JuliaLang / julia / #37919

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#37919

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fix rawbigints OOB issues (#55917)

Fixes issues introduced in #50691 and found in #55906:
* use `@inbounds` and `@boundscheck` macros in rawbigints, for catching
OOB with `--check-bounds=yes`
* fix OOB in `truncate`

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1287 existing lines in 41 files now uncovered.

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90.24
/stdlib/Random/src/normal.jl
1 
# This file is a part of Julia. License is MIT: https://julialang.org/license
2 









3



# Normally distributed random numbers using Ziggurat algorithm










4












5



# The Ziggurat Method for generating random variables - Marsaglia and Tsang










6



# Paper and reference code: https://www.jstatsoft.org/v05/i08/










7












8



# randmtzig (covers also exponential variates)










9












10



## randn










11












12



"""










13



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










14












15



Generate a normally-distributed random number of type `T`










16



with mean 0 and standard deviation 1.










17



Given the optional `dims` argument(s), generate an array of size `dims` of such numbers.










18



Julia's standard library supports `randn` for any floating-point type










19



that implements [`rand`](@ref), e.g. the `Base` types










20



[`Float16`](@ref), [`Float32`](@ref), [`Float64`](@ref) (the default), and [`BigFloat`](@ref),










21



along with their [`Complex`](@ref) counterparts.










22












23



(When `T` is complex, the values are drawn










24



from the circularly symmetric complex normal distribution of variance 1, corresponding to real and imaginary parts










25



having independent normal distribution with mean zero and variance `1/2`).










26












27



See also [`randn!`](@ref) to act in-place.










28












29



# Examples










30












31



Generating a single random number (with the default `Float64` type):










32












33



```julia-repl










34



julia> randn()










35



-0.942481877315864










36



```










37












38



Generating a matrix of normal random numbers (with the default `Float64` type):










39












40



```julia-repl










41



julia> randn(2,3)










42



2×3 Matrix{Float64}:










43



  1.18786   -0.678616   1.49463










44



 -0.342792  -0.134299  -1.45005










45



```










46












47



Setting up of the random number generator `rng` with a user-defined seed (for reproducible numbers)










48



and using it to generate a random `Float32` number or a matrix of `ComplexF32` random numbers:










49












50



```jldoctest










51



julia> using Random










52












53



julia> rng = Xoshiro(123);










54












55



julia> randn(rng, Float32)










56



-0.6457307f0










57












58



julia> randn(rng, ComplexF32, (2, 3))










59



2×3 Matrix{ComplexF32}:










60



  -1.03467-1.14806im  0.693657+0.056538im   0.291442+0.419454im










61



 -0.153912+0.34807im    1.0954-0.948661im  -0.543347-0.0538589im










62



```










63



"""










64



@inline function randn(rng::AbstractRNG=default_rng())




103,687✔





65



    #=










66



    When defining










67



    `@inline randn(rng::AbstractRNG=default_rng()) = _randn(rng, rand(rng, UInt52Raw()))`










68



    the function call to `_randn` is currently not inlined, resulting in slightly worse










69



    performance for scalar random normal numbers than repeating the code of `_randn`










70



    inside the following function.










71



    =#










72



    @inbounds begin




11,099,338✔





73



        r = rand(rng, UInt52())




11,089,050✔





74












75



        # the following code is identical to the one in `_randn(rng::AbstractRNG, r::UInt64)`










76



        rabs = Int64(r>>1) # One bit for the sign




11,089,050✔





77



        idx = rabs & 0xFF




11,089,050✔





78



        x = ifelse(r % Bool, -rabs, rabs)*wi[idx+1]




11,089,050✔





79



        rabs < ki[idx+1] && return x # 99.3% of the time we return here 1st try




11,089,050✔





80



        return randn_unlikely(rng, idx, rabs, x)




165,177✔





81



    end










82



end










83












84



@inline function _randn(rng::AbstractRNG, r::UInt64)










85



    @inbounds begin




4,528,062✔





86



        r &= 0x000fffffffffffff




4,528,062✔





87



        rabs = Int64(r>>1) # One bit for the sign




4,528,062✔





88



        idx = rabs & 0xFF




4,528,062✔





89



        x = ifelse(r % Bool, -rabs, rabs)*wi[idx+1]




4,528,062✔





90



        rabs < ki[idx+1] && return x # 99.3% of the time we return here 1st try




4,528,062✔





91



        return randn_unlikely(rng, idx, rabs, x)




66,999✔





92



    end










93



end










94












95



# this unlikely branch is put in a separate function for better efficiency










96



@noinline function randn_unlikely(rng, idx, rabs, x)




232,176✔





97



    @inbounds if idx == 0




232,176✔





98



        while true




4,290✔





99



            xx = -ziggurat_nor_inv_r*log1p(-rand(rng))




4,290✔





100



            yy = -log1p(-rand(rng))




4,290✔





101



            yy+yy > xx*xx &&




4,290✔





102



                return (rabs >> 8) % Bool ? -ziggurat_nor_r-xx : ziggurat_nor_r+xx










103



        end




256✔





104



    elseif (fi[idx] - fi[idx+1])*rand(rng) + fi[idx+1] < exp(-0.5*x*x)




228,142✔





105



        return x # return from the triangular area




124,478✔





106



    else










107



        return randn(rng)




103,664✔





108



    end










109



end










110












111



### complex randn










112












113



Base.@irrational SQRT_HALF 0.7071067811865475244008  sqrt(big(0.5))










114












115



randn(rng::AbstractRNG, ::Type{Complex{T}}) where {T<:AbstractFloat} =




148,358✔





116



    Complex{T}(SQRT_HALF * randn(rng, T), SQRT_HALF * randn(rng, T))










117












118












119



### fallback randn for float types defining rand:










120



function randn(rng::AbstractRNG, ::Type{T}) where {T<:AbstractFloat}




1✔





121



    # Marsaglia polar variant of Box–Muller transform:










122



    while true




1✔





123



        x, y = 2rand(rng, T)-1, 2rand(rng, T)-1




1✔






UNCOV

124



        0 < (s = x^2 + y^2) < 1 && return x * sqrt(-2log(s)/s)




×








UNCOV

125



    end




×







126



end










127












128



## randexp










129












130



"""










131



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










132












133



Generate a random number of type `T` according to the










134



exponential distribution with scale 1.










135



Optionally generate an array of such random numbers.










136



The `Base` module currently provides an implementation for the types










137



[`Float16`](@ref), [`Float32`](@ref), and [`Float64`](@ref) (the default).










