opensimplex2_noise.jl

abstract type Orientation end
struct OrientStandard <: Orientation end
struct OrientX <: Orientation end
struct OrientXY <: Orientation end
struct OrientXZ <: Orientation end
struct OrientXYZ <: Orientation end

struct OpenSimplex2{N,O<:Orientation} <: NoiseSampler{N}
    random_state::RandomState
    simplex_state::SimplexState
    table::Vector{Float64}
end

@inline function _opensimplex2(dims, seed, table_size, gradients, orientation, smooth)
    rs = RandomState(seed)
    orientation = os2_orientation_type(Val(orientation))
    table = Iterators.take(Iterators.cycle(gradients), table_size) |> collect
    T = OpenSimplex2{dims,orientation}
    T(rs, SimplexState(T, Val(smooth)), table)
end

@inline os2_orientation_type(::Val{nothing}) = OrientStandard
@inline os2_orientation_type(::Val{:x}) = OrientX
@inline os2_orientation_type(::Val{:xy}) = OrientXY
@inline os2_orientation_type(::Val{:xz}) = OrientXZ
@inline os2_orientation_type(::Val{:xyz}) = OrientXYZ

SimplexState(::Type{<:OpenSimplex2{2}}, ::Val) = SimplexState(0.5, 1.0)
SimplexState(::Type{<:OpenSimplex2{3}}, ::Val{true}) = SimplexState(0.5, 2.6142672496001165)
SimplexState(::Type{<:OpenSimplex2{3}}, ::Val{false}) = SimplexState(0.6, 1.0)
SimplexState(::Type{<:OpenSimplex2{4}}, ::Val{true}) = SimplexState(0.5, 2.323520023997645)
SimplexState(::Type{<:OpenSimplex2{4}}, ::Val{false}) = SimplexState(0.6, 1.0)

# 2D

const OS2_SKEW_2D = 0.366025403784439
const OS2_UNSKEW_2D = -0.21132486540518713
const OS2_NUM_GRADIENTS_EXP_2D = 7
const OS2_NUM_GRADIENTS_2D = 1 << OS2_NUM_GRADIENTS_EXP_2D
const OS2_GRADIENTS_NORMALIZED_2D = [
    0.38268343236509, 0.923879532511287, 0.923879532511287, 0.38268343236509,
    0.923879532511287, -0.38268343236509, 0.38268343236509, -0.923879532511287,
    -0.38268343236509, -0.923879532511287, -0.923879532511287, -0.38268343236509,
    -0.923879532511287, 0.38268343236509, -0.38268343236509, 0.923879532511287,
    0.130526192220052, 0.99144486137381, 0.608761429008721, 0.793353340291235,
    0.793353340291235, 0.608761429008721, 0.99144486137381, 0.130526192220051,
    0.99144486137381, -0.130526192220051, 0.793353340291235, -0.60876142900872,
    0.608761429008721, -0.793353340291235, 0.130526192220052, -0.99144486137381,
    -0.130526192220052, -0.99144486137381, -0.608761429008721, -0.793353340291235,
    -0.793353340291235, -0.608761429008721, -0.99144486137381, -0.130526192220052,
    -0.99144486137381, 0.130526192220051, -0.793353340291235, 0.608761429008721,
    -0.608761429008721, 0.793353340291235, -0.130526192220052, 0.99144486137381]

"""
    opensimplex2_2d(; seed=nothing, orient=nothing)

Construct a sampler that outputs 2-dimensional OpenSimplex2 noise when it is sampled from.

# Arguments

  - `seed`: An unsigned integer used to seed the random number generator for this sampler, or
    `nothing` for non-deterministic results.

