [S0|S1];
%%
%% Traditional integer triplet seed
exsplus_seed({A1, A2, A3}) ->
    {_, R1} = exsp_next(
                [?MASK(58, (A1 * 4294967197) + 1)|
                 ?MASK(58, (A2 * 4294967231) + 1)]),
    {_, R2} = exsp_next(
                [?MASK(58, (A3 * 4294967279) + 1)|
                 tl(R1)]),
    R2.

-dialyzer({no_improper_lists, exsss_seed/1}).

exsss_seed(L) when is_list(L) ->
    [S0,S1] = seed58_nz(2, L),
    [S0|S1];
exsss_seed(X) when is_integer(X) ->
    [S0,S1] = seed58(2, X),
    [S0|S1];
%%
%% Seed from traditional integer triple - mix into splitmix
exsss_seed({A1, A2, A3}) ->
    {_, X0} = seed58(A1),
    {S0, X1} = seed58(A2 bxor X0),
    {S1, _} = seed58(A3 bxor X1),
    [S0|S1].

%% Advance Xorshift116 state one step
-define(
   exs_next(S0, S1, S1_b),
   begin
       S1_b = ?MASK(58, S1) bxor ?BSL(58, S1, 24),
       S1_b bxor S0 bxor (S1_b bsr 11) bxor (S0 bsr 41)
   end).

-define(
   scramble_starstar(S, V_a, V_b),
   begin
       %% The multiply by add shifted trick avoids creating bignums
       %% which improves performance significantly
       %%
       %% Scramble ** (all operations modulo word size)
       %% ((S * 5) rotl 7) * 9
       %%
       V_a = S + ?BSL(58, S, 2),                             % * 5
       V_b = ?BSL(58, V_a, 7) bor ?MASK(7, V_a bsr (58-7)),  % rotl 7
       ?MASK(58, V_b + ?BSL(58, V_b, 3))                     % * 9
   end).

%% Advance state and generate 58bit unsigned integer
%%
-dialyzer({no_improper_lists, exsp_next/1}).
-doc """
Generate an Xorshift116+ random integer and new algorithm state.

From the specified [`AlgState`](`t:exsplus_state/0`),
generates a random 58-bit integer [`X`](`t:uint58/0`)
and a new algorithm state [`NewAlgState`](`t:exsplus_state/0`),
according to the Xorshift116+ algorithm.

This is an API function exposing the internal implementation of the
[`exsp`](#algorithms) algorithm that enables using it without the
overhead of the plug-in framework, which might be useful for time critial
applications. On a typical 64 bit Erlang VM this approach executes
in just above 30% (1/3) of the time for the default algorithm through
this module's normal plug-in framework.

To seed this generator use [`{_, AlgState} = rand:seed_s(exsp)`](`seed_s/1`)
or [`{_, AlgState} = rand:seed_s(exsp, Seed)`](`seed_s/1`)
with a specific [`Seed`](`t:seed/0`).

> #### Note {: .info }
>
> This function offers no help in generating a number on a selected range,
> nor in generating floating point numbers.  It is easy to accidentally
> mess up the statistical properties of this generator or to loose
> the performance advantage when doing either.
> See the recepies at the start of this
> [Niche algorithms API](#niche-algorithms-api) description.
>
> Note also the caveat about weak low bits that this generator suffers from.
>
> The generator is exported in this form primarily for performance reasons.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec exsp_next(AlgState :: exsplus_state()) ->
                       {X :: uint58(), NewAlgState :: exsplus_state()}.
exsp_next([S1|S0]) ->
    %% Note: members s0 and s1 are swapped here
    S0_1 = ?MASK(58, S0),
    NewS1 = ?exs_next(S0_1, S1, S1_b),
    %% Scramble + (all operations modulo word size)
    %% S0 + NewS1
    {?MASK(58, S0_1 + NewS1), [S0_1|NewS1]}.

-dialyzer({no_improper_lists, exsss_next/1}).

-spec exsss_next(exsplus_state()) -> {uint58(), exsplus_state()}.
exsss_next([S1|S0]) ->
    %% Note: members s0 and s1 are swapped here
    S0_1 = ?MASK(58, S0),
    NewS1 = ?exs_next(S0_1, S1, S1_b),
    {?scramble_starstar(S0_1, V_1, V_2), [S0_1|NewS1]}.

exsp_uniform({AlgHandler, R0}) ->
    {I, R1} = exsp_next(R0),
    %% Waste the lowest bit since it is of lower
    %% randomness quality than the others
    {(I bsr (58-53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}}.

exsss_uniform({AlgHandler, R0}) ->
    {I, R1} = exsss_next(R0),
    {(I bsr (58-53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}}.

exsp_uniform(Range, {AlgHandler, R}) ->
    {V, R1} = exsp_next(R),
    MaxMinusRange = ?BIT(58) - Range,
    ?uniform_range(Range, AlgHandler, R1, V, MaxMinusRange, I).

exsss_uniform(Range, {AlgHandler, R}) ->
    {V, R1} = exsss_next(R),
    MaxMinusRange = ?BIT(58) - Range,
    ?uniform_range(Range, AlgHandler, R1, V, MaxMinusRange, I).