138












139



# Examples










140



```jldoctest










141



julia> rng = Xoshiro(123);










142












143



julia> randexp(rng, Float32)










144



1.1757717f0










145












146



julia> randexp(rng, 3, 3)










147



3×3 Matrix{Float64}:










148



 1.37766  0.456653  0.236418










149



 3.40007  0.229917  0.0684921










150



 0.48096  0.577481  0.71835










151



```










152



"""










153



randexp(rng::AbstractRNG=default_rng()) = _randexp(rng, rand(rng, UInt52Raw()))




119,327✔





154












155



function _randexp(rng::AbstractRNG, ri::UInt64)




119,327✔





156



    @inbounds begin




119,327✔





157



        ri &= 0x000fffffffffffff




119,327✔





158



        idx = ri & 0xFF




119,327✔





159



        x = ri*we[idx+1]




119,327✔





160



        ri < ke[idx+1] && return x # 98.9% of the time we return here 1st try




119,327✔





161



        return randexp_unlikely(rng, idx, x)




2,498✔





162



    end










163



end










164












165



@noinline function randexp_unlikely(rng, idx, x)




2,498✔





166



    @inbounds if idx == 0




2,498✔





167



        return ziggurat_exp_r - log1p(-rand(rng))




66✔





168



    elseif (fe[idx] - fe[idx+1])*rand(rng) + fe[idx+1] < exp(-x)




2,432✔





169



        return x # return from the triangular area




1,062✔





170



    else










171



        return randexp(rng)




1,370✔





172



    end










173



end










174












175



### fallback randexp for float types defining rand:











UNCOV

176



randexp(rng::AbstractRNG, ::Type{T}) where {T<:AbstractFloat} =




×







177



    -log1p(-rand(rng, T))










178












179



## arrays & other scalar methods










180












181



"""










182



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










183












184



Fill the array `A` with normally-distributed (mean 0, standard deviation 1) random numbers.










185



Also see the [`rand`](@ref) function.










186












187



# Examples










188



```jldoctest










189



julia> randn!(Xoshiro(123), zeros(5))










190



5-element Vector{Float64}:










191



 -0.6457306721039767










192



 -1.4632513788889214










193



 -1.6236037455860806










194



 -0.21766510678354617










195



  0.4922456865251828










196



```










197



"""










198



function randn! end










199












200



"""










201



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










202












203



Fill the array `A` with random numbers following the exponential distribution










204



(with scale 1).










205












206



# Examples










207



```jldoctest










208



julia> randexp!(Xoshiro(123), zeros(5))










209



5-element Vector{Float64}:










210



 1.1757716836348473










211



 1.758884569451514










212



 1.0083623637301151










213



 0.3510644315565272










214



 0.6348266443720407










215



```










216



"""










217



function randexp! end










218












219



for randfun in [:randn, :randexp]










220



    randfun! = Symbol(randfun, :!)










221



    _randfun = Symbol(:_, randfun)










222



    @eval begin










223



        # scalars










224



        $randfun(rng::AbstractRNG, T::BitFloatType) = convert(T, $randfun(rng))




10,974,979✔





225



        $randfun(::Type{T}) where {T} = $randfun(default_rng(), T)




327,732✔





226












227



        # filling arrays










228



        function $randfun!(rng::AbstractRNG, A::AbstractArray{T}) where T




3✔





229



            for i in eachindex(A)




529✔





230



                @inbounds A[i] = $randfun(rng, T)




10,168,994✔





231



            end




20,337,493✔





232



            A




527✔





233



        end










234












235



        # optimization for MersenneTwister, which randomizes natively Array{Float64}










236



        function $randfun!(rng::MersenneTwister, A::Array{Float64})




2✔





237



            if length(A) < 13




2✔






UNCOV

238



                for i in eachindex(A)




×








UNCOV

239



                    @inbounds A[i] = $randfun(rng, Float64)




×








UNCOV

240



                end




×







241



            else










242



                rand!(rng, A, CloseOpen12())




2✔





243



                for i in eachindex(A)




2✔





244



                    @inbounds A[i] = $_randfun(rng, reinterpret(UInt64, A[i]))




190✔





245



                end




190✔





246



            end










247



            A




2✔





248



        end










249












250



        # optimization for Xoshiro, which randomizes natively Array{UInt64}










251



        function $randfun!(rng::Union{Xoshiro, TaskLocalRNG}, A::Array{Float64})




3,978✔





252



            if length(A) < 7




3,978✔





253



                for i in eachindex(A)




596✔





254



                    @inbounds A[i] = $randfun(rng, Float64)




2,215✔





255



                end




3,871✔





256



            else










257



                GC.@preserve A rand!(rng, UnsafeView{UInt64}(pointer(A), length(A)))




3,382✔





258












259



                for i in eachindex(A)




3,382✔





260



                    @inbounds A[i] = $_randfun(rng, reinterpret(UInt64, A[i]) >>> 12)




4,527,872✔





261



                end




4,527,872✔





262



            end










263



            A




3,978✔





264



        end










265












266



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




1✔





267












268



        # generating arrays










269



        $randfun(rng::AbstractRNG, ::Type{T}, dims::Dims                     ) where {T} = $randfun!(rng, Array{T}(undef, dims))




36✔





270



        # Note that this method explicitly does not define $randfun(rng, T),










271



        # in order to prevent an infinite recursion.










272



        $randfun(rng::AbstractRNG, ::Type{T}, dim1::Integer, dims::Integer...) where {T} = $randfun!(rng, Array{T}(undef, dim1, dims...))




10,020,605✔






UNCOV

273



        $randfun(                  ::Type{T}, dims::Dims                     ) where {T} = $randfun(default_rng(), T, dims)




×







274



        $randfun(                  ::Type{T}, dims::Integer...               ) where {T} = $randfun(default_rng(), T, dims...)




10,000,709✔





275



        $randfun(rng::AbstractRNG,            dims::Dims                     )           = $randfun(rng, Float64, dims)




×







276



        $randfun(rng::AbstractRNG,            dims::Integer...               )           = $randfun(rng, Float64, dims...)




308✔





277



        $randfun(                             dims::Dims                     )           = $randfun(default_rng(), Float64, dims)




36✔





278



        $randfun(                             dims::Integer...               )           = $randfun(default_rng(), Float64, dims...)




