random.jl

# This file is a part of Julia. License is MIT: https://julialang.org/license

module Random

using Base.dSFMT
using Base.GMP: Limb, MPZ
import Base: copymutable, copy, copy!, ==

export srand,
       rand, rand!,
       randn, randn!,
       randexp, randexp!,
       bitrand,
       randstring,
       randsubseq,randsubseq!,
       shuffle,shuffle!,
       randperm, randcycle,
       AbstractRNG, MersenneTwister, RandomDevice,
       GLOBAL_RNG, randjump


abstract type AbstractRNG end

abstract type FloatInterval end
mutable struct CloseOpen <: FloatInterval end
mutable struct Close1Open2 <: FloatInterval end


## RandomDevice

if is_windows()

    struct RandomDevice <: AbstractRNG
        buffer::Vector{UInt128}

        RandomDevice() = new(Vector{UInt128}(1))
    end

    function rand{T<:Union{Bool, Base.BitInteger}}(rd::RandomDevice, ::Type{T})
        win32_SystemFunction036!(rd.buffer)
        @inbounds return rd.buffer[1] % T
    end

    rand!(rd::RandomDevice, A::Array{<:Union{Bool, Base.BitInteger}}) = (win32_SystemFunction036!(A); A)
else # !windows
    struct RandomDevice <: AbstractRNG
        file::IOStream
        unlimited::Bool

        RandomDevice(unlimited::Bool=true) = new(open(unlimited ? "/dev/urandom" : "/dev/random"), unlimited)
    end

    rand{T<:Union{Bool, Base.BitInteger}}(rd::RandomDevice, ::Type{T}) = read( rd.file, T)
    rand!(rd::RandomDevice, A::Array{<:Union{Bool, Base.BitInteger}})  = read!(rd.file, A)
end # os-test


"""
    RandomDevice()

Create a `RandomDevice` RNG object. Two such objects will always generate different streams of random numbers.
"""
RandomDevice


rand(rng::RandomDevice, ::Type{Close1Open2}) =
    reinterpret(Float64, 0x3ff0000000000000 | rand(rng, UInt64) & 0x000fffffffffffff)

rand(rng::RandomDevice, ::Type{CloseOpen}) = rand(rng, Close1Open2) - 1.0


## MersenneTwister

const MTCacheLength = dsfmt_get_min_array_size()

mutable struct MersenneTwister <: AbstractRNG
    seed::Vector{UInt32}
    state::DSFMT_state
    vals::Vector{Float64}
    idx::Int

    function MersenneTwister(seed, state, vals, idx)
        length(vals) == MTCacheLength &&  0 <= idx <= MTCacheLength || throw(DomainError())
        new(seed, state, vals, idx)
    end
end

MersenneTwister(seed::Vector{UInt32}, state::DSFMT_state) =
    MersenneTwister(seed, state, zeros(Float64, MTCacheLength), MTCacheLength)

"""
    MersenneTwister(seed)

Create a `MersenneTwister` RNG object. Different RNG objects can have their own seeds, which
may be useful for generating different streams of random numbers.
"""
MersenneTwister(seed) = srand(MersenneTwister(Vector{UInt32}(), DSFMT_state()), seed)

function copy!(dst::MersenneTwister, src::MersenneTwister)
    copy!(resize!(dst.seed, length(src.seed)), src.seed)
    copy!(dst.state, src.state)
    copy!(dst.vals, src.vals)
    dst.idx = src.idx
    dst
end

copy(src::MersenneTwister) =
    MersenneTwister(copy(src.seed), copy(src.state), copy(src.vals), src.idx)

==(r1::MersenneTwister, r2::MersenneTwister) =
    r1.seed == r2.seed && r1.state == r2.state && isequal(r1.vals, r2.vals) && r1.idx == r2.idx


## Low level API for MersenneTwister

@inline mt_avail(r::MersenneTwister) = MTCacheLength - r.idx
@inline mt_empty(r::MersenneTwister) = r.idx == MTCacheLength
@inline mt_setfull!(r::MersenneTwister) = r.idx = 0
@inline mt_setempty!(r::MersenneTwister) = r.idx = MTCacheLength
@inline mt_pop!(r::MersenneTwister) = @inbounds return r.vals[r.idx+=1]

function gen_rand(r::MersenneTwister)
    dsfmt_fill_array_close1_open2!(r.state, pointer(r.vals), length(r.vals))
    mt_setfull!(r)
end

@inline reserve_1(r::MersenneTwister) = mt_empty(r) && gen_rand(r)
# `reserve` allows one to call `rand_inbounds` n times
# precondition: n <= MTCacheLength
@inline reserve(r::MersenneTwister, n::Int) = mt_avail(r) < n && gen_rand(r)

# precondition: !mt_empty(r)
@inline rand_inbounds(r::MersenneTwister, ::Type{Close1Open2}) = mt_pop!(r)
@inline rand_inbounds(r::MersenneTwister, ::Type{CloseOpen}) = rand_inbounds(r, Close1Open2) - 1.0
@inline rand_inbounds(r::MersenneTwister) = rand_inbounds(r, CloseOpen)

# produce Float64 values
@inline rand{I<:FloatInterval}(r::MersenneTwister, ::Type{I}) = (reserve_1(r); rand_inbounds(r, I))

@inline rand_ui52_raw_inbounds(r::MersenneTwister) = reinterpret(UInt64, rand_inbounds(r, Close1Open2))
@inline rand_ui52_raw(r::MersenneTwister) = (reserve_1(r); rand_ui52_raw_inbounds(r))
@inline rand_ui2x52_raw(r::MersenneTwister) = rand_ui52_raw(r) % UInt128 << 64 | rand_ui52_raw(r)

function srand(r::MersenneTwister, seed::Vector{UInt32})
    copy!(resize!(r.seed, length(seed)), seed)
    dsfmt_init_by_array(r.state, r.seed)
    mt_setempty!(r)
    return r
end

# MersenneTwister jump

"""
    randjump(r::MersenneTwister, jumps::Integer, [jumppoly::AbstractString=dSFMT.JPOLY1e21]) -> Vector{MersenneTwister}

Create an array of the size `jumps` of initialized `MersenneTwister` RNG objects. The
first RNG object given as a parameter and following `MersenneTwister` RNGs in the array are
initialized such that a state of the RNG object in the array would be moved forward (without
generating numbers) from a previous RNG object array element on a particular number of steps
encoded by the jump polynomial `jumppoly`.

Default jump polynomial moves forward `MersenneTwister` RNG state by `10^20` steps.
"""
function randjump(mt::MersenneTwister, jumps::Integer, jumppoly::AbstractString)
    mts = MersenneTwister[]
    push!(mts, mt)
    for i in 1:jumps-1
        cmt = mts[end]
        push!(mts, MersenneTwister(copy(cmt.seed), dSFMT.dsfmt_jump(cmt.state, jumppoly)))
    end
    return mts
end
randjump(r::MersenneTwister, jumps::Integer) = randjump(r, jumps, dSFMT.JPOLY1e21)

## initialization

function __init__()
    try
        srand()
    catch ex
        Base.showerror_nostdio(ex,
            "WARNING: Error during initialization of module Random")
    end
end


## make_seed()
# make_seed methods produce values of type Array{UInt32}, suitable for MersenneTwister seeding

function make_seed()
    try
        return rand(RandomDevice(), UInt32, 4)
    catch
        println(STDERR, "Entropy pool not available to seed RNG; using ad-hoc entropy sources.")
        seed = reinterpret(UInt64, time())
        seed = hash(seed, UInt64(getpid()))
        try
        seed = hash(seed, parse(UInt64, readstring(pipeline(`ifconfig`, `sha1sum`))[1:40], 16))
        end
        return make_seed(seed)
    end
end

function make_seed(n::Integer)
    n < 0 && throw(DomainError())
    seed = UInt32[]
    while true
        push!(seed, n & 0xffffffff)
        n >>= 32
        if n == 0
            return seed
        end
    end
end

## srand()

"""
    srand([rng=GLOBAL_RNG], seed) -> rng
    srand([rng=GLOBAL_RNG]) -> rng

Reseed the random number generator. If a `seed` is provided, the RNG will give a
reproducible sequence of numbers, otherwise Julia will get entropy from the system. For
`MersenneTwister`, the `seed` may be a non-negative integer or a vector of [`UInt32`](@ref)
integers. `RandomDevice` does not support seeding.
"""
srand(r::MersenneTwister) = srand(r, make_seed())
srand(r::MersenneTwister, n::Integer) = srand(r, make_seed(n))


function dsfmt_gv_srand()
    # Temporary fix for #8874 and #9124: update global RNG for Rmath
    dsfmt_gv_init_by_array(GLOBAL_RNG.seed+UInt32(1))
    return GLOBAL_RNG
end

function srand()
    srand(GLOBAL_RNG)
    dsfmt_gv_srand()
end

function srand(seed::Union{Integer, Vector{UInt32}})
    srand(GLOBAL_RNG, seed)
    dsfmt_gv_srand()
end

## Global RNG

const GLOBAL_RNG = MersenneTwister(0)
globalRNG() = GLOBAL_RNG

# rand: a non-specified RNG defaults to GLOBAL_RNG

"""
    rand([rng=GLOBAL_RNG], [S], [dims...])

