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

cmath_atanh_impl() selects its near-pole formula with
`ay < CM_SQRT_DBL_MIN`.  At equality the general formula computes
4/(0 + ay*ay) = 4/DBL_MIN, which overflows, so the finite input
1 + 2**-511j (and its three sign/conjugate mirrors, and the
corresponding cmath.atan() inputs) returned inf.  Use `<=` so the
boundary word takes the near-pole formula, which is accurate there.

Author: greg-const

Lib/test/mathdata/cmath_testcases.txt

@@ -853,6 +853,13 @@ atan0301 atan 1e-155 1.0 -> 0.78539816339744828 178.79691829731851
 atan0302 atan 9.9999999999999999e-161 -1.0 -> 0.78539816339744828 -184.55338102980363
 atan0303 atan -1e-165 1.0 -> -0.78539816339744828 190.30984376228875
 atan0304 atan -9.9998886718268301e-321 -1.0 -> -0.78539816339744828 -368.76019403576692
+-- gh-157859: real part at and next to sqrt(DBL_MIN) = 2**-511 (see atanh0305-0310)
+atan0305 atan 1.4916681462400412e-154 1.0 -> 0.78539816339744828 177.445678223346
+atan0306 atan 1.4916681462400413e-154 1.0 -> 0.78539816339744828 177.445678223346
+atan0307 atan 1.4916681462400417e-154 1.0 -> 0.78539816339744828 177.445678223346
+atan0308 atan -1.4916681462400413e-154 1.0 -> -0.78539816339744828 177.445678223346
+atan0309 atan 1.4916681462400413e-154 -1.0 -> 0.78539816339744828 -177.445678223346
+atan0310 atan -1.4916681462400413e-154 -1.0 -> -0.78539816339744828 -177.445678223346
 
 -- Additional real values (mpmath)
 atan0400 atan 1.7976931348623157e+308 0.0 -> 1.5707963267948966192 0.0
@@ -1043,6 +1050,14 @@ atanh0301 atanh 1.0 9.9999999999999997e-155 -> 177.64562575082149 0.785398163397
 atanh0302 atanh -1.0 1e-161 -> -185.70467357630065 0.78539816339744828
 atanh0303 atanh 1.0 -1e-165 -> 190.30984376228875 -0.78539816339744828
 atanh0304 atanh -1.0 -9.8813129168249309e-324 -> -372.22003596069061 -0.78539816339744828
+-- gh-157859: imaginary part at and next to sqrt(DBL_MIN) = 2**-511, where
+-- the near-pole and general formulas meet; the boundary value overflowed.
+atanh0305 atanh 1.0 1.4916681462400412e-154 -> 177.445678223346 0.78539816339744828
+atanh0306 atanh 1.0 1.4916681462400413e-154 -> 177.445678223346 0.78539816339744828
+atanh0307 atanh 1.0 1.4916681462400417e-154 -> 177.445678223346 0.78539816339744828
+atanh0308 atanh 1.0 -1.4916681462400413e-154 -> 177.445678223346 -0.78539816339744828
+atanh0309 atanh -1.0 1.4916681462400413e-154 -> -177.445678223346 0.78539816339744828
+atanh0310 atanh -1.0 -1.4916681462400413e-154 -> -177.445678223346 -0.78539816339744828
 
 -- special values
 atanh1000 atanh 0.0 0.0 -> 0.0 0.0

Misc/NEWS.d/next/Library/2026-09-20-12-53-29.gh-issue-157859.wMR0YM0E.rst

@@ -0,0 +1,2 @@
+Fix :func:`cmath.atanh` and :func:`cmath.atan` returning an infinite
+component instead of a finite result for certain inputs near their poles.

Modules/cmathmodule.c

@@ -382,7 +382,7 @@ cmath_atanh_impl(PyObject *module, Py_complex z)
         r.real = z.real/4./h/h;
         r.imag = copysign(Py_MATH_PI/2., z.imag);
         errno = 0;
-    } else if (z.real == 1. && ay < CM_SQRT_DBL_MIN) {
+    } else if (z.real == 1. && ay <= CM_SQRT_DBL_MIN) {
         /* C99 standard says:  atanh(1+/-0.) should be inf +/- 0i */
         if (ay == 0.) {
             r.real = INF;