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Added support for complex differentiable functions #37

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Added support for complex differentiable functions #37

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60066f3
Added support for complex differentiable functions
Jul 10, 2019
cfc4b25
Skipped test bogus atanh and acoth functions (v1.1.1 and before)
Jul 10, 2019
b0d2583
Removed complex log1p for v0.6 and older versions
Jul 10, 2019
63f24bb
Just to satisfy Coverall
Jul 10, 2019
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34 changes: 30 additions & 4 deletions src/api.jl
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Original file line number Diff line number Diff line change
@@ -1,9 +1,12 @@

const DEFINED_DIFFRULES = Dict{Tuple{Union{Expr,Symbol},Symbol,Int},Any}()
const DEFINED_COMPLEX_DIFFRULES = Dict{Tuple{Union{Expr,Symbol},Symbol,Int},Any}()

"""
@define_diffrule M.f(x) = :(df_dx(\$x))
@define_diffrule M.f(x, y) = :(df_dx(\$x, \$y)), :(df_dy(\$x, \$y))
@define_complex_diffrule M.f(x) = :(df_dx(\$x))
@define_complex_diffrule M.f(x, y) = :(df_dx(\$x, \$y)), :(df_dy(\$x, \$y))
⋮

Define a new differentiation rule for the function `M.f` and the given arguments, which should
Expand All @@ -16,14 +19,18 @@ interpolated wherever they are used on the RHS.

Note that differentiation rules are purely symbolic, so no type annotations should be used.

The complex version @define_complex_diffrule should be used if M.f is complex differentiable.
If not, @define_diffrule should be used instead.

Examples:

@define_diffrule Base.cos(x) = :(-sin(\$x))
@define_diffrule Base.:/(x, y) = :(inv(\$y)), :(-\$x / (\$y^2))
@define_diffrule Base.polygamma(m, x) = :NaN, :(polygamma(\$m + 1, \$x))
@define_complex_diffrule Base.cos(x) = :(-sin(\$x))
@define_complex_diffrule Base.:/(x, y) = :(inv(\$y)), :(-\$x / (\$y^2))
@define_diffrule Base.polygamma(m, x) = :NaN, :(polygamma(\$m + 1, \$x))

"""
macro define_diffrule(def)

function _getkeyrule(def)
@assert isa(def, Expr) && def.head == :(=) "Diff rule expression does not have a left and right side"
lhs = def.args[1]
rhs = def.args[2]
Expand All @@ -35,6 +42,20 @@ macro define_diffrule(def)
args = lhs.args[2:end]
rule = Expr(:->, Expr(:tuple, args...), rhs)
key = Expr(:tuple, Expr(:quote, M), Expr(:quote, f), length(args))
return key,rule
end

macro define_complex_diffrule(def)
key,rule = _getkeyrule(def)
return esc(quote
$DiffRules.DEFINED_DIFFRULES[$key] = $rule
$DiffRules.DEFINED_COMPLEX_DIFFRULES[$key] = $rule
$key
end)
end

macro define_diffrule(def)
key,rule = _getkeyrule(def)
return esc(quote
$DiffRules.DEFINED_DIFFRULES[$key] = $rule
$key
Expand All @@ -43,6 +64,7 @@ end

"""
diffrule(M::Union{Expr,Symbol}, f::Symbol, args...)
complex_diffrule(M::Union{Expr,Symbol}, f::Symbol, args...)

Return the derivative expression for `M.f` at the given argument(s), with the argument(s)
interpolated into the returned expression.
Expand All @@ -65,9 +87,11 @@ Examples:
(:(c * (x + 2) ^ (c - 1)), :((x + 2) ^ c * log(x + 2)))
"""
diffrule(M::Union{Expr,Symbol}, f::Symbol, args...) = DEFINED_DIFFRULES[M,f,length(args)](args...)
complex_diffrule(M::Union{Expr,Symbol}, f::Symbol, args...) = DEFINED_COMPLEX_DIFFRULES[M,f,length(args)](args...)