Pick a random element or array of random elements from the set of values specified by `S`; `S` can be

* an indexable collection (for example `1:n` or `['x','y','z']`), or
* a `Dict`, a `Set` or an `IntSet`, or
* a type: the set of values to pick from is then equivalent to `typemin(S):typemax(S)` for
  integers (this is not applicable to [`BigInt`](@ref)), and to ``[0, 1)`` for floating
  point numbers;

`S` defaults to [`Float64`](@ref).

```julia-repl
julia> rand(Int, 2)
2-element Array{Int64,1}:
 1339893410598768192
 1575814717733606317

julia> rand(MersenneTwister(0), Dict(1=>2, 3=>4))
1=>2
```
"""
@inline rand() = rand(GLOBAL_RNG, CloseOpen)
@inline rand(T::Type) = rand(GLOBAL_RNG, T)
rand(dims::Dims) = rand(GLOBAL_RNG, dims)
rand(dims::Integer...) = rand(convert(Tuple{Vararg{Int}}, dims))
rand(T::Type, dims::Dims) = rand(GLOBAL_RNG, T, dims)
rand(T::Type, d1::Integer, dims::Integer...) = rand(T, tuple(Int(d1), convert(Tuple{Vararg{Int}}, dims)...))
rand!(A::AbstractArray) = rand!(GLOBAL_RNG, A)

rand(r::AbstractArray) = rand(GLOBAL_RNG, r)

"""
    rand!([rng=GLOBAL_RNG], A, [coll])

Populate the array `A` with random values. If the collection `coll` is specified,
the values are picked randomly from `coll`. This is equivalent to `copy!(A, rand(rng, coll, size(A)))`
or `copy!(A, rand(rng, eltype(A), size(A)))` but without allocating a new array.
"""
rand!(A::AbstractArray, r::AbstractArray) = rand!(GLOBAL_RNG, A, r)

rand(r::AbstractArray, dims::Dims) = rand(GLOBAL_RNG, r, dims)
rand(r::AbstractArray, dims::Integer...) = rand(GLOBAL_RNG, r, convert(Tuple{Vararg{Int}}, dims))

## random floating point values

@inline rand(r::AbstractRNG) = rand(r, CloseOpen)

# MersenneTwister & RandomDevice
@inline rand(r::Union{RandomDevice,MersenneTwister}, ::Type{Float64}) = rand(r, CloseOpen)

rand_ui10_raw(r::MersenneTwister) = rand_ui52_raw(r)
rand_ui23_raw(r::MersenneTwister) = rand_ui52_raw(r)
rand_ui10_raw(r::AbstractRNG)    = rand(r, UInt16)
rand_ui23_raw(r::AbstractRNG)    = rand(r, UInt32)

rand(r::Union{RandomDevice,MersenneTwister}, ::Type{Float16}) =
    Float16(reinterpret(Float32, (rand_ui10_raw(r) % UInt32 << 13) & 0x007fe000 | 0x3f800000) - 1)

rand(r::Union{RandomDevice,MersenneTwister}, ::Type{Float32}) =
    reinterpret(Float32, rand_ui23_raw(r) % UInt32 & 0x007fffff | 0x3f800000) - 1


## random integers

@inline rand_ui52_raw(r::AbstractRNG) = reinterpret(UInt64, rand(r, Close1Open2))
@inline rand_ui52(r::AbstractRNG) = rand_ui52_raw(r) & 0x000fffffffffffff

# MersenneTwister

@inline rand{T<:Union{Bool, Int8, UInt8, Int16, UInt16, Int32, UInt32}}(r::MersenneTwister, ::Type{T}) = rand_ui52_raw(r) % T

function rand(r::MersenneTwister, ::Type{UInt64})
    reserve(r, 2)
    rand_ui52_raw_inbounds(r) << 32 ⊻ rand_ui52_raw_inbounds(r)
end

function rand(r::MersenneTwister, ::Type{UInt128})
    reserve(r, 3)
    xor(rand_ui52_raw_inbounds(r) % UInt128 << 96,
        rand_ui52_raw_inbounds(r) % UInt128 << 48,
        rand_ui52_raw_inbounds(r))
end

rand(r::MersenneTwister, ::Type{Int64})   = reinterpret(Int64,  rand(r, UInt64))
rand(r::MersenneTwister, ::Type{Int128})  = reinterpret(Int128, rand(r, UInt128))

## random Complex values

rand{T<:Real}(r::AbstractRNG, ::Type{Complex{T}}) = complex(rand(r, T), rand(r, T))

# random Char values
# returns a random valid Unicode scalar value (i.e. 0 - 0xd7ff, 0xe000 - # 0x10ffff)
function rand(r::AbstractRNG, ::Type{Char})
    c = rand(r, 0x00000000:0x0010f7ff)
    (c < 0xd800) ? Char(c) : Char(c+0x800)
end

# random values from Dict, Set, IntSet (for efficiency)
function rand(r::AbstractRNG, t::Dict)
    isempty(t) && throw(ArgumentError("collection must be non-empty"))
    rg = RangeGenerator(1:length(t.slots))
    while true
        i = rand(r, rg)
        Base.isslotfilled(t, i) && @inbounds return (t.keys[i] => t.vals[i])
    end
end
rand(t::Dict) = rand(GLOBAL_RNG, t)
rand(r::AbstractRNG, s::Set) = rand(r, s.dict).first
rand(s::Set) = rand(GLOBAL_RNG, s)

function rand(r::AbstractRNG, s::IntSet)
    isempty(s) && throw(ArgumentError("collection must be non-empty"))
    # s can be empty while s.bits is not, so we cannot rely on the
    # length check in RangeGenerator below
    rg = RangeGenerator(1:length(s.bits))
    while true
        n = rand(r, rg)
        @inbounds b = s.bits[n]
        b && return n
    end
end

rand(s::IntSet) = rand(GLOBAL_RNG, s)

## Arrays of random numbers

rand(r::AbstractRNG, dims::Dims) = rand(r, Float64, dims)
rand(r::AbstractRNG, dims::Integer...) = rand(r, convert(Tuple{Vararg{Int}}, dims))

rand(r::AbstractRNG, T::Type, dims::Dims) = rand!(r, Array{T}(dims))
rand(r::AbstractRNG, T::Type, d1::Integer, dims::Integer...) = rand(r, T, tuple(Int(d1), convert(Tuple{Vararg{Int}}, dims)...))
# note: the above method would trigger an ambiguity warning if d1 was not separated out:
# rand(r, ()) would match both this method and rand(r, dims::Dims)
# moreover, a call like rand(r, NotImplementedType()) would be an infinite loop

function rand!{T}(r::AbstractRNG, A::AbstractArray{T})
    for i in eachindex(A)
        @inbounds A[i] = rand(r, T)
    end
    A
end

function rand!(r::AbstractRNG, A::AbstractArray, s::Union{Dict,Set,IntSet})
    for i in eachindex(A)
        @inbounds A[i] = rand(r, s)
    end
    A
end

rand!(A::AbstractArray, s::Union{Dict,Set,IntSet}) = rand!(GLOBAL_RNG, A, s)

rand(r::AbstractRNG, s::Dict{K,V}, dims::Dims) where {K,V} = rand!(r, Array{Pair{K,V}}(dims), s)
rand(r::AbstractRNG, s::Set{T}, dims::Dims) where {T} = rand!(r, Array{T}(dims), s)
rand(r::AbstractRNG, s::IntSet, dims::Dims) = rand!(r, Array{Int}(dims), s)
rand(r::AbstractRNG, s::Union{Dict,Set,IntSet}, dims::Integer...) = rand(r, s, convert(Dims, dims))
rand(s::Union{Dict,Set,IntSet}, dims::Integer...) = rand(GLOBAL_RNG, s, dims)
rand(s::Union{Dict,Set,IntSet}, dims::Dims) = rand(GLOBAL_RNG, s, dims)

# MersenneTwister

function rand_AbstractArray_Float64!{I<:FloatInterval}(r::MersenneTwister, A::AbstractArray{Float64}, n=length(A), ::Type{I}=CloseOpen)
    # what follows is equivalent to this simple loop but more efficient:
    # for i=1:n
    #     @inbounds A[i] = rand(r, I)
    # end
    m = 0
    while m < n
        s = mt_avail(r)
        if s == 0
            gen_rand(r)
            s = mt_avail(r)
        end
        m2 = min(n, m+s)
        for i=m+1:m2
            @inbounds A[i] = rand_inbounds(r, I)
        end
        m = m2
    end
    A
end

rand!(r::MersenneTwister, A::AbstractArray{Float64}) = rand_AbstractArray_Float64!(r, A)

fill_array!(s::DSFMT_state, A::Ptr{Float64}, n::Int, ::Type{CloseOpen}) = dsfmt_fill_array_close_open!(s, A, n)
fill_array!(s::DSFMT_state, A::Ptr{Float64}, n::Int, ::Type{Close1Open2}) = dsfmt_fill_array_close1_open2!(s, A, n)

function rand!{I<:FloatInterval}(r::MersenneTwister, A::Array{Float64}, n::Int=length(A), ::Type{I}=CloseOpen)
    # depending on the alignment of A, the data written by fill_array! may have
    # to be left-shifted by up to 15 bytes (cf. unsafe_copy! below) for
    # reproducibility purposes;
    # so, even for well aligned arrays, fill_array! is used to generate only
    # the n-2 first values (or n-3 if n is odd), and the remaining values are
    # generated by the scalar version of rand
    if n > length(A)
        throw(BoundsError(A,n))
    end
    n2 = (n-2) ÷ 2 * 2
    if n2 < dsfmt_get_min_array_size()
        rand_AbstractArray_Float64!(r, A, n, I)
    else
        pA = pointer(A)
        align = Csize_t(pA) % 16
        if align > 0
            pA2 = pA + 16 - align
            fill_array!(r.state, pA2, n2, I) # generate the data in-place, but shifted
            unsafe_copy!(pA, pA2, n2) # move the data to the beginning of the array
        else
            fill_array!(r.state, pA, n2, I)
        end
        for i=n2+1:n
            @inbounds A[i] = rand(r, I)
        end
    end
    A
end

@inline mask128(u::UInt128, ::Type{Float16}) = (u & 0x03ff03ff03ff03ff03ff03ff03ff03ff) | 0x3c003c003c003c003c003c003c003c00
@inline mask128(u::UInt128, ::Type{Float32}) = (u & 0x007fffff007fffff007fffff007fffff) | 0x3f8000003f8000003f8000003f800000

function rand!{T<:Union{Float16, Float32}}(r::MersenneTwister, A::Array{T}, ::Type{Close1Open2})
    n = length(A)
    n128 = n * sizeof(T) ÷ 16
    rand!(r, unsafe_wrap(Array, convert(Ptr{Float64}, pointer(A)), 2*n128), 2*n128, Close1Open2)
    A128 = unsafe_wrap(Array, convert(Ptr{UInt128}, pointer(A)), n128)
    @inbounds for i in 1:n128
        u = A128[i]
        u ⊻= u << 26
        # at this point, the 64 low bits of u, "k" being the k-th bit of A128[i] and "+" the bit xor, are:
        # [..., 58+32,..., 53+27, 52+26, ..., 33+7, 32+6, ..., 27+1, 26, ..., 1]
        # the bits needing to be random are
        # [1:10, 17:26, 33:42, 49:58] (for Float16)
        # [1:23, 33:55] (for Float32)
        # this is obviously satisfied on the 32 low bits side, and on the high side, the entropy comes
        # from bits 33:52 of A128[i] and then from bits 27:32 (which are discarded on the low side)
        # this is similar for the 64 high bits of u
        A128[i] = mask128(u, T)
    end
    for i in 16*n128÷sizeof(T)+1:n
        @inbounds A[i] = rand(r, T) + oneunit(T)
    end
    A
end

function rand!{T<:Union{Float16, Float32}}(r::MersenneTwister, A::Array{T}, ::Type{CloseOpen})
    rand!(r, A, Close1Open2)
    I32 = one(Float32)
    for i in eachindex(A)
        @inbounds A[i] = T(Float32(A[i])-I32) # faster than "A[i] -= one(T)" for T==Float16
    end
    A
end

rand!(r::MersenneTwister, A::Array{<:Union{Float16, Float32}}) = rand!(r, A, CloseOpen)


function rand!(r::MersenneTwister, A::Array{UInt128}, n::Int=length(A))
    if n > length(A)
        throw(BoundsError(A,n))
    end
    Af = unsafe_wrap(Array, convert(Ptr{Float64}, pointer(A)), 2n)
    i = n
    while true
        rand!(r, Af, 2i, Close1Open2)
        n < 5 && break
        i = 0
        @inbounds while n-i >= 5
            u = A[i+=1]
            A[n]    ⊻= u << 48
            A[n-=1] ⊻= u << 36
            A[n-=1] ⊻= u << 24
            A[n-=1] ⊻= u << 12
            n-=1
        end
    end
    if n > 0
        u = rand_ui2x52_raw(r)
        for i = 1:n
            @inbounds A[i] ⊻= u << 12*i
        end
    end
    A
end

function rand!{T<:Union{Int8, UInt8, Int16, UInt16, Int32, UInt32, Int64, UInt64, Int128}}(r::MersenneTwister, A::Array{T})
    n=length(A)
    n128 = n * sizeof(T) ÷ 16
    rand!(r, unsafe_wrap(Array, convert(Ptr{UInt128}, pointer(A)), n128))
    for i = 16*n128÷sizeof(T)+1:n
        @inbounds A[i] = rand(r, T)
    end
    A
end

