faster inv for normal sized ComplexF64 (#47255)

* faster `inv` for normal sized `ComplexF64`

base/complex.jl

@@ -472,9 +472,13 @@ function inv(z::Complex{T}) where T<:Union{Float16,Float32}
 end
 function inv(w::ComplexF64)
     c, d = reim(w)
-    (isinf(c) | isinf(d)) && return complex(copysign(0.0, c), flipsign(-0.0, d))
     absc, absd = abs(c), abs(d)
-    cd = ifelse(absc>absd, absc, absd) # cheap `max`: don't need sign- and nan-checks here
+    cd, dc = ifelse(absc>absd, (absc, absd), (absd, absc))
+    # no overflow from abs2
+    if sqrt(floatmin(Float64)/2) <= cd <= sqrt(floatmax(Float64)/2)
+        return conj(w) / muladd(cd, cd, dc*dc)
+    end
+    (isinf(c) | isinf(d)) && return complex(copysign(0.0, c), flipsign(-0.0, d))
 
     ϵ  = eps(Float64)
     bs = 2/(ϵ*ϵ)
@@ -493,12 +497,13 @@ function inv(w::ComplexF64)
     else
         q, p = robust_cinv(-d, -c)
     end
-    return ComplexF64(p*s, q*s) # undo scaling
+    return ComplexF64(p*s, q*s)
 end
 function robust_cinv(c::Float64, d::Float64)
     r = d/c
-    p = inv(muladd(d, r, c))
-    q = -r*p
+    z = muladd(d, r, c)
+    p = 1.0/z
+    q = -r/z
     return p, q
 end
 

doc/src/manual/complex-and-rational-numbers.md

@@ -48,7 +48,7 @@ julia> 3(2 - 5im)^2
 -63 - 60im
 
 julia> 3(2 - 5im)^-1.0
-0.20689655172413796 + 0.5172413793103449im
+0.20689655172413793 + 0.5172413793103449im
 ```
 
 The promotion mechanism ensures that combinations of operands of different types just work: