Over/underflow in complex power function

[jwmerrill]

Julia's base complex^complex power function may produce NaN for some arguments when the result should be representable because of underflow/overflow at intermediate stages of the computation.

Example:

julia> a=exp(1+im)
1.4686939399158851 + 2.2873552871788423im

julia> b=1e5*(1+im)
100000.0 + 100000.0im

julia> abs(a^b)
NaN

In this case, the naive implementation of complex powers produces a correct result

julia> abs(exp(b*log(a)))
1.0

I suspect that this issue was introduced by #24570.

The example is constructed to cause overflow/underflow in

julia/base/complex.jl

Line 860 in a5d6b50

rᵖ = r^pᵣ * exp(-pᵢ*θ)

by making r^pᵣ very large and exp(-pᵢ*θ) very small.

Version info

julia> versioninfo()
Julia Version 1.10.4
Commit 48d4fd48430 (2024-06-04 10:41 UTC)
Build Info:
  Official https://julialang.org/ release
Platform Info:
  OS: macOS (arm64-apple-darwin22.4.0)
  CPU: 12 × Apple M2 Pro
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-15.0.7 (ORCJIT, apple-m1)
Threads: 1 default, 0 interactive, 1 GC (on 8 virtual cores)