## Generate random integer within a range

# remainder function according to Knuth, where rem_knuth(a, 0) = a
rem_knuth(a::UInt, b::UInt) = a % (b + (b == 0)) + a * (b == 0)
rem_knuth(a::T, b::T) where {T<:Unsigned} = b != 0 ? a % b : a

# maximum multiple of k <= 2^bits(T) decremented by one,
# that is 0xFFFF...FFFF if k = typemax(T) - typemin(T) with intentional underflow
# see http:https://stackoverflow.com/questions/29182036/integer-arithmetic-add-1-to-uint-max-and-divide-by-n-without-overflow
maxmultiple(k::T) where {T<:Unsigned} = (div(typemax(T) - k + oneunit(k), k + (k == 0))*k + k - oneunit(k))::T

# maximum multiple of k within 1:2^32 or 1:2^64 decremented by one, depending on size
maxmultiplemix(k::UInt64) = if k >> 32 != 0; maxmultiple(k); else (div(0x0000000100000000, k + (k == 0))*k - oneunit(k))::UInt64; end

abstract type RangeGenerator end

struct RangeGeneratorInt{T<:Integer,U<:Unsigned} <: RangeGenerator
    a::T   # first element of the range
    k::U   # range length or zero for full range
    u::U   # rejection threshold
end
# generators with 32, 128 bits entropy
RangeGeneratorInt(a::T, k::U) where {T,U<:Union{UInt32,UInt128}} = RangeGeneratorInt{T,U}(a, k, maxmultiple(k))
# mixed 32/64 bits entropy generator
RangeGeneratorInt(a::T, k::UInt64) where {T} = RangeGeneratorInt{T,UInt64}(a, k, maxmultiplemix(k))
# generator for ranges
function RangeGenerator(r::UnitRange{T}) where T<:Unsigned
    if isempty(r)
        throw(ArgumentError("range must be non-empty"))
    end
    RangeGeneratorInt(first(r), last(r) - first(r) + oneunit(T))
end

# specialized versions
for (T, U) in [(UInt8, UInt32), (UInt16, UInt32),
               (Int8, UInt32), (Int16, UInt32), (Int32, UInt32), (Int64, UInt64), (Int128, UInt128),
               (Bool, UInt32)]

    @eval RangeGenerator(r::UnitRange{$T}) = begin
        if isempty(r)
            throw(ArgumentError("range must be non-empty"))
        end
        RangeGeneratorInt(first(r), convert($U, unsigned(last(r) - first(r)) + one($U))) # overflow ok
    end
end

struct RangeGeneratorBigInt <: RangeGenerator
    a::BigInt             # first
    m::BigInt             # range length - 1
    nlimbs::Int           # number of limbs in generated BigInt's
    mask::Limb            # applied to the highest limb
end


function RangeGenerator(r::UnitRange{BigInt})
    m = last(r) - first(r)
    m < 0 && throw(ArgumentError("range must be non-empty"))
    nd = ndigits(m, 2)
    nlimbs, highbits = divrem(nd, 8*sizeof(Limb))
    highbits > 0 && (nlimbs += 1)
    mask = highbits == 0 ? ~zero(Limb) : one(Limb)<<highbits - one(Limb)
    return RangeGeneratorBigInt(first(r), m, nlimbs, mask)
end


# this function uses 32 bit entropy for small ranges of length <= typemax(UInt32) + 1
# RangeGeneratorInt is responsible for providing the right value of k
function rand{T<:Union{UInt64, Int64}}(rng::AbstractRNG, g::RangeGeneratorInt{T,UInt64})
    local x::UInt64
    if (g.k - 1) >> 32 == 0
        x = rand(rng, UInt32)
        while x > g.u
            x = rand(rng, UInt32)
        end
    else
        x = rand(rng, UInt64)
        while x > g.u
            x = rand(rng, UInt64)
        end
    end
    return reinterpret(T, reinterpret(UInt64, g.a) + rem_knuth(x, g.k))
end

function rand{T<:Integer, U<:Unsigned}(rng::AbstractRNG, g::RangeGeneratorInt{T,U})
    x = rand(rng, U)
    while x > g.u
        x = rand(rng, U)
    end
    (unsigned(g.a) + rem_knuth(x, g.k)) % T
end

function rand(rng::AbstractRNG, g::RangeGeneratorBigInt)
    x = MPZ.realloc2(g.nlimbs*8*sizeof(Limb))
    limbs = unsafe_wrap(Array, x.d, g.nlimbs)
    while true
        rand!(rng, limbs)
        @inbounds limbs[end] &= g.mask
        MPZ.mpn_cmp(x, g.m, g.nlimbs) <= 0 && break
    end
    # adjust x.size (normally done by mpz_limbs_finish, in GMP version >= 6)
    x.size = g.nlimbs
    while x.size > 0
        @inbounds limbs[x.size] != 0 && break
        x.size -= 1
    end
    MPZ.add!(x, g.a)
end

rand(rng::AbstractRNG, r::UnitRange{<:Union{Signed,Unsigned,BigInt,Bool}}) = rand(rng, RangeGenerator(r))


# Randomly draw a sample from an AbstractArray r
# (e.g. r is a range 0:2:8 or a vector [2, 3, 5, 7])
rand(rng::AbstractRNG, r::AbstractArray) = @inbounds return r[rand(rng, 1:length(r))]

function rand!(rng::AbstractRNG, A::AbstractArray, g::RangeGenerator)
    for i in eachindex(A)
        @inbounds A[i] = rand(rng, g)
    end
    return A
end

rand!(rng::AbstractRNG, A::AbstractArray, r::UnitRange{<:Union{Signed,Unsigned,BigInt,Bool,Char}}) = rand!(rng, A, RangeGenerator(r))

function rand!(rng::AbstractRNG, A::AbstractArray, r::AbstractArray)
    g = RangeGenerator(1:(length(r)))
    for i in eachindex(A)
        @inbounds A[i] = r[rand(rng, g)]
    end
    return A
end

rand{T}(rng::AbstractRNG, r::AbstractArray{T}, dims::Dims) = rand!(rng, Array{T}(dims), r)
rand(rng::AbstractRNG, r::AbstractArray, dims::Int...) = rand(rng, r, dims)

## random BitArrays (AbstractRNG)

function rand!(rng::AbstractRNG, B::BitArray)
    isempty(B) && return B
    Bc = B.chunks
    rand!(rng, Bc)
    Bc[end] &= Base._msk_end(B)
    return B
end

"""
    bitrand([rng=GLOBAL_RNG], [dims...])

Generate a `BitArray` of random boolean values.
"""
bitrand(r::AbstractRNG, dims::Dims)   = rand!(r, BitArray(dims))
bitrand(r::AbstractRNG, dims::Int...) = rand!(r, BitArray(dims))

bitrand(dims::Dims)   = rand!(BitArray(dims))
bitrand(dims::Int...) = rand!(BitArray(dims))

## randn() - Normally distributed random numbers using Ziggurat algorithm

# The Ziggurat Method for generating random variables - Marsaglia and Tsang
# Paper and reference code: http:https://www.jstatsoft.org/v05/i08/

