Describe the issue:

It seems, that numpy uses Smith's algorithm to do complex true division. In some cases (see below) it produces (nan+nanj), but meaningful components could be recovered. See e.g. C11 Annex G.5.2, _Cdivd()'s example.

FYI: see also CPython's issue python/cpython#119372

Reproduce the code example:

>>> 1/np.complex128('inf')
0j
>>> 1/np.complex128('infj')
-0j
>>> 1/np.complex128('(inf+infj)')  # oops, should be 0j
<stdin>:1: RuntimeWarning: invalid value encountered in scalar divide
(nan+nanj)

Error message:

No response

Python and NumPy Versions:

>>> np.__version__
'1.26.4'

Runtime Environment:

No response

Context for the issue:

No response