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FA-16396 / Floating-point arithmetic / Open access

Complex division forgets the ratio factor in the dominant-imaginary denominator · case 01

Complex division forgets the ratio factor in the dominant-imaginary denominator.

Verified by executionVariant 1 · 12 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Complex division forgets the ratio factor in the dominant-imaginary denominator. The faulty expression is den=d+c.

VERIFIED REPAIR

Apply the contract at this fault site using den=d+c*r.

Unsuccessful approach: The attempted local correction den=d still violates the explicit regression fixtures.

Case contract

Divide finite complex a+ib by nonzero c+id using a denominator-ratio branch. Fixtures bound products and exercise huge or tiny denominators whose naive squares overflow or underflow. Return rendered real and imaginary components. 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,d):
    try:
        if c==0 and d==0: return 'zero-denominator'
        if abs(c)>=abs(d):
            r=d/c
            den=c+d*r
            real=(a+b*r)/den
            imag=(b-a*r)/den
        else:
            r=c/d
            den=d+c
            real=(a*r+b)/den
            imag=(b*r-a)/den
        return [render(real),render(imag)]
    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('asymmetric numerator', solve(3.0,2.0,4.0,1.0), [render(14/17),render(5/17)])
check('asymmetric imaginary', solve(3.0,2.0,1.0,4.0), [render(11/17),render(-10/17)])
check('negative imaginary ratio', solve(3.0,2.0,-1.0,4.0), [render(5/17),render(-14/17)])
check('negative extreme ratio', solve(float(N),2.0,-1e200,1e-200), [render(-N/1e200),render(-2e-200)])
check('large real denominator', solve(float(N),2.0,1e200,1e199), [render((N+0.2)/1.01e200),render((2-N*0.1)/1.01e200)])
check('tiny imaginary denominator', solve(float(N),2.0,1e-201,1e-200), [render((N*0.1+2)/1.01e-200),render((0.2-N)/1.01e-200)])
check('normal real branch', solve(float(N),2.0,4.0,1.0), [render((4*N+2)/17),render((8-N)/17)])
check('normal imaginary branch', solve(float(N),2.0,1.0,4.0), [render((N+8)/17),render((2-4*N)/17)])
check('negative denominator', solve(float(N),2.0,-4.0,1.0), [render((-4*N+2)/17),render((-8-N)/17)])
check('pure real', solve(float(N),2.0,2.0,0.0), [render(N/2),"1"])
check('pure imaginary', solve(float(N),2.0,0.0,2.0), ["1",render(-N/2)])
check('zero denominator', solve(float(N),2.0,0.0,0.0), "zero-denominator")
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 fixtureActualExpectedOutcome
asymmetric numerator['0.82352941176', '0.29411764706']['0.82352941176', '0.29411764706']Passed
asymmetric imaginary['0.55', '-0.5']['0.64705882353', '-0.58823529412']Failed
negative imaginary ratio['0.41666666667', '-1.1666666667']['0.29411764706', '-0.82352941176']Failed
negative extreme ratio['-1e-200', '-2e-200']['-1e-200', '-2e-200']Passed
large real denominator['1.1881188119e-200', '1.8811881188e-200']['1.1881188119e-200', '1.8811881188e-200']Passed
tiny imaginary denominator['1.9090909091e+200', '-7.2727272727e+199']['2.0792079208e+200', '-7.9207920792e+199']Failed
normal real branch['0.35294117647', '0.41176470588']['0.35294117647', '0.41176470588']Passed
normal imaginary branch['0.45', '-0.1']['0.52941176471', '-0.11764705882']Failed
negative denominator['-0.11764705882', '-0.52941176471']['-0.11764705882', '-0.52941176471']Passed
pure real['0.5', '1']['0.5', '1']Passed
pure imaginary['1', '-0.5']['1', '-0.5']Passed
zero denominatorzero-denominatorzero-denominatorPassed