# randmtzig (covers also exponential variates)
## Tables for normal variates
const ki =
    UInt64[0x0007799ec012f7b2,0x0000000000000000,0x0006045f4c7de363,0x0006d1aa7d5ec0a5,
           0x000728fb3f60f777,0x0007592af4e9fbc0,0x000777a5c0bf655d,0x00078ca3857d2256,
           0x00079bf6b0ffe58b,0x0007a7a34ab092ad,0x0007b0d2f20dd1cb,0x0007b83d3aa9cb52,
           0x0007be597614224d,0x0007c3788631abe9,0x0007c7d32bc192ee,0x0007cb9263a6e86d,
           0x0007ced483edfa84,0x0007d1b07ac0fd39,0x0007d437ef2da5fc,0x0007d678b069aa6e,
           0x0007d87db38c5c87,0x0007da4fc6a9ba62,0x0007dbf611b37f3b,0x0007dd7674d0f286,
           0x0007ded5ce8205f6,0x0007e018307fb62b,0x0007e141081bd124,0x0007e2533d712de8,
           0x0007e3514bbd7718,0x0007e43d54944b52,0x0007e5192f25ef42,0x0007e5e67481118d,
           0x0007e6a6897c1ce2,0x0007e75aa6c7f64c,0x0007e803df8ee498,0x0007e8a326eb6272,
           0x0007e93954717a28,0x0007e9c727f8648f,0x0007ea4d4cc85a3c,0x0007eacc5c4907a9,
           0x0007eb44e0474cf6,0x0007ebb754e47419,0x0007ec242a3d8474,0x0007ec8bc5d69645,
           0x0007ecee83d3d6e9,0x0007ed4cb8082f45,0x0007eda6aee0170f,0x0007edfcae2dfe68,
           0x0007ee4ef5dccd3e,0x0007ee9dc08c394e,0x0007eee9441a17c7,0x0007ef31b21b4fb1,
           0x0007ef773846a8a7,0x0007efba00d35a17,0x0007effa32ccf69f,0x0007f037f25e1278,
           0x0007f0736112d12c,0x0007f0ac9e145c25,0x0007f0e3c65e1fcc,0x0007f118f4ed8e54,
           0x0007f14c42ed0dc8,0x0007f17dc7daa0c3,0x0007f1ad99aac6a5,0x0007f1dbcce80015,
           0x0007f20874cf56bf,0x0007f233a36a3b9a,0x0007f25d69a604ad,0x0007f285d7694a92,
           0x0007f2acfba75e3b,0x0007f2d2e4720909,0x0007f2f79f09c344,0x0007f31b37ec883b,
           0x0007f33dbae36abc,0x0007f35f330f08d5,0x0007f37faaf2fa79,0x0007f39f2c805380,
           0x0007f3bdc11f4f1c,0x0007f3db71b83850,0x0007f3f846bba121,0x0007f4144829f846,
           0x0007f42f7d9a8b9d,0x0007f449ee420432,0x0007f463a0f8675e,0x0007f47c9c3ea77b,
           0x0007f494e643cd8e,0x0007f4ac84e9c475,0x0007f4c37dc9cd50,0x0007f4d9d638a432,
           0x0007f4ef934a5b6a,0x0007f504b9d5f33d,0x0007f5194e78b352,0x0007f52d55994a96,
           0x0007f540d36aba0c,0x0007f553cbef0e77,0x0007f56642f9ec8f,0x0007f5783c32f31e,
           0x0007f589bb17f609,0x0007f59ac2ff1525,0x0007f5ab5718b15a,0x0007f5bb7a71427c,
           0x0007f5cb2ff31009,0x0007f5da7a67cebe,0x0007f5e95c7a24e7,0x0007f5f7d8b7171e,
           0x0007f605f18f5ef4,0x0007f613a958ad0a,0x0007f621024ed7e9,0x0007f62dfe94f8cb,
           0x0007f63aa036777a,0x0007f646e928065a,0x0007f652db488f88,0x0007f65e786213ff,
           0x0007f669c22a7d8a,0x0007f674ba446459,0x0007f67f623fc8db,0x0007f689bb9ac294,
           0x0007f693c7c22481,0x0007f69d881217a6,0x0007f6a6fdd6ac36,0x0007f6b02a4c61ee,
           0x0007f6b90ea0a7f4,0x0007f6c1abf254c0,0x0007f6ca03521664,0x0007f6d215c2db82,
           0x0007f6d9e43a3559,0x0007f6e16fa0b329,0x0007f6e8b8d23729,0x0007f6efc09e4569,
           0x0007f6f687c84cbf,0x0007f6fd0f07ea09,0x0007f703570925e2,0x0007f709606cad03,
           0x0007f70f2bc8036f,0x0007f714b9a5b292,0x0007f71a0a85725d,0x0007f71f1edc4d9e,
           0x0007f723f714c179,0x0007f728938ed843,0x0007f72cf4a03fa0,0x0007f7311a945a16,
           0x0007f73505ac4bf8,0x0007f738b61f03bd,0x0007f73c2c193dc0,0x0007f73f67bd835c,
           0x0007f74269242559,0x0007f745305b31a1,0x0007f747bd666428,0x0007f74a103f12ed,
           0x0007f74c28d414f5,0x0007f74e0709a42d,0x0007f74faab939f9,0x0007f75113b16657,
           0x0007f75241b5a155,0x0007f753347e16b8,0x0007f753ebb76b7c,0x0007f75467027d05,
           0x0007f754a5f4199d,0x0007f754a814b207,0x0007f7546ce003ae,0x0007f753f3c4bb29,
           0x0007f7533c240e92,0x0007f75245514f41,0x0007f7510e91726c,0x0007f74f971a9012,
           0x0007f74dde135797,0x0007f74be2927971,0x0007f749a39e051c,0x0007f747202aba8a,
           0x0007f744571b4e3c,0x0007f741473f9efe,0x0007f73def53dc43,0x0007f73a4dff9bff,
           0x0007f73661d4deaf,0x0007f732294f003f,0x0007f72da2d19444,0x0007f728cca72bda,
           0x0007f723a5000367,0x0007f71e29f09627,0x0007f7185970156b,0x0007f7123156c102,
           0x0007f70baf5c1e2c,0x0007f704d1150a23,0x0007f6fd93f1a4e5,0x0007f6f5f53b10b6,
           0x0007f6edf211023e,0x0007f6e587671ce9,0x0007f6dcb2021679,0x0007f6d36e749c64,
           0x0007f6c9b91bf4c6,0x0007f6bf8e1c541b,0x0007f6b4e95ce015,0x0007f6a9c68356ff,
           0x0007f69e20ef5211,0x0007f691f3b517eb,0x0007f6853997f321,0x0007f677ed03ff19,
           0x0007f66a08075bdc,0x0007f65b844ab75a,0x0007f64c5b091860,0x0007f63c8506d4bc,
           0x0007f62bfa8798fe,0x0007f61ab34364b0,0x0007f608a65a599a,0x0007f5f5ca4737e8,
           0x0007f5e214d05b48,0x0007f5cd7af7066e,0x0007f5b7f0e4c2a1,0x0007f5a169d68fcf,
           0x0007f589d80596a5,0x0007f5712c8d0174,0x0007f557574c912b,0x0007f53c46c77193,
           0x0007f51fe7feb9f2,0x0007f5022646ecfb,0x0007f4e2eb17ab1d,0x0007f4c21dd4a3d1,
           0x0007f49fa38ea394,0x0007f47b5ebb62eb,0x0007f4552ee27473,0x0007f42cf03d58f5,
           0x0007f4027b48549f,0x0007f3d5a44119df,0x0007f3a63a8fb552,0x0007f37408155100,
           0x0007f33ed05b55ec,0x0007f3064f9c183e,0x0007f2ca399c7ba1,0x0007f28a384bb940,
           0x0007f245ea1b7a2b,0x0007f1fcdffe8f1b,0x0007f1ae9af758cd,0x0007f15a8917f27e,
           0x0007f10001ccaaab,0x0007f09e413c418a,0x0007f034627733d7,0x0007efc15815b8d5,
           0x0007ef43e2bf7f55,0x0007eeba84e31dfe,0x0007ee237294df89,0x0007ed7c7c170141,
           0x0007ecc2f0d95d3a,0x0007ebf377a46782,0x0007eb09d6deb285,0x0007ea00a4f17808,
           0x0007e8d0d3da63d6,0x0007e771023b0fcf,0x0007e5d46c2f08d8,0x0007e3e937669691,
           0x0007e195978f1176,0x0007deb2c0e05c1c,0x0007db0362002a19,0x0007d6202c151439,
           0x0007cf4b8f00a2cb,0x0007c4fd24520efd,0x0007b362fbf81816,0x00078d2d25998e24]
const wi =
    [1.7367254121602630e-15,9.5586603514556339e-17,1.2708704834810623e-16,
     1.4909740962495474e-16,1.6658733631586268e-16,1.8136120810119029e-16,
     1.9429720153135588e-16,2.0589500628482093e-16,2.1646860576895422e-16,
     2.2622940392218116e-16,2.3532718914045892e-16,2.4387234557428771e-16,
     2.5194879829274225e-16,2.5962199772528103e-16,2.6694407473648285e-16,
     2.7395729685142446e-16,2.8069646002484804e-16,2.8719058904113930e-16,
     2.9346417484728883e-16,2.9953809336782113e-16,3.0543030007192440e-16,
     3.1115636338921572e-16,3.1672988018581815e-16,3.2216280350549905e-16,
     3.2746570407939751e-16,3.3264798116841710e-16,3.3771803417353232e-16,
     3.4268340353119356e-16,3.4755088731729758e-16,3.5232663846002031e-16,
     3.5701624633953494e-16,3.6162480571598339e-16,3.6615697529653540e-16,
     3.7061702777236077e-16,3.7500889278747798e-16,3.7933619401549554e-16,
     3.8360228129677279e-16,3.8781025861250247e-16,3.9196300853257678e-16,
     3.9606321366256378e-16,4.0011337552546690e-16,4.0411583124143332e-16,
     4.0807276830960448e-16,4.1198623774807442e-16,4.1585816580828064e-16,
     4.1969036444740733e-16,4.2348454071520708e-16,4.2724230518899761e-16,
     4.3096517957162941e-16,4.3465460355128760e-16,4.3831194100854571e-16,
     4.4193848564470665e-16,4.4553546609579137e-16,4.4910405058828750e-16,
     4.5264535118571397e-16,4.5616042766900381e-16,4.5965029108849407e-16,
     4.6311590702081647e-16,4.6655819856008752e-16,4.6997804906941950e-16,
     4.7337630471583237e-16,4.7675377680908526e-16,4.8011124396270155e-16,
     4.8344945409350080e-16,4.8676912627422087e-16,4.9007095245229938e-16,
     4.9335559904654139e-16,4.9662370843221783e-16,4.9987590032409088e-16,
     5.0311277306593187e-16,5.0633490483427195e-16,5.0954285476338923e-16,
     5.1273716399787966e-16,5.1591835667857364e-16,5.1908694086703434e-16,
     5.2224340941340417e-16,5.2538824077194543e-16,5.2852189976823820e-16,
     5.3164483832166176e-16,5.3475749612647295e-16,5.3786030129452348e-16,
     5.4095367096239933e-16,5.4403801186554671e-16,5.4711372088173611e-16,
     5.5018118554603362e-16,5.5324078453927836e-16,5.5629288815190902e-16,
     5.5933785872484621e-16,5.6237605106900435e-16,5.6540781286489604e-16,
     5.6843348504368141e-16,5.7145340215092040e-16,5.7446789269419609e-16,
     5.7747727947569648e-16,5.8048187991076857e-16,5.8348200633338921e-16,
     5.8647796628943653e-16,5.8947006281858718e-16,5.9245859472561339e-16,
     5.9544385684180598e-16,5.9842614027720281e-16,6.0140573266426640e-16,
     6.0438291839361250e-16,6.0735797884236057e-16,6.1033119259564394e-16,
     6.1330283566179110e-16,6.1627318168165963e-16,6.1924250213258470e-16,
     6.2221106652737879e-16,6.2517914260879998e-16,6.2814699653988953e-16,
     6.3111489309056042e-16,6.3408309582080600e-16,6.3705186726088149e-16,
     6.4002146908880247e-16,6.4299216230548961e-16,6.4596420740788321e-16,
     6.4893786456033965e-16,6.5191339376461587e-16,6.5489105502874154e-16,
     6.5787110853507413e-16,6.6085381480782587e-16,6.6383943488035057e-16,
     6.6682823046247459e-16,6.6982046410815579e-16,6.7281639938375311e-16,
     6.7581630103719006e-16,6.7882043516829803e-16,6.8182906940062540e-16,
     6.8484247305500383e-16,6.8786091732516637e-16,6.9088467545571690e-16,
     6.9391402292275690e-16,6.9694923761748294e-16,6.9999060003307640e-16,
     7.0303839345521508e-16,7.0609290415654822e-16,7.0915442159548734e-16,
     7.1222323861967788e-16,7.1529965167453030e-16,7.1838396101720629e-16,
     7.2147647093647067e-16,7.2457748997883870e-16,7.2768733118146927e-16,
     7.3080631231227429e-16,7.3393475611774048e-16,7.3707299057898310e-16,
     7.4022134917657997e-16,7.4338017116476479e-16,7.4654980185558890e-16,
     7.4973059291369793e-16,7.5292290266240584e-16,7.5612709640179217e-16,
     7.5934354673958895e-16,7.6257263393567558e-16,7.6581474626104873e-16,
     7.6907028037219191e-16,7.7233964170182985e-16,7.7562324486711744e-16,
     7.7892151409638524e-16,7.8223488367564108e-16,7.8556379841610841e-16,
     7.8890871414417552e-16,7.9227009821522709e-16,7.9564843005293662e-16,
     7.9904420171571300e-16,8.0245791849212591e-16,8.0589009952726568e-16,
     8.0934127848215009e-16,8.1281200422845008e-16,8.1630284158098775e-16,
     8.1981437207065329e-16,8.2334719476060504e-16,8.2690192710884700e-16,
     8.3047920588053737e-16,8.3407968811366288e-16,8.3770405214202216e-16,
     8.4135299867980282e-16,8.4502725197240968e-16,8.4872756101861549e-16,
     8.5245470086955962e-16,8.5620947401062333e-16,8.5999271183276646e-16,
     8.6380527620052589e-16,8.6764806112455816e-16,8.7152199454736980e-16,
     8.7542804025171749e-16,8.7936719990210427e-16,8.8334051523084080e-16,
     8.8734907038131345e-16,8.9139399442240861e-16,8.9547646404950677e-16,
     8.9959770648910994e-16,9.0375900262601175e-16,9.0796169037400680e-16,
     9.1220716831348461e-16,9.1649689962191353e-16,9.2083241632623076e-16,
     9.2521532390956933e-16,9.2964730630864167e-16,9.3413013134252651e-16,
     9.3866565661866598e-16,9.4325583596767065e-16,9.4790272646517382e-16,
     9.5260849610662787e-16,9.5737543220974496e-16,9.6220595062948384e-16,
     9.6710260588230542e-16,9.7206810229016259e-16,9.7710530627072088e-16,
     9.8221725991905411e-16,9.8740719604806711e-16,9.9267855488079765e-16,
     9.9803500261836449e-16,1.0034804521436181e-15,1.0090190861637457e-15,
     1.0146553831467086e-15,1.0203941464683124e-15,1.0262405372613567e-15,
     1.0322001115486456e-15,1.0382788623515399e-15,1.0444832676000471e-15,
     1.0508203448355195e-15,1.0572977139009890e-15,1.0639236690676801e-15,
     1.0707072623632994e-15,1.0776584002668106e-15,1.0847879564403425e-15,
     1.0921079038149563e-15,1.0996314701785628e-15,1.1073733224935752e-15,
     1.1153497865853155e-15,1.1235791107110833e-15,1.1320817840164846e-15,
     1.1408809242582780e-15,1.1500027537839792e-15,1.1594771891449189e-15,
     1.1693385786910960e-15,1.1796266352955801e-15,1.1903876299282890e-15,
     1.2016759392543819e-15,1.2135560818666897e-15,1.2261054417450561e-15,
     1.2394179789163251e-15,1.2536093926602567e-15,1.2688244814255010e-15,
     1.2852479319096109e-15,1.3031206634689985e-15,1.3227655770195326e-15,
     1.3446300925011171e-15,1.3693606835128518e-15,1.3979436672775240e-15,
     1.4319989869661328e-15,1.4744848603597596e-15,1.5317872741611144e-15,
     1.6227698675312968e-15]
const fi =
    [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]