  - `orient`: Either the symbol `:x` or the value `nothing`:

      + `nothing`: Use the standard orientation.
      + `:x`: The noise space will be re-oriented with the Y axis pointing down the main diagonal to
        improve visual isotropy.
"""
function opensimplex2_2d(; seed=nothing, orient=nothing)
    size = OS2_NUM_GRADIENTS_2D * 2
    gradients = OS2_GRADIENTS_NORMALIZED_2D ./ 0.01001634121365712
    _opensimplex2(2, seed, size, gradients, orient, false)
end

@inline orient(::Type{OpenSimplex2{2,OrientStandard}}, x, y) = (x, y) .+ OS2_SKEW_2D .* (x + y)

@inline function orient(::Type{OpenSimplex2{2,OrientX}}, x, y)
    xx = x * ROOT_2_OVER_2
    yy = y * ROOT_2_OVER_2 * (2OS2_SKEW_2D + 1)
    (yy + xx, yy - xx)
end

@inline function grad(table, seed, X, Y, x, y)
    hash = (seed ⊻ X ⊻ Y) * HASH_MULTIPLIER
    hash ⊻= hash >> (64 - OS2_NUM_GRADIENTS_EXP_2D + 1)
    i = trunc(hash) & ((OS2_NUM_GRADIENTS_2D - 1) << 1)
    @inbounds t = (table[i+1], table[(i|1)+1])
    sum((t .* (x, y)))
end

function sample(sampler::S, x::T, y::T) where {O,S<:OpenSimplex2{2,O},T<:Real}
    seed = sampler.random_state.seed
    table = sampler.table
    state = sampler.simplex_state
    falloff = state.falloff
    primes = (PRIME_X, PRIME_Y)
    tr = orient(S, x, y)
    XY = floor.(Int, tr)
    vtr = tr .- XY
    t = sum(vtr) * OS2_UNSKEW_2D
    X1, Y1 = XY .* primes
    X2, Y2 = (X1, Y1) .+ primes
    x1, y1 = vtr .+ t
    us1 = 2OS2_UNSKEW_2D + 1
    result = 0.0
    a1 = falloff - x1^2 - y1^2
    if a1 > 0
        result += pow4(a1) * grad(table, seed, X1, Y1, x1, y1)
    end
    a2 = 2us1 * (1 / OS2_UNSKEW_2D + 2) * t + -2us1^2 + a1
    if a2 > 0
        x, y = (x1, y1) .- 2OS2_UNSKEW_2D .- 1
        result += pow4(a2) * grad(table, seed, X2, Y2, x, y)
    end
    if y1 > x1
        x = x1 - OS2_UNSKEW_2D
        y = y1 - OS2_UNSKEW_2D - 1
        a3 = falloff - x^2 - y^2
        if a3 > 0
            result += pow4(a3) * grad(table, seed, X1, Y2, x, y)
        end
    else
        x = x1 - OS2_UNSKEW_2D - 1
        y = y1 - OS2_UNSKEW_2D
        a4 = falloff - x^2 - y^2
        if a4 > 0
            result += pow4(a4) * grad(table, seed, X2, Y1, x, y)
        end
    end
    result * state.scale_factor
end

