From 738a82ced671cc364c2c30a49a733ef5e34dd3d6 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Leandro=20Mart=C3=ADnez?= <31046348+lmiq@users.noreply.github.com> Date: Tue, 21 Nov 2023 10:51:20 -0300 Subject: [PATCH 1/2] Add simpler example to `randn` function. As noted by https://discourse.julialang.org/t/how-do-we-julians-win-big-when-the-situation-is-so-unfair/106433/63?u=lmiq the example provided is too complex and does not directly indicate the most typical usage of the `randn` function. --- stdlib/Random/src/normal.jl | 17 ++++++++++++++++- 1 file changed, 16 insertions(+), 1 deletion(-) diff --git a/stdlib/Random/src/normal.jl b/stdlib/Random/src/normal.jl index bbc1b03e50b5d..bac4f42894527 100644 --- a/stdlib/Random/src/normal.jl +++ b/stdlib/Random/src/normal.jl @@ -23,8 +23,23 @@ from the circularly symmetric complex normal distribution of variance 1 (corresp See also [`randn!`](@ref) to act in-place. # Examples + +1. Generating a single random number, by default of type `Float64`: + +```julia-repl +julia> using Random + +julia> julia> randn() +-0.942481877315864 +``` + +2. Setting up of the random number generator with a user-defined seed, and the +generation of a random number with a `ComplexF32` type: + ```jldoctest -julia> using Random; rng = Xoshiro(123); +julia> using Random + +julia> rng = Xoshiro(123); julia> randn(rng, ComplexF64) -0.45660053706486897 - 1.0346749725929225im From 7218013b081d337d137ab4bde031c0d422e9d9bf Mon Sep 17 00:00:00 2001 From: Leandro Martinez Date: Tue, 21 Nov 2023 11:07:01 -0300 Subject: [PATCH 2/2] remove duplicated julia prompt --- stdlib/Random/src/normal.jl | 996 ++++++++++++++++++------------------ 1 file changed, 498 insertions(+), 498 deletions(-) diff --git a/stdlib/Random/src/normal.jl b/stdlib/Random/src/normal.jl index bac4f42894527..7ec11f9449ee5 100644 --- a/stdlib/Random/src/normal.jl +++ b/stdlib/Random/src/normal.jl @@ -29,7 +29,7 @@ See also [`randn!`](@ref) to act in-place. ```julia-repl julia> using Random -julia> julia> randn() +julia> randn() -0.942481877315864 ``` @@ -62,9 +62,9 @@ julia> randn(rng, ComplexF32, (2, 3)) r = rand(rng, UInt52()) # the following code is identical to the one in `_randn(rng::AbstractRNG, r::UInt64)` - rabs = Int64(r>>1) # One bit for the sign + rabs = Int64(r >> 1) # One bit for the sign idx = rabs & 0xFF - x = ifelse(r % Bool, -rabs, rabs)*wi[idx+1] + 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 @@ -73,9 +73,9 @@ end @inline function _randn(rng::AbstractRNG, r::UInt64) @inbounds begin r &= 0x000fffffffffffff - rabs = Int64(r>>1) # One bit for the sign + rabs = Int64(r >> 1) # One bit for the sign idx = rabs & 0xFF - x = ifelse(r % Bool, -rabs, rabs)*wi[idx+1] + 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 @@ -85,12 +85,12 @@ end @noinline function randn_unlikely(rng, idx, rabs, x) @inbounds if idx == 0 while true - xx = -ziggurat_nor_inv_r*log1p(-rand(rng)) + xx = -ziggurat_nor_inv_r * log1p(-rand(rng)) yy = -log1p(-rand(rng)) - yy+yy > xx*xx && - return (rabs >> 8) % Bool ? -ziggurat_nor_r-xx : ziggurat_nor_r+xx + 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) + 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) @@ -99,7 +99,7 @@ end ### complex randn -Base.