%% This is the jump function for the exs... generators,
%% i.e the Xorshift116 generators,  equivalent
%% to 2^64 calls to next/1; it can be used to generate 2^52
%% non-overlapping subsequences for parallel computations.
%% Note: the jump function takes 116 times of the execution time of
%% next/1.
%%
%% #include <stdint.h>
%%
%% void jump(void) {
%%   static const uint64_t JUMP[] = { 0x02f8ea6bc32c797,
%% 				   0x345d2a0f85f788c };
%%   int i, b;
%%   uint64_t s0 = 0;
%%   uint64_t s1 = 0;
%%   for(i = 0; i < sizeof JUMP / sizeof *JUMP; i++)
%%     for(b = 0; b < 58; b++) {
%%       if (JUMP[i] & 1ULL << b) {
%% 	s0 ^= s[0];
%% 	s1 ^= s[1];
%%       }
%%       next();
%%     }
%%   s[0] = s0;
%%   s[1] = s1;
%% }
%%
%% -define(JUMPCONST, 16#000d174a83e17de2302f8ea6bc32c797).
%% split into 58-bit chunks
%% and two iterative executions

-define(JUMPCONST1, 16#02f8ea6bc32c797).
-define(JUMPCONST2, 16#345d2a0f85f788c).
-define(JUMPELEMLEN, 58).

-dialyzer({no_improper_lists, exsplus_jump/1}).
-spec exsplus_jump({alg_handler(), exsplus_state()}) ->
          {alg_handler(), exsplus_state()}.
exsplus_jump({AlgHandler, S}) ->
    {AlgHandler, exsp_jump(S)}.

-dialyzer({no_improper_lists, exsp_jump/1}).
-doc """
Jump the generator state forward.

Performs a [`State`](`t:state/0`) jump calculation
that is equvalent to a 2^64 state iterations.

Returns the [`NewState`](`t:state/0`).

This feature can be used to create many non-overlapping
random number sequences from one start state.

See the description of jump functions at the top of this module description.

See `exsp_next/1` about why this internal implementation function
has been exposed.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec exsp_jump(AlgState :: exsplus_state()) ->
          NewAlgState :: exsplus_state().
exsp_jump(S) ->
    {S1, AS1} = exsplus_jump(S, [0|0], ?JUMPCONST1, ?JUMPELEMLEN),
    {_,  AS2} = exsplus_jump(S1, AS1,  ?JUMPCONST2, ?JUMPELEMLEN),
    AS2.

-dialyzer({no_improper_lists, exsplus_jump/4}).
exsplus_jump(S, AS, _, 0) ->
    {S, AS};
exsplus_jump(S, [AS0|AS1], J, N) ->
    {_, NS} = exsp_next(S),
    case ?MASK(1, J) 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
%% Reference URL: http://xorshift.di.unimi.it/
%% =====================================================================

-doc "Algorithm specific internal state".
-opaque exs1024_state() :: {list(uint64()), list(uint64())}.

exs1024_seed(L) when is_list(L) ->
    {seed64_nz(16, L), []};
exs1024_seed(X) when is_integer(X) ->
    {seed64(16, X), []};
%%
%% Seed from traditional triple, remain backwards compatible
exs1024_seed({A1, A2, A3}) ->
    B1 = ?MASK(21, (?MASK(21, A1) + 1) * 2097131),
    B2 = ?MASK(21, (?MASK(21, A2) + 1) * 2097133),
    B3 = ?MASK(21, (?MASK(21, A3) + 1) * 2097143),
    {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 ?BSL(64, S1, 31),
    S12 = S11 bxor (S11 bsr 11),
    S01 = S0 bxor (S0 bsr 30),
    NS1 = S01 bxor S12,
    {?MASK(64, NS1 * 1181783497276652981), 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, []}).


%% 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({alg_handler(), exs1024_state()}) ->
                          {alg_handler(), exs1024_state()}.
exs1024_jump({AlgHandler, {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),
    {AlgHandler, {ASL, lists:reverse(ASR)}}.

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 ?MASK(1, J) 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.

%% =====================================================================
%% exro928ss PRNG: Xoroshiro928**
%%
%% Reference URL: http://vigna.di.unimi.it/ftp/papers/ScrambledLinear.pdf
%% i.e the Xoroshiro1024 generator with ** scrambler
%% with {S, R, T} = {5, 7, 9} as recommended in the paper.
%%
%% {A, B, C} were tried out and selected as {44, 9, 45}
%% and the jump coefficients calculated.
%%
%% Standard jump function pseudocode:
%% 
%%     Jump constant j = 0xb10773cb...44085302f77130ca
%%     Generator state: s
%%     New generator state: t = 0
%%     foreach bit in j, low to high:
%%         if the bit is one:
%%             t ^= s
%%         next s
%%     s = t
%%
%% Generator used for reference value calculation:
%%
%%     #include <stdint.h>
%%     #include <stdio.h>
%%     
%%     int p = 0;
%%     uint64_t s[16];
%%     
%%     #define MASK(x) ((x) & ((UINT64_C(1) << 58) - 1))
%%     static __inline uint64_t rotl(uint64_t x, int n) {
%%         return MASK(x << n) | (x >> (58 - n));
%%     }
%%     
%%     uint64_t next() {
%%         const int q = p;
%%         const uint64_t s0 = s[p = (p + 1) & 15];
%%         uint64_t s15 = s[q];
%%     
%%         const uint64_t result_starstar = MASK(rotl(MASK(s0 * 5), 7) * 9);
%%     
%%         s15 ^= s0;
%%         s[q] = rotl(s0, 44) ^ s15 ^ MASK(s15 << 9);
%%         s[p] = rotl(s15, 45);
%%     
%%         return result_starstar;
%%     }
%%
%%     static const uint64_t jump_2pow512[15] =
%%         { 0x44085302f77130ca, 0xba05381fdfd14902, 0x10a1de1d7d6813d2,
%%           0xb83fe51a1eb3be19, 0xa81b0090567fd9f0, 0x5ac26d5d20f9b49f,
%%           0x4ddd98ee4be41e01, 0x0657e19f00d4b358, 0xf02f778573cf0f0a,
%%           0xb45a3a8a3cef3cc0, 0x6e62a33cc2323831, 0xbcb3b7c4cc049c53,
%%           0x83f240c6007e76ce, 0xe19f5fc1a1504acd, 0x00000000b10773cb };
%%
%%     static const uint64_t jump_2pow20[15] =
%%         { 0xbdb966a3daf905e6, 0x644807a56270cf78, 0xda90f4a806c17e9e,
%%           0x4a426866bfad3c77, 0xaf699c306d8e7566, 0x8ebc73c700b8b091,
%%           0xc081a7bf148531fb, 0xdc4d3af15f8a4dfd, 0x90627c014098f4b6,
%%           0x06df2eb1feaf0fb6, 0x5bdeb1a5a90f2e6b, 0xa480c5878c3549bd,
%%           0xff45ef33c82f3d48, 0xa30bebc15fefcc78, 0x00000000cb3d181c };
%%
%%     void jump(const uint64_t *jump) {
%%         uint64_t j, t[16] = {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0};
%%         int m, n, k;
%%         for (m = 0;  m < 15;  m++, jump++) {
%%             for (n = 0, j = *jump;  n < 64;  n++, j >>= 1) {
%%                 if ((j & 1) != 0) {
%%                     for (k = 0;  k < 16;  k++) {
%%                         t[k] ^= s[(p + k) & 15];
%%                     }
%%                 }
%%                 next();
%%             }
%%         }
%%         for (k = 0;  k < 16;  k++) {
%%             s[(p + k) & 15] = t[k];
%%         }
%%     }
%%
%% =====================================================================