# 3D

const OS2_SEED_FLIP_3D = -0x52d547b2e96ed629
const OS2_FALLBACK_ROTATE_3D = 2 / 3
const OS2_ROTATE_3D_ORTHONORMALIZER = OS2_UNSKEW_2D
const OS2_NUM_GRADIENTS_EXP_3D = 8
const OS2_NUM_GRADIENTS_3D = 1 << OS2_NUM_GRADIENTS_EXP_3D
const OS2_GRADIENTS_NORMALIZED_3D = [
    2.22474487139, 2.22474487139, -1.0, 0.0,
    2.22474487139, 2.22474487139, 1.0, 0.0,
    3.0862664687972017, 1.1721513422464978, 0.0, 0.0,
    1.1721513422464978, 3.0862664687972017, 0.0, 0.0,
    -2.22474487139, 2.22474487139, -1.0, 0.0,
    -2.22474487139, 2.22474487139, 1.0, 0.0,
    -1.1721513422464978, 3.0862664687972017, 0.0, 0.0,
    -3.0862664687972017, 1.1721513422464978, 0.0, 0.0,
    -1.0, -2.22474487139, -2.22474487139, 0.0,
    1.0, -2.22474487139, -2.22474487139, 0.0,
    0.0, -3.0862664687972017, -1.1721513422464978, 0.0,
    0.0, -1.1721513422464978, -3.0862664687972017, 0.0,
    -1.0, -2.22474487139, 2.22474487139, 0.0,
    1.0, -2.22474487139, 2.22474487139, 0.0,
    0.0, -1.1721513422464978, 3.0862664687972017, 0.0,
    0.0, -3.0862664687972017, 1.1721513422464978, 0.0,
    -2.22474487139, -2.22474487139, -1.0, 0.0,
    -2.22474487139, -2.22474487139, 1.0, 0.0,
    -3.0862664687972017, -1.1721513422464978, 0.0, 0.0,
    -1.1721513422464978, -3.0862664687972017, 0.0, 0.0,
    -2.22474487139, -1.0, -2.22474487139, 0.0,
    -2.22474487139, 1.0, -2.22474487139, 0.0,
    -1.1721513422464978, 0.0, -3.0862664687972017, 0.0,
    -3.0862664687972017, 0.0, -1.1721513422464978, 0.0,
    -2.22474487139, -1.0, 2.22474487139, 0.0,
    -2.22474487139, 1.0, 2.22474487139, 0.0,
    -3.0862664687972017, 0.0, 1.1721513422464978, 0.0,
    -1.1721513422464978, 0.0, 3.0862664687972017, 0.0,
    -1.0, 2.22474487139, -2.22474487139, 0.0,
    1.0, 2.22474487139, -2.22474487139, 0.0,
    0.0, 1.1721513422464978, -3.0862664687972017, 0.0,
    0.0, 3.0862664687972017, -1.1721513422464978, 0.0,
    -1.0, 2.22474487139, 2.22474487139, 0.0,
    1.0, 2.22474487139, 2.22474487139, 0.0,
    0.0, 3.0862664687972017, 1.1721513422464978, 0.0,
    0.0, 1.1721513422464978, 3.0862664687972017, 0.0,
    2.22474487139, -2.22474487139, -1.0, 0.0,
    2.22474487139, -2.22474487139, 1.0, 0.0,
    1.1721513422464978, -3.0862664687972017, 0.0, 0.0,
    3.0862664687972017, -1.1721513422464978, 0.0, 0.0,
    2.22474487139, -1.0, -2.22474487139, 0.0,
    2.22474487139, 1.0, -2.22474487139, 0.0,
    3.0862664687972017, 0.0, -1.1721513422464978, 0.0,
    1.1721513422464978, 0.0, -3.0862664687972017, 0.0,
    2.22474487139, -1.0, 2.22474487139, 0.0,
    2.22474487139, 1.0, 2.22474487139, 0.0,
    1.1721513422464978, 0.0, 3.0862664687972017, 0.0,
    3.0862664687972017, 0.0, 1.1721513422464978, 0.0]

"""
    opensimplex2_3d(; seed=nothing, smooth=false, orient=nothing)

Construct a sampler that outputs 3-dimensional OpenSimplex2 noise when it is sampled from.

# Arguments

  - `seed`: An unsigned integer used to seed the random number generator for this sampler, or
    `nothing` for non-deterministic results.

  - `smooth`: Specify whether to have continuous gradients.
    Simplex variants, even the original Simplex noise by Ken Perlin, overshoot the radial extent for
    the signal reconstruction kernel in order to improve the visual of the noise. Normally this is
    okay, especially if layering multiple octaves of the noise. However, in some applications, such
    as creating height or bump maps, this will produce discontinuities visually identified by
    jarring creases in the generated noise.

    This option changes the falloff in order to produce smooth continuous noise, however, the
    resulting noise may look quite different than the non-smooth option, depending on the Simplex
    variant.

    The default value is `false`, in order to be true to the original implementation.

  - `orient`: Either the symbol `:x` or the value `nothing`:

      + `nothing`: Use the standard orientation.
      + `:x`: The noise space will be re-oriented with the Y axis pointing down the main diagonal to
        improve visual isotropy.
      + `:xy`: Re-orient the noise space to have better visual isotropy in the XY plane.
      + `:xz`: Re-orient the noise space to have better visual isotropy in the XZ plane.
"""
function opensimplex2_3d(; seed=nothing, orient=nothing, smooth=false)
    size = OS2_NUM_GRADIENTS_3D * 4
    gradients = OS2_GRADIENTS_NORMALIZED_3D ./ 0.07969837668935331
    _opensimplex2(3, seed, size, gradients, orient, smooth)
end

@inline function orient(::Type{OpenSimplex2{3,OrientStandard}}, x, y, z)
    OS2_FALLBACK_ROTATE_3D * (x + y + z) .- (x, y, z)
end

@inline function orient(::Type{OpenSimplex2{3,OrientXY}}, x, y, z)
    xy = x + y
    zz = z * ROOT_3_OVER_3
    xr, yr = (x, y) .+ xy .* OS2_ROTATE_3D_ORTHONORMALIZER .+ zz
    zr = xy * -ROOT_3_OVER_3 + zz
    (xr, yr, zr)
end

@inline function orient(::Type{OpenSimplex2{3,OrientXZ}}, x, y, z)
    orient(OpenSimplex2{3,OrientXY}, x, z, y)
end

@inline function grad(table, seed, X, Y, Z, x, y, z)
    hash = ((seed ⊻ X) ⊻ (Y ⊻ Z)) * HASH_MULTIPLIER
    hash ⊻= hash >> (64 - OS2_NUM_GRADIENTS_EXP_3D + 2)
    i = trunc(hash) & ((OS2_NUM_GRADIENTS_3D - 1) << 2)
    @inbounds t = (table[i+1], table[(i|1)+1], table[(i|2)+1])
    sum((t .* (x, y, z)))
end

@inline function os2_contribute1(seed, table, a, X, Y, Z, x1, y1, z1, x2, y2, z2, xs, ys, zs)
    result = 0.0
    if a > 0
        result += pow4(a) * grad(table, seed, X, Y, Z, x1, y1, z1)
    end
    if x2 ≥ y2 && x2 ≥ z2
        result += os2_contribute2(seed, table, a + 2x2, X - xs * PRIME_X, Y, Z, x1 + xs, y1, z1)
    elseif y2 ≥ x2 && y2 ≥ z2
        result += os2_contribute2(seed, table, a + 2y2, X, Y - ys * PRIME_Y, Z, x1, y1 + ys, z1)
    else
        result += os2_contribute2(seed, table, a + 2z2, X, Y, Z - zs * PRIME_Z, x1, y1, z1 + zs)
    end
    result
end

@inline function os2_contribute2(seed, table, a, args...)
    a > 1 ? pow4(a - 1) * grad(table, seed, args...) : 0.0
end

function sample(sampler::S, x::T, y::T, z::T) where {O,S<:OpenSimplex2{3,O},T<:Real}
    seed = sampler.random_state.seed
    table = sampler.table
    state = sampler.simplex_state
    falloff = state.falloff
    primes = (PRIME_X, PRIME_Y, PRIME_Z)
    tr = orient(S, x, y, z)
    V = round.(Int, tr)
    XYZ = V .* primes
    x1, y1, z1 = tr .- V
    s = trunc.(Int, -1 .- (x1, y1, z1)) .| 1
    XYZ2 = XYZ .+ s .>> 1 .& primes
    xyz2 = s .* .-((x1, y1, z1))
    x4, y4, z4 = 0.5 .- xyz2
    xyz3 = s .* (x4, y4, z4)
    a1 = falloff - x1^2 - (y1^2 + z1^2)
    c1 = os2_contribute1(seed, table, a1, XYZ..., x1, y1, z1, xyz2..., s...)
    a2 = a1 + 0.75 - x4 - (y4 + z4)
    c2 = os2_contribute1(seed ⊻ OS2_SEED_FLIP_3D, table, a2, XYZ2..., xyz3..., x4, y4, z4, .-s...)
    (c1 + c2) * state.scale_factor
end

