const BESSEL_ORDER_INFO = """
derivatives of Bessel functions with respect to the order are not implemented currently:
https://github.com/JuliaMath/SpecialFunctions.jl/issues/160
"""

const INCOMPLETE_GAMMA_INFO = """
derivatives of the incomplete Gamma functions with respect to parameter `a` are not
implemented currently:
https://github.com/JuliaMath/SpecialFunctions.jl/issues/317
"""

const INCOMPLETE_EXPINT_INFO = """
derivatives of the exponential integral with respect to parameter `ν` are not implemented
currently:
https://github.com/JuliaMath/SpecialFunctions.jl/issues/321
"""

ChainRulesCore.@scalar_rule(airyai(x), airyaiprime(x))
ChainRulesCore.@scalar_rule(airyaix(x), airyaiprimex(x) + sqrt(x) * Ω)
ChainRulesCore.@scalar_rule(airyaiprime(x), x * airyai(x))
ChainRulesCore.@scalar_rule(airyaiprimex(x), x * airyaix(x) + sqrt(x) * Ω)
ChainRulesCore.@scalar_rule(airybi(x), airybiprime(x))
ChainRulesCore.@scalar_rule(airybiprime(x), x * airybi(x))
ChainRulesCore.@scalar_rule(besselj0(x), -besselj1(x))
ChainRulesCore.@scalar_rule(
    besselj1(x),
    (besselj0(x) - besselj(2, x)) / 2,
)
ChainRulesCore.@scalar_rule(bessely0(x), -bessely1(x))
ChainRulesCore.@scalar_rule(
    bessely1(x),
    (bessely0(x) - bessely(2, x)) / 2,
)
ChainRulesCore.@scalar_rule(dawson(x), 1 - (2 * x * Ω))
ChainRulesCore.@scalar_rule(digamma(x), trigamma(x))

# TODO: use `invsqrtπ` if it is added to IrrationalConstants
ChainRulesCore.@scalar_rule(erf(x), (2 * exp(-x^2)) / sqrtπ)
ChainRulesCore.@scalar_rule(erf(x, y), (- (2 * exp(-x^2)) / sqrtπ, (2 * exp(-y^2)) / sqrtπ))
ChainRulesCore.@scalar_rule(erfc(x), - (2 * exp(-x^2)) / sqrtπ)
ChainRulesCore.@scalar_rule(logerfc(x), - (2 * exp(-x^2 - Ω)) / sqrtπ)
ChainRulesCore.@scalar_rule(erfcinv(x), - (sqrtπ * (exp(Ω^2) / 2)))
ChainRulesCore.@scalar_rule(erfcx(x), 2 * (x * Ω - inv(oftype(Ω, sqrtπ))))
ChainRulesCore.@scalar_rule(logerfcx(x), 2 * (x - exp(-Ω) / sqrtπ))
ChainRulesCore.@scalar_rule(erfi(x), (2 * exp(x^2)) / sqrtπ)
ChainRulesCore.@scalar_rule(erfinv(x), sqrtπ * (exp(Ω^2) / 2))

ChainRulesCore.@scalar_rule(gamma(x), Ω * digamma(x))
ChainRulesCore.@scalar_rule(
    gamma(a, x),
    (
        ChainRulesCore.@not_implemented(INCOMPLETE_GAMMA_INFO),
        - exp(-x) * x^(a - 1),
    ),
)
ChainRulesCore.@scalar_rule(
    gamma_inc(a, x, IND),
    @setup(z = exp(-x) * x^(a - 1) / gamma(a)),
    (
        ChainRulesCore.@not_implemented(INCOMPLETE_GAMMA_INFO),
        z,
        ChainRulesCore.NoTangent(),
    ),
    (
        ChainRulesCore.@not_implemented(INCOMPLETE_GAMMA_INFO),
        -z,
        ChainRulesCore.NoTangent(),
    ),
)
ChainRulesCore.@scalar_rule(
    invdigamma(x),
    inv(trigamma(invdigamma(x))),
)
ChainRulesCore.@scalar_rule(trigamma(x), polygamma(2, x))