-doc "Algorithm specific internal state".
-opaque exro928_state() :: {list(uint58()), list(uint58())}.

-doc false.
-spec exro928_seed(
        list(uint58()) | integer() | {integer(), integer(), integer()}) ->
                          exro928_state().
exro928_seed(L) when is_list(L) ->
    {seed58_nz(16, L), []};
exro928_seed(X) when is_integer(X) ->
    {seed58(16, X), []};
%%
%% Seed from traditional integer triple - mix into splitmix
exro928_seed({A1, A2, A3}) ->
    {S0, X0} = seed58(A1),
    {S1, X1} = seed58(A2 bxor X0),
    {S2, X2} = seed58(A3 bxor X1),
    {[S0,S1,S2|seed58(13, X2)], []}.


%% Update the state and calculate output word
-spec exro928ss_next(exro928_state()) -> {uint58(), exro928_state()}.
exro928ss_next({[S15,S0|Ss], Rs}) ->
    SR = exro928_next_state(Ss, Rs, S15, S0),
    %%
    %% {S, R, T} = {5, 7, 9}
    %% const uint64_t result_starstar = rotl(s0 * S, R) * T;
    %%
    {?scramble_starstar(S0, V_0, V_1), SR};
exro928ss_next({[S15], Rs}) ->
    exro928ss_next({[S15|lists:reverse(Rs)], []}).

-doc false.
-spec exro928_next(exro928_state()) -> {{uint58(),uint58()}, exro928_state()}.
exro928_next({[S15,S0|Ss], Rs}) ->
    SR = exro928_next_state(Ss, Rs, S15, S0),
    {{S15,S0}, SR};
exro928_next({[S15], Rs}) ->
    exro928_next({[S15|lists:reverse(Rs)], []}).

%% Just update the state
-doc false.
-spec exro928_next_state(exro928_state()) -> exro928_state().
exro928_next_state({[S15,S0|Ss], Rs}) ->
    exro928_next_state(Ss, Rs, S15, S0);
exro928_next_state({[S15], Rs}) ->
    [S0|Ss] = lists:reverse(Rs),
    exro928_next_state(Ss, [], S15, S0).

exro928_next_state(Ss, Rs, S15, S0) ->
    %% {A, B, C} = {44, 9, 45},
    %% s15 ^= s0;
    %% NewS15: s[q] = rotl(s0, A) ^ s15 ^ (s15 << B);
    %% NewS0: s[p] = rotl(s15, C);
    %%
    S0_1 = ?MASK(58, S0),
    Q = ?MASK(58, S15) bxor S0_1,
    NewS15 = ?ROTL(58, S0_1, 44) bxor Q bxor ?BSL(58, Q, 9),
    NewS0 = ?ROTL(58, Q, 45),
    {[NewS0|Ss], [NewS15|Rs]}.


exro928ss_uniform({AlgHandler, SR}) ->
    {V, NewSR} = exro928ss_next(SR),
    {(V bsr (58-53)) * ?TWO_POW_MINUS53, {AlgHandler, NewSR}}.

exro928ss_uniform(Range, {AlgHandler, SR}) ->
    {V, NewSR} = exro928ss_next(SR),
    MaxMinusRange = ?BIT(58) - Range,
    ?uniform_range(Range, AlgHandler, NewSR, V, MaxMinusRange, I).


-spec exro928_jump({alg_handler(), exro928_state()}) ->
                          {alg_handler(), exro928_state()}.
exro928_jump({AlgHandler, SR}) ->
    {AlgHandler,exro928_jump_2pow512(SR)}.

-doc false.
-spec exro928_jump_2pow512(exro928_state()) -> exro928_state().
exro928_jump_2pow512(SR) ->
    polyjump(
      SR, fun exro928_next_state/1,
      %% 2^512
      [16#4085302F77130CA, 16#54E07F7F4524091,
       16#5E1D7D6813D2BA0, 16#4687ACEF8644287,
       16#4567FD9F0B83FE5, 16#43E6D27EA06C024,
       16#641E015AC26D5D2, 16#6CD61377663B92F,
       16#70A0657E19F00D4, 16#43C0BDDE15CF3C3,
       16#745A3A8A3CEF3CC, 16#58A8CF308C8E0C6,
       16#7B7C4CC049C536E, 16#431801F9DB3AF2C,
       16#41A1504ACD83F24, 16#6C41DCF2F867D7F]).

