FA-17256 / Floating-point arithmetic / Open access

Reciprocal drops sign after magnitude normalization · case 01

Reciprocal drops sign after magnitude normalization.

Verified by executionVariant 1 · 9 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Reciprocal drops sign after magnitude normalization. The faulty expression is return render(result).

VERIFIED REPAIR

Apply the contract at this fault site using return render(math.copysign(result,x)).

Unsuccessful approach: The attempted local correction return render(-result) still violates the explicit regression fixtures.

Case contract

Compute the reciprocal of finite binary64 x using exponent decomposition. Signed zeros return signed infinity; overflow returns signed infinity and representable subnormal reciprocals are retained. 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 '-infinity' if math.copysign(1.0,x)<0 else '+infinity'
        m,e=math.frexp(abs(x))
        fraction=1/m
        try:
            result=math.ldexp(fraction,-e)
        except OverflowError:
            return '-infinity' if x<0 else '+infinity'
        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('positive ordinary', solve(float(N)), render(1/N))
check('negative ordinary', solve(-float(N)), render(-1/N))
check('huge', solve(1e308), render(1e-308))
check('negative huge', solve(-1e308), render(-1e-308))
check('tiny overflow', solve(math.ldexp(1.0,-1074)), "+infinity")
check('negative tiny', solve(-math.ldexp(1.0,-1074)), "-infinity")
check('positive zero', solve(0.0), "+infinity")
check('negative zero', solve(-0.0), "-infinity")
check('fraction', solve(0.125), "8")
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
positive ordinary	`1`	`1`	Passed
negative ordinary	`1`	`-1`	Failed
huge	`1e-308`	`1e-308`	Passed
negative huge	`1e-308`	`-1e-308`	Failed
tiny overflow	`+infinity`	`+infinity`	Passed
negative tiny	`-infinity`	`-infinity`	Passed
positive zero	`+infinity`	`+infinity`	Passed
negative zero	`-infinity`	`-infinity`	Passed
fraction	`8`	`8`	Passed

SHA-256 / bbed0da9b02f9cbceea4602fc6e64fb472d5e58e504f9c6ea6be2e2f4bb48437

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 '-infinity' if math.copysign(1.0,x)<0 else '+infinity'
        m,e=math.frexp(abs(x))
        fraction=1/m
        try:
            result=math.ldexp(fraction,-e)
        except OverflowError:
            return '-infinity' if x<0 else '+infinity'
        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('positive ordinary', solve(float(N)), render(1/N))
check('negative ordinary', solve(-float(N)), render(-1/N))
check('huge', solve(1e308), render(1e-308))
check('negative huge', solve(-1e308), render(-1e-308))
check('tiny overflow', solve(math.ldexp(1.0,-1074)), "+infinity")
check('negative tiny', solve(-math.ldexp(1.0,-1074)), "-infinity")
check('positive zero', solve(0.0), "+infinity")
check('negative zero', solve(-0.0), "-infinity")
check('fraction', solve(0.125), "8")
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
positive ordinary	`-1`	`1`	Failed
negative ordinary	`-1`	`-1`	Passed
huge	`-1e-308`	`1e-308`	Failed
negative huge	`-1e-308`	`-1e-308`	Passed
tiny overflow	`+infinity`	`+infinity`	Passed
negative tiny	`-infinity`	`-infinity`	Passed
positive zero	`+infinity`	`+infinity`	Passed
negative zero	`-infinity`	`-infinity`	Passed
fraction	`-8`	`8`	Failed

SHA-256 / 698f1ddcb0613a53b740948ba23a84e91f131939d7766399e32655adfb76410f

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 '-infinity' if math.copysign(1.0,x)<0 else '+infinity'
        m,e=math.frexp(abs(x))
        fraction=1/m
        try:
            result=math.ldexp(fraction,-e)
        except OverflowError:
            return '-infinity' if x<0 else '+infinity'
        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('positive ordinary', solve(float(N)), render(1/N))
check('negative ordinary', solve(-float(N)), render(-1/N))
check('huge', solve(1e308), render(1e-308))
check('negative huge', solve(-1e308), render(-1e-308))
check('tiny overflow', solve(math.ldexp(1.0,-1074)), "+infinity")
check('negative tiny', solve(-math.ldexp(1.0,-1074)), "-infinity")
check('positive zero', solve(0.0), "+infinity")
check('negative zero', solve(-0.0), "-infinity")
check('fraction', solve(0.125), "8")
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
positive ordinary	`1`	`1`	Passed
negative ordinary	`-1`	`-1`	Passed
huge	`1e-308`	`1e-308`	Passed
negative huge	`-1e-308`	`-1e-308`	Passed
tiny overflow	`+infinity`	`+infinity`	Passed
negative tiny	`-infinity`	`-infinity`	Passed
positive zero	`+infinity`	`+infinity`	Passed
negative zero	`-infinity`	`-infinity`	Passed
fraction	`8`	`8`	Passed

SHA-256 / 82643796da93f08648e3a833d9bd6056f8ef38de0bbe127c8c746d13d50f4515

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 2026-09-29T14:39:44.411605+00:00.

Case digest / b619f6a6ab7ded92dba5ee58180756b7484c85f07381fc343566de4a8e148065