FA-16336 / Floating-point arithmetic / Open access

Sinc series uses x instead of x squared · case 01

Sinc series uses x instead of x squared.

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

ROOT CAUSE

Sinc series uses x instead of x squared. The faulty expression is t=x.

VERIFIED REPAIR

Apply the contract at this fault site using t=x*x.

Unsuccessful approach: The attempted local correction t=abs(x) still violates the explicit regression fixtures.

Case contract

Evaluate sin(x)/x on finite x with continuous value one at zero. Tiny values use the fourth-degree even series. 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 '1'
        if abs(x)<1e-3:
            t=x
            result=1-t/6+t*t/120
        else:
            result=math.sin(x)/x
        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('zero', solve(0.0), "1")
check('negative zero', solve(-0.0), "1")
check('small', solve(N*1e-4), render(math.sin(N*1e-4)/(N*1e-4)))
check('negative small', solve(-N*1e-4), render(math.sin(-N*1e-4)/(-N*1e-4)))
check('normal', solve(float(N)), render(math.sin(N)/N))
check('negative normal', solve(-float(N)), render(math.sin(-N)/(-N)))
check('phase crossing', solve(math.pi+0.125*N), render(math.sin(math.pi+0.125*N)/(math.pi+0.125*N)))
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
zero	`1`	`1`	Passed
negative zero	`1`	`1`	Passed
small	`0.99998333342`	`0.99999999833`	Failed
negative small	`1.0000166668`	`0.99999999833`	Failed
normal	`0.84147098481`	`0.84147098481`	Passed
negative normal	`0.84147098481`	`0.84147098481`	Passed
phase crossing	`-0.038166599453`	`-0.038166599453`	Passed

SHA-256 / 153ceb66e821eea4aeb9ef2e5322ae2cddf86680da702dfaabbb716eec7d7c9a

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 '1'
        if abs(x)<1e-3:
            t=abs(x)
            result=1-t/6+t*t/120
        else:
            result=math.sin(x)/x
        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('zero', solve(0.0), "1")
check('negative zero', solve(-0.0), "1")
check('small', solve(N*1e-4), render(math.sin(N*1e-4)/(N*1e-4)))
check('negative small', solve(-N*1e-4), render(math.sin(-N*1e-4)/(-N*1e-4)))
check('normal', solve(float(N)), render(math.sin(N)/N))
check('negative normal', solve(-float(N)), render(math.sin(-N)/(-N)))
check('phase crossing', solve(math.pi+0.125*N), render(math.sin(math.pi+0.125*N)/(math.pi+0.125*N)))
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
zero	`1`	`1`	Passed
negative zero	`1`	`1`	Passed
small	`0.99998333342`	`0.99999999833`	Failed
negative small	`0.99998333342`	`0.99999999833`	Failed
normal	`0.84147098481`	`0.84147098481`	Passed
negative normal	`0.84147098481`	`0.84147098481`	Passed
phase crossing	`-0.038166599453`	`-0.038166599453`	Passed

SHA-256 / a71c16c4565c67c01b66700cebad83fb2002339663cbebff91ba284972257d29

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 '1'
        if abs(x)<1e-3:
            t=x*x
            result=1-t/6+t*t/120
        else:
            result=math.sin(x)/x
        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('zero', solve(0.0), "1")
check('negative zero', solve(-0.0), "1")
check('small', solve(N*1e-4), render(math.sin(N*1e-4)/(N*1e-4)))
check('negative small', solve(-N*1e-4), render(math.sin(-N*1e-4)/(-N*1e-4)))
check('normal', solve(float(N)), render(math.sin(N)/N))
check('negative normal', solve(-float(N)), render(math.sin(-N)/(-N)))
check('phase crossing', solve(math.pi+0.125*N), render(math.sin(math.pi+0.125*N)/(math.pi+0.125*N)))
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
zero	`1`	`1`	Passed
negative zero	`1`	`1`	Passed
small	`0.99999999833`	`0.99999999833`	Passed
negative small	`0.99999999833`	`0.99999999833`	Passed
normal	`0.84147098481`	`0.84147098481`	Passed
negative normal	`0.84147098481`	`0.84147098481`	Passed
phase crossing	`-0.038166599453`	`-0.038166599453`	Passed

SHA-256 / ae22c0a2889ac818972e70cc72be1b702de4e30ecdff2ffa50f781c55900d8e9

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 2026-09-29T14:39:35.358147+00:00.

Case digest / d73f4db646a46af539af16286ef0451d306a0edb5889a9dd6c416372cb4a5b05