-doc false.
-spec exro928_jump_2pow20(exro928_state()) -> exro928_state().
exro928_jump_2pow20(SR) ->
    polyjump(
      SR, fun exro928_next_state/1,
      %% 2^20
      [16#5B966A3DAF905E6, 16#601E9589C33DE2F,
       16#74A806C17E9E644, 16#59AFEB4F1DF6A43,
       16#46D8E75664A4268, 16#42E2C246BDA670C,
       16#4531FB8EBC73C70, 16#537F702069EFC52,
       16#4B6DC4D3AF15F8A, 16#5A4189F0050263D,
       16#46DF2EB1FEAF0FB, 16#77AC696A43CB9AC,
       16#4C5878C3549BD5B, 16#7CCF20BCF522920,
       16#415FEFCC78FF45E, 16#72CF460728C2FAF]).

%% =====================================================================
%% exrop PRNG: Xoroshiro116+
%%
%% Reference URL: http://xorshift.di.unimi.it/
%%
%% 58 bits fits into an immediate on 64bits Erlang and is thus much faster.
%% In fact, an immediate number is 60 bits signed in Erlang so you can
%% add two positive 58 bit numbers and get a 59 bit number that still is
%% a positive immediate, which is a property we utilize here...
%%
%% Modification of the original Xororhiro128+ algorithm to 116 bits
%% by Sebastiano Vigna.  A lot of thanks for his help and work.
%% =====================================================================
%% (a, b, c) = (24, 2, 35)
%% JUMP Polynomial = 0x9863200f83fcd4a11293241fcb12a (116 bit)
%%
%% From http://xoroshiro.di.unimi.it/xoroshiro116plus.c:
%% ---------------------------------------------------------------------
%% /* Written in 2017 by Sebastiano Vigna (vigna@acm.org).
%%
%% To the extent possible under law, the author has dedicated all copyright
%% and related and neighboring rights to this software to the public domain
%% worldwide. This software is distributed without any warranty.
%%
%% See <http://creativecommons.org/publicdomain/zero/1.0/>. */
%%
%% #include <stdint.h>
%%
%% #define UINT58MASK (uint64_t)((UINT64_C(1) << 58) - 1)
%%
%% uint64_t s[2];
%%
%% static inline uint64_t rotl58(const uint64_t x, int k) {
%%     return (x << k) & UINT58MASK | (x >> (58 - k));
%% }
%% 
%% uint64_t next(void) {
%%     uint64_t s1 = s[1];
%%     const uint64_t s0 = s[0];
%%     const uint64_t result = (s0 + s1) & UINT58MASK;
%%
%%     s1 ^= s0;
%%     s[0] = rotl58(s0, 24) ^ s1 ^ ((s1 << 2) & UINT58MASK); // a, b
%%     s[1] = rotl58(s1, 35); // c
%%     return result;
%% }
%%
%% void jump(void) {
%%     static const uint64_t JUMP[] =
%%         { 0x4a11293241fcb12a, 0x0009863200f83fcd };
%%
%%     uint64_t s0 = 0;
%%     uint64_t s1 = 0;
%%     for(int i = 0; i < sizeof JUMP / sizeof *JUMP; i++)
%%         for(int b = 0; b < 64; b++) {
%%             if (JUMP[i] & UINT64_C(1) << b) {
%%                 s0 ^= s[0];
%% 	           s1 ^= s[1];
%% 	       }
%% 	       next();
%% 	   }
%%     s[0] = s0;
%%     s[1] = s1;
%% }

-doc "Algorithm specific internal state".
-opaque exrop_state() :: nonempty_improper_list(uint58(), uint58()).

-dialyzer({no_improper_lists, exrop_seed/1}).

exrop_seed(L) when is_list(L) ->
    [S0,S1] = seed58_nz(2, L),
    [S0|S1];
exrop_seed(X) when is_integer(X) ->
    [S0,S1] = seed58(2, X),
    [S0|S1];
%%
%% Traditional integer triplet seed
exrop_seed({A1, A2, A3}) ->
    [_|S1] =
        exrop_next_s(
          ?MASK(58, (A1 * 4294967197) + 1),
          ?MASK(58, (A2 * 4294967231) + 1)),
    exrop_next_s(?MASK(58, (A3 * 4294967279) + 1), S1).

-dialyzer({no_improper_lists, exrop_next_s/2}).
%% Advance xoroshiro116+ state one step
%% [a, b, c] = [24, 2, 35]
-define(
   exrop_next_s(S0, S1, S1_a),
   begin
       S1_a = S1 bxor S0,
       [?ROTL(58, S0, 24) bxor S1_a bxor ?BSL(58, S1_a, 2)| % a, b
        ?ROTL(58, S1_a, 35)] % c
   end).
exrop_next_s(S0, S1) ->
    ?exrop_next_s(S0, S1, S1_a).

-dialyzer({no_improper_lists, exrop_next/1}).
%% Advance xoroshiro116+ state one step, generate 58 bit unsigned integer,
%% and waste the lowest bit since it is of lower randomness quality
exrop_next([S0|S1]) ->
    {?MASK(58, S0 + S1), ?exrop_next_s(S0, S1, S1_a)}.

exrop_uniform({AlgHandler, R}) ->
    {V, R1} = exrop_next(R),
    %% Waste the lowest bit since it is of lower
    %% randomness quality than the others
    {(V bsr (58-53)) * ?TWO_POW_MINUS53, {AlgHandler, R1}}.

exrop_uniform(Range, {AlgHandler, R}) ->
    {V, R1} = exrop_next(R),
    MaxMinusRange = ?BIT(58) - Range,
    ?uniform_range(Range, AlgHandler, R1, V, MaxMinusRange, I).

