cmath.atanh() returns infinity for complex(1.0, 2.0**-511)

[greg-const]

Bug report

Bug description:

cmath.atanh() returns an infinite real part for a finite input whose
result should be finite:

>>> import cmath
>>> import math
>>> y = 2.0**-511
>>> cmath.atanh(complex(1.0, y))
(inf+0.7853981633974483j)

The expected result is approximately
(177.445678223346+0.7853981633974483j). The real part is approximately
256 * math.log(2).

The immediately adjacent floating-point inputs both give finite results:

>>> cmath.atanh(complex(1.0, math.nextafter(y, 0.0)))
(177.445678223346+0.7853981633974483j)
>>> cmath.atanh(complex(1.0, math.nextafter(y, math.inf)))
(177.445678223346+0.7853981633974483j)

The same problem occurs for all four sign combinations of the real and
imaginary parts.

In Modules/cmathmodule.c, cmath_atanh_impl() selects the near-pole
formula using ay < CM_SQRT_DBL_MIN. At equality, the general formula
instead forms 4.0 / DBL_MIN, which overflows before the logarithm is
evaluated. Changing the comparison to <= includes the boundary in the
existing near-pole branch.

This also affects cmath.atan(), which uses the same implementation:

>>> cmath.atan(complex(y, 1.0))
(0.7853981633974483+infj)

Here the expected result is approximately
(0.7853981633974483+177.445678223346j).

Reproduced on CPython 3.14.7, 3.14.4, and main. x86-64 Linux (glibc).

CPython versions tested on:

CPython main branch, 3.14

Operating systems tested on:

Linux

Linked PRs

• gh-157859: Fix cmath.atanh() returning inf at ±1 ± sqrt(DBL_MIN)*1j #157860

[skirpichev]

Indeed, here is glibc's libm:

Python 3.16.0a0 (heads/main:c1df6843d36, Sep 20 2026, 04:13:21) [GCC 14.2.0] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> import cmath
>>> y = 2.0**-511
>>> cmath.atanh(complex(1.0, y))
(inf+0.7853981633974483j)
>>> import ctypes
>>> libm = ctypes.CDLL('libm.so.6')
>>> import ctypes.util
>>> @ctypes.util.wrap_dll_function(libm)
... def catanh(z: ctypes.c_double_complex) -> ctypes.c_double_complex:
...     pass
... 
>>> catanh(complex(1.0, y))
(177.445678223346+0.7853981633974483j)

and numpy:

Python 3.14.7 (tags/v3.14.7:823f0323ee6, Aug 12 2026, 13:12:17) [GCC 14.2.0] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> import numpy as np
>>> y = 2.0**-511
>>> np.atanh(complex(1, y))
np.complex128(177.445678223346+0.7853981633974483j)