Division of complex finite values by infinite complex values should always return zero

[giordano]

Currently

julia> complex(1.0) / (Inf + Inf*im)
NaN + NaN*im

but the result should be a complex 0. According to the comments, division between ComplexF64 is based on https://arxiv.org/abs/1210.4539, but I'm not sure how to fix the algorithm to deal with this.

This is related to #29840 (but that issue has been hidden by using inv in PR #32252). This is not a duplicate of #23134 as that issue was about the case where the numerator is real.

[DilumAluthge]

Fascinating. It gives a complex zero for Float16, Float32, and BigFloat. But for Float64 it gives the NaN's.

julia> complex(Float16(1.0)) / (Float16(Inf) + Float16(Inf)*im)
Float16(0.0) + Float16(0.0)im

julia> complex(Float32(1.0)) / (Float32(Inf) + Float32(Inf)*im)
0.0f0 + 0.0f0im

julia> complex(BigFloat(1.0)) / (BigFloat(Inf) + BigFloat(Inf)*im)
0.0 - 0.0im

julia> complex(Float64(1.0)) / (Float64(Inf) + Float64(Inf)*im)
NaN + NaN*im

[giordano]

That's because complex division for Complex{Union{Float16,Float32}} is defined as

/(z::Complex{T}, w::Complex{T}) where {T<:Union{Float16,Float32}} =
    oftype(z, widen(z)*inv(widen(w)))

and inv can deal with Inf denominator since PR #32050. Division for Complex{BigFloat} has yet another method. Note that the sign of the complex part is different.

Edit: I suspect the sign of the complex part for Complex{Union{Float16,Float32}} is wrong 🤔

julia> inv(Float32(Inf) + Float32(Inf)*im)
0.0f0 - 0.0f0im

julia> complex(Float32(1.0)) * inv(Float32(Inf) + Float32(Inf)*im)
0.0f0 + 0.0f0im