# 4D

const OS2_SEED_OFFSET_4D = 0xe83dc3e0da7164d
const OS2_SKEW_4D = -0.138196601125011
const OS2_UNSKEW_4D = 0.309016994374947
const OS2_LATTICE_STEP_4D = 0.2
const OS2_NUM_GRADIENTS_EXP_4D = 9
const OS2_NUM_GRADIENTS_4D = 1 << OS2_NUM_GRADIENTS_EXP_4D
const OS2_GRADIENTS_NORMALIZED_4D = [
    -0.6740059517812944, -0.3239847771997537, -0.3239847771997537, 0.5794684678643381,
    -0.7504883828755602, -0.4004672082940195, 0.15296486218853164, 0.5029860367700724,
    -0.7504883828755602, 0.15296486218853164, -0.4004672082940195, 0.5029860367700724,
    -0.8828161875373585, 0.08164729285680945, 0.08164729285680945, 0.4553054119602712,
    -0.4553054119602712, -0.08164729285680945, -0.08164729285680945, 0.8828161875373585,
    -0.5029860367700724, -0.15296486218853164, 0.4004672082940195, 0.7504883828755602,
    -0.5029860367700724, 0.4004672082940195, -0.15296486218853164, 0.7504883828755602,
    -0.5794684678643381, 0.3239847771997537, 0.3239847771997537, 0.6740059517812944,
    -0.6740059517812944, -0.3239847771997537, 0.5794684678643381, -0.3239847771997537,
    -0.7504883828755602, -0.4004672082940195, 0.5029860367700724, 0.15296486218853164,
    -0.7504883828755602, 0.15296486218853164, 0.5029860367700724, -0.4004672082940195,
    -0.8828161875373585, 0.08164729285680945, 0.4553054119602712, 0.08164729285680945,
    -0.4553054119602712, -0.08164729285680945, 0.8828161875373585, -0.08164729285680945,
    -0.5029860367700724, -0.15296486218853164, 0.7504883828755602, 0.4004672082940195,
    -0.5029860367700724, 0.4004672082940195, 0.7504883828755602, -0.15296486218853164,
    -0.5794684678643381, 0.3239847771997537, 0.6740059517812944, 0.3239847771997537,
    -0.6740059517812944, 0.5794684678643381, -0.3239847771997537, -0.3239847771997537,
    -0.7504883828755602, 0.5029860367700724, -0.4004672082940195, 0.15296486218853164,
    -0.7504883828755602, 0.5029860367700724, 0.15296486218853164, -0.4004672082940195,
    -0.8828161875373585, 0.4553054119602712, 0.08164729285680945, 0.08164729285680945,
    -0.4553054119602712, 0.8828161875373585, -0.08164729285680945, -0.08164729285680945,
    -0.5029860367700724, 0.7504883828755602, -0.15296486218853164, 0.4004672082940195,
    -0.5029860367700724, 0.7504883828755602, 0.4004672082940195, -0.15296486218853164,
    -0.5794684678643381, 0.6740059517812944, 0.3239847771997537, 0.3239847771997537,
    0.5794684678643381, -0.6740059517812944, -0.3239847771997537, -0.3239847771997537,
    0.5029860367700724, -0.7504883828755602, -0.4004672082940195, 0.15296486218853164,
    0.5029860367700724, -0.7504883828755602, 0.15296486218853164, -0.4004672082940195,
    0.4553054119602712, -0.8828161875373585, 0.08164729285680945, 0.08164729285680945,
    0.8828161875373585, -0.4553054119602712, -0.08164729285680945, -0.08164729285680945,
    0.7504883828755602, -0.5029860367700724, -0.15296486218853164, 0.4004672082940195,
    0.7504883828755602, -0.5029860367700724, 0.4004672082940195, -0.15296486218853164,
    0.6740059517812944, -0.5794684678643381, 0.3239847771997537, 0.3239847771997537,
    -0.753341017856078, -0.37968289875261624, -0.37968289875261624, -0.37968289875261624,
    -0.7821684431180708, -0.4321472685365301, -0.4321472685365301, 0.12128480194602098,
    -0.7821684431180708, -0.4321472685365301, 0.12128480194602098, -0.4321472685365301,
    -0.7821684431180708, 0.12128480194602098, -0.4321472685365301, -0.4321472685365301,
    -0.8586508742123365, -0.508629699630796, 0.044802370851755174, 0.044802370851755174,
    -0.8586508742123365, 0.044802370851755174, -0.508629699630796, 0.044802370851755174,
    -0.8586508742123365, 0.044802370851755174, 0.044802370851755174, -0.508629699630796,
    -0.9982828964265062, -0.03381941603233842, -0.03381941603233842, -0.03381941603233842,
    -0.37968289875261624, -0.753341017856078, -0.37968289875261624, -0.37968289875261624,
    -0.4321472685365301, -0.7821684431180708, -0.4321472685365301, 0.12128480194602098,
    -0.4321472685365301, -0.7821684431180708, 0.12128480194602098, -0.4321472685365301,
    0.12128480194602098, -0.7821684431180708, -0.4321472685365301, -0.4321472685365301,
