Correct chainrules for abs2, abs, conj and angle

src/rulesets/Base/fastmath_able.jl

@@ -37,9 +37,16 @@ let
 
         # Unary complex functions
         @scalar_rule abs(x::Real) sign(x)
-        @scalar_rule abs2(x) 2x
+        @scalar_rule abs2(x::Real) 2x
         @scalar_rule angle(x::Real) Zero()
         @scalar_rule conj(x::Real) One()
+        function frule((Δx,), abs, x::ComplexF64)
+            Ω = abs(x)
+            return Ω, (real(x) * real(Δx) + imag(x) * imag(Δx)) / Ω
+        end
+        function frule((Δx,), abs2, x::ComplexF64)
+            return abs2(x), 2 * (real(x) * real(Δx) + imag(x) * imag(Δx))
+        end        
 
         # Binary functions
         @scalar_rule hypot(x::Real, y::Real) (x / Ω, y / Ω)