%% Split a 116 bit constant into two 58 bit words,
%% a top '1' marks the end of the low word.
-define(
   JUMP_116(Jump),
   [?BIT(58) bor ?MASK(58, (Jump)),(Jump) bsr 58]).
%%
exrop_jump({AlgHandler,S}) ->
    [J|Js] = ?JUMP_116(16#9863200f83fcd4a11293241fcb12a),
    {AlgHandler, exrop_jump(S, 0, 0, J, Js)}.
%%
-dialyzer({no_improper_lists, exrop_jump/5}).
exrop_jump(_S, S0, S1, 0, []) -> % End of jump constant
    [S0|S1];
exrop_jump(S, S0, S1, 1, [J|Js]) -> % End of word
    exrop_jump(S, S0, S1, J, Js);
exrop_jump([S__0|S__1] = _S, S0, S1, J, Js) ->
    case ?MASK(1, J) of
        1 ->
            NewS = exrop_next_s(S__0, S__1),
            exrop_jump(NewS, S0 bxor S__0, S1 bxor S__1, J bsr 1, Js);
        0 ->
            NewS = exrop_next_s(S__0, S__1),
            exrop_jump(NewS, S0, S1, J bsr 1, Js)
    end.

%% =====================================================================
%% dummy "PRNG": Benchmark dummy overhead reference
%%
%% As fast as possible - return something daft and update state;
%% to measure plug-in framework overhead.
%%
%% =====================================================================

-doc "Algorithm specific internal state".
-type dummy_state() :: uint58().

dummy_uniform(_Range, {AlgHandler,R}) ->
    {1, {AlgHandler,(R bxor ?MASK(58))}}. % 1 is always in Range
dummy_next(R) ->
    {R, R bxor ?MASK(58)}.
dummy_uniform({AlgHandler,R}) ->
    {0.5, {AlgHandler,(R bxor ?MASK(58))}}. % Perfect mean value

%% Serious looking seed, to avoid rand_SUITE seed test failure
%%
dummy_seed(L) when is_list(L) ->
    case L of
        [] ->
            erlang:error(zero_seed);
        [X] when is_integer(X) ->
            ?MASK(58, X);
        [X|_] when is_integer(X) ->
            erlang:error(too_many_seed_integers);
        [_|_] ->
            erlang:error(non_integer_seed)
    end;
dummy_seed(X) when is_integer(X) ->
    {Z1, _} = splitmix64_next(X),
    ?MASK(58, Z1);
dummy_seed({A1, A2, A3}) ->
    {_, X1} = splitmix64_next(A1),
    {_, X2} = splitmix64_next(A2 bxor X1),
    {Z3, _} = splitmix64_next(A3 bxor X2),
    ?MASK(58, Z3).


%% =====================================================================
%% mcg58 PRNG: Multiply With Carry generator
%%
%% Parameters deduced in collaboration with
%% Prof. Sebastiano Vigna of the University of Milano.
%%
%% X = CX0 & (2^B - 1)  % Low B bits - digit
%% C = CX0 >> B         % High bits  - carry
%% CX1 = A * X0 + C0
%%
%% An MWC generator is an efficient alternative implementation of
%% a Multiplicative Congruential Generator, that is, the generator
%% CX1 = (CX0 * 2^B) rem P
%% where P is the safe prime (A * 2^B - 1), that generates
%% the same sequence in the reverse order.  The generator
%% CX1 = (A * CX0) rem P
%% that uses the multiplicative inverse mod P is, indeed,
%% an exact equivalent to the corresponding MWC generator.
%%
%% An MWC generator has, due to the power of two multiplier
%% in the corresponding MCG, got known statistical weaknesses
%% in the spectral score for 3 dimensions, so it should be used
%% with a scrambler that hides the flaws.  The scramblers
%% have been tried out in the PractRand and TestU01 frameworks
%% and settled for a single Xorshift to get B good bits,
%% and a double Xorshift to get all bits good enough.
%%
%% The chosen parameters are:
%% A = 16#7fa6502
%% B = 32
%% Single Xorshift: 8
%% Double Xorshift: 4, 27
%%
%% These parameters gives the MWC "digit" size 32 bits
%% which gives them theoretical statistical guarantees,
%% and keeps the state in 59 bits.
%%
%% The state should only be used to mask or rem out low bits.
%% The scramblers return 58 bits from which a number should
%% be masked or rem:ed out.
%%
%% =====================================================================
-define(MWC59_A, (16#7fa6502)).
-define(MWC59_B, (32)).
-define(MWC59_P, ((?MWC59_A bsl ?MWC59_B) - 1)).

-define(MWC59_XS, 8).
-define(MWC59_XS1, 4).
-define(MWC59_XS2, 27).

-doc """
`1 .. (16#1ffb072 bsl 29) - 2`
""".
-type mwc59_state() :: 1..?MWC59_P-1.

-doc """
Generate a new MWC59 state.

From the specified generator state [`CX0`](`t:mwc59_state/0`) generate
a new state [`CX1`](`t:mwc59_state/0`), according to a Multiply With Carry
generator, which is an efficient implementation of
a Multiplicative Congruential Generator with a power of 2 multiplier
and a prime modulus.

