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Support FFTs
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‎.gitignore‎

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# committed for packages, but should be committed for applications that require a static
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# environment.
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Manifest.toml
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.DS_Store

‎Project.toml‎

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name = "FastTransformsForwardDiff"
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uuid = "77fa7db0-1c81-401d-9fde-3592fc42b8bc"
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authors = ["Sheehan Olver <solver@mac.com>"]
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version = "0.0.1"
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[deps]
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AbstractFFTs = "621f4979-c628-5d54-868e-fcf4e3e8185c"
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ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
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[compat]
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AbstractFFTs = "1"
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FFTW = "1"
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ForwardDiff = "0.10"
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[extras]
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Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
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FFTW = "7a1cc6ca-52ef-59f5-83cd-3a7055c09341"
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[targets]
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test = ["Test", "FFTW"]

‎README.md‎

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# FastTransformsForwardDiff.jl
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A Julia package to support forward-mode auto-differentiation for fast transforms
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[![Build Status](https://github.com/JuliaApproximation/FastTransformsForwardDiff.jl/workflows/CI/badge.svg)](https://github.com/JuliaApproximation/FastTransformsForwardDiff.jl/actions)
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[![codecov](https://codecov.io/gh/JuliaApproximation/FastTransformsForwardDiff.jl/branch/master/graph/badge.svg)](https://codecov.io/gh/JuliaApproximation/FastTransformsForwardDiff.jl)
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A package for forward-mode auto-differentiation for fast transforms. Currently supports the fft:
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```julia
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julia> using FastTransformsForwardDiff: derivative
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julia> θ = range(0,2π; length=n+1)[1:end-1];
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julia> derivative(ω -> fft(exp.(ω .* cos.(θ)))[1]/n, 1)
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0.5651591039924849 + 0.0im
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```

‎src/FastTransformsForwardDiff.jl‎

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module FastTransformsForwardDiff
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using ForwardDiff
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import AbstractFFTs
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import ForwardDiff: value, partials, npartials, Dual, tagtype, derivative, jacobian, gradient
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@inline tagtype(::Complex{T}) where T = tagtype(T)
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@inline tagtype(::Type{Complex{T}}) where T = tagtype(T)
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include("fft.jl")
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end # module FastTransformsForwardDiff

‎src/fft.jl‎

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value(x::Complex{<:Dual}) = Complex(x.re.value, x.im.value)
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partials(x::Complex{<:Dual}, n::Int) = Complex(partials(x.re, n), partials(x.im, n))
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npartials(x::Complex{<:Dual{T,V,N}}) where {T,V,N} = N
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npartials(::Type{<:Complex{<:Dual{T,V,N}}}) where {T,V,N} = N
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# AbstractFFTs.complexfloat(x::AbstractArray{<:Dual}) = float.(x .+ 0im)
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AbstractFFTs.complexfloat(x::AbstractArray{<:Dual}) = AbstractFFTs.complexfloat.(x)
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AbstractFFTs.complexfloat(d::Dual{T,V,N}) where {T,V,N} = convert(Dual{T,float(V),N}, d) + 0im
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AbstractFFTs.realfloat(x::AbstractArray{<:Dual}) = AbstractFFTs.realfloat.(x)
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AbstractFFTs.realfloat(d::Dual{T,V,N}) where {T,V,N} = convert(Dual{T,float(V),N}, d)
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for plan in [:plan_fft, :plan_ifft, :plan_bfft]
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@eval begin
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AbstractFFTs.$plan(x::AbstractArray{<:Dual}, region=1:ndims(x)) =
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AbstractFFTs.$plan(value.(x), region)
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AbstractFFTs.$plan(x::AbstractArray{<:Complex{<:Dual}}, region=1:ndims(x)) =
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AbstractFFTs.$plan(value.(x), region)
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end
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end
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# rfft only accepts real arrays
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AbstractFFTs.plan_rfft(x::AbstractArray{<:Dual}, region=1:ndims(x)) =
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AbstractFFTs.plan_rfft(value.(x), region)
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for plan in [:plan_irfft, :plan_brfft] # these take an extra argument, only when complex?
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@eval begin
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AbstractFFTs.$plan(x::AbstractArray{<:Dual}, region=1:ndims(x)) =
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AbstractFFTs.$plan(value.(x), region)
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AbstractFFTs.$plan(x::AbstractArray{<:Complex{<:Dual}}, d::Integer, region=1:ndims(x)) =
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AbstractFFTs.$plan(value.(x), d, region)
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end
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end
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# for f in (:dct, :idct)
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# pf = Symbol("plan_", f)
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# @eval begin
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# AbstractFFTs.$f(x::AbstractArray{<:Dual}) = $pf(x) * x
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# AbstractFFTs.$f(x::AbstractArray{<:Dual}, region) = $pf(x, region) * x
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# AbstractFFTs.$pf(x::AbstractArray{<:Dual}, region; kws...) = $pf(value.(x), region; kws...)
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# AbstractFFTs.$pf(x::AbstractArray{<:Complex}, region; kws...) = $pf(value.(x), region; kws...)
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# end
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# end
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for P in [:Plan, :ScaledPlan] # need ScaledPlan to avoid ambiguities
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@eval begin
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Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:Dual}) =
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_apply_plan(p, x)
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Base.:*(p::AbstractFFTs.$P, x::AbstractArray{<:Complex{<:Dual}}) =
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_apply_plan(p, x)
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end
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end
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function _apply_plan(p::AbstractFFTs.Plan, x::AbstractArray)
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xtil = p * value.(x)
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dxtils = ntuple(npartials(eltype(x))) do n
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p * partials.(x, n)
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end
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__apply_plan(tagtype(eltype(x)), xtil, dxtils)
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end
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function __apply_plan(T, xtil, dxtils)
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map(xtil, dxtils...) do val, parts...
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Complex(
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Dual{T}(real(val), map(real, parts)),
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Dual{T}(imag(val), map(imag, parts)),
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)
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end
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end