3,913✔





279



    end










280



end










281












282



## Tables for normal variates










283












284



const ki =










285



    UInt64[0x0007799ec012f7b2,0x0000000000000000,0x0006045f4c7de363,0x0006d1aa7d5ec0a5,










286



           0x000728fb3f60f777,0x0007592af4e9fbc0,0x000777a5c0bf655d,0x00078ca3857d2256,










287



           0x00079bf6b0ffe58b,0x0007a7a34ab092ad,0x0007b0d2f20dd1cb,0x0007b83d3aa9cb52,










288



           0x0007be597614224d,0x0007c3788631abe9,0x0007c7d32bc192ee,0x0007cb9263a6e86d,










289



           0x0007ced483edfa84,0x0007d1b07ac0fd39,0x0007d437ef2da5fc,0x0007d678b069aa6e,










290



           0x0007d87db38c5c87,0x0007da4fc6a9ba62,0x0007dbf611b37f3b,0x0007dd7674d0f286,










291



           0x0007ded5ce8205f6,0x0007e018307fb62b,0x0007e141081bd124,0x0007e2533d712de8,










292



           0x0007e3514bbd7718,0x0007e43d54944b52,0x0007e5192f25ef42,0x0007e5e67481118d,










293



           0x0007e6a6897c1ce2,0x0007e75aa6c7f64c,0x0007e803df8ee498,0x0007e8a326eb6272,










294



           0x0007e93954717a28,0x0007e9c727f8648f,0x0007ea4d4cc85a3c,0x0007eacc5c4907a9,










295



           0x0007eb44e0474cf6,0x0007ebb754e47419,0x0007ec242a3d8474,0x0007ec8bc5d69645,










296



           0x0007ecee83d3d6e9,0x0007ed4cb8082f45,0x0007eda6aee0170f,0x0007edfcae2dfe68,










297



           0x0007ee4ef5dccd3e,0x0007ee9dc08c394e,0x0007eee9441a17c7,0x0007ef31b21b4fb1,










298



           0x0007ef773846a8a7,0x0007efba00d35a17,0x0007effa32ccf69f,0x0007f037f25e1278,










299



           0x0007f0736112d12c,0x0007f0ac9e145c25,0x0007f0e3c65e1fcc,0x0007f118f4ed8e54,










300



           0x0007f14c42ed0dc8,0x0007f17dc7daa0c3,0x0007f1ad99aac6a5,0x0007f1dbcce80015,










301



           0x0007f20874cf56bf,0x0007f233a36a3b9a,0x0007f25d69a604ad,0x0007f285d7694a92,










302



           0x0007f2acfba75e3b,0x0007f2d2e4720909,0x0007f2f79f09c344,0x0007f31b37ec883b,










303



           0x0007f33dbae36abc,0x0007f35f330f08d5,0x0007f37faaf2fa79,0x0007f39f2c805380,










304



           0x0007f3bdc11f4f1c,0x0007f3db71b83850,0x0007f3f846bba121,0x0007f4144829f846,










305



           0x0007f42f7d9a8b9d,0x0007f449ee420432,0x0007f463a0f8675e,0x0007f47c9c3ea77b,










306



           0x0007f494e643cd8e,0x0007f4ac84e9c475,0x0007f4c37dc9cd50,0x0007f4d9d638a432,










307



           0x0007f4ef934a5b6a,0x0007f504b9d5f33d,0x0007f5194e78b352,0x0007f52d55994a96,










308



           0x0007f540d36aba0c,0x0007f553cbef0e77,0x0007f56642f9ec8f,0x0007f5783c32f31e,










309



           0x0007f589bb17f609,0x0007f59ac2ff1525,0x0007f5ab5718b15a,0x0007f5bb7a71427c,










310



           0x0007f5cb2ff31009,0x0007f5da7a67cebe,0x0007f5e95c7a24e7,0x0007f5f7d8b7171e,










311



           0x0007f605f18f5ef4,0x0007f613a958ad0a,0x0007f621024ed7e9,0x0007f62dfe94f8cb,










312



           0x0007f63aa036777a,0x0007f646e928065a,0x0007f652db488f88,0x0007f65e786213ff,










313



           0x0007f669c22a7d8a,0x0007f674ba446459,0x0007f67f623fc8db,0x0007f689bb9ac294,










314



           0x0007f693c7c22481,0x0007f69d881217a6,0x0007f6a6fdd6ac36,0x0007f6b02a4c61ee,










315



           0x0007f6b90ea0a7f4,0x0007f6c1abf254c0,0x0007f6ca03521664,0x0007f6d215c2db82,










316



           0x0007f6d9e43a3559,0x0007f6e16fa0b329,0x0007f6e8b8d23729,0x0007f6efc09e4569,










317



           0x0007f6f687c84cbf,0x0007f6fd0f07ea09,0x0007f703570925e2,0x0007f709606cad03,










318



           0x0007f70f2bc8036f,0x0007f714b9a5b292,0x0007f71a0a85725d,0x0007f71f1edc4d9e,










319



           0x0007f723f714c179,0x0007f728938ed843,0x0007f72cf4a03fa0,0x0007f7311a945a16,










320



           0x0007f73505ac4bf8,0x0007f738b61f03bd,0x0007f73c2c193dc0,0x0007f73f67bd835c,










321



           0x0007f74269242559,0x0007f745305b31a1,0x0007f747bd666428,0x0007f74a103f12ed,










322



           0x0007f74c28d414f5,0x0007f74e0709a42d,0x0007f74faab939f9,0x0007f75113b16657,










323



           0x0007f75241b5a155,0x0007f753347e16b8,0x0007f753ebb76b7c,0x0007f75467027d05,










324



           0x0007f754a5f4199d,0x0007f754a814b207,0x0007f7546ce003ae,0x0007f753f3c4bb29,










325



           0x0007f7533c240e92,0x0007f75245514f41,0x0007f7510e91726c,0x0007f74f971a9012,










326



           0x0007f74dde135797,0x0007f74be2927971,0x0007f749a39e051c,0x0007f747202aba8a,










327



           0x0007f744571b4e3c,0x0007f741473f9efe,0x0007f73def53dc43,0x0007f73a4dff9bff,










328



           0x0007f73661d4deaf,0x0007f732294f003f,0x0007f72da2d19444,0x0007f728cca72bda,










329



           0x0007f723a5000367,0x0007f71e29f09627,0x0007f7185970156b,0x0007f7123156c102,










330



           0x0007f70baf5c1e2c,0x0007f704d1150a23,0x0007f6fd93f1a4e5,0x0007f6f5f53b10b6,










331



           0x0007f6edf211023e,0x0007f6e587671ce9,0x0007f6dcb2021679,0x0007f6d36e749c64,










332



           0x0007f6c9b91bf4c6,0x0007f6bf8e1c541b,0x0007f6b4e95ce015,0x0007f6a9c68356ff,










333



           0x0007f69e20ef5211,0x0007f691f3b517eb,0x0007f6853997f321,0x0007f677ed03ff19,










334



           0x0007f66a08075bdc,0x0007f65b844ab75a,0x0007f64c5b091860,0x0007f63c8506d4bc,










335



           0x0007f62bfa8798fe,0x0007f61ab34364b0,0x0007f608a65a599a,0x0007f5f5ca4737e8,










336



           0x0007f5e214d05b48,0x0007f5cd7af7066e,0x0007f5b7f0e4c2a1,0x0007f5a169d68fcf,










337



           0x0007f589d80596a5,0x0007f5712c8d0174,0x0007f557574c912b,0x0007f53c46c77193,










338



           0x0007f51fe7feb9f2,0x0007f5022646ecfb,0x0007f4e2eb17ab1d,0x0007f4c21dd4a3d1,










339



           0x0007f49fa38ea394,0x0007f47b5ebb62eb,0x0007f4552ee27473,0x0007f42cf03d58f5,










340



           0x0007f4027b48549f,0x0007f3d5a44119df,0x0007f3a63a8fb552,0x0007f37408155100,










341



           0x0007f33ed05b55ec,0x0007f3064f9c183e,0x0007f2ca399c7ba1,0x0007f28a384bb940,










342



           0x0007f245ea1b7a2b,0x0007f1fcdffe8f1b,0x0007f1ae9af758cd,0x0007f15a8917f27e,










343



           0x0007f10001ccaaab,0x0007f09e413c418a,0x0007f034627733d7,0x0007efc15815b8d5,










344



           0x0007ef43e2bf7f55,0x0007eeba84e31dfe,0x0007ee237294df89,0x0007ed7c7c170141,










345



           0x0007ecc2f0d95d3a,0x0007ebf377a46782,0x0007eb09d6deb285,0x0007ea00a4f17808,










346



           0x0007e8d0d3da63d6,0x0007e771023b0fcf,0x0007e5d46c2f08d8,0x0007e3e937669691,










347



           0x0007e195978f1176,0x0007deb2c0e05c1c,0x0007db0362002a19,0x0007d6202c151439,










348



           0x0007cf4b8f00a2cb,0x0007c4fd24520efd,0x0007b362fbf81816,0x00078d2d25998e24]










349



const wi =










350



    [1.7367254121602630e-15,9.5586603514556339e-17,1.2708704834810623e-16,










351



     1.4909740962495474e-16,1.6658733631586268e-16,1.8136120810119029e-16,










352



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353



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354



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355



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356



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357



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358



     3.2746570407939751e-16,3.3264798116841710e-16,3.3771803417353232e-16,










359



     3.4268340353119356e-16,3.4755088731729758e-16,3.5232663846002031e-16,










360



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361



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362



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363



     3.9606321366256378e-16,4.0011337552546690e-16,4.0411583124143332e-16,










364



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365



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366



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367



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368



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369



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370



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371



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372



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373



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374



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375



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376



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377



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378



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379



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380



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381



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382



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383



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384



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385



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386



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387



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388