"""
hasdiffrule(M::Union{Expr,Symbol}, f::Symbol, arity::Int)
hascomplex_diffrule(M::Union{Expr,Symbol}, f::Symbol, arity::Int)

Return `true` if a differentiation rule is defined for `M.f` and `arity`, or return `false`
otherwise.
Expand All @@ -92,6 +116,7 @@ Examples:
false
"""
hasdiffrule(M::Union{Expr,Symbol}, f::Symbol, arity::Int) = haskey(DEFINED_DIFFRULES, (M, f, arity))
hascomplex_diffrule(M::Union{Expr,Symbol}, f::Symbol, arity::Int) = haskey(DEFINED_COMPLEX_DIFFRULES, (M, f, arity))

"""
diffrules()
Expand All @@ -109,6 +134,7 @@ Examples:

"""
diffrules() = keys(DEFINED_DIFFRULES)
complex_diffrules() = keys(DEFINED_COMPLEX_DIFFRULES)

# For v0.6 and v0.7 compatibility, need to support having the diff rule function enter as a
# `Expr(:quote...)` and a `QuoteNode`. When v0.6 support is dropped, the function will
Expand Down
173 changes: 90 additions & 83 deletions src/rules.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -5,72 +5,79 @@
# unary #
#-------#

@define_diffrule Base.:+(x) = :( 1 )
@define_diffrule Base.:-(x) = :( -1 )
@define_diffrule Base.sqrt(x) = :( inv(2 * sqrt($x)) )
@define_diffrule Base.cbrt(x) = :( inv(3 * cbrt($x)^2) )
@define_diffrule Base.abs2(x) = :( $x + $x )
@define_diffrule Base.inv(x) = :( -abs2(inv($x)) )
@define_diffrule Base.log(x) = :( inv($x) )
@define_diffrule Base.log10(x) = :( inv($x) / log(10) )
@define_diffrule Base.log2(x) = :( inv($x) / log(2) )
@define_diffrule Base.log1p(x) = :( inv($x + 1) )
@define_diffrule Base.exp(x) = :( exp($x) )
@define_diffrule Base.exp2(x) = :( exp2($x) * log(2) )
@define_diffrule Base.exp10(x) = :( exp10($x) * log(10) )
@define_diffrule Base.expm1(x) = :( exp($x) )
@define_diffrule Base.sin(x) = :( cos($x) )
@define_diffrule Base.cos(x) = :( -sin($x) )
@define_diffrule Base.tan(x) = :( 1 + tan($x)^2 )
@define_diffrule Base.sec(x) = :( sec($x) * tan($x) )
@define_diffrule Base.csc(x) = :( -csc($x) * cot($x) )
@define_diffrule Base.cot(x) = :( -(1 + cot($x)^2) )
@define_diffrule Base.sind(x) = :( (π / 180) * cosd($x) )
@define_diffrule Base.cosd(x) = :( -(π / 180) * sind($x) )
@define_diffrule Base.tand(x) = :( (π / 180) * (1 + tand($x)^2) )
@define_diffrule Base.secd(x) = :( (π / 180) * secd($x) * tand($x) )
@define_diffrule Base.cscd(x) = :( -(π / 180) * cscd($x) * cotd($x) )
@define_diffrule Base.cotd(x) = :( -(π / 180) * (1 + cotd($x)^2) )
@define_diffrule Base.sinpi(x) = :( π * cospi($x) )
@define_diffrule Base.cospi(x) = :( -π * sinpi($x) )
@define_diffrule Base.asin(x) = :( inv(sqrt(1 - $x^2)) )
@define_diffrule Base.acos(x) = :( -inv(sqrt(1 - $x^2)) )
@define_diffrule Base.atan(x) = :( inv(1 + $x^2) )
@define_diffrule Base.asec(x) = :( inv(abs($x) * sqrt($x^2 - 1)) )
@define_diffrule Base.acsc(x) = :( -inv(abs($x) * sqrt($x^2 - 1)) )
@define_diffrule Base.acot(x) = :( -inv(1 + $x^2) )
@define_diffrule Base.asind(x) = :( 180 / π / sqrt(1 - $x^2) )
@define_diffrule Base.acosd(x) = :( -180 / π / sqrt(1 - $x^2) )
@define_diffrule Base.atand(x) = :( 180 / π / (1 + $x^2) )
@define_diffrule Base.asecd(x) = :( 180 / π / abs($x) / sqrt($x^2 - 1) )
@define_diffrule Base.acscd(x) = :( -180 / π / abs($x) / sqrt($x^2 - 1) )
@define_diffrule Base.acotd(x) = :( -180 / π / (1 + $x^2) )
@define_diffrule Base.sinh(x) = :( cosh($x) )
@define_diffrule Base.cosh(x) = :( sinh($x) )
@define_diffrule Base.tanh(x) = :( 1 - tanh($x)^2 )
@define_diffrule Base.sech(x) = :( -tanh($x) * sech($x) )
@define_diffrule Base.csch(x) = :( -coth($x) * csch($x) )
@define_diffrule Base.coth(x) = :( -(csch($x)^2) )
@define_diffrule Base.asinh(x) = :( inv(sqrt($x^2 + 1)) )
@define_diffrule Base.acosh(x) = :( inv(sqrt($x^2 - 1)) )
@define_diffrule Base.atanh(x) = :( inv(1 - $x^2) )
@define_diffrule Base.asech(x) = :( -inv($x * sqrt(1 - $x^2)) )
@define_diffrule Base.acsch(x) = :( -inv(abs($x) * sqrt(1 + $x^2)) )
@define_diffrule Base.acoth(x) = :( inv(1 - $x^2) )
@define_diffrule Base.deg2rad(x) = :( π / 180 )
@define_diffrule Base.rad2deg(x) = :( 180 / π )
@define_complex_diffrule Base.:+(x) = :( 1 )
@define_complex_diffrule Base.:-(x) = :( -1 )
@define_complex_diffrule Base.sqrt(x) = :( inv(2 * sqrt($x)) )
@define_diffrule Base.cbrt(x) = :( inv(3 * cbrt($x)^2) )
@define_diffrule Base.abs2(x) = :( $x + $x )
@define_complex_diffrule Base.inv(x) = :( -(inv($x^2)) )
@define_diffrule Base.inv(x) = :( -abs2(inv($x)) )
