statistics.NormalDist.overlap returns incorrect coefficient for large finite standard deviations

[augusto-rehfeldt]

Bug report

Bug description:

Documented behaviour: NormalDist.overlap documentation: "Compute the overlapping coefficient (OVL) between two normal distributions." The overlapping coefficient is the integral of the minimum of the two probability density functions.

Expected: Approximately 0.6773254311652314, within absolute tolerance 1e-12, without raising.

Actual: 1.0 for NormalDist(0.0, 1e150).overlap(NormalDist(0.0, 2e150)).

import math
from statistics import NormalDist

p = ((0.0, 1e150), (0.0, 2e150))
if not all(math.isfinite(m) and math.isfinite(s) and s > 0 for m, s in p):
    print("REFUTATION REJECTED:", "invalid normal-distribution parameters")
else:
    # Integrate the narrower density outside the crossings, wider inside.
    t = math.sqrt(4 * math.log(2) / 3)
    expected = 1 + math.erf(t / 2) - math.erf(t)
    try:
        actual = NormalDist(*p[0]).overlap(NormalDist(*p[1]))
    except Exception as e:
        print("REFUTATION CONFIRMED:", p, "actual =", repr(e), "expected =", expected)
    else:
        if not math.isfinite(actual) or abs(actual - expected) > 1e-12:
            print("REFUTATION CONFIRMED:", p, "actual =", actual, "expected =", expected)
        else:
            print("REFUTATION REJECTED:", "actual agrees with the independent integral")

Output on Python 3.14.6 (Windows-11-10.0.26220-SP0), standard library statistics:

REFUTATION CONFIRMED: ((0.0, 1e+150), (0.0, 2e+150)) actual = 1.0 expected = 0.6773254311652314

This report was found and written by an automated property-testing tool I run (bugforge). The reproducer above was executed and its output is pasted unedited; no person reviewed the report before it was filed. The search script is in https://github.com/augusto-rehfeldt/bugforge-results/tree/main/statistics-20261003-124541-c4

CPython versions tested on:

3.14

Operating systems tested on:

Windows

[ZeroIntensity]

See #158748 (comment).