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389



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390



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391



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392



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393



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394



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395



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396



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397



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398



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399



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400



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401



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402



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403



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404



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405



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406



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407



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408



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409



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410



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411



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412



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413



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414



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415



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416



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417



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418



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419



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420



     9.8221725991905411e-16,9.8740719604806711e-16,9.9267855488079765e-16,










421



     9.9803500261836449e-16,1.0034804521436181e-15,1.0090190861637457e-15,










422



     1.0146553831467086e-15,1.0203941464683124e-15,1.0262405372613567e-15,










423



     1.0322001115486456e-15,1.0382788623515399e-15,1.0444832676000471e-15,










424



     1.0508203448355195e-15,1.0572977139009890e-15,1.0639236690676801e-15,










425



     1.0707072623632994e-15,1.0776584002668106e-15,1.0847879564403425e-15,










426



     1.0921079038149563e-15,1.0996314701785628e-15,1.1073733224935752e-15,










427



     1.1153497865853155e-15,1.1235791107110833e-15,1.1320817840164846e-15,










428



     1.1408809242582780e-15,1.1500027537839792e-15,1.1594771891449189e-15,










429



     1.1693385786910960e-15,1.1796266352955801e-15,1.1903876299282890e-15,










430



     1.2016759392543819e-15,1.2135560818666897e-15,1.2261054417450561e-15,










431



     1.2394179789163251e-15,1.2536093926602567e-15,1.2688244814255010e-15,










432



     1.2852479319096109e-15,1.3031206634689985e-15,1.3227655770195326e-15,










433



     1.3446300925011171e-15,1.3693606835128518e-15,1.3979436672775240e-15,










434



     1.4319989869661328e-15,1.4744848603597596e-15,1.5317872741611144e-15,










435



     1.6227698675312968e-15]










436



const fi =










437



    [1.0000000000000000e+00,9.7710170126767082e-01,9.5987909180010600e-01,










438



     9.4519895344229909e-01,9.3206007595922991e-01,9.1999150503934646e-01,










439



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440



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441



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442



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443



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444



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445



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446



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447



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448



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449



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450



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451



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452



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453



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454



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455



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456



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457



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458



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459



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460



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461



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462



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463



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464



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465



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466



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467



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468



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469



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470



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471



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472



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473



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474



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475



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476



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477



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478



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479



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480



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481



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482



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483



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484



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485



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486



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487



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488



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489



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490



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491



     2.0640540639788071e-01,2.0364274931033485e-01,2.0089182249465656e-01,










492



     1.9815258654577511e-01,1.9542500351413428e-01,1.9270903690358912e-01,










493



     1.9000465167046496e-01,1.8731181422380025e-01,1.8463049242679927e-01,










494



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495



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496



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497



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498



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499



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500



     1.3584147657125373e-01,1.3338602969166913e-01,1.3094177717364430e-01,










501



     1.2850872227999952e-01,1.2608687022018586e-01,1.2367622820159654e-01,










502



     1.2127680548479021e-01,1.1888861344290998e-01,1.1651166562561080e-01,










503



     1.1414597782783835e-01,1.1179156816383801e-01,1.0944845714681163e-01,










504



     1.0711666777468364e-01,1.0479622562248690e-01,1.0248715894193508e-01,










505



     1.0018949876880981e-01,9.7903279038862284e-02,9.5628536713008819e-02,










506



     9.3365311912690860e-02,9.1113648066373634e-02,8.8873592068275789e-02,










507



     8.6645194450557961e-02,8.4428509570353374e-02,8.2223595813202863e-02,










508



     8.0030515814663056e-02,7.7849336702096039e-02,7.5680130358927067e-02,










509



     7.3522973713981268e-02,7.1377949058890375e-02,6.9245144397006769e-02,










510



     6.7124653827788497e-02,6.5016577971242842e-02,6.2921024437758113e-02,










511



     6.0838108349539864e-02,5.8767952920933758e-02,5.6710690106202902e-02,










512



     5.4666461324888914e-02,5.2635418276792176e-02,5.0617723860947761e-02,










513



     4.8613553215868521e-02,4.6623094901930368e-02,4.4646552251294443e-02,










514



     4.2684144916474431e-02,4.0736110655940933e-02,3.8802707404526113e-02,










515



     3.6884215688567284e-02,3.4980941461716084e-02,3.3093219458578522e-02,










516



     3.1221417191920245e-02,2.9365939758133314e-02,2.7527235669603082e-02,










517



     2.5705804008548896e-02,2.3902203305795882e-02,2.2117062707308864e-02,










518



     2.0351096230044517e-02,1.8605121275724643e-02,1.6880083152543166e-02,










519



     1.5177088307935325e-02,1.3497450601739880e-02,1.1842757857907888e-02,










520



     1.0214971439701471e-02,8.6165827693987316e-03,7.0508754713732268e-03,










521



     5.5224032992509968e-03,4.0379725933630305e-03,2.6090727461021627e-03,










522



     1.2602859304985975e-03]