## Tables for exponential variates
const ke =
    UInt64[0x000e290a13924be3,0x0000000000000000,0x0009beadebce18bf,0x000c377ac71f9e08,
           0x000d4ddb99075857,0x000de893fb8ca23e,0x000e4a8e87c4328d,0x000e8dff16ae1cb9,
           0x000ebf2deab58c59,0x000ee49a6e8b9638,0x000f0204efd64ee4,0x000f19bdb8ea3c1b,
           0x000f2d458bbe5bd1,0x000f3da104b78236,0x000f4b86d784571f,0x000f577ad8a7784f,
           0x000f61de83da32ab,0x000f6afb7843cce7,0x000f730a57372b44,0x000f7a37651b0e68,
           0x000f80a5bb6eea52,0x000f867189d3cb5b,0x000f8bb1b4f8fbbd,0x000f9079062292b8,
           0x000f94d70ca8d43a,0x000f98d8c7dcaa99,0x000f9c8928abe083,0x000f9ff175b734a6,
           0x000fa319996bc47d,0x000fa6085f8e9d07,0x000fa8c3a62e1991,0x000fab5084e1f660,
           0x000fadb36c84cccb,0x000faff041086846,0x000fb20a6ea22bb9,0x000fb404fb42cb3c,
           0x000fb5e295158173,0x000fb7a59e99727a,0x000fb95038c8789d,0x000fbae44ba684eb,
           0x000fbc638d822e60,0x000fbdcf89209ffa,0x000fbf29a303cfc5,0x000fc0731df1089c,
           0x000fc1ad1ed6c8b1,0x000fc2d8b02b5c89,0x000fc3f6c4d92131,0x000fc5083ac9ba7d,
           0x000fc60ddd1e9cd6,0x000fc7086622e825,0x000fc7f881009f0b,0x000fc8decb41ac70,
           0x000fc9bbd623d7ec,0x000fca9027c5b26d,0x000fcb5c3c319c49,0x000fcc20864b4449,
           0x000fccdd70a35d40,0x000fcd935e34bf80,0x000fce42ab0db8bd,0x000fceebace7ec01,
           0x000fcf8eb3b0d0e7,0x000fd02c0a049b60,0x000fd0c3f59d199c,0x000fd156b7b5e27e,
           0x000fd1e48d670341,0x000fd26daff73551,0x000fd2f2552684be,0x000fd372af7233c1,
           0x000fd3eeee528f62,0x000fd4673e73543a,0x000fd4dbc9e72ff7,0x000fd54cb856dc2c,
           0x000fd5ba2f2c4119,0x000fd62451ba02c2,0x000fd68b415fcff4,0x000fd6ef1dabc160,
           0x000fd75004790eb6,0x000fd7ae120c583f,0x000fd809612dbd09,0x000fd8620b40effa,
           0x000fd8b8285b78fd,0x000fd90bcf594b1d,0x000fd95d15efd425,0x000fd9ac10bfa70c,
           0x000fd9f8d364df06,0x000fda437086566b,0x000fda8bf9e3c9fe,0x000fdad28062fed5,
           0x000fdb17141bff2c,0x000fdb59c4648085,0x000fdb9a9fda83cc,0x000fdbd9b46e3ed4,
           0x000fdc170f6b5d04,0x000fdc52bd81a3fb,0x000fdc8ccacd07ba,0x000fdcc542dd3902,
           0x000fdcfc30bcb793,0x000fdd319ef77143,0x000fdd6597a0f60b,0x000fdd98245a48a2,
           0x000fddc94e575271,0x000fddf91e64014f,0x000fde279ce914ca,0x000fde54d1f0a06a,
           0x000fde80c52a47cf,0x000fdeab7def394e,0x000fded50345eb35,0x000fdefd5be59fa0,
           0x000fdf248e39b26f,0x000fdf4aa064b4af,0x000fdf6f98435894,0x000fdf937b6f30ba,
           0x000fdfb64f414571,0x000fdfd818d48262,0x000fdff8dd07fed8,0x000fe018a08122c4,
           0x000fe03767adaa59,0x000fe05536c58a13,0x000fe07211ccb4c5,0x000fe08dfc94c532,
           0x000fe0a8fabe8ca1,0x000fe0c30fbb87a5,0x000fe0dc3ecf3a5a,0x000fe0f48b107521,
           0x000fe10bf76a82ef,0x000fe122869e41ff,0x000fe1383b4327e1,0x000fe14d17c83187,
           0x000fe1611e74c023,0x000fe1745169635a,0x000fe186b2a09176,0x000fe19843ef4e07,
           0x000fe1a90705bf63,0x000fe1b8fd6fb37c,0x000fe1c828951443,0x000fe1d689ba4bfd,
           0x000fe1e4220099a4,0x000fe1f0f26655a0,0x000fe1fcfbc726d4,0x000fe2083edc2830,
           0x000fe212bc3bfeb4,0x000fe21c745adfe3,0x000fe225678a8895,0x000fe22d95fa23f4,
           0x000fe234ffb62282,0x000fe23ba4a800d9,0x000fe2418495fddc,0x000fe2469f22bffb,
           0x000fe24af3cce90d,0x000fe24e81ee9858,0x000fe25148bcda19,0x000fe253474703fe,
           0x000fe2547c75fdc6,0x000fe254e70b754f,0x000fe25485a0fd1a,0x000fe25356a71450,
           0x000fe2515864173a,0x000fe24e88f316f1,0x000fe24ae64296fa,0x000fe2466e132f60,
           0x000fe2411df611bd,0x000fe23af34b6f73,0x000fe233eb40bf41,0x000fe22c02cee01b,
           0x000fe22336b81710,0x000fe2198385e5cc,0x000fe20ee586b707,0x000fe20358cb5dfb,
           0x000fe1f6d92465b1,0x000fe1e9621f2c9e,0x000fe1daef02c8da,0x000fe1cb7accb0a6,
           0x000fe1bb002d22c9,0x000fe1a9798349b8,0x000fe196e0d9140c,0x000fe1832fdebc44,
           0x000fe16e5fe5f931,0x000fe15869dccfcf,0x000fe1414647fe78,0x000fe128ed3cf8b2,
           0x000fe10f565b69cf,0x000fe0f478c633ab,0x000fe0d84b1bdd9e,0x000fe0bac36e6688,
           0x000fe09bd73a6b5b,0x000fe07b7b5d920a,0x000fe059a40c26d2,0x000fe03644c5d7f8,
           0x000fe011504979b2,0x000fdfeab887b95c,0x000fdfc26e94a447,0x000fdf986297e305,
           0x000fdf6c83bb8663,0x000fdf3ec0193eed,0x000fdf0f04a5d30a,0x000fdedd3d1aa204,
           0x000fdea953dcfc13,0x000fde7331e3100d,0x000fde3abe9626f2,0x000fddffdfb1dbd5,
           0x000fddc2791ff351,0x000fdd826cd068c6,0x000fdd3f9a8d3856,0x000fdcf9dfc95b0c,
           0x000fdcb1176a55fe,0x000fdc65198ba50b,0x000fdc15bb3b2daa,0x000fdbc2ce2dc4ae,
           0x000fdb6c206aaaca,0x000fdb117becb4a1,0x000fdab2a6379bf0,0x000fda4f5fdfb4e9,
           0x000fd9e76401f3a3,0x000fd97a67a9ce1f,0x000fd90819221429,0x000fd8901f2d4b02,
           0x000fd812182170e1,0x000fd78d98e23cd3,0x000fd7022bb3f082,0x000fd66f4edf96b9,
           0x000fd5d473200305,0x000fd530f9ccff94,0x000fd48432b7b351,0x000fd3cd59a8469e,
           0x000fd30b9368f90a,0x000fd23dea45f500,0x000fd16349e2e04a,0x000fd07a7a3ef98a,
           0x000fcf8219b5df05,0x000fce7895bcfcde,0x000fcd5c220ad5e2,0x000fcc2aadbc17dc,
           0x000fcae1d5e81fbc,0x000fc97ed4e778f9,0x000fc7fe6d4d720e,0x000fc65ccf39c2fc,
           0x000fc4957623cb03,0x000fc2a2fc826dc7,0x000fc07ee19b01cd,0x000fbe213c1cf493,
           0x000fbb8051ac1566,0x000fb890078d120e,0x000fb5411a5b9a95,0x000fb18000547133,
           0x000fad334827f1e2,0x000fa839276708b9,0x000fa263b32e37ed,0x000f9b72d1c52cd1,
           0x000f930a1a281a05,0x000f889f023d820a,0x000f7b577d2be5f3,0x000f69c650c40a8f,
           0x000f51530f0916d8,0x000f2cb0e3c5933e,0x000eeefb15d605d8,0x000e6da6ecf27460]