This generator uses the multiplier `2^32` and the modulus
`16#7fa6502 * 2^32 - 1`, which have been selected, in collaboration with
Sebastiano Vigna, to avoid bignum operations and still get
good statistical quality. It has been named "MWC59" and can be written as:

```erlang
C = CX0 bsr 32
X = CX0 band ((1 bsl 32)-1))
CX1 = 16#7fa6502 * X + C
```

Because the generator uses a multiplier that is a power of 2 it gets
statistical flaws for collision tests and birthday spacings tests
in 2 and 3 dimensions, and these caveats apply even when looking
only at the MWC "digit", that is the low 32 bits (the multiplier)
of the generator state.  The higher bits of the state are worse.

The quality of the output value improves much by using a scrambler,
instead of just taking the low bits.
Function [`mwc59_value32`](`mwc59_value32/1`) is a fast scrambler
that returns a decent 32-bit number. The slightly slower
[`mwc59_value`](`mwc59_value/1`) scrambler returns 59 bits of
very good quality, and [`mwc59_float`](`mwc59_float/1`) returns
a `t:float/0` of very good quality.

The low bits of the base generator are surprisingly good, so the lowest
16 bits actually pass fairly strict PRNG tests, despite the generator's
weaknesses that lie in the high bits of the 32-bit MWC "digit".
It is recommended to use `rem` on the the generator state, or bit mask
extracting the lowest bits to produce numbers in a range 16 bits or less.
See the recepies at the start of this
[Niche algorithms API](#niche-algorithms-api) description.

On a typical 64 bit Erlang VM this generator executes in below 8% (1/13)
of the time for the default algorithm in the
[plug-in framework API](#plug-in-framework-api) of this module.
With the [`mwc59_value32`](`mwc59_value32/1`) scrambler the total time
becomes 16% (1/6), and with [`mwc59_value`](`mwc59_value/1`)
it becomes 20% (1/5) of the time for the default algorithm.
With [`mwc59_float`](`mwc59_float/1`) the total time
is 60% of the time for the default algorithm generating a `t:float/0`.

> #### Note {: .info }
>
> This generator is a niche generator for high speed applications.
> It has a much shorter period than the default generator, which in itself
> is a quality concern, although when used with the value scramblers
> it passes strict PRNG tests.  The generator is much faster than
> `exsp_next/1` but with a bit lower quality and much shorter period.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec mwc59(CX0 :: mwc59_state()) -> CX1 :: mwc59_state().
mwc59(CX) when is_integer(CX), 1 =< CX, CX < ?MWC59_P ->
    C = CX bsr ?MWC59_B,
    X = ?MASK(?MWC59_B, CX),
    ?MWC59_A * X + C.

%%% %% Verification by equivalent MCG generator
%%% mwc59_r(CX1) ->
%%%     (CX1 bsl ?MWC59_B) rem ?MWC59_P. % Reverse
%%% %%%     (CX1 * ?MWC59_A) rem ?MWC59_P. % Forward
%%%
%%% mwc59(CX0, 0) ->
%%%     CX0;
%%% mwc59(CX0, N) ->
%%%     CX1 = mwc59(CX0),
%%%     CX0 = mwc59_r(CX1),
%%%     mwc59(CX1, N - 1).

-doc """
Calculate a 32-bit scrambled value from a [MWC59 state](`t:mwc59_state/0`).

Returns a 32-bit value [`V`](`t:integer/0`) from a generator state `CX`.
The generator state is scrambled using an 8-bit xorshift which masks
the statistical imperfecions of the base generator [`mwc59`](`mwc59/1`)
enough to produce numbers of decent quality. Still some problems
in 2- and 3-dimensional birthday spacing and collision tests show through.

When using this scrambler it is in general better to use the high bits of the
value than the low. The lowest 8 bits are of good quality and are passed
right through from the base generator. They are combined with the next 8
in the xorshift making the low 16 good quality, but in the range
16..31 bits there are weaker bits that should not become high bits
of the generated values.

Therefore it is in general safer to shift out low bits. See the recepies
at the start of this [Niche algorithms API](#niche-algorithms-api)
description.

For a non power of 2 range less than about 16 bits (to not get
too much bias and to avoid bignums) truncated multiplication can be used,
that is: `(Range*V) bsr 32`, which is much faster than using `rem`.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec mwc59_value32(CX :: mwc59_state()) -> V :: 0..?MASK(32).
mwc59_value32(CX1) when is_integer(CX1), 1 =< CX1, CX1 < ?MWC59_P ->
    CX = ?MASK(32, CX1),
    CX bxor ?BSL(32, CX, ?MWC59_XS).

-doc """
Calculate a 59-bit scrambled value from a [MWC59 state](`t:mwc59_state/0`).

Returns a 59-bit value [`V`](`t:integer/0`) from a generator state `CX`.
The generator state is scrambled using an 4-bit followed by a 27-bit xorshift,
which masks the statistical imperfecions of the [MWC59](`mwc59/1`)
base generator enough that all 59 bits are of very good quality.

Be careful to not accidentaly create a bignum when handling the value `V`.

It is in general general better to use the high bits from this scrambler than
the low. See the recepies at the start of this
[Niche algorithms API](#niche-algorithms-api) description.

For a non power of 2 range less than about 29 bits (to not get
too much bias and to avoid bignums) truncated multiplication can be used,
which is much faster than using `rem`. Example for range 1'000'000'000;
the range is 30 bits, we use 29 bits from the generator,
adding up to 59 bits, which is not a bignum (on a 64-bit VM ):
`(1000000000 * (V bsr (59-29))) bsr 29`.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec mwc59_value(CX :: mwc59_state()) -> V :: 0..?MASK(59).
mwc59_value(CX) when is_integer(CX), 1 =< CX, CX < ?MWC59_P ->
    CX2 = CX bxor ?BSL(59, CX, ?MWC59_XS1),
    CX2 bxor ?BSL(59, CX2, ?MWC59_XS2).