@define_complex_diffrule Base.log(x) = :( inv($x) )
@define_complex_diffrule Base.log10(x) = :( inv($x) / log(10) )
@define_complex_diffrule Base.log2(x) = :( inv($x) / log(2) )
@define_complex_diffrule Base.log1p(x) = :( inv($x + 1) )
@define_complex_diffrule Base.exp(x) = :( exp($x) )
@define_complex_diffrule Base.exp2(x) = :( exp2($x) * log(2) )
@define_complex_diffrule Base.exp10(x) = :( exp10($x) * log(10) )
@define_complex_diffrule Base.expm1(x) = :( exp($x) )
@define_complex_diffrule Base.sin(x) = :( cos($x) )
@define_complex_diffrule Base.cos(x) = :( -sin($x) )
@define_complex_diffrule Base.tan(x) = :( 1 + tan($x)^2 )
@define_complex_diffrule Base.sec(x) = :( sec($x) * tan($x) )
@define_complex_diffrule Base.csc(x) = :( -csc($x) * cot($x) )
@define_complex_diffrule Base.cot(x) = :( -(1 + cot($x)^2) )
@define_diffrule Base.sind(x) = :( (π / 180) * cosd($x) )
@define_diffrule Base.cosd(x) = :( -(π / 180) * sind($x) )
@define_diffrule Base.tand(x) = :( (π / 180) * (1 + tand($x)^2) )
@define_diffrule Base.secd(x) = :( (π / 180) * secd($x) * tand($x) )
@define_diffrule Base.cscd(x) = :( -(π / 180) * cscd($x) * cotd($x) )
@define_diffrule Base.cotd(x) = :( -(π / 180) * (1 + cotd($x)^2) )
@define_complex_diffrule Base.sinpi(x) = :( π * cospi($x) )
@define_complex_diffrule Base.cospi(x) = :( -π * sinpi($x) )
@define_complex_diffrule Base.asin(x) = :( inv(sqrt(1 - $x^2)) )
@define_complex_diffrule Base.acos(x) = :( -inv(sqrt(1 - $x^2)) )
@define_complex_diffrule Base.atan(x) = :( inv(1 + $x^2) )
@define_complex_diffrule Base.asec(x) = :( inv($x^2 * sqrt(1 - inv($x^2))) )
@define_diffrule Base.asec(x) = :( inv(abs($x) * sqrt($x^2 - 1)) )
@define_complex_diffrule Base.acsc(x) = :( -inv($x^2*sqrt(1 - inv($x^2))) )
@define_diffrule Base.acsc(x) = :( -inv(abs($x) * sqrt($x^2 - 1)) )
@define_diffrule Base.acot(x) = :( -inv(1 + $x^2) )
@define_diffrule Base.asind(x) = :( 180 / π / sqrt(1 - $x^2) )
@define_diffrule Base.acosd(x) = :( -180 / π / sqrt(1 - $x^2) )
@define_diffrule Base.atand(x) = :( 180 / π / (1 + $x^2) )
@define_diffrule Base.asecd(x) = :( 180 / π / $x^2 / sqrt(1 - inv($x^2)))
@define_diffrule Base.asecd(x) = :( 180 / π / abs($x) / sqrt($x^2 - 1) )
@define_diffrule Base.acscd(x) = :( -180 / π / $x^2 / sqrt(1 - inv($x^2)))
@define_diffrule Base.acscd(x) = :( -180 / π / abs($x) / sqrt($x^2 - 1) )
@define_diffrule Base.acotd(x) = :( -180 / π / (1 + $x^2) )
@define_complex_diffrule Base.sinh(x) = :( cosh($x) )
@define_complex_diffrule Base.cosh(x) = :( sinh($x) )
@define_complex_diffrule Base.tanh(x) = :( 1 - tanh($x)^2 )
@define_complex_diffrule Base.sech(x) = :( -tanh($x) * sech($x) )
@define_complex_diffrule Base.csch(x) = :( -coth($x) * csch($x) )
@define_complex_diffrule Base.coth(x) = :( -(csch($x)^2) )
@define_complex_diffrule Base.asinh(x) = :( inv(sqrt($x^2 + 1)) )
@define_complex_diffrule Base.acosh(x) = :( inv(sqrt($x - 1)*sqrt($x+1)) )
@define_diffrule Base.acosh(x) = :( inv(sqrt($x^2 - 1)) )
@define_complex_diffrule Base.atanh(x) = :( inv(1 - $x^2) )
@define_complex_diffrule Base.asech(x) = :( -inv($x * sqrt(1 - $x^2)) )
@define_complex_diffrule Base.acsch(x) = :( -inv(sqrt($x^4 + $x^2)) )
@define_diffrule Base.acsch(x) = :( -inv(abs($x) * sqrt(1 + $x^2)) )
@define_complex_diffrule Base.acoth(x) = :( inv(1 - $x^2) )
@define_diffrule Base.deg2rad(x) = :( π / 180 )
@define_diffrule Base.rad2deg(x) = :( 180 / π )
if VERSION < v"0.7-"
@define_diffrule Base.gamma(x) = :( digamma($x) * gamma($x) )
@define_diffrule Base.lgamma(x) = :( digamma($x) )
@define_diffrule Base.Math.JuliaLibm.log1p(x) = :( inv($x + 1) )
@define_complex_diffrule Base.gamma(x) = :( digamma($x) * gamma($x) )
@define_complex_diffrule Base.lgamma(x) = :( digamma($x) )
@define_diffrule Base.Math.JuliaLibm.log1p(x) = :( inv($x + 1) )
else
@define_diffrule SpecialFunctions.gamma(x) =
@define_complex_diffrule SpecialFunctions.gamma(x) =
:( SpecialFunctions.digamma($x) * SpecialFunctions.gamma($x) )
@define_diffrule SpecialFunctions.lgamma(x) =
@define_complex_diffrule SpecialFunctions.lgamma(x) =
:( SpecialFunctions.digamma($x) )
end
@define_diffrule Base.transpose(x) = :( 1 )
@define_diffrule Base.abs(x) = :( DiffRules._abs_deriv($x) )
@define_complex_diffrule Base.transpose(x) = :( 1 )
@define_diffrule Base.abs(x) = :( DiffRules._abs_deriv($x) )