@irrational SQRT_HALF 0.7071067811865475244008 sqrt(big(0.5)) +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)) @@ -109,8 +109,8 @@ randn(rng::AbstractRNG, ::Type{Complex{T}}) where {T<:AbstractFloat} = function randn(rng::AbstractRNG, ::Type{T}) where {T<:AbstractFloat} # Marsaglia polar variant of Box–Muller transform: while true - x, y = 2rand(rng, T)-1, 2rand(rng, T)-1 - 0 < (s = x^2 + y^2) < 1 && return x * sqrt(-2log(s)/s) + x, y = 2rand(rng, T) - 1, 2rand(rng, T) - 1 + 0 < (s = x^2 + y^2) < 1 && return x * sqrt(-2log(s) / s) end end @@ -145,7 +145,7 @@ function _randexp(rng::AbstractRNG, ri::UInt64) @inbounds begin ri &= 0x000fffffffffffff idx = ri & 0xFF - x = ri*we[idx+1] + 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 @@ -154,7 +154,7 @@ end @noinline function randexp_unlikely(rng, idx, x) @inbounds if idx == 0 return ziggurat_exp_r - log1p(-rand(rng)) - elseif (fe[idx] - fe[idx+1])*rand(rng) + fe[idx+1] < exp(-x) + elseif (fe[idx] - fe[idx+1]) * rand(rng) + fe[idx+1] < exp(-x) return x # return from the triangular area else return randexp(rng) @@ -214,7 +214,7 @@ for randfun in [:randn, :randexp] $randfun(::Type{T}) where {T} = $randfun(default_rng(), T) # filling arrays - function $randfun!(rng::AbstractRNG, A::AbstractArray{T}) where T + function $randfun!(rng::AbstractRNG, A::AbstractArray{T}) where {T} for i in eachindex(A) @inbounds A[i] = $randfun(rng, T) end @@ -237,7 +237,7 @@ for randfun in [:randn, :randexp] end # optimization for Xoshiro, which randomizes natively Array{UInt64} - function $randfun!(rng::Union{Xoshiro, TaskLocalRNG}, A::Array{Float64}) + function $randfun!(rng::Union{Xoshiro,TaskLocalRNG}, A::Array{Float64}) if length(A) < 7 for i in eachindex(A) @inbounds A[i] = $randfun(rng, Float64) @@ -255,506 +255,506 @@ for randfun in [:randn, :randexp] $randfun!(A::AbstractArray) = $randfun!(default_rng(), A) # generating arrays - $randfun(rng::AbstractRNG, ::Type{T}, dims::Dims ) where {T} = $randfun!(rng, Array{T}(undef, dims)) + $randfun(rng::AbstractRNG, ::Type{T}, dims::Dims) where {T} = $randfun!(rng, Array{T}(undef, 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}(undef, dim1, dims...)) - $randfun( ::Type{T}, dims::Dims ) where {T} = $randfun(default_rng(), T, dims) - $randfun( ::Type{T}, dims::Integer... ) where {T} = $randfun(default_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(default_rng(), Float64, dims) - $randfun( dims::Integer... ) = $randfun(default_rng(), Float64, dims...) + $randfun(::Type{T}, dims::Dims) where {T} = $randfun(default_rng(), T, dims) + $randfun(::Type{T}, dims::Integer...) where {T} = $randfun(default_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(default_rng(), Float64, dims) + $randfun(dims::Integer...) = $randfun(default_rng(), Float64, dims...) end end ## 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] + 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] + [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, 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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] + 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, - 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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] ## Constants -const ziggurat_nor_r = 3.6541528853610087963519472518 -const ziggurat_nor_inv_r = inv(ziggurat_nor_r) -const ziggurat_exp_r = 7.6971174701310497140446280481 +const ziggurat_nor_r = 3.6541528853610087963519472518 +const ziggurat_nor_inv_r = inv(ziggurat_nor_r) +const ziggurat_exp_r = 7.6971174701310497140446280481