const we =
    [1.9311480126418366e-15,1.4178028487910829e-17,2.3278824993382448e-17,
     3.0487830247064320e-17,3.6665697714474878e-17,4.2179302189289733e-17,
     4.7222561556862764e-17,5.1911915446217879e-17,5.6323471083955047e-17,
     6.0510082606427647e-17,6.4510165096727506e-17,6.8352646803700541e-17,
     7.2059939574689050e-17,7.5649815537392981e-17,7.9136643961951065e-17,
     8.2532235563518929e-17,8.5846436168850513e-17,8.9087554865647428e-17,
     9.2262679629663719e-17,9.5377914505292719e-17,9.8438560874559257e-17,
     1.0144925809006294e-16,1.0441409405585343e-16,1.0733669323436384e-16,
     1.1022028745670189e-16,1.1306777346479334e-16,1.1588176009705533e-16,
     1.1866460730417886e-16,1.2141845865694359e-16,1.2414526862326387e-16,
     1.2684682560606153e-16,1.2952477151912284e-16,1.3218061851538810e-16,
     1.3481576335745444e-16,1.3743149982367625e-16,1.4002902946807859e-16,
     1.4260947099321287e-16,1.4517386844829297e-16,1.4772319842763584e-16,
     1.5025837641447456e-16,1.5278026239101652e-16,1.5528966581595696e-16,
     1.5778735005459581e-16,1.6027403633350909e-16,1.6275040728083524e-16,
     1.6521711010420076e-16,1.6767475945078279e-16,1.7012393998770646e-16,
     1.7256520873568226e-16,1.7499909718432365e-16,1.7742611321380505e-16,
     1.7984674284430714e-16,1.8226145183195818e-16,1.8467068712763576e-16,
     1.8707487821298258e-16,1.8947443832625899e-16,1.9186976558915995e-16,
     1.9426124404443042e-16,1.9664924461299023e-16,1.9903412597830144e-16,
     2.0141623540485899e-16,2.0379590949693882e-16,2.0617347490308439e-16,
     2.0854924897123771e-16,2.1092354035891528e-16,2.1329664960238294e-16,
     2.1566886964838970e-16,2.1804048635167009e-16,2.2041177894111562e-16,
     2.2278302045723950e-16,2.2515447816331350e-16,2.2752641393233694e-16,
     2.2989908461180186e-16,2.3227274236804366e-16,2.3464763501180916e-16,
     2.3702400630653389e-16,2.3940209626069303e-16,2.4178214140547710e-16,
     2.4416437505894123e-16,2.4654902757768304e-16,2.4893632659702250e-16,
     2.5132649726057970e-16,2.5371976244007951e-16,2.5611634294614988e-16,
     2.5851645773082391e-16,2.6092032408240577e-16,2.6332815781331452e-16,
     2.6574017344147618e-16,2.6815658436579989e-16,2.7057760303623509e-16,
     2.7300344111887955e-16,2.7543430965657619e-16,2.7787041922541278e-16,
     2.8031198008751431e-16,2.8275920234049704e-16,2.8521229606393309e-16,
     2.8767147146315804e-16,2.9013693901073754e-16,2.9260890958589514e-16,
     2.9508759461219033e-16,2.9757320619372521e-16,3.0006595725014739e-16,
     3.0256606165070789e-16,3.0507373434762511e-16,3.0758919150899939e-16,
     3.1011265065151543e-16,3.1264433077316750e-16,3.1518445248623523e-16,
     3.1773323815073683e-16,3.2029091200858335e-16,3.2285770031865573e-16,
     3.2543383149302610e-16,3.2801953623454359e-16,3.3061504767600738e-16,
     3.3322060152114841e-16,3.3583643618764577e-16,3.3846279295240445e-16,
     3.4109991609932597e-16,3.4374805306980633e-16,3.4640745461620167e-16,
     3.4907837495850680e-16,3.5176107194449828e-16,3.5445580721360130e-16,
     3.5716284636474652e-16,3.5988245912849274e-16,3.6261491954370031e-16,
     3.6536050613905045e-16,3.6811950211971757e-16,3.7089219555951389e-16,
     3.7367887959883854e-16,3.7647985264877841e-16,3.7929541860172334e-16,
     3.8212588704887531e-16,3.8497157350504876e-16,3.8783279964117988e-16,
     3.9070989352498183e-16,3.9360318987020748e-16,3.9651303029500381e-16,
     3.9943976358986842e-16,4.0238374599574693e-16,4.0534534149283966e-16,
     4.0832492210071775e-16,4.1132286819038357e-16,4.1433956880894741e-16,
     4.1737542201763194e-16,4.2043083524385856e-16,4.2350622564821518e-16,
     4.2660202050715582e-16,4.2971865761233266e-16,4.3285658568752094e-16,
     4.3601626482415681e-16,4.3919816693657415e-16,4.4240277623809919e-16,
     4.4563058973923611e-16,4.4888211776926172e-16,4.5215788452263475e-16,
     4.5545842863172421e-16,4.5878430376746227e-16,4.6213607926964266e-16,
     4.6551434080870692e-16,4.6891969108099157e-16,4.7235275053955480e-16,
     4.7581415816285534e-16,4.7930457226372470e-16,4.8282467134125866e-16,
     4.8637515497845119e-16,4.8995674478861404e-16,4.9357018541385775e-16,
     4.9721624557917034e-16,5.0089571920591141e-16,5.0460942658884340e-16,
     5.0835821564116245e-16,5.1214296321235415e-16,5.1596457648410618e-16,
     5.1982399444994938e-16,5.2372218948478484e-16,5.2766016901098856e-16,
     5.3163897726836902e-16,5.3565969719590503e-16,5.3972345243389779e-16,
     5.4383140945596370e-16,5.4798477984116296e-16,5.5218482269752343e-16,
     5.5643284724928722e-16,5.6073021560139669e-16,5.6507834569605064e-16,
     5.6947871447763482e-16,5.7393286128396354e-16,5.7844239148359912e-16,
     5.8300898038105864e-16,5.8763437741400573e-16,5.9232041066909314e-16,
     5.9706899174600906e-16,6.0188212100252363e-16,6.0676189321700068e-16,
     6.1171050370897217e-16,6.1673025496306200e-16,6.2182356380685327e-16,
     6.2699296919933262e-16,6.3224114069342115e-16,6.3757088764394262e-16,
     6.4298516924135947e-16,6.4848710546189033e-16,6.5407998903644809e-16,
     6.5976729855445663e-16,6.6555271283433428e-16,6.7144012671064882e-16,
     6.7743366840910103e-16,6.8353771870512740e-16,6.8975693209068478e-16,
     6.9609626020748846e-16,7.0256097784459588e-16,7.0915671184495837e-16,
     7.1588947332085531e-16,7.2276569364381212e-16,7.2979226475290851e-16,
     7.3697658441912426e-16,7.4432660721604146e-16,7.5185090208325131e-16,
     7.5955871753377488e-16,7.6746005575784274e-16,7.7556575712157906e-16,
     7.8388759686228577e-16,7.9243839615735500e-16,8.0123215021130834e-16,
     8.1028417659131464e-16,8.1961128778061250e-16,8.2923199285818092e-16,
     8.3916673441467979e-16,8.4943816836487701e-16,8.6007149633349414e-16,
     8.7109486293879040e-16,8.8253983380721398e-16,8.9444197485198646e-16,
     9.0684155971316690e-16,9.1978444098118649e-16,9.3332313294229516e-16,
     9.4751817065249841e-16,9.6243983456584759e-16,9.7817036547844198e-16,
     9.9480684723838795e-16,1.0124650144288319e-15,1.0312843657756166e-15,
     1.0514351604044550e-15,1.0731281954224043e-15,1.0966288068517408e-15,
     1.1222774909350319e-15,1.1505212963006663e-15,1.1819635283304206e-15,
     1.2174462832361815e-15,1.2581958069755114e-15,1.3060984107128082e-15,
     1.3642786158057857e-15,1.4384889932178723e-15,1.5412190700064194e-15,
     1.7091034077168055e-15]
const fe =
    [1.0000000000000000e+00,9.3814368086217470e-01,9.0046992992574648e-01,
     8.7170433238120359e-01,8.4778550062398961e-01,8.2699329664305032e-01,
     8.0842165152300838e-01,7.9152763697249562e-01,7.7595685204011555e-01,
     7.6146338884989628e-01,7.4786862198519510e-01,7.3503809243142348e-01,
     7.2286765959357202e-01,7.1127476080507601e-01,7.0019265508278816e-01,
     6.8956649611707799e-01,6.7935057226476536e-01,6.6950631673192473e-01,
     6.6000084107899970e-01,6.5080583341457110e-01,6.4189671642726609e-01,
     6.3325199421436607e-01,6.2485273870366598e-01,6.1668218091520766e-01,
     6.0872538207962201e-01,6.0096896636523223e-01,5.9340090169173343e-01,
     5.8601031847726803e-01,5.7878735860284503e-01,5.7172304866482582e-01,
     5.6480919291240017e-01,5.5803828226258745e-01,5.5140341654064129e-01,
     5.4489823767243961e-01,5.3851687200286191e-01,5.3225388026304332e-01,
     5.2610421398361973e-01,5.2006317736823360e-01,5.1412639381474856e-01,
     5.0828977641064288e-01,5.0254950184134772e-01,4.9690198724154955e-01,
     4.9134386959403253e-01,4.8587198734188491e-01,4.8048336393045421e-01,
     4.7517519303737737e-01,4.6994482528395998e-01,4.6478975625042618e-01,
     4.5970761564213769e-01,4.5469615747461550e-01,4.4975325116275500e-01,
     4.4487687341454851e-01,4.4006510084235390e-01,4.3531610321563657e-01,
     4.3062813728845883e-01,4.2599954114303434e-01,4.2142872899761658e-01,
     4.1691418643300288e-01,4.1245446599716118e-01,4.0804818315203240e-01,
     4.0369401253053028e-01,3.9939068447523107e-01,3.9513698183329016e-01,
     3.9093173698479711e-01,3.8677382908413765e-01,3.8266218149600983e-01,
     3.7859575940958079e-01,3.7457356761590216e-01,3.7059464843514600e-01,
     3.6665807978151416e-01,3.6276297335481777e-01,3.5890847294874978e-01,
     3.5509375286678746e-01,3.5131801643748334e-01,3.4758049462163698e-01,
     3.4388044470450241e-01,3.4021714906678002e-01,3.3658991402867761e-01,
     3.3299806876180899e-01,3.2944096426413633e-01,3.2591797239355619e-01,
     3.2242848495608917e-01,3.1897191284495724e-01,3.1554768522712895e-01,
     3.1215524877417955e-01,3.0879406693456019e-01,3.0546361924459026e-01,
     3.0216340067569353e-01,2.9889292101558179e-01,2.9565170428126120e-01,
     2.9243928816189257e-01,2.8925522348967775e-01,2.8609907373707683e-01,
     2.8297041453878075e-01,2.7986883323697292e-01,2.7679392844851736e-01,
     2.7374530965280297e-01,2.7072259679906002e-01,2.6772541993204479e-01,
     2.6475341883506220e-01,2.6180624268936298e-01,2.5888354974901623e-01,
     2.5598500703041538e-01,2.5311029001562946e-01,2.5025908236886230e-01,
     2.4743107566532763e-01,2.4462596913189211e-01,2.4184346939887721e-01,
     2.3908329026244918e-01,2.3634515245705964e-01,2.3362878343743335e-01,
     2.3093391716962741e-01,2.2826029393071670e-01,2.2560766011668407e-01,
     2.2297576805812019e-01,2.2036437584335949e-01,2.1777324714870053e-01,
     2.1520215107537868e-01,2.1265086199297828e-01,2.1011915938898826e-01,
     2.0760682772422204e-01,2.0511365629383771e-01,2.0263943909370902e-01,
     2.0018397469191127e-01,1.9774706610509887e-01,1.9532852067956322e-01,
     1.9292814997677135e-01,1.9054576966319539e-01,1.8818119940425432e-01,
     1.8583426276219711e-01,1.8350478709776746e-01,1.8119260347549629e-01,
     1.7889754657247831e-01,1.7661945459049488e-01,1.7435816917135349e-01,
     1.7211353531532006e-01,1.6988540130252766e-01,1.6767361861725019e-01,
     1.6547804187493600e-01,1.6329852875190182e-01,1.6113493991759203e-01,
     1.5898713896931421e-01,1.5685499236936523e-01,1.5473836938446808e-01,
     1.5263714202744286e-01,1.5055118500103989e-01,1.4848037564386679e-01,
     1.4642459387834494e-01,1.4438372216063478e-01,1.4235764543247220e-01,
     1.4034625107486245e-01,1.3834942886358020e-01,1.3636707092642886e-01,
     1.3439907170221363e-01,1.3244532790138752e-01,1.3050573846833077e-01,
     1.2858020454522817e-01,1.2666862943751067e-01,1.2477091858083096e-01,
     1.2288697950954514e-01,1.2101672182667483e-01,1.1916005717532768e-01,
     1.1731689921155557e-01,1.1548716357863353e-01,1.1367076788274431e-01,
     1.1186763167005630e-01,1.1007767640518538e-01,1.0830082545103380e-01,
     1.0653700405000166e-01,1.0478613930657017e-01,1.0304816017125772e-01,
     1.0132299742595363e-01,9.9610583670637132e-02,9.7910853311492199e-02,
     9.6223742550432798e-02,9.4549189376055859e-02,9.2887133556043541e-02,
     9.1237516631040155e-02,8.9600281910032858e-02,8.7975374467270218e-02,
     8.6362741140756913e-02,8.4762330532368119e-02,8.3174093009632383e-02,
     8.1597980709237419e-02,8.0033947542319905e-02,7.8481949201606421e-02,
     7.6941943170480503e-02,7.5413888734058410e-02,7.3897746992364746e-02,
     7.2393480875708738e-02,7.0901055162371829e-02,6.9420436498728755e-02,
     6.7951593421936601e-02,6.6494496385339774e-02,6.5049117786753749e-02,
     6.3615431999807334e-02,6.2193415408540995e-02,6.0783046445479633e-02,
     5.9384305633420266e-02,5.7997175631200659e-02,5.6621641283742877e-02,
     5.5257689676697037e-02,5.3905310196046087e-02,5.2564494593071692e-02,
     5.1235237055126281e-02,4.9917534282706372e-02,4.8611385573379497e-02,
     4.7316792913181548e-02,4.6033761076175170e-02,4.4762297732943282e-02,
     4.3502413568888183e-02,4.2254122413316234e-02,4.1017441380414819e-02,
     3.9792391023374125e-02,3.8578995503074857e-02,3.7377282772959361e-02,
     3.6187284781931423e-02,3.5009037697397410e-02,3.3842582150874330e-02,
     3.2687963508959535e-02,3.1545232172893609e-02,3.0414443910466604e-02,
     2.9295660224637393e-02,2.8188948763978636e-02,2.7094383780955800e-02,
     2.6012046645134217e-02,2.4942026419731783e-02,2.3884420511558171e-02,
     2.2839335406385240e-02,2.1806887504283581e-02,2.0787204072578117e-02,
     1.9780424338009743e-02,1.8786700744696030e-02,1.7806200410911362e-02,
     1.6839106826039948e-02,1.5885621839973163e-02,1.4945968011691148e-02,
     1.4020391403181938e-02,1.3109164931254991e-02,1.2212592426255381e-02,
     1.1331013597834597e-02,1.0464810181029979e-02,9.6144136425022099e-03,
     8.7803149858089753e-03,7.9630774380170400e-03,7.1633531836349839e-03,
     6.3819059373191791e-03,5.6196422072054830e-03,4.8776559835423923e-03,
     4.1572951208337953e-03,3.4602647778369040e-03,2.7887987935740761e-03,
     2.1459677437189063e-03,1.5362997803015724e-03,9.6726928232717454e-04,
     4.5413435384149677e-04]