-doc """
Calculate a scrambled `t:float/0` from a [MWC59 state](`t:mwc59_state/0`).

Returns a value `V ::` `t:float/0` from a generator state `CX`,
in the range `0.0 =< V < 1.0` like for example `uniform_s/1`.

The generator state is scrambled as with
[`mwc59_value/1`](`mwc59_value/1`) before converted to a `t:float/0`.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec mwc59_float(CX :: mwc59_state()) -> V :: float().
mwc59_float(CX1) when is_integer(CX1), 1 =< CX1, CX1 < ?MWC59_P ->
    CX = ?MASK(53, CX1),
    CX2 = CX bxor ?BSL(53, CX, ?MWC59_XS1),
    CX3 = CX2 bxor ?BSL(53, CX2, ?MWC59_XS2),
    CX3 * ?TWO_POW_MINUS53.

-doc """
Create a [MWC59 generator state](`t:mwc59_state/0`).

Like `mwc59_seed/1` but it hashes the default seed value
of [`seed_s(atom())`](`seed_s/1`).
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec mwc59_seed() -> CX :: mwc59_state().
mwc59_seed() ->
    {A1, A2, A3} = default_seed(),
    X1 = hash58(A1),
    X2 = hash58(A2),
    X3 = hash58(A3),
    (X1 bxor X2 bxor X3) + 1.

-doc """
Create a [MWC59 generator state](`t:mwc59_state/0`).

Returns a generator state [`CX`](`t:mwc59_state/0`).
The 58-bit seed value `S` is hashed to create the generator state,
to avoid that similar seeds create similar sequences.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec mwc59_seed(S :: 0..?MASK(58)) -> CX :: mwc59_state().
mwc59_seed(S) when is_integer(S), 0 =< S, S =< ?MASK(58) ->
    hash58(S) + 1.

%% Constants a'la SplitMix64, MurMurHash, etc.
%% Not that critical, just mix the bits using bijections
%% (reversible mappings) to not have any two user input seeds
%% become the same generator start state.
%%
hash58(X) ->
    X0 = ?MASK(58, X),
    X1 = ?MASK(58, (X0 bxor (X0 bsr 29)) * 16#351afd7ed558ccd),
    X2 = ?MASK(58, (X1 bxor (X1 bsr 29)) * 16#0ceb9fe1a85ec53),
    X2 bxor (X2 bsr 29).


%% =====================================================================
%% Mask and fill state list, ensure not all zeros
%% =====================================================================

seed58_nz(N, Ss) ->
    seed_nz(N, Ss, 58, false).

seed64_nz(N, Ss) ->
    seed_nz(N, Ss, 64, false).

seed_nz(_N, [], _M, false) ->
    erlang:error(zero_seed);
seed_nz(0, [_|_], _M, _NZ) ->
    erlang:error(too_many_seed_integers);
seed_nz(0, [], _M, _NZ) ->
    [];
seed_nz(N, [], M, true) ->
    [0|seed_nz(N - 1, [], M, true)];
seed_nz(N, [S|Ss], M, NZ) ->
    if
	is_integer(S) ->
	    R = ?MASK(M, S),
	    [R|seed_nz(N - 1, Ss, M, NZ orelse R =/= 0)];
	true ->
	    erlang:error(non_integer_seed)
    end.

%% =====================================================================
%% Splitmix seeders, lowest bits of SplitMix64, zeros skipped
%% =====================================================================

-doc false.
-spec seed58(non_neg_integer(), uint64()) -> list(uint58()).
seed58(0, _X) ->
    [];
seed58(N, X) ->
    {Z,NewX} = seed58(X),
    [Z|seed58(N - 1, NewX)].
%%
seed58(X_0) ->
    {Z0,X} = splitmix64_next(X_0),
    case ?MASK(58, Z0) of
	0 ->
	    seed58(X);
	Z ->
	    {Z,X}
    end.

-spec seed64(non_neg_integer(), uint64()) -> list(uint64()).
seed64(0, _X) ->
    [];
seed64(N, X) ->
    {Z,NewX} = seed64(X),
    [Z|seed64(N - 1, NewX)].
%%
seed64(X_0) ->
    {Z,X} = ZX = splitmix64_next(X_0),
    if
	Z =:= 0 ->
	    seed64(X);
	true ->
	    ZX
    end.

%% =====================================================================
%% The SplitMix64 generator:
%%
%% uint64_t splitmix64_next() {
%% 	uint64_t z = (x += 0x9e3779b97f4a7c15);
%% 	z = (z ^ (z >> 30)) * 0xbf58476d1ce4e5b9;
%% 	z = (z ^ (z >> 27)) * 0x94d049bb133111eb;
%% 	return z ^ (z >> 31);
%% }
%%

-doc "Algorithm specific state".
-type splitmix64_state() :: uint64().

-doc """
Generate a SplitMix64 random 64-bit integer and new algorithm state.

From the specified `AlgState` generates a random 64-bit integer
[`X`](`t:uint64/0`) and a new generator state
[`NewAlgState`](`t:splitmix64_state/0`),
according to the SplitMix64 algorithm.

