FA-16731 / Floating-point arithmetic / Open access

Degenerate quadratic equation applies a quadratic denominator · case 01

Degenerate quadratic equation applies a quadratic denominator.

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

ROOT CAUSE

Degenerate quadratic equation applies a quadratic denominator. The faulty expression is return [render(-c/(2*b))] if b!=0 else ("all" if c==0 else "none").

THE FAILURE

Degenerate quadratic equation applies a quadratic denominator. The faulty expression is return [render(-c/(2*b))] if b!=0 else ("all" if c==0 else "none").

Unsuccessful approach: The attempted local correction return [render(c/b)] if b!=0 else ("all" if c==0 else "none") still violates the explicit regression fixtures.

Case contract

Real roots of a*x*x+b*x+c=0 for finite coefficients with bounded discriminant products. Use sign-aware q to avoid subtractive cancellation and c/q for the complementary root. Return sorted rendered roots, repeated root once, or explicit degenerate/no-real markers. 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(a,b,c):
    try:
        if a==0:
            return [render(-c/(2*b))] if b!=0 else ("all" if c==0 else "none")
        d=b*b-4*a*c
        if d<0: return 'no-real'
        if d==0: return [render(-b/(2*a))]
        s=math.sqrt(d)
        q=-0.5*(b+math.copysign(s,b))
        r1=q/a
        r2=c/q
        return [render(v) for v in sorted([r1,r2])]
    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('large positive b', solve(1.0,1e16*N,1.0), [render(-1e16*N),render(-1/(1e16*N))])
check('large negative b', solve(1.0,-1e16*N,1.0), [render(1/(1e16*N)),render(1e16*N)])
check('normal roots', solve(1.0,-3.0,2.0), ["1","2"])
check('repeated', solve(1.0,-2.0*N,float(N*N)), [render(float(N))])
check('linear', solve(0.0,2.0,-2.0*N), [render(float(N))])
check('no real', solve(1.0,0.0,float(N)), "no-real")
check('all', solve(0.0,0.0,0.0), "all")
check('inconsistent', solve(0.0,0.0,float(N)), "none")
check('zero constant', solve(1.0,-float(N),0.0), ["0",render(float(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
large positive b	`['-1e+16', '-1e-16']`	`['-1e+16', '-1e-16']`	Passed
large negative b	`['1e-16', '1e+16']`	`['1e-16', '1e+16']`	Passed
normal roots	`['1', '2']`	`['1', '2']`	Passed
repeated	`['1']`	`['1']`	Passed
linear	`['0.5']`	`['1']`	Failed
no real	`no-real`	`no-real`	Passed
all	`all`	`all`	Passed
inconsistent	`none`	`none`	Passed
zero constant	`['0', '1']`	`['0', '1']`	Passed

SHA-256 / 03e0aa82d17a51e4105bac538f25eb1353b6c7ad59d5db08aedfcdd09af43880

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(a,b,c):
    try:
        if a==0:
            return [render(c/b)] if b!=0 else ("all" if c==0 else "none")
        d=b*b-4*a*c
        if d<0: return 'no-real'
        if d==0: return [render(-b/(2*a))]
        s=math.sqrt(d)
        q=-0.5*(b+math.copysign(s,b))
        r1=q/a
        r2=c/q
        return [render(v) for v in sorted([r1,r2])]
    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('large positive b', solve(1.0,1e16*N,1.0), [render(-1e16*N),render(-1/(1e16*N))])
check('large negative b', solve(1.0,-1e16*N,1.0), [render(1/(1e16*N)),render(1e16*N)])
check('normal roots', solve(1.0,-3.0,2.0), ["1","2"])
check('repeated', solve(1.0,-2.0*N,float(N*N)), [render(float(N))])
check('linear', solve(0.0,2.0,-2.0*N), [render(float(N))])
check('no real', solve(1.0,0.0,float(N)), "no-real")
check('all', solve(0.0,0.0,0.0), "all")
check('inconsistent', solve(0.0,0.0,float(N)), "none")
check('zero constant', solve(1.0,-float(N),0.0), ["0",render(float(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
large positive b	`['-1e+16', '-1e-16']`	`['-1e+16', '-1e-16']`	Passed
large negative b	`['1e-16', '1e+16']`	`['1e-16', '1e+16']`	Passed
normal roots	`['1', '2']`	`['1', '2']`	Passed
repeated	`['1']`	`['1']`	Passed
linear	`['-1']`	`['1']`	Failed
no real	`no-real`	`no-real`	Passed
all	`all`	`all`	Passed
inconsistent	`none`	`none`	Passed
zero constant	`['0', '1']`	`['0', '1']`	Passed

SHA-256 / 116ae50d826bceaab86a560c3d37afd5f339145cbcb59137928add7d6da159eb

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 9 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:39:39.466454+00:00.

Case digest / c820416eb9f1d4d9d05908c18c7f490317190ab0cf9f58d4609e2f7382b3b022