FA-16246 / Floating-point arithmetic / Open access

Inverse sine rationalizes with the wrong square-root denominator · case 01

Inverse sine rationalizes with the wrong square-root denominator.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Inverse sine rationalizes with the wrong square-root denominator. The faulty expression is result=math.log1p(a+aa/(math.sqrt(1+aa))).

VERIFIED REPAIR

Apply the contract at this fault site using result=math.log1p(a+a*a/(1+math.sqrt(1+a*a))).

Unsuccessful approach: The attempted local correction result=math.log1p(a+a*a/(1+a)) still violates the explicit regression fixtures.

Case contract

Evaluate inverse hyperbolic sine for finite binary64 values with tiny-value preservation and a large-value logarithmic branch. Finite results are rendered to eleven significant decimal digits; modeled domain violations and arithmetic errors are explicit strings.

Why this case matters

An offline floating representation model isolates a reproducible arithmetic fault.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
def render(x):
    if math.isnan(x): return 'nan'
    if math.isinf(x): return '-infinity' if x<0 else '+infinity'
    return format(x,'.11g')

N = 1
observations = []
def solve(x):
    try:
        a=abs(x)
        if a<1e-8: return render(x)
        if a>1e150:
            result=math.log(a)+math.log(2)
        else:
            result=math.log1p(a+a*a/(math.sqrt(1+a*a)))
        return render(math.copysign(result,x))
    except (ValueError, OverflowError, ZeroDivisionError, TypeError):
        return "arithmetic-error"
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('tiny', solve(N*2.0**-60), render(N*2.0**-60))
check('negative tiny', solve(-N*2.0**-60), render(-N*2.0**-60))
check('large', solve(N*1e300), render(math.asinh(N*1e300)))
check('negative large', solve(-N*1e300), render(math.asinh(-N*1e300)))
check('normal', solve(float(N)), render(math.asinh(N)))
check('negative normal', solve(-float(N)), render(math.asinh(-N)))
check('negative zero', solve(-0.0), "-0")
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
tiny	`8.6736173799e-19`	`8.6736173799e-19`	Passed
negative tiny	`-8.6736173799e-19`	`-8.6736173799e-19`	Passed
large	`691.46867508`	`691.46867508`	Passed
negative large	`-691.46867508`	`-691.46867508`	Passed
normal	`0.99588045617`	`0.88137358702`	Failed
negative normal	`-0.99588045617`	`-0.88137358702`	Failed
negative zero	`-0`	`-0`	Passed

SHA-256 / 07c2ddbd0f4e01f5579a8cd5db6d0f8d0908882c1232b2e6b863f41e6ff95aa3

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
def render(x):
    if math.isnan(x): return 'nan'
    if math.isinf(x): return '-infinity' if x<0 else '+infinity'
    return format(x,'.11g')

N = 1
observations = []
def solve(x):
    try:
        a=abs(x)
        if a<1e-8: return render(x)
        if a>1e150:
            result=math.log(a)+math.log(2)
        else:
            result=math.log1p(a+a*a/(1+a))
        return render(math.copysign(result,x))
    except (ValueError, OverflowError, ZeroDivisionError, TypeError):
        return "arithmetic-error"
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('tiny', solve(N*2.0**-60), render(N*2.0**-60))
check('negative tiny', solve(-N*2.0**-60), render(-N*2.0**-60))
check('large', solve(N*1e300), render(math.asinh(N*1e300)))
check('negative large', solve(-N*1e300), render(math.asinh(-N*1e300)))
check('normal', solve(float(N)), render(math.asinh(N)))
check('negative normal', solve(-float(N)), render(math.asinh(-N)))
check('negative zero', solve(-0.0), "-0")
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
tiny	`8.6736173799e-19`	`8.6736173799e-19`	Passed
negative tiny	`-8.6736173799e-19`	`-8.6736173799e-19`	Passed
large	`691.46867508`	`691.46867508`	Passed
negative large	`-691.46867508`	`-691.46867508`	Passed
normal	`0.91629073187`	`0.88137358702`	Failed
negative normal	`-0.91629073187`	`-0.88137358702`	Failed
negative zero	`-0`	`-0`	Passed

SHA-256 / 1577e9fb84c1bc9ddd14e01e623626b61619fa0c5a5344e75b7137c8251e8148

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
def render(x):
    if math.isnan(x): return 'nan'
    if math.isinf(x): return '-infinity' if x<0 else '+infinity'
    return format(x,'.11g')

N = 1
observations = []
def solve(x):
    try:
        a=abs(x)
        if a<1e-8: return render(x)
        if a>1e150:
            result=math.log(a)+math.log(2)
        else:
            result=math.log1p(a+a*a/(1+math.sqrt(1+a*a)))
        return render(math.copysign(result,x))
    except (ValueError, OverflowError, ZeroDivisionError, TypeError):
        return "arithmetic-error"
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('tiny', solve(N*2.0**-60), render(N*2.0**-60))
check('negative tiny', solve(-N*2.0**-60), render(-N*2.0**-60))
check('large', solve(N*1e300), render(math.asinh(N*1e300)))
check('negative large', solve(-N*1e300), render(math.asinh(-N*1e300)))
check('normal', solve(float(N)), render(math.asinh(N)))
check('negative normal', solve(-float(N)), render(math.asinh(-N)))
check('negative zero', solve(-0.0), "-0")
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)

Boundary fixture	Actual	Expected	Outcome
tiny	`8.6736173799e-19`	`8.6736173799e-19`	Passed
negative tiny	`-8.6736173799e-19`	`-8.6736173799e-19`	Passed
large	`691.46867508`	`691.46867508`	Passed
negative large	`-691.46867508`	`-691.46867508`	Passed
normal	`0.88137358702`	`0.88137358702`	Passed
negative normal	`-0.88137358702`	`-0.88137358702`	Passed
negative zero	`-0`	`-0`	Passed

SHA-256 / 63b2caefba5e573952c603b31c9fbd968c653bd31fa97b5a6acc0ceaedc2a138

Verification & scope

Controlled binary64 or explicitly stipulated miniature format; no hardware exception flags or platform floating environment are modeled. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.

Observations recorded using Python 3.12.14 at 2026-09-29T14:39:34.673819+00:00.

Case digest / 3d14b4fffe1660089347681d117aae9bf590f26b4ef58342795c0da44e8dede3