‎test/runtests.jl‎

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using FastTransformsForwardDiff, FFTW, Test
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using ForwardDiff: Dual, valtype, value, partials, derivative
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using AbstractFFTs: complexfloat, realfloat
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@testset "fft and rfft" begin
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x1 = Dual.(1:4.0, 2:5, 3:6)
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@test value.(x1) == 1:4
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@test partials.(x1, 1) == 2:5
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@test complexfloat(x1)[1] === complexfloat(x1[1]) === Dual(1.0, 2.0, 3.0) + 0im
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@test realfloat(x1)[1] === realfloat(x1[1]) === Dual(1.0, 2.0, 3.0)
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@test fft(x1, 1)[1] isa Complex{<:Dual}
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@testset "$f" for f in [fft, ifft, rfft, bfft]
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@test value.(f(x1)) == f(value.(x1))
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@test partials.(f(x1), 1) == f(partials.(x1, 1))
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end
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f = x -> real(fft([x; 0; 0])[1])
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@test derivative(f,0.1) ≈ 1
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r = x -> real(rfft([x; 0; 0])[1])
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@test derivative(r,0.1) ≈ 1
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n = 100
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θ = range(0,2π; length=n+1)[1:end-1]
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# emperical from Mathematical
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@test derivative(ω -> fft(exp.(ω .* cos.(θ)))[1]/n, 1) ≈ 0.565159103992485
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# c = x -> dct([x; 0; 0])[1]
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# @test derivative(c,0.1) ≈ 1
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end

‎workflows/CompatHelper.yml‎

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name: CompatHelper
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on:
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schedule:
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- cron: 0 0 * * *
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workflow_dispatch:
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permissions:
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contents: write
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pull-requests: write
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jobs:
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CompatHelper:
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runs-on: ubuntu-latest
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steps:
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- name: Check if Julia is already available in the PATH
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id: julia_in_path
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run: which julia
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continue-on-error: true
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- name: Install Julia, but only if it is not already available in the PATH
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uses: julia-actions/setup-julia@v1
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with:
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version: '1'
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arch: ${{ runner.arch }}
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if: steps.julia_in_path.outcome != 'success'
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- name: "Add the General registry via Git"
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run: |
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import Pkg
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ENV["JULIA_PKG_SERVER"] = ""
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Pkg.Registry.add("General")
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shell: julia --color=yes {0}
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- name: "Install CompatHelper"
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run: |
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import Pkg
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name = "CompatHelper"
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uuid = "aa819f21-2bde-4658-8897-bab36330d9b7"
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version = "3"
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Pkg.add(; name, uuid, version)
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shell: julia --color=yes {0}
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- name: "Run CompatHelper"
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run: |
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import CompatHelper
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CompatHelper.main()
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shell: julia --color=yes {0}
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env:
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GITHUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
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COMPATHELPER_PRIV: ${{ secrets.DOCUMENTER_KEY }}
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# COMPATHELPER_PRIV: ${{ secrets.COMPATHELPER_PRIV }}

‎workflows/TagBot.yml‎

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name: TagBot
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on:
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issue_comment: # THIS BIT IS NEW
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types:
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- created
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workflow_dispatch:
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jobs:
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TagBot:
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# THIS 'if' LINE IS NEW
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if: github.event_name == 'workflow_dispatch' || github.actor == 'JuliaTagBot'
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# NOTHING BELOW HAS CHANGED
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runs-on: ubuntu-latest
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steps:
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- uses: JuliaRegistries/TagBot@v1
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with:
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token: ${{ secrets.GITHUB_TOKEN }}
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ssh: ${{ secrets.DOCUMENTER_KEY }}

‎workflows/ci.yml‎

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name: CI
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on:
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push:
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branches:
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- master
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pull_request:
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jobs:
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test:
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name: Julia ${{ matrix.version }} - ${{ matrix.os }} - ${{ matrix.arch }} - ${{ github.event_name }}
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runs-on: ${{ matrix.os }}
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strategy:
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fail-fast: false
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matrix:
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version:
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- '1.6'
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- '1' # Latest stable release
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os:
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- ubuntu-latest
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- macOS-latest
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- windows-latest
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arch:
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- x64
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steps:
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- uses: actions/checkout@v2
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- uses: julia-actions/setup-julia@v1
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with:
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version: ${{ matrix.version }}
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arch: ${{ matrix.arch }}
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- uses: actions/cache@v1
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env:
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cache-name: cache-artifacts
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with:
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path: ~/.julia/artifacts
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key: ${{ runner.os }}-test-${{ env.cache-name }}-${{ hashFiles('**/Project.toml') }}
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restore-keys: |
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${{ runner.os }}-test-${{ env.cache-name }}-
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${{ runner.os }}-test-
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${{ runner.os }}-
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- uses: julia-actions/julia-buildpkg@v1
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- uses: julia-actions/julia-runtest@v1
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- uses: julia-actions/julia-processcoverage@v1
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- uses: codecov/codecov-action@v1
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with:
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file: lcov.info

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