FA-16371 / Floating-point arithmetic / Open access

Cosine decrement returns a positive half-angle square · case 01

Cosine decrement returns a positive half-angle square.

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

ROOT CAUSE

Cosine decrement returns a positive half-angle square. The faulty expression is result=2ss.

VERIFIED REPAIR

Apply the contract at this fault site using result=-2*s*s.

Unsuccessful approach: The attempted local correction result=-s*s still violates the explicit regression fixtures.

Case contract

Evaluate cos(x)-1 for finite x without cancellation using the half-angle identity. 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:
        if x==0: return '-0'
        s=math.sin(x/2)
        result=2*s*s
        return render(result)
    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**-30), render(-0.5*(N*2.0**-30)**2))
check('negative tiny', solve(-N*2.0**-30), render(-0.5*(N*2.0**-30)**2))
check('normal', solve(float(N)), render(-2*math.sin(N/2)**2))
check('negative normal', solve(-float(N)), render(-2*math.sin(N/2)**2))
check('zero', solve(0.0), "-0")
check('pi', solve(math.pi), "-2")
check('half pi', solve(math.pi/2), "-1")
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	`4.3368086899e-19`	`-4.3368086899e-19`	Failed
negative tiny	`4.3368086899e-19`	`-4.3368086899e-19`	Failed
normal	`0.45969769413`	`-0.45969769413`	Failed
negative normal	`0.45969769413`	`-0.45969769413`	Failed
zero	`-0`	`-0`	Passed
pi	`2`	`-2`	Failed
half pi	`1`	`-1`	Failed

SHA-256 / 1cadae1ad04f419cb84a057beab0523b0099787e81c99dbf286864fceb956751

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:
        if x==0: return '-0'
        s=math.sin(x/2)
        result=-s*s
        return render(result)
    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**-30), render(-0.5*(N*2.0**-30)**2))
check('negative tiny', solve(-N*2.0**-30), render(-0.5*(N*2.0**-30)**2))
check('normal', solve(float(N)), render(-2*math.sin(N/2)**2))
check('negative normal', solve(-float(N)), render(-2*math.sin(N/2)**2))
check('zero', solve(0.0), "-0")
check('pi', solve(math.pi), "-2")
check('half pi', solve(math.pi/2), "-1")
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	`-2.168404345e-19`	`-4.3368086899e-19`	Failed
negative tiny	`-2.168404345e-19`	`-4.3368086899e-19`	Failed
normal	`-0.22984884707`	`-0.45969769413`	Failed
negative normal	`-0.22984884707`	`-0.45969769413`	Failed
zero	`-0`	`-0`	Passed
pi	`-1`	`-2`	Failed
half pi	`-0.5`	`-1`	Failed

SHA-256 / 311e68eeef9f78e3cb3159cd261281fb38389c385e3cb0b7f57bd0fb47a1c587

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:
        if x==0: return '-0'
        s=math.sin(x/2)
        result=-2*s*s
        return render(result)
    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**-30), render(-0.5*(N*2.0**-30)**2))
check('negative tiny', solve(-N*2.0**-30), render(-0.5*(N*2.0**-30)**2))
check('normal', solve(float(N)), render(-2*math.sin(N/2)**2))
check('negative normal', solve(-float(N)), render(-2*math.sin(N/2)**2))
check('zero', solve(0.0), "-0")
check('pi', solve(math.pi), "-2")
check('half pi', solve(math.pi/2), "-1")
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	`-4.3368086899e-19`	`-4.3368086899e-19`	Passed
negative tiny	`-4.3368086899e-19`	`-4.3368086899e-19`	Passed
normal	`-0.45969769413`	`-0.45969769413`	Passed
negative normal	`-0.45969769413`	`-0.45969769413`	Passed
zero	`-0`	`-0`	Passed
pi	`-2`	`-2`	Passed
half pi	`-1`	`-1`	Passed

SHA-256 / 574c04d51a1b28a816c764b6b202eef5b0e3c80f7a3240d643a5f67654f8271c

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 2026-09-29T14:39:35.663490+00:00.

Case digest / b6a08fde77c38a14a5d38553cb8f73e0623f890afa8d91b0e5e52596e74e63d8