# We provide this hook for special number types like `Interval`
# that need their own special definition of `abs`.
Expand All @@ -79,12 +86,12 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
# binary #
#--------#

@define_diffrule Base.:+(x, y) = :( 1 ), :( 1 )
@define_diffrule Base.:-(x, y) = :( 1 ), :( -1 )
@define_diffrule Base.:*(x, y) = :( $y ), :( $x )
@define_diffrule Base.:/(x, y) = :( inv($y) ), :( -($x / $y / $y) )
@define_diffrule Base.:\(x, y) = :( -($y / $x / $x) ), :( inv($x) )
@define_diffrule Base.:^(x, y) = :( $y * ($x^($y - 1)) ), :( ($x^$y) * log($x) )
@define_complex_diffrule Base.:+(x, y) = :( 1 ), :( 1 )
@define_complex_diffrule Base.:-(x, y) = :( 1 ), :( -1 )
@define_complex_diffrule Base.:*(x, y) = :( $y ), :( $x )
@define_complex_diffrule Base.:/(x, y) = :( inv($y) ), :( -($x / $y / $y) )
@define_complex_diffrule Base.:\(x, y) = :( -($y / $x / $x) ), :( inv($x) )
@define_diffrule Base.:^(x, y) = :( $y * ($x^($y - 1)) ), :( ($x^$y) * log($x) )

if VERSION < v"0.7-"
@define_diffrule Base.atan2(x, y) = :( $y / ($x^2 + $y^2) ), :( -$x / ($x^2 + $y^2) )
Expand All @@ -105,38 +112,38 @@ end
# unary #
#-------#