523












524



## Tables for exponential variates










525












526



const ke =










527



    UInt64[0x000e290a13924be3,0x0000000000000000,0x0009beadebce18bf,0x000c377ac71f9e08,










528



           0x000d4ddb99075857,0x000de893fb8ca23e,0x000e4a8e87c4328d,0x000e8dff16ae1cb9,










529



           0x000ebf2deab58c59,0x000ee49a6e8b9638,0x000f0204efd64ee4,0x000f19bdb8ea3c1b,










530



           0x000f2d458bbe5bd1,0x000f3da104b78236,0x000f4b86d784571f,0x000f577ad8a7784f,










531



           0x000f61de83da32ab,0x000f6afb7843cce7,0x000f730a57372b44,0x000f7a37651b0e68,










532



           0x000f80a5bb6eea52,0x000f867189d3cb5b,0x000f8bb1b4f8fbbd,0x000f9079062292b8,










533



           0x000f94d70ca8d43a,0x000f98d8c7dcaa99,0x000f9c8928abe083,0x000f9ff175b734a6,










534



           0x000fa319996bc47d,0x000fa6085f8e9d07,0x000fa8c3a62e1991,0x000fab5084e1f660,










535



           0x000fadb36c84cccb,0x000faff041086846,0x000fb20a6ea22bb9,0x000fb404fb42cb3c,










536



           0x000fb5e295158173,0x000fb7a59e99727a,0x000fb95038c8789d,0x000fbae44ba684eb,










537



           0x000fbc638d822e60,0x000fbdcf89209ffa,0x000fbf29a303cfc5,0x000fc0731df1089c,










538



           0x000fc1ad1ed6c8b1,0x000fc2d8b02b5c89,0x000fc3f6c4d92131,0x000fc5083ac9ba7d,










539



           0x000fc60ddd1e9cd6,0x000fc7086622e825,0x000fc7f881009f0b,0x000fc8decb41ac70,










540



           0x000fc9bbd623d7ec,0x000fca9027c5b26d,0x000fcb5c3c319c49,0x000fcc20864b4449,










541



           0x000fccdd70a35d40,0x000fcd935e34bf80,0x000fce42ab0db8bd,0x000fceebace7ec01,










542



           0x000fcf8eb3b0d0e7,0x000fd02c0a049b60,0x000fd0c3f59d199c,0x000fd156b7b5e27e,










543



           0x000fd1e48d670341,0x000fd26daff73551,0x000fd2f2552684be,0x000fd372af7233c1,










544



           0x000fd3eeee528f62,0x000fd4673e73543a,0x000fd4dbc9e72ff7,0x000fd54cb856dc2c,










545



           0x000fd5ba2f2c4119,0x000fd62451ba02c2,0x000fd68b415fcff4,0x000fd6ef1dabc160,










546



           0x000fd75004790eb6,0x000fd7ae120c583f,0x000fd809612dbd09,0x000fd8620b40effa,










547



           0x000fd8b8285b78fd,0x000fd90bcf594b1d,0x000fd95d15efd425,0x000fd9ac10bfa70c,










548



           0x000fd9f8d364df06,0x000fda437086566b,0x000fda8bf9e3c9fe,0x000fdad28062fed5,










549



           0x000fdb17141bff2c,0x000fdb59c4648085,0x000fdb9a9fda83cc,0x000fdbd9b46e3ed4,










550



           0x000fdc170f6b5d04,0x000fdc52bd81a3fb,0x000fdc8ccacd07ba,0x000fdcc542dd3902,










551



           0x000fdcfc30bcb793,0x000fdd319ef77143,0x000fdd6597a0f60b,0x000fdd98245a48a2,










552



           0x000fddc94e575271,0x000fddf91e64014f,0x000fde279ce914ca,0x000fde54d1f0a06a,










553



           0x000fde80c52a47cf,0x000fdeab7def394e,0x000fded50345eb35,0x000fdefd5be59fa0,










554



           0x000fdf248e39b26f,0x000fdf4aa064b4af,0x000fdf6f98435894,0x000fdf937b6f30ba,










555



           0x000fdfb64f414571,0x000fdfd818d48262,0x000fdff8dd07fed8,0x000fe018a08122c4,










556



           0x000fe03767adaa59,0x000fe05536c58a13,0x000fe07211ccb4c5,0x000fe08dfc94c532,










557



           0x000fe0a8fabe8ca1,0x000fe0c30fbb87a5,0x000fe0dc3ecf3a5a,0x000fe0f48b107521,










558



