FA-16121 / Floating-point arithmetic / Open access

Square-root increment loses the sign of exact negative zero · case 01

Square-root increment loses the sign of exact negative zero.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Square-root increment loses the sign of exact negative zero. The faulty expression is if x == 0: return '0'.

VERIFIED REPAIR

Apply the contract at this fault site using if x == 0: return render(x).

Unsuccessful approach: The attempted local correction if x == 0: return '-0' still violates the explicit regression fixtures.

Case contract

Evaluate sqrt(1+x)-1 stably for finite x >= -1. 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 < -1: return 'domain'
        if x == -1: return '-1'
        if x == 0: return '0'
        result=x/(math.sqrt(1+x)+1)
        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 positive', solve(N*2.0**-60), render(N*2.0**-61))
check('tiny negative', solve(-N*2.0**-60), render(-N*2.0**-61))
check('normal', solve(3.0), "1")
check('endpoint', solve(-1.0), "-1")
check('domain', solve(-2.0), "domain")
check('negative zero', solve(-0.0), "-0")
check('positive zero', solve(0.0), "0")
check('large', solve(1e300), render(1e150))
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 positive	`4.3368086899e-19`	`4.3368086899e-19`	Passed
tiny negative	`-4.3368086899e-19`	`-4.3368086899e-19`	Passed
normal	`1`	`1`	Passed
endpoint	`-1`	`-1`	Passed
domain	`domain`	`domain`	Passed
negative zero	`0`	`-0`	Failed
positive zero	`0`	`0`	Passed
large	`1e+150`	`1e+150`	Passed

SHA-256 / 30b22ff69eba68c79a719d757bd86d64d5c7a2486c6623a06ebc3318c84dd417

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 < -1: return 'domain'
        if x == -1: return '-1'
        if x == 0: return '-0'
        result=x/(math.sqrt(1+x)+1)
        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 positive', solve(N*2.0**-60), render(N*2.0**-61))
check('tiny negative', solve(-N*2.0**-60), render(-N*2.0**-61))
check('normal', solve(3.0), "1")
check('endpoint', solve(-1.0), "-1")
check('domain', solve(-2.0), "domain")
check('negative zero', solve(-0.0), "-0")
check('positive zero', solve(0.0), "0")
check('large', solve(1e300), render(1e150))
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 positive	`4.3368086899e-19`	`4.3368086899e-19`	Passed
tiny negative	`-4.3368086899e-19`	`-4.3368086899e-19`	Passed
normal	`1`	`1`	Passed
endpoint	`-1`	`-1`	Passed
domain	`domain`	`domain`	Passed
negative zero	`-0`	`-0`	Passed
positive zero	`-0`	`0`	Failed
large	`1e+150`	`1e+150`	Passed

SHA-256 / fccdefae29c1568dcff87673a7a17c1d4e92e5810ba208d06c9d0788990b92b3

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 < -1: return 'domain'
        if x == -1: return '-1'
        if x == 0: return render(x)
        result=x/(math.sqrt(1+x)+1)
        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 positive', solve(N*2.0**-60), render(N*2.0**-61))
check('tiny negative', solve(-N*2.0**-60), render(-N*2.0**-61))
check('normal', solve(3.0), "1")
check('endpoint', solve(-1.0), "-1")
check('domain', solve(-2.0), "domain")
check('negative zero', solve(-0.0), "-0")
check('positive zero', solve(0.0), "0")
check('large', solve(1e300), render(1e150))
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 positive	`4.3368086899e-19`	`4.3368086899e-19`	Passed
tiny negative	`-4.3368086899e-19`	`-4.3368086899e-19`	Passed
normal	`1`	`1`	Passed
endpoint	`-1`	`-1`	Passed
domain	`domain`	`domain`	Passed
negative zero	`-0`	`-0`	Passed
positive zero	`0`	`0`	Passed
large	`1e+150`	`1e+150`	Passed

SHA-256 / b8a95528e63549f4f6ce109bdd8a9d3b31dfad76fa72cd3670fdbcd994108370

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 2026-09-29T14:39:33.389952+00:00.

Case digest / 769266e37784104e77c4b296041a42b905322d2bb10e2a685c7f65d24bd37990