_dot to _realdot

src/AbstractDifferentiation.jl

@@ -221,28 +221,26 @@ _zero(::AbstractVector, d::AbstractMatrix) = zero(similar(d, size(d, 2)))
 _zero(::AbstractMatrix, d::AbstractMatrix) = zero(d)
 _zero(::Any, d::Any) = zero(d)
 
-@inline _dot(x, y) = dot(x, y)
-@inline function _dot(x::AbstractVector, y::UniformScaling)
+# support pullbacks for complex numbers
+@inline _realdot(x, y) = real(dot(x, y))
+@inline _realdot(x::Number, y::Number) = muladd(real(x), real(y), imag(x) * imag(y))
+@inline _realdot(x::Real, y::Number) = x * real(y)
+@inline _realdot(x::Number, y::Real) = real(x) * y
+@inline _realdot(x::Real, y::Real) = x * y
+@inline function _realdot(x::AbstractVector, y::UniformScaling)
     @assert length(x) == 1
-    return @inbounds dot(x[1], y.λ)
+    return @inbounds _realdot(x[1], y.λ)
 end
-@inline function _dot(x::AbstractVector, y::AbstractMatrix)
+@inline function _realdot(x::AbstractVector, y::AbstractMatrix)
     @assert size(y, 2) == 1
-    return dot(x, y)
+    return _realdot(x, vec(y))
+end
+@inline function _realdot(xs::NTuple{N}, ys::NTuple{N}) where {N}
+    return sum(Base.splat(_realdot), zip(xs, ys))
 end
 
 function pullback_function(ab::AbstractBackend, f, xs...)
-    return (ws) -> begin
-        return gradient(lowest(ab), (xs...,) -> begin
-            vs = f(xs...)
-            if ws isa Tuple
-                @assert length(vs) == length(ws)
-                return sum(Base.splat(_dot), zip(ws, vs))
-            else
-                return _dot(vs, ws)
-            end
-        end, xs...)
-    end
+    return (ws) -> gradient(lowest(ab), (xs...,) -> _realdot(ws, f(xs...)), xs...)
 end
 function value_and_pullback_function(
     ab::AbstractBackend,