           0x000fe10bf76a82ef,0x000fe122869e41ff,0x000fe1383b4327e1,0x000fe14d17c83187,










559



           0x000fe1611e74c023,0x000fe1745169635a,0x000fe186b2a09176,0x000fe19843ef4e07,










560



           0x000fe1a90705bf63,0x000fe1b8fd6fb37c,0x000fe1c828951443,0x000fe1d689ba4bfd,










561



           0x000fe1e4220099a4,0x000fe1f0f26655a0,0x000fe1fcfbc726d4,0x000fe2083edc2830,










562



           0x000fe212bc3bfeb4,0x000fe21c745adfe3,0x000fe225678a8895,0x000fe22d95fa23f4,










563



           0x000fe234ffb62282,0x000fe23ba4a800d9,0x000fe2418495fddc,0x000fe2469f22bffb,










564



           0x000fe24af3cce90d,0x000fe24e81ee9858,0x000fe25148bcda19,0x000fe253474703fe,










565



           0x000fe2547c75fdc6,0x000fe254e70b754f,0x000fe25485a0fd1a,0x000fe25356a71450,










566



           0x000fe2515864173a,0x000fe24e88f316f1,0x000fe24ae64296fa,0x000fe2466e132f60,










567



           0x000fe2411df611bd,0x000fe23af34b6f73,0x000fe233eb40bf41,0x000fe22c02cee01b,










568



           0x000fe22336b81710,0x000fe2198385e5cc,0x000fe20ee586b707,0x000fe20358cb5dfb,










569



           0x000fe1f6d92465b1,0x000fe1e9621f2c9e,0x000fe1daef02c8da,0x000fe1cb7accb0a6,










570



           0x000fe1bb002d22c9,0x000fe1a9798349b8,0x000fe196e0d9140c,0x000fe1832fdebc44,










571



           0x000fe16e5fe5f931,0x000fe15869dccfcf,0x000fe1414647fe78,0x000fe128ed3cf8b2,










572



           0x000fe10f565b69cf,0x000fe0f478c633ab,0x000fe0d84b1bdd9e,0x000fe0bac36e6688,










573



           0x000fe09bd73a6b5b,0x000fe07b7b5d920a,0x000fe059a40c26d2,0x000fe03644c5d7f8,










574



           0x000fe011504979b2,0x000fdfeab887b95c,0x000fdfc26e94a447,0x000fdf986297e305,










575



           0x000fdf6c83bb8663,0x000fdf3ec0193eed,0x000fdf0f04a5d30a,0x000fdedd3d1aa204,










576



           0x000fdea953dcfc13,0x000fde7331e3100d,0x000fde3abe9626f2,0x000fddffdfb1dbd5,










577



           0x000fddc2791ff351,0x000fdd826cd068c6,0x000fdd3f9a8d3856,0x000fdcf9dfc95b0c,










578



           0x000fdcb1176a55fe,0x000fdc65198ba50b,0x000fdc15bb3b2daa,0x000fdbc2ce2dc4ae,










579



           0x000fdb6c206aaaca,0x000fdb117becb4a1,0x000fdab2a6379bf0,0x000fda4f5fdfb4e9,










580



           0x000fd9e76401f3a3,0x000fd97a67a9ce1f,0x000fd90819221429,0x000fd8901f2d4b02,










581



           0x000fd812182170e1,0x000fd78d98e23cd3,0x000fd7022bb3f082,0x000fd66f4edf96b9,










582



           0x000fd5d473200305,0x000fd530f9ccff94,0x000fd48432b7b351,0x000fd3cd59a8469e,










583



           0x000fd30b9368f90a,0x000fd23dea45f500,0x000fd16349e2e04a,0x000fd07a7a3ef98a,










584



           0x000fcf8219b5df05,0x000fce7895bcfcde,0x000fcd5c220ad5e2,0x000fcc2aadbc17dc,










585



           0x000fcae1d5e81fbc,0x000fc97ed4e778f9,0x000fc7fe6d4d720e,0x000fc65ccf39c2fc,










586



           0x000fc4957623cb03,0x000fc2a2fc826dc7,0x000fc07ee19b01cd,0x000fbe213c1cf493,










587



           0x000fbb8051ac1566,0x000fb890078d120e,0x000fb5411a5b9a95,0x000fb18000547133,










588



           0x000fad334827f1e2,0x000fa839276708b9,0x000fa263b32e37ed,0x000f9b72d1c52cd1,










589



           0x000f930a1a281a05,0x000f889f023d820a,0x000f7b577d2be5f3,0x000f69c650c40a8f,










590



           0x000f51530f0916d8,0x000f2cb0e3c5933e,0x000eeefb15d605d8,0x000e6da6ecf27460]