    -0.508629699630796, -0.8586508742123365, 0.044802370851755174, 0.044802370851755174,
    0.044802370851755174, -0.8586508742123365, -0.508629699630796, 0.044802370851755174,
    0.044802370851755174, -0.8586508742123365, 0.044802370851755174, -0.508629699630796,
    -0.03381941603233842, -0.9982828964265062, -0.03381941603233842, -0.03381941603233842,
    -0.37968289875261624, -0.37968289875261624, -0.753341017856078, -0.37968289875261624,
    -0.4321472685365301, -0.4321472685365301, -0.7821684431180708, 0.12128480194602098,
    -0.4321472685365301, 0.12128480194602098, -0.7821684431180708, -0.4321472685365301,
    0.12128480194602098, -0.4321472685365301, -0.7821684431180708, -0.4321472685365301,
    -0.508629699630796, 0.044802370851755174, -0.8586508742123365, 0.044802370851755174,
    0.044802370851755174, -0.508629699630796, -0.8586508742123365, 0.044802370851755174,
    0.044802370851755174, 0.044802370851755174, -0.8586508742123365, -0.508629699630796,
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    0.4004672082940195, -0.5029860367700724, 0.7504883828755602, -0.15296486218853164,
    0.3239847771997537, -0.5794684678643381, 0.6740059517812944, 0.3239847771997537,
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    0.15296486218853164, -0.4004672082940195, 0.5029860367700724, -0.7504883828755602,
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    -0.08164729285680945, -0.08164729285680945, 0.8828161875373585, -0.4553054119602712,
    -0.15296486218853164, 0.4004672082940195, 0.7504883828755602, -0.5029860367700724,
    0.4004672082940195, -0.15296486218853164, 0.7504883828755602, -0.5029860367700724,
    0.3239847771997537, 0.3239847771997537, 0.6740059517812944, -0.5794684678643381,
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    0.4004672082940195, 0.7504883828755602, -0.5029860367700724, -0.15296486218853164,
    0.3239847771997537, 0.6740059517812944, -0.5794684678643381, 0.3239847771997537,
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    0.3239847771997537, 0.6740059517812944, 0.3239847771997537, -0.5794684678643381,
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    0.4553054119602712, 0.08164729285680945, -0.8828161875373585, 0.08164729285680945,
    0.8828161875373585, -0.08164729285680945, -0.4553054119602712, -0.08164729285680945,
    0.7504883828755602, -0.15296486218853164, -0.5029860367700724, 0.4004672082940195,
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    0.6740059517812944, 0.3239847771997537, -0.5794684678643381, 0.3239847771997537,
    0.5794684678643381, -0.3239847771997537, -0.3239847771997537, -0.6740059517812944,
    0.5029860367700724, -0.4004672082940195, 0.15296486218853164, -0.7504883828755602,
    0.5029860367700724, 0.15296486218853164, -0.4004672082940195, -0.7504883828755602,
    0.4553054119602712, 0.08164729285680945, 0.08164729285680945, -0.8828161875373585,
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    0.6740059517812944, 0.3239847771997537, 0.3239847771997537, -0.5794684678643381,
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    0.4321472685365301, 0.4321472685365301, -0.12128480194602098, 0.7821684431180708,
    0.37968289875261624, 0.37968289875261624, 0.37968289875261624, 0.753341017856078,
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    -0.12128480194602098, 0.4321472685365301, 0.7821684431180708, 0.4321472685365301,
    0.508629699630796, -0.044802370851755174, 0.8586508742123365, -0.044802370851755174,
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    0.4321472685365301, 0.4321472685365301, 0.7821684431180708, -0.12128480194602098,
    0.37968289875261624, 0.37968289875261624, 0.753341017856078, 0.37968289875261624,
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    0.7821684431180708, -0.12128480194602098, 0.4321472685365301, 0.4321472685365301,
    0.8586508742123365, 0.508629699630796, -0.044802370851755174, -0.044802370851755174,
    0.7821684431180708, 0.4321472685365301, -0.12128480194602098, 0.4321472685365301,
    0.7821684431180708, 0.4321472685365301, 0.4321472685365301, -0.12128480194602098,
    0.753341017856078, 0.37968289875261624, 0.37968289875261624, 0.37968289875261624]