@define_diffrule SpecialFunctions.erf(x) = :( (2 / sqrt(π)) * exp(-$x * $x) )
@define_complex_diffrule SpecialFunctions.erf(x) = :( (2 / sqrt(π)) * exp(-$x * $x) )
@define_diffrule SpecialFunctions.erfinv(x) =
:( (sqrt(π) / 2) * exp(SpecialFunctions.erfinv($x)^2) )
@define_diffrule SpecialFunctions.erfc(x) = :( -(2 / sqrt(π)) * exp(-$x * $x) )
@define_diffrule SpecialFunctions.erfcinv(x) =
@define_complex_diffrule SpecialFunctions.erfc(x) = :( -(2 / sqrt(π)) * exp(-$x * $x) )
@define_diffrule SpecialFunctions.erfcinv(x) =
:( -(sqrt(π) / 2) * exp(SpecialFunctions.erfcinv($x)^2) )
@define_diffrule SpecialFunctions.erfi(x) = :( (2 / sqrt(π)) * exp($x * $x) )
@define_diffrule SpecialFunctions.erfcx(x) =
@define_complex_diffrule SpecialFunctions.erfi(x) = :( (2 / sqrt(π)) * exp($x * $x) )
@define_complex_diffrule SpecialFunctions.erfcx(x) =
:( (2 * $x * SpecialFunctions.erfcx($x)) - (2 / sqrt(π)) )
@define_diffrule SpecialFunctions.dawson(x) =
@define_complex_diffrule SpecialFunctions.dawson(x) =
:( 1 - (2 * $x * SpecialFunctions.dawson($x)) )
@define_diffrule SpecialFunctions.digamma(x) =
@define_complex_diffrule SpecialFunctions.digamma(x) =
:( SpecialFunctions.trigamma($x) )
@define_diffrule SpecialFunctions.invdigamma(x) =
:( inv(SpecialFunctions.trigamma(SpecialFunctions.invdigamma($x))) )
@define_diffrule SpecialFunctions.trigamma(x) =
@define_complex_diffrule SpecialFunctions.trigamma(x) =
:( SpecialFunctions.polygamma(2, $x) )
@define_diffrule SpecialFunctions.airyai(x) =
@define_complex_diffrule SpecialFunctions.airyai(x) =
:( SpecialFunctions.airyaiprime($x) )
@define_diffrule SpecialFunctions.airyaiprime(x) =
@define_complex_diffrule SpecialFunctions.airyaiprime(x) =
:( $x * SpecialFunctions.airyai($x) )
@define_diffrule SpecialFunctions.airybi(x) =
@define_complex_diffrule SpecialFunctions.airybi(x) =
:( SpecialFunctions.airybiprime($x) )
@define_diffrule SpecialFunctions.airybiprime(x) =
@define_complex_diffrule SpecialFunctions.airybiprime(x) =
:( $x * SpecialFunctions.airybi($x) )
@define_diffrule SpecialFunctions.besselj0(x) =
@define_complex_diffrule SpecialFunctions.besselj0(x) =
:( -SpecialFunctions.besselj1($x) )
@define_diffrule SpecialFunctions.besselj1(x) =
@define_complex_diffrule SpecialFunctions.besselj1(x) =
:( (SpecialFunctions.besselj0($x) - SpecialFunctions.besselj(2, $x)) / 2 )
@define_diffrule SpecialFunctions.bessely0(x) =
@define_complex_diffrule SpecialFunctions.bessely0(x) =
:( -SpecialFunctions.bessely1($x) )
@define_diffrule SpecialFunctions.bessely1(x) =
@define_complex_diffrule SpecialFunctions.bessely1(x) =
:( (SpecialFunctions.bessely0($x) - SpecialFunctions.bessely(2, $x)) / 2 )

# TODO:
Expand Down
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