591












592



const we =










593



    [1.9311480126418366e-15,1.4178028487910829e-17,2.3278824993382448e-17,










594



     3.0487830247064320e-17,3.6665697714474878e-17,4.2179302189289733e-17,










595



     4.7222561556862764e-17,5.1911915446217879e-17,5.6323471083955047e-17,










596



     6.0510082606427647e-17,6.4510165096727506e-17,6.8352646803700541e-17,










597



     7.2059939574689050e-17,7.5649815537392981e-17,7.9136643961951065e-17,










598



     8.2532235563518929e-17,8.5846436168850513e-17,8.9087554865647428e-17,










599



     9.2262679629663719e-17,9.5377914505292719e-17,9.8438560874559257e-17,










600



     1.0144925809006294e-16,1.0441409405585343e-16,1.0733669323436384e-16,










601



     1.1022028745670189e-16,1.1306777346479334e-16,1.1588176009705533e-16,










602



     1.1866460730417886e-16,1.2141845865694359e-16,1.2414526862326387e-16,










603



     1.2684682560606153e-16,1.2952477151912284e-16,1.3218061851538810e-16,










604



     1.3481576335745444e-16,1.3743149982367625e-16,1.4002902946807859e-16,










605



     1.4260947099321287e-16,1.4517386844829297e-16,1.4772319842763584e-16,










606



     1.5025837641447456e-16,1.5278026239101652e-16,1.5528966581595696e-16,










607



     1.5778735005459581e-16,1.6027403633350909e-16,1.6275040728083524e-16,










608



     1.6521711010420076e-16,1.6767475945078279e-16,1.7012393998770646e-16,










609



     1.7256520873568226e-16,1.7499909718432365e-16,1.7742611321380505e-16,










610



     1.7984674284430714e-16,1.8226145183195818e-16,1.8467068712763576e-16,










611



     1.8707487821298258e-16,1.8947443832625899e-16,1.9186976558915995e-16,










612



     1.9426124404443042e-16,1.9664924461299023e-16,1.9903412597830144e-16,










613



     2.0141623540485899e-16,2.0379590949693882e-16,2.0617347490308439e-16,










614



     2.0854924897123771e-16,2.1092354035891528e-16,2.1329664960238294e-16,










615



     2.1566886964838970e-16,2.1804048635167009e-16,2.2041177894111562e-16,










616



     2.2278302045723950e-16,2.2515447816331350e-16,2.2752641393233694e-16,










617



     2.2989908461180186e-16,2.3227274236804366e-16,2.3464763501180916e-16,










618



     2.3702400630653389e-16,2.3940209626069303e-16,2.4178214140547710e-16,










619



     2.4416437505894123e-16,2.4654902757768304e-16,2.4893632659702250e-16,










620



     2.5132649726057970e-16,2.5371976244007951e-16,2.5611634294614988e-16,










621



     2.5851645773082391e-16,2.6092032408240577e-16,2.6332815781331452e-16,










622



     2.6574017344147618e-16,2.6815658436579989e-16,2.7057760303623509e-16,










623



     2.7300344111887955e-16,2.7543430965657619e-16,2.7787041922541278e-16,










624



     2.8031198008751431e-16,2.8275920234049704e-16,2.8521229606393309e-16,










625



     2.8767147146315804e-16,2.9013693901073754e-16,2.9260890958589514e-16,










626



     2.9508759461219033e-16,2.9757320619372521e-16,3.0006595725014739e-16,










627



     3.0256606165070789e-16,3.0507373434762511e-16,3.0758919150899939e-16,










628



     3.1011265065151543e-16,3.1264433077316750e-16,3.1518445248623523e-16,










629



     3.1773323815073683e-16,3.2029091200858335e-16,3.2285770031865573e-16,










630



     3.2543383149302610e-16,3.2801953623454359e-16,3.3061504767600738e-16,










631



     3.3322060152114841e-16,3.3583643618764577e-16,3.3846279295240445e-16,










632



     3.4109991609932597e-16,3.4374805306980633e-16,3.4640745461620167e-16,










633



     3.4907837495850680e-16,3.5176107194449828e-16,3.5445580721360130e-16,










634



     3.5716284636474652e-16,3.5988245912849274e-16,3.6261491954370031e-16,










635



     3.6536050613905045e-16,3.6811950211971757e-16,3.7089219555951389e-16,










636



     3.7367887959883854e-16,3.7647985264877841e-16,3.7929541860172334e-16,










637



     3.8212588704887531e-16,3.8497157350504876e-16,3.8783279964117988e-16,










638



     3.9070989352498183e-16,3.9360318987020748e-16,3.9651303029500381e-16,










639



     3.9943976358986842e-16,4.0238374599574693e-16,4.0534534149283966e-16,










640



     4.0832492210071775e-16,4.1132286819038357e-16,4.1433956880894741e-16,










641



     4.1737542201763194e-16,4.2043083524385856e-16,4.2350622564821518e-16,










642



     4.2660202050715582e-16,4.2971865761233266e-16,4.3285658568752094e-16,










643



     4.3601626482415681e-16,4.3919816693657415e-16,4.4240277623809919e-16,










644



     4.4563058973923611e-16,4.4888211776926172e-16,4.5215788452263475e-16,










645



     4.5545842863172421e-16,4.5878430376746227e-16,4.6213607926964266e-16,










646



     4.6551434080870692e-16,4.6891969108099157e-16,4.7235275053955480e-16,










647



     4.7581415816285534e-16,4.7930457226372470e-16,4.8282467134125866e-16,










648



     4.8637515497845119e-16,4.8995674478861404e-16,4.9357018541385775e-16,










649



     4.9721624557917034e-16,5.0089571920591141e-16,5.0460942658884340e-16,










650



     5.0835821564116245e-16,5.1214296321235415e-16,5.1596457648410618e-16,










651



     5.1982399444994938e-16,5.2372218948478484e-16,5.2766016901098856e-16,










652



     5.3163897726836902e-16,5.3565969719590503e-16,5.3972345243389779e-16,










653



     5.4383140945596370e-16,5.4798477984116296e-16,5.5218482269752343e-16,










654



     5.5643284724928722e-16,5.6073021560139669e-16,5.6507834569605064e-16,










655



     5.6947871447763482e-16,5.7393286128396354e-16,5.7844239148359912e-16,










656



     5.8300898038105864e-16,5.8763437741400573e-16,5.9232041066909314e-16,










657



     5.9706899174600906e-16,6.0188212100252363e-16,6.0676189321700068e-16,










658



     6.1171050370897217e-16,6.1673025496306200e-16,6.2182356380685327e-16,










659



     6.2699296919933262e-16,6.3224114069342115e-16,6.3757088764394262e-16,










660



     6.4298516924135947e-16,6.4848710546189033e-16,6.5407998903644809e-16,










661



     6.5976729855445663e-16,6.6555271283433428e-16,6.7144012671064882e-16,










662



     6.7743366840910103e-16,6.8353771870512740e-16,6.8975693209068478e-16,










663



     6.9609626020748846e-16,7.0256097784459588e-16,7.0915671184495837e-16,










664



     7.1588947332085531e-16,7.2276569364381212e-16,7.2979226475290851e-16,










665



     7.3697658441912426e-16,7.4432660721604146e-16,7.5185090208325131e-16,










666



     7.5955871753377488e-16,7.6746005575784274e-16,7.7556575712157906e-16,










667



     7.8388759686228577e-16,7.9243839615735500e-16,8.0123215021130834e-16,










668



     8.1028417659131464e-16,8.1961128778061250e-16,8.2923199285818092e-16,










669



     8.3916673441467979e-16,8.4943816836487701e-16,8.6007149633349414e-16,










670



     8.7109486293879040e-16,8.8253983380721398e-16,8.9444197485198646e-16,










671



     9.0684155971316690e-16,9.1978444098118649e-16,9.3332313294229516e-16,










672



     9.4751817065249841e-16,9.6243983456584759e-16,9.7817036547844198e-16,










673



     9.9480684723838795e-16,1.0124650144288319e-15,1.0312843657756166e-15,










674



     1.0514351604044550e-15,1.0731281954224043e-15,1.0966288068517408e-15,










675



     1.1222774909350319e-15,1.1505212963006663e-15,1.1819635283304206e-15,










676



     1.2174462832361815e-15,1.2581958069755114e-15,1.3060984107128082e-15,










677



     1.3642786158057857e-15,1.4384889932178723e-15,1.5412190700064194e-15,










678



     1.7091034077168055e-15]