"""
    opensimplex2_4d(; seed=nothing, smooth=false, orient=nothing)

Construct a sampler that outputs 4-dimensional OpenSimplex2 noise when it is sampled from.

# Arguments

  - `seed`: An unsigned integer used to seed the random number generator for this sampler, or
    `nothing` for non-deterministic results.

  - `smooth`: Specify whether to have continuous gradients.
    Simplex variants, even the original Simplex noise by Ken Perlin, overshoot the radial extent for
    the signal reconstruction kernel in order to improve the visual of the noise. Normally this is
    okay, especially if layering multiple octaves of the noise. However, in some applications, such
    as creating height or bump maps, this will produce discontinuities visually identified by
    jarring creases in the generated noise.

    This option changes the falloff in order to produce smooth continuous noise, however, the
    resulting noise may look quite different than the non-smooth option, depending on the Simplex
    variant.

    The default value is `false`, in order to be true to the original implementation.

  - `orient`: Either the symbol `:x` or the value `nothing`:

      + `nothing`: Use the standard orientation.
      + `:x`: The noise space will be re-oriented with the Y axis pointing down the main diagonal to
        improve visual isotropy.
      + `:xy`: Re-orient the noise space to have better visual isotropy in the XY plane.
      + `:xz`: Re-orient the noise space to have better visual isotropy in the XZ plane.
      + `:xyz`: Re-orient the noise space to be better suited for time-varied animations, where
        the W axis is time.
"""
function opensimplex2_4d(; seed=nothing, orient=nothing, smooth=false)
    size = OS2_NUM_GRADIENTS_4D * 4
    gradients = OS2_GRADIENTS_NORMALIZED_4D ./ 0.0220065933241897
    _opensimplex2(4, seed, size, gradients, orient, smooth)
end

@inline function orient(::Type{OpenSimplex2{4,OrientStandard}}, x, y, z, w)
    (x, y, z, w) .+ OS2_SKEW_4D .* (x + y + z + w)
end

@inline function orient(::Type{OpenSimplex2{4,OrientXY}}, x, y, z, w)
    xy = x + y
    ww = w * 0.2236067977499788
    zw = z * 0.28867513459481294226 + ww
    xr, yr = (x, y) .+ zw .+ xy .* -0.21132486540518699998
    zr = xy * -0.57735026918962599998 + zw
    wr = z * -0.866025403784439 + ww
    (xr, yr, zr, wr)
end