This generator is used internally in the `rand` module for seeding other
generators since it is of a quite different breed which reduces
the probability for creating an accidentally bad seed.
""".
-doc(#{title => <<"Niche algorithms API">>,since => <<"OTP 25.0">>}).
-spec splitmix64_next(AlgState :: integer()) ->
                             {X :: uint64(), NewAlgState :: splitmix64_state()}.
splitmix64_next(X_0) ->
    X = ?MASK(64, X_0 + 16#9e3779b97f4a7c15),
    Z_0 = ?MASK(64, (X bxor (X bsr 30)) * 16#bf58476d1ce4e5b9),
    Z_1 = ?MASK(64, (Z_0 bxor (Z_0 bsr 27)) * 16#94d049bb133111eb),
    {?MASK(64, Z_1 bxor (Z_1 bsr 31)),X}.

%% =====================================================================
%% Polynomial jump with a jump constant word list,
%% high bit in each word marking top of word,
%% SR is a {Forward, Reverse} queue tuple with Forward never empty
%% =====================================================================

polyjump({Ss, Rs} = SR, NextState, JumpConst) ->
    %% Create new state accumulator T
    Ts = lists:duplicate(length(Ss) + length(Rs), 0),
    polyjump(SR, NextState, JumpConst, Ts).
%%
%% Foreach jump word
polyjump(_SR, _NextState, [], Ts) ->
    %% Return new calculated state
    {Ts, []};
polyjump(SR, NextState, [J|Js], Ts) ->
    polyjump(SR, NextState, Js, Ts, J).
%%
%% Foreach bit in jump word until top bit
polyjump(SR, NextState, Js, Ts, 1) ->
    polyjump(SR, NextState, Js, Ts);
polyjump({Ss, Rs} = SR, NextState, Js, Ts, J) when J =/= 0 ->
    NewSR = NextState(SR),
    NewJ = J bsr 1,
    case ?MASK(1, J) of
        0 ->
            polyjump(NewSR, NextState, Js, Ts, NewJ);
        1 ->
            %% Xor this state onto T
            polyjump(NewSR, NextState, Js, xorzip_sr(Ts, Ss, Rs), NewJ)
    end.

xorzip_sr([], [], undefined) ->
    [];
xorzip_sr(Ts, [], Rs) ->
    xorzip_sr(Ts, lists:reverse(Rs), undefined);
xorzip_sr([T|Ts], [S|Ss], Rs) ->
    [T bxor S|xorzip_sr(Ts, Ss, Rs)].

%% =====================================================================

-doc false.
format_jumpconst58(String) ->
    ReOpts = [{newline,any},{capture,all_but_first,binary},global],
    {match,Matches} = re:run(String, "0x([a-zA-Z0-9]+)", ReOpts),
    format_jumcons58_matches(lists:reverse(Matches), 0).

format_jumcons58_matches([], J) ->
    format_jumpconst58_value(J);
format_jumcons58_matches([[Bin]|Matches], J) ->
    NewJ = (J bsl 64) bor binary_to_integer(Bin, 16),
    format_jumcons58_matches(Matches, NewJ).

format_jumpconst58_value(0) ->
    ok;
format_jumpconst58_value(J) ->
    io:format("16#~s,~n", [integer_to_list(?MASK(58, J) bor ?BIT(58), 16)]),
    format_jumpconst58_value(J bsr 58).

%% =====================================================================
%% Ziggurat cont
%% =====================================================================
-define(NOR_R, 3.6541528853610087963519472518).
-define(NOR_INV_R, 1/?NOR_R).

%% return a {sign, Random51bits, State}
get_52({#{bits:=Bits, next:=Next} = AlgHandler, S0}) ->
    %% Use the high bits
    {Int,S1} = Next(S0),
    {?BIT(Bits - 51 - 1) band Int, Int bsr (Bits - 51), {AlgHandler, S1}};
get_52({#{next:=Next} = AlgHandler, S0}) ->
    {Int,S1} = Next(S0),
    {?BIT(51) band Int, ?MASK(51, Int), {AlgHandler, 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}).

%%%bitcount64(0) -> 0;
%%%bitcount64(V) -> 1 + bitcount(V, 64).
%%%
%%%-define(
%%%   BITCOUNT(V, N),
%%%   bitcount(V, N) ->
%%%       if
%%%           (1 bsl ((N) bsr 1)) =< (V) ->
%%%               ((N) bsr 1) + bitcount((V) bsr ((N) bsr 1), ((N) bsr 1));
%%%           true ->
%%%               bitcount((V), ((N) bsr 1))
%%%       end).
%%%?BITCOUNT(V, 64);
%%%?BITCOUNT(V, 32);
%%%?BITCOUNT(V, 16);
%%%?BITCOUNT(V, 8);
%%%?BITCOUNT(V, 4);
%%%?BITCOUNT(V, 2);
%%%bitcount(_, 1) -> 0.

-doc false.
bc64(V) -> ?BC(V, 64).

%% Linear from high bit - higher probability first gives faster execution
bc(V, B, N) when B =< V -> N;
bc(V, B, N) -> bc(V, B bsr 1, N - 1).


%%% %% Non-negative rem
%%% mod(Q, X) when 0 =< X, X < Q ->
%%%     X;
%%% mod(Q, X) ->
%%%     Y = X rem Q,
%%%     if
%%%         Y < 0 ->
%%%             Y + Q;
%%%         true ->
%%%             Y
%%%     end.


-doc false.
make_float(S, E, M) ->
    <<F/float>> = <<S:1, E:11, M:52>>,
    F.

-doc false.
float2str(N) ->
    <<S:1, E:11, M:52>> = <<(float(N))/float>>,
    lists:flatten(
      io_lib:format(
      "~c~c.~13.16.0bE~b",
      [case S of 1 -> $-; 0 -> $+ end,
       case E of 0 -> $0; _ -> $1 end,
       M, E - 16#3ff])).