679



const fe =










680



    [1.0000000000000000e+00,9.3814368086217470e-01,9.0046992992574648e-01,










681



     8.7170433238120359e-01,8.4778550062398961e-01,8.2699329664305032e-01,










682



     8.0842165152300838e-01,7.9152763697249562e-01,7.7595685204011555e-01,










683



     7.6146338884989628e-01,7.4786862198519510e-01,7.3503809243142348e-01,










684



     7.2286765959357202e-01,7.1127476080507601e-01,7.0019265508278816e-01,










685



     6.8956649611707799e-01,6.7935057226476536e-01,6.6950631673192473e-01,










686



     6.6000084107899970e-01,6.5080583341457110e-01,6.4189671642726609e-01,










687



     6.3325199421436607e-01,6.2485273870366598e-01,6.1668218091520766e-01,










688



     6.0872538207962201e-01,6.0096896636523223e-01,5.9340090169173343e-01,










689



     5.8601031847726803e-01,5.7878735860284503e-01,5.7172304866482582e-01,










690



     5.6480919291240017e-01,5.5803828226258745e-01,5.5140341654064129e-01,










691



     5.4489823767243961e-01,5.3851687200286191e-01,5.3225388026304332e-01,










692



     5.2610421398361973e-01,5.2006317736823360e-01,5.1412639381474856e-01,










693



     5.0828977641064288e-01,5.0254950184134772e-01,4.9690198724154955e-01,










694



     4.9134386959403253e-01,4.8587198734188491e-01,4.8048336393045421e-01,










695



     4.7517519303737737e-01,4.6994482528395998e-01,4.6478975625042618e-01,










696



     4.5970761564213769e-01,4.5469615747461550e-01,4.4975325116275500e-01,










697



     4.4487687341454851e-01,4.4006510084235390e-01,4.3531610321563657e-01,










698



     4.3062813728845883e-01,4.2599954114303434e-01,4.2142872899761658e-01,










699



     4.1691418643300288e-01,4.1245446599716118e-01,4.0804818315203240e-01,










700



     4.0369401253053028e-01,3.9939068447523107e-01,3.9513698183329016e-01,










701



     3.9093173698479711e-01,3.8677382908413765e-01,3.8266218149600983e-01,










702



     3.7859575940958079e-01,3.7457356761590216e-01,3.7059464843514600e-01,










703



     3.6665807978151416e-01,3.6276297335481777e-01,3.5890847294874978e-01,










704



     3.5509375286678746e-01,3.5131801643748334e-01,3.4758049462163698e-01,










705



     3.4388044470450241e-01,3.4021714906678002e-01,3.3658991402867761e-01,










706



     3.3299806876180899e-01,3.2944096426413633e-01,3.2591797239355619e-01,










707



     3.2242848495608917e-01,3.1897191284495724e-01,3.1554768522712895e-01,










708



     3.1215524877417955e-01,3.0879406693456019e-01,3.0546361924459026e-01,










709



     3.0216340067569353e-01,2.9889292101558179e-01,2.9565170428126120e-01,










710



     2.9243928816189257e-01,2.8925522348967775e-01,2.8609907373707683e-01,










711



     2.8297041453878075e-01,2.7986883323697292e-01,2.7679392844851736e-01,










712



     2.7374530965280297e-01,2.7072259679906002e-01,2.6772541993204479e-01,










713



     2.6475341883506220e-01,2.6180624268936298e-01,2.5888354974901623e-01,










714



     2.5598500703041538e-01,2.5311029001562946e-01,2.5025908236886230e-01,










715



     2.4743107566532763e-01,2.4462596913189211e-01,2.4184346939887721e-01,










716



     2.3908329026244918e-01,2.3634515245705964e-01,2.3362878343743335e-01,










717



     2.3093391716962741e-01,2.2826029393071670e-01,2.2560766011668407e-01,










718



     2.2297576805812019e-01,2.2036437584335949e-01,2.1777324714870053e-01,










719



     2.1520215107537868e-01,2.1265086199297828e-01,2.1011915938898826e-01,










720



     2.0760682772422204e-01,2.0511365629383771e-01,2.0263943909370902e-01,










721



     2.0018397469191127e-01,1.9774706610509887e-01,1.9532852067956322e-01,










722



     1.9292814997677135e-01,1.9054576966319539e-01,1.8818119940425432e-01,










723



     1.8583426276219711e-01,1.8350478709776746e-01,1.8119260347549629e-01,










724



     1.7889754657247831e-01,1.7661945459049488e-01,1.7435816917135349e-01,










725



     1.7211353531532006e-01,1.6988540130252766e-01,1.6767361861725019e-01,










726



     1.6547804187493600e-01,1.6329852875190182e-01,1.6113493991759203e-01,










727



     1.5898713896931421e-01,1.5685499236936523e-01,1.5473836938446808e-01,










728



     1.5263714202744286e-01,1.5055118500103989e-01,1.4848037564386679e-01,










729



     1.4642459387834494e-01,1.4438372216063478e-01,1.4235764543247220e-01,










730



     1.4034625107486245e-01,1.3834942886358020e-01,1.3636707092642886e-01,










731



     1.3439907170221363e-01,1.3244532790138752e-01,1.3050573846833077e-01,










732



     1.2858020454522817e-01,1.2666862943751067e-01,1.2477091858083096e-01,










733



     1.2288697950954514e-01,1.2101672182667483e-01,1.1916005717532768e-01,










734



     1.1731689921155557e-01,1.1548716357863353e-01,1.1367076788274431e-01,










735



     1.1186763167005630e-01,1.1007767640518538e-01,1.0830082545103380e-01,










736



     1.0653700405000166e-01,1.0478613930657017e-01,1.0304816017125772e-01,










737



     1.0132299742595363e-01,9.9610583670637132e-02,9.7910853311492199e-02,










738



     9.6223742550432798e-02,9.4549189376055859e-02,9.2887133556043541e-02,










739



     9.1237516631040155e-02,8.9600281910032858e-02,8.7975374467270218e-02,










740



     8.6362741140756913e-02,8.4762330532368119e-02,8.3174093009632383e-02,










741



     8.1597980709237419e-02,8.0033947542319905e-02,7.8481949201606421e-02,










742



     7.6941943170480503e-02,7.5413888734058410e-02,7.3897746992364746e-02,










743



     7.2393480875708738e-02,7.0901055162371829e-02,6.9420436498728755e-02,










744



     6.7951593421936601e-02,6.6494496385339774e-02,6.5049117786753749e-02,










745



     6.3615431999807334e-02,6.2193415408540995e-02,6.0783046445479633e-02,










746



     5.9384305633420266e-02,5.7997175631200659e-02,5.6621641283742877e-02,










747



     5.5257689676697037e-02,5.3905310196046087e-02,5.2564494593071692e-02,










748



     5.1235237055126281e-02,4.9917534282706372e-02,4.8611385573379497e-02,










749



     4.7316792913181548e-02,4.6033761076175170e-02,4.4762297732943282e-02,










750



     4.3502413568888183e-02,4.2254122413316234e-02,4.1017441380414819e-02,










751



     3.9792391023374125e-02,3.8578995503074857e-02,3.7377282772959361e-02,










752



     3.6187284781931423e-02,3.5009037697397410e-02,3.3842582150874330e-02,










753



     3.2687963508959535e-02,3.1545232172893609e-02,3.0414443910466604e-02,










754



     2.9295660224637393e-02,2.8188948763978636e-02,2.7094383780955800e-02,










755



     2.6012046645134217e-02,2.4942026419731783e-02,2.3884420511558171e-02,










756



     2.2839335406385240e-02,2.1806887504283581e-02,2.0787204072578117e-02,










757



     1.9780424338009743e-02,1.8786700744696030e-02,1.7806200410911362e-02,










758



     1.6839106826039948e-02,1.5885621839973163e-02,1.4945968011691148e-02,










759



     1.4020391403181938e-02,1.3109164931254991e-02,1.2212592426255381e-02,










760



     1.1331013597834597e-02,1.0464810181029979e-02,9.6144136425022099e-03,










761



     8.7803149858089753e-03,7.9630774380170400e-03,7.1633531836349839e-03,










762



     6.3819059373191791e-03,5.6196422072054830e-03,4.8776559835423923e-03,










763



     4.1572951208337953e-03,3.4602647778369040e-03,2.7887987935740761e-03,










764



     2.1459677437189063e-03,1.5362997803015724e-03,9.6726928232717454e-04,










765



     4.5413435384149677e-04]










766












767



## Constants










768












769



const ziggurat_nor_r      = 3.6541528853610087963519472518










770



const ziggurat_nor_inv_r  = inv(ziggurat_nor_r)










771



const ziggurat_exp_r      = 7.6971174701310497140446280481
























    

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