@inline function orient(::Type{OpenSimplex2{4,OrientXZ}}, x, y, z, w)
    orient(OpenSimplex2{4,OrientXY}, x, z, y, w)
end

@inline function orient(::Type{OpenSimplex2{4,OrientXYZ}}, x, y, z, w)
    xyz = -(x + y + z)
    ww = w * 0.2236067977499788
    s = xyz / 6 + ww
    xs, ys, zs = (x, y, z) .+ s
    ws = xyz * 0.5 + ww
    (xs, ys, zs, ws)
end

@inline function grad(table, seed, X, Y, Z, W, x, y, z, w)
    hash = seed ⊻ (X ⊻ Y) ⊻ (Z ⊻ W) * HASH_MULTIPLIER
    hash ⊻= hash >> (64 - OS2_NUM_GRADIENTS_EXP_4D + 2)
    i = trunc(hash) & ((OS2_NUM_GRADIENTS_4D - 1) << 2)
    @inbounds t = (table[i+1], table[(i|1)+1], table[(i|2)+1], table[(i|3)+1])
    sum((t .* (x, y, z, w)))
end

function sample(sampler::S, x::T, y::T, z::T, w::T) where {O,S<:OpenSimplex2{4,O},T<:Real}
    seed = sampler.random_state.seed
    table = sampler.table
    state = sampler.simplex_state
    falloff = state.falloff
    primes = (PRIME_X, PRIME_Y, PRIME_Z, PRIME_W)
    tr = orient(S, x, y, z, w)
    X1, Y1, Z1, W1 = floor.(Int, tr)
    X2, Y2, Z2, W2 = (X1, Y1, Z1, W1) .* primes
    v = tr .- (X1, Y1, Z1, W1)
    sv = sum(v)
    lattice = trunc(Int, sv * 1.25)
    lattice_offset = lattice * -OS2_LATTICE_STEP_4D
    x1, y1, z1, w1 = v .+ lattice_offset
    ssi = (sv + 4lattice_offset) * OS2_UNSKEW_4D
    seed += lattice * OS2_SEED_OFFSET_4D
    result = 0.0
    for i in 0:4
        score = 1 + ssi * (-1 / OS2_UNSKEW_4D)
        if x1 ≥ y1 && x1 ≥ z1 && x1 ≥ w1 && x1 ≥ score
            X2 += PRIME_X
            x1 -= 1
            ssi -= OS2_UNSKEW_4D
        elseif y1 > x1 && y1 ≥ z1 && y1 ≥ w1 && y1 ≥ score
            Y2 += PRIME_Y
            y1 -= 1
            ssi -= OS2_UNSKEW_4D
        elseif z1 > x1 && z1 > y1 && z1 ≥ w1 && z1 ≥ score
            Z2 += PRIME_Z
            z1 -= 1
            ssi -= OS2_UNSKEW_4D
        elseif w1 > x1 && w1 > y1 && w1 > z1 && w1 ≥ score
            W2 += PRIME_W
            w1 -= 1
            ssi -= OS2_UNSKEW_4D
        end
        xyzw = (x1, y1, z1, w1) .+ ssi
        a = sum(xyzw .^ 2)
        if a < falloff
            result += pow4(a - falloff) * grad(table, seed, X2, Y2, Z2, W2, xyzw...)
        end
        if i !== 4
            x1, y1, z1, w1 = (x1, y1, z1, w1) .+ OS2_LATTICE_STEP_4D
            ssi += OS2_LATTICE_STEP_4D * 4OS2_UNSKEW_4D
            seed -= OS2_SEED_OFFSET_4D
            if i == lattice
                X2, Y2, Z2, W2 = (X2, Y2, Z2, W2) .- primes
                seed += 5OS2_SEED_OFFSET_4D
            end
        end
    end
    result * state.scale_factor
end