const ziggurat_nor_r      = 3.6541528853610087963519472518
const ziggurat_nor_inv_r  = inv(ziggurat_nor_r)
const ziggurat_exp_r      = 7.6971174701310497140446280481

"""
    randn([rng=GLOBAL_RNG], [T=Float64], [dims...])

Generate a normally-distributed random number of type `T` with mean 0 and standard deviation 1.
Optionally generate an array of normally-distributed random numbers.
The `Base` module currently provides an implementation for the types
[`Float16`](@ref), [`Float32`](@ref), and [`Float64`](@ref) (the default), and their
[`Complex`](@ref) counterparts. When the type argument is complex, the values are drawn
from the circularly symmetric complex normal distribution.
"""
@inline function randn(rng::AbstractRNG=GLOBAL_RNG)
    @inbounds begin
        r = rand_ui52(rng)
        rabs = Int64(r>>1) # One bit for the sign
        idx = rabs & 0xFF
        x = ifelse(r % Bool, -rabs, rabs)*wi[idx+1]
        rabs < ki[idx+1] && return x # 99.3% of the time we return here 1st try
        return randn_unlikely(rng, idx, rabs, x)
    end
end

# this unlikely branch is put in a separate function for better efficiency
function randn_unlikely(rng, idx, rabs, x)
    @inbounds if idx == 0
        while true
            xx = -ziggurat_nor_inv_r*log(rand(rng))
            yy = -log(rand(rng))
            yy+yy > xx*xx && return (rabs >> 8) % Bool ? -ziggurat_nor_r-xx : ziggurat_nor_r+xx
        end
    elseif (fi[idx] - fi[idx+1])*rand(rng) + fi[idx+1] < exp(-0.5*x*x)
        return x # return from the triangular area
    else
        return randn(rng)
    end
end

"""
    randexp([rng=GLOBAL_RNG], [T=Float64], [dims...])

Generate a random number of type `T` according to the exponential distribution with scale 1.
Optionally generate an array of such random numbers.
The `Base` module currently provides an implementation for the types
[`Float16`](@ref), [`Float32`](@ref), and [`Float64`](@ref) (the default).
"""
@inline function randexp(rng::AbstractRNG=GLOBAL_RNG)
    @inbounds begin
        ri = rand_ui52(rng)
        idx = ri & 0xFF
        x = ri*we[idx+1]
        ri < ke[idx+1] && return x # 98.9% of the time we return here 1st try
        return randexp_unlikely(rng, idx, x)
    end
end

function randexp_unlikely(rng, idx, x)
    @inbounds if idx == 0
        return ziggurat_exp_r - log(rand(rng))
    elseif (fe[idx] - fe[idx+1])*rand(rng) + fe[idx+1] < exp(-x)
        return x # return from the triangular area
    else
        return randexp(rng)
    end
end

"""
    randn!([rng=GLOBAL_RNG], A::AbstractArray) -> A

Fill the array `A` with normally-distributed (mean 0, standard deviation 1) random numbers.
Also see the [`rand`](@ref) function.
"""
function randn! end

"""
    randexp!([rng=GLOBAL_RNG], A::AbstractArray) -> A

Fill the array `A` with random numbers following the exponential distribution (with scale 1).
"""
function randexp! end

let Floats = Union{Float16,Float32,Float64}
    for randfun in [:randn, :randexp]
        randfun! = Symbol(randfun, :!)
        @eval begin
            # scalars
            $randfun(rng::AbstractRNG, ::Type{T}) where {T<:$Floats} = convert(T, $randfun(rng))
            $randfun(::Type{T}) where {T} = $randfun(GLOBAL_RNG, T)