# Bessel functions
ChainRulesCore.@scalar_rule(
    besselj(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (besselj(ν - 1, x) - besselj(ν + 1, x)) / 2
    ),
)
ChainRulesCore.@scalar_rule(
    besseli(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (besseli(ν - 1, x) + besseli(ν + 1, x)) / 2,
    ),
)
ChainRulesCore.@scalar_rule(
    bessely(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (bessely(ν - 1, x) - bessely(ν + 1, x)) / 2,
    ),
)
ChainRulesCore.@scalar_rule(
    besselk(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        -(besselk(ν - 1, x) + besselk(ν + 1, x)) / 2,
    ),
)
ChainRulesCore.@scalar_rule(
    besselkx(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        -(besselkx(ν - 1, x) + besselkx(ν + 1, x)) / 2 + Ω,
    ),
)
ChainRulesCore.@scalar_rule(
    hankelh1(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (hankelh1(ν - 1, x) - hankelh1(ν + 1, x)) / 2,
    ),
)
ChainRulesCore.@scalar_rule(
    hankelh1x(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (hankelh1x(ν - 1, x) - hankelh1x(ν + 1, x)) / 2 - im * Ω,
    ),
)
ChainRulesCore.@scalar_rule(
    hankelh2(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (hankelh2(ν - 1, x) - hankelh2(ν + 1, x)) / 2,
    ),
)
ChainRulesCore.@scalar_rule(
    hankelh2x(ν, x),
    (
        ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO),
        (hankelh2x(ν - 1, x) - hankelh2x(ν + 1, x)) / 2 + im * Ω,
    ),
)

ChainRulesCore.@scalar_rule(
    polygamma(m, x),
    (
        ChainRulesCore.NoTangent(),
        polygamma(m + 1, x),
    ),
)
# todo: setup for common expr
ChainRulesCore.@scalar_rule(
    beta(a, b),
    (Ω*(digamma(a) - digamma(a + b)),
     Ω*(digamma(b) - digamma(a + b)),)
)
ChainRulesCore.@scalar_rule(
    logbeta(a, b),
    (digamma(a) - digamma(a + b),
     digamma(b) - digamma(a + b),)
)

# actually is the absolute value of the logorithm of gamma paired with sign gamma
ChainRulesCore.@scalar_rule(
    logabsgamma(x),
    digamma(x),
    ChainRulesCore.ZeroTangent()
)

ChainRulesCore.@scalar_rule(loggamma(x), digamma(x))
ChainRulesCore.@scalar_rule(
    loggamma(a, x),
    (
        ChainRulesCore.@not_implemented(INCOMPLETE_GAMMA_INFO),
        -exp(- (x + Ω)) * x^(a - 1),
    )
)

# exponential integrals
ChainRulesCore.@scalar_rule(expint(z), - exp(-z) / z)
ChainRulesCore.@scalar_rule(
    expint(ν, z),
    (
        ChainRulesCore.@not_implemented(INCOMPLETE_EXPINT_INFO),
        - expint(ν - 1, z),
    )
)
ChainRulesCore.@scalar_rule(expintx(z), Ω - inv(z))
ChainRulesCore.@scalar_rule(
    expintx(ν, z),
    (
        ChainRulesCore.@not_implemented(INCOMPLETE_EXPINT_INFO),
        Ω - expintx(ν - 1, z),
    )
)
ChainRulesCore.@scalar_rule(expinti(x), exp(x) / x)
ChainRulesCore.@scalar_rule(sinint(x), sinc(invπ * x))
ChainRulesCore.@scalar_rule(cosint(x), cos(x) / x)