            # filling arrays
            function $randfun!(rng::AbstractRNG, A::AbstractArray{T}) where T
                for i in eachindex(A)
                    @inbounds A[i] = $randfun(rng, T)
                end
                A
            end

            $randfun!(A::AbstractArray) = $randfun!(GLOBAL_RNG, A)

            # generating arrays
            $randfun(rng::AbstractRNG, ::Type{T}, dims::Dims                     ) where {T} = $randfun!(rng, Array{T}(dims))
            # Note that this method explicitly does not define $randfun(rng, T), in order to prevent an infinite recursion.
            $randfun(rng::AbstractRNG, ::Type{T}, dim1::Integer, dims::Integer...) where {T} = $randfun!(rng, Array{T}(dim1, dims...))
            $randfun(                  ::Type{T}, dims::Dims                     ) where {T} = $randfun(GLOBAL_RNG, T, dims)
            $randfun(                  ::Type{T}, dims::Integer...               ) where {T} = $randfun(GLOBAL_RNG, T, dims...)
            $randfun(rng::AbstractRNG,            dims::Dims                     )           = $randfun(rng, Float64, dims)
            $randfun(rng::AbstractRNG,            dims::Integer...               )           = $randfun(rng, Float64, dims...)
            $randfun(                             dims::Dims                     )           = $randfun(GLOBAL_RNG, Float64, dims)
            $randfun(                             dims::Integer...               )           = $randfun(GLOBAL_RNG, Float64, dims...)
        end
    end
end

# complex randn
Base.@irrational SQRT_HALF 0.7071067811865475244008  sqrt(big(0.5))
randn(rng::AbstractRNG, ::Type{Complex{T}}) where {T <: AbstractFloat} =
    Complex{T}(SQRT_HALF * randn(rng, T), SQRT_HALF * randn(rng, T))

## random UUID generation

struct UUID
    value::UInt128

    UUID(u::UInt128) = new(u)
end

"""
    uuid1([rng::AbstractRNG=GLOBAL_RNG]) -> UUID

Generates a version 1 (time-based) universally unique identifier (UUID), as specified
by RFC 4122. Note that the Node ID is randomly generated (does not identify the host)
according to section 4.5 of the RFC.
"""
function uuid1(rng::AbstractRNG=GLOBAL_RNG)
    u = rand(rng, UInt128)

    # mask off clock sequence and node
    u &= 0x00000000000000003fffffffffffffff

    # set the unicast/multicast bit and version
    u |= 0x00000000000010000000010000000000

    # 0x01b21dd213814000 is the number of 100 nanosecond intervals
    # between the UUID epoch and Unix epoch
    timestamp = round(UInt64, time() * 1e7) + 0x01b21dd213814000
    ts_low = timestamp & typemax(UInt32)
    ts_mid = (timestamp >> 32) & typemax(UInt16)
    ts_hi = (timestamp >> 48) & 0x0fff

    u |= UInt128(ts_low) << 96
    u |= UInt128(ts_mid) << 80
    u |= UInt128(ts_hi) << 64

    UUID(u)
end

"""
    uuid4([rng::AbstractRNG=GLOBAL_RNG]) -> UUID

Generates a version 4 (random or pseudo-random) universally unique identifier (UUID),
as specified by RFC 4122.
"""
function uuid4(rng::AbstractRNG=GLOBAL_RNG)
    u = rand(rng, UInt128)
    u &= 0xffffffffffff0fff3fffffffffffffff
    u |= 0x00000000000040008000000000000000
    UUID(u)
end

"""
    uuid_version(u::UUID) -> Integer

Inspects the given UUID and returns its version (see RFC 4122).
"""
function uuid_version(u::UUID)
    Int((u.value >> 76) & 0xf)
end

Base.convert(::Type{UInt128}, u::UUID) = u.value

function Base.convert(::Type{UUID}, s::AbstractString)
    s = lowercase(s)

    if !ismatch(r"^[0-9a-f]{8}(?:-[0-9a-f]{4}){3}-[0-9a-f]{12}$", s)
        throw(ArgumentError("Malformed UUID string"))
    end

    u = UInt128(0)
    for i in [1:8; 10:13; 15:18; 20:23; 25:36]
        u <<= 4
        d = s[i]-'0'
        u |= 0xf & (d-39*(d>9))
    end
    return UUID(u)
end

function Base.repr(u::UUID)
    u = u.value
    a = Vector{UInt8}(36)
    for i = [36:-1:25; 23:-1:20; 18:-1:15; 13:-1:10; 8:-1:1]
        d = u & 0xf
        a[i] = '0'+d+39*(d>9)
        u >>= 4
    end
    a[[24,19,14,9]] = '-'

    return String(a)
end

Base.show(io::IO, u::UUID) = write(io, Base.repr(u))

# return a random string (often useful for temporary filenames/dirnames)
let b = UInt8['0':'9';'A':'Z';'a':'z']
    global randstring
    randstring(r::AbstractRNG, n::Int) = String(b[rand(r, 1:length(b), n)])
    randstring(r::AbstractRNG) = randstring(r,8)
    randstring(n::Int) = randstring(GLOBAL_RNG, n)
    randstring() = randstring(GLOBAL_RNG)
end


# Fill S (resized as needed) with a random subsequence of A, where
# each element of A is included in S with independent probability p.
# (Note that this is different from the problem of finding a random
#  size-m subset of A where m is fixed!)
function randsubseq!(r::AbstractRNG, S::AbstractArray, A::AbstractArray, p::Real)
    0 <= p <= 1 || throw(ArgumentError("probability $p not in [0,1]"))
    n = length(A)
    p == 1 && return copy!(resize!(S, n), A)
    empty!(S)
    p == 0 && return S
    nexpected = p * length(A)
    sizehint!(S, round(Int,nexpected + 5*sqrt(nexpected)))
    if p > 0.15 # empirical threshold for trivial O(n) algorithm to be better
        for i = 1:n
            rand(r) <= p && push!(S, A[i])
        end
    else
        # Skip through A, in order, from each element i to the next element i+s
        # included in S. The probability that the next included element is
        # s==k (k > 0) is (1-p)^(k-1) * p, and hence the probability (CDF) that
        # s is in {1,...,k} is 1-(1-p)^k = F(k).   Thus, we can draw the skip s
        # from this probability distribution via the discrete inverse-transform
        # method: s = ceil(F^{-1}(u)) where u = rand(), which is simply
        # s = ceil(log(rand()) / log1p(-p)).
        # -log(rand()) is an exponential variate, so can use randexp().
        L = -1 / log1p(-p) # L > 0
        i = 0
        while true
            s = randexp(r) * L
            s >= n - i && return S # compare before ceil to avoid overflow
            push!(S, A[i += ceil(Int,s)])
        end
        # [This algorithm is similar in spirit to, but much simpler than,
        #  the one by Vitter for a related problem in "Faster methods for
        #  random sampling," Comm. ACM Magazine 7, 703-718 (1984).]
    end
    return S
end
randsubseq!(S::AbstractArray, A::AbstractArray, p::Real) = randsubseq!(GLOBAL_RNG, S, A, p)

randsubseq(r::AbstractRNG, A::AbstractArray{T}, p::Real) where {T} = randsubseq!(r, T[], A, p)

"""
    randsubseq(A, p) -> Vector

Return a vector consisting of a random subsequence of the given array `A`, where each
element of `A` is included (in order) with independent probability `p`. (Complexity is
linear in `p*length(A)`, so this function is efficient even if `p` is small and `A` is
large.) Technically, this process is known as "Bernoulli sampling" of `A`.
"""
randsubseq(A::AbstractArray, p::Real) = randsubseq(GLOBAL_RNG, A, p)

"Return a random `Int` (masked with `mask`) in ``[0, n)``, when `n <= 2^52`."
@inline function rand_lt(r::AbstractRNG, n::Int, mask::Int=nextpow2(n)-1)
    # this duplicates the functionality of RangeGenerator objects,
    # to optimize this special case
    while true
        x = (rand_ui52_raw(r) % Int) & mask
        x < n && return x
    end
end

"""
    shuffle!([rng=GLOBAL_RNG,] v)

In-place version of [`shuffle`](@ref): randomly permute the array `v` in-place,
optionally supplying the random-number generator `rng`.
"""
function shuffle!(r::AbstractRNG, a::AbstractVector)
    n = length(a)
    @assert n <= Int64(2)^52
    mask = nextpow2(n) - 1
    for i = n:-1:2
        (mask >> 1) == i && (mask >>= 1)
        j = 1 + rand_lt(r, i, mask)
        a[i], a[j] = a[j], a[i]
    end
    return a
end

shuffle!(a::AbstractVector) = shuffle!(GLOBAL_RNG, a)

"""
    shuffle([rng=GLOBAL_RNG,] v)

Return a randomly permuted copy of `v`. The optional `rng` argument specifies a random
number generator (see [Random Numbers](@ref)).
To permute `v` in-place, see [`shuffle!`](@ref).  To obtain randomly permuted
indices, see [`randperm`](@ref).
"""
shuffle(r::AbstractRNG, a::AbstractVector) = shuffle!(r, copymutable(a))
shuffle(a::AbstractVector) = shuffle(GLOBAL_RNG, a)

"""
    randperm([rng=GLOBAL_RNG,] n::Integer)

Construct a random permutation of length `n`. The optional `rng` argument specifies a random
number generator (see [Random Numbers](@ref)).
To randomly permute a arbitrary vector, see [`shuffle`](@ref)
or [`shuffle!`](@ref).
"""
function randperm(r::AbstractRNG, n::Integer)
    a = Vector{typeof(n)}(n)
    @assert n <= Int64(2)^52
    if n == 0
       return a
    end
    a[1] = 1
    mask = 3
    @inbounds for i = 2:Int(n)
        j = 1 + rand_lt(r, i, mask)
        if i != j # a[i] is uninitialized (and could be #undef)
            a[i] = a[j]
        end
        a[j] = i
        i == 1+mask && (mask = 2mask + 1)
    end
    return a
end
randperm(n::Integer) = randperm(GLOBAL_RNG, n)

"""
    randcycle([rng=GLOBAL_RNG,] n::Integer)

Construct a random cyclic permutation of length `n`. The optional `rng`
argument specifies a random number generator, see [Random Numbers](@ref).
"""
function randcycle(r::AbstractRNG, n::Integer)
    a = Vector{typeof(n)}(n)
    n == 0 && return a
    @assert n <= Int64(2)^52
    a[1] = 1
    mask = 3
    @inbounds for i = 2:Int(n)
        j = 1 + rand_lt(r, i-1, mask)
        a[i] = a[j]
        a[j] = i
        i == 1+mask && (mask = 2mask + 1)
    end
    return a
end
randcycle(n::Integer) = randcycle(GLOBAL_RNG, n)

end # module