# non-holomorphic functions
function ChainRulesCore.frule((_, Δν, Δx), ::typeof(besselix), ν::Number, x::Number)
    # primal
    Ω = besselix(ν, x)

    # derivative
    ∂Ω_∂ν = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
    a = (besselix(ν - 1, x) + besselix(ν + 1, x)) / 2
    ΔΩ = if Δx isa Real
        muladd(muladd(-sign(real(x)), Ω, a), Δx, ∂Ω_∂ν * Δν)
    else
        muladd(a, Δx, muladd(-sign(real(x)) * real(Δx), Ω, ∂Ω_∂ν * Δν))
    end

    return Ω, ΔΩ
end
function ChainRulesCore.rrule(::typeof(besselix), ν::Number, x::Number)
    Ω = besselix(ν, x)
    project_x = ChainRulesCore.ProjectTo(x)
    function besselix_pullback(ΔΩ)
        ν̄ = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
        a = (besselix(ν - 1, x) + besselix(ν + 1, x)) / 2
        x̄ = project_x(muladd(conj(a), ΔΩ, - sign(real(x)) * real(conj(Ω) * ΔΩ)))
        return ChainRulesCore.NoTangent(), ν̄, x̄
    end
    return Ω, besselix_pullback
end

function ChainRulesCore.frule((_, Δν, Δx), ::typeof(besseljx), ν::Number, x::Number)
    # primal
    Ω = besseljx(ν, x)

    # derivative
    ∂Ω_∂ν = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
    a = (besseljx(ν - 1, x) - besseljx(ν + 1, x)) / 2
    ΔΩ = if Δx isa Real
        muladd(a, Δx, ∂Ω_∂ν * Δν)
    else
        muladd(a, Δx, muladd(-sign(imag(x)) * imag(Δx), Ω, ∂Ω_∂ν * Δν))
    end

    return Ω, ΔΩ
end
function ChainRulesCore.rrule(::typeof(besseljx), ν::Number, x::Number)
    Ω = besseljx(ν, x)
    project_x = ChainRulesCore.ProjectTo(x)
    function besseljx_pullback(ΔΩ)
        ν̄ = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
        a = (besseljx(ν - 1, x) - besseljx(ν + 1, x)) / 2
        x̄ = if x isa Real
            project_x(a * ΔΩ)
        else
            project_x(muladd(conj(a), ΔΩ, - sign(imag(x)) * real(conj(Ω) * ΔΩ) * im))
        end
        return ChainRulesCore.NoTangent(), ν̄, x̄
    end
    return Ω, besseljx_pullback
end

function ChainRulesCore.frule((_, Δν, Δx), ::typeof(besselyx), ν::Number, x::Number)
    # primal
    Ω = besselyx(ν, x)

    # derivative
    ∂Ω_∂ν = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
    a = (besselyx(ν - 1, x) - besselyx(ν + 1, x)) / 2
    ΔΩ = if Δx isa Real
        muladd(a, Δx, ∂Ω_∂ν * Δν)
    else
        muladd(a, Δx, muladd(-sign(imag(x)) * imag(Δx), Ω, ∂Ω_∂ν * Δν))
    end

    return Ω, ΔΩ
end
function ChainRulesCore.rrule(::typeof(besselyx), ν::Number, x::Number)
    Ω = besselyx(ν, x)
    project_x = ChainRulesCore.ProjectTo(x)
    function besselyx_pullback(ΔΩ)
        ν̄ = ChainRulesCore.@not_implemented(BESSEL_ORDER_INFO)
        a = (besselyx(ν - 1, x) - besselyx(ν + 1, x)) / 2
        x̄ = if x isa Real
            project_x(a * ΔΩ)
        else
            project_x(muladd(conj(a), ΔΩ, - sign(imag(x)) * real(conj(Ω) * ΔΩ) * im))
        end
        return ChainRulesCore.NoTangent(), ν̄, x̄
    end
    return Ω, besselyx_pullback
end