SHA-256 / d803ac2e9d6dda3432cf171837e9e7e176602a08ddb18738983044b7ad9f746e

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,d):
    try:
        if c==0 and d==0: return 'zero-denominator'
        if abs(c)>=abs(d):
            r=d/c
            den=c+d*r
            real=(a+b*r)/den
            imag=(b-a*r)/den
        else:
            r=c/d
            den=d
            real=(a*r+b)/den
            imag=(b*r-a)/den
        return [render(real),render(imag)]
    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('asymmetric numerator', solve(3.0,2.0,4.0,1.0), [render(14/17),render(5/17)])
check('asymmetric imaginary', solve(3.0,2.0,1.0,4.0), [render(11/17),render(-10/17)])
check('negative imaginary ratio', solve(3.0,2.0,-1.0,4.0), [render(5/17),render(-14/17)])
check('negative extreme ratio', solve(float(N),2.0,-1e200,1e-200), [render(-N/1e200),render(-2e-200)])
check('large real denominator', solve(float(N),2.0,1e200,1e199), [render((N+0.2)/1.01e200),render((2-N*0.1)/1.01e200)])
check('tiny imaginary denominator', solve(float(N),2.0,1e-201,1e-200), [render((N*0.1+2)/1.01e-200),render((0.2-N)/1.01e-200)])
check('normal real branch', solve(float(N),2.0,4.0,1.0), [render((4*N+2)/17),render((8-N)/17)])
check('normal imaginary branch', solve(float(N),2.0,1.0,4.0), [render((N+8)/17),render((2-4*N)/17)])
check('negative denominator', solve(float(N),2.0,-4.0,1.0), [render((-4*N+2)/17),render((-8-N)/17)])
check('pure real', solve(float(N),2.0,2.0,0.0), [render(N/2),"1"])
check('pure imaginary', solve(float(N),2.0,0.0,2.0), ["1",render(-N/2)])
check('zero denominator', solve(float(N),2.0,0.0,0.0), "zero-denominator")
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 fixtureActualExpectedOutcome
asymmetric numerator['0.82352941176', '0.29411764706']['0.82352941176', '0.29411764706']Passed
asymmetric imaginary['0.6875', '-0.625']['0.64705882353', '-0.58823529412']Failed
negative imaginary ratio['0.3125', '-0.875']['0.29411764706', '-0.82352941176']Failed
negative extreme ratio['-1e-200', '-2e-200']['-1e-200', '-2e-200']Passed
large real denominator['1.1881188119e-200', '1.8811881188e-200']['1.1881188119e-200', '1.8811881188e-200']Passed
tiny imaginary denominator['2.1e+200', '-8e+199']['2.0792079208e+200', '-7.9207920792e+199']Failed
normal real branch['0.35294117647', '0.41176470588']['0.35294117647', '0.41176470588']Passed
normal imaginary branch['0.5625', '-0.125']['0.52941176471', '-0.11764705882']Failed
negative denominator['-0.11764705882', '-0.52941176471']['-0.11764705882', '-0.52941176471']Passed
pure real['0.5', '1']['0.5', '1']Passed
pure imaginary['1', '-0.5']['1', '-0.5']Passed
zero denominatorzero-denominatorzero-denominatorPassed

SHA-256 / e34dd5feee649698da0e68e7297a7d9e6a005505dee4d3213b6d0a21ebeabf57

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(a,b,c,d):
    try:
        if c==0 and d==0: return 'zero-denominator'
        if abs(c)>=abs(d):
            r=d/c
            den=c+d*r
            real=(a+b*r)/den
            imag=(b-a*r)/den
        else:
            r=c/d
            den=d+c*r
            real=(a*r+b)/den
            imag=(b*r-a)/den
        return [render(real),render(imag)]
    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('asymmetric numerator', solve(3.0,2.0,4.0,1.0), [render(14/17),render(5/17)])
check('asymmetric imaginary', solve(3.0,2.0,1.0,4.0), [render(11/17),render(-10/17)])
check('negative imaginary ratio', solve(3.0,2.0,-1.0,4.0), [render(5/17),render(-14/17)])
check('negative extreme ratio', solve(float(N),2.0,-1e200,1e-200), [render(-N/1e200),render(-2e-200)])
check('large real denominator', solve(float(N),2.0,1e200,1e199), [render((N+0.2)/1.01e200),render((2-N*0.1)/1.01e200)])
check('tiny imaginary denominator', solve(float(N),2.0,1e-201,1e-200), [render((N*0.1+2)/1.01e-200),render((0.2-N)/1.01e-200)])
check('normal real branch', solve(float(N),2.0,4.0,1.0), [render((4*N+2)/17),render((8-N)/17)])
check('normal imaginary branch', solve(float(N),2.0,1.0,4.0), [render((N+8)/17),render((2-4*N)/17)])
check('negative denominator', solve(float(N),2.0,-4.0,1.0), [render((-4*N+2)/17),render((-8-N)/17)])
check('pure real', solve(float(N),2.0,2.0,0.0), [render(N/2),"1"])
check('pure imaginary', solve(float(N),2.0,0.0,2.0), ["1",render(-N/2)])
check('zero denominator', solve(float(N),2.0,0.0,0.0), "zero-denominator")
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 fixtureActualExpectedOutcome
asymmetric numerator['0.82352941176', '0.29411764706']['0.82352941176', '0.29411764706']Passed
asymmetric imaginary['0.64705882353', '-0.58823529412']['0.64705882353', '-0.58823529412']Passed
negative imaginary ratio['0.29411764706', '-0.82352941176']['0.29411764706', '-0.82352941176']Passed
negative extreme ratio['-1e-200', '-2e-200']['-1e-200', '-2e-200']Passed
large real denominator['1.1881188119e-200', '1.8811881188e-200']['1.1881188119e-200', '1.8811881188e-200']Passed
tiny imaginary denominator['2.0792079208e+200', '-7.9207920792e+199']['2.0792079208e+200', '-7.9207920792e+199']Passed
normal real branch['0.35294117647', '0.41176470588']['0.35294117647', '0.41176470588']Passed
normal imaginary branch['0.52941176471', '-0.11764705882']['0.52941176471', '-0.11764705882']Passed
negative denominator['-0.11764705882', '-0.52941176471']['-0.11764705882', '-0.52941176471']Passed
pure real['0.5', '1']['0.5', '1']Passed
pure imaginary['1', '-0.5']['1', '-0.5']Passed
zero denominatorzero-denominatorzero-denominatorPassed

SHA-256 / ea0c34e8a04d95e0092f14e87c50abcf9d046b26c24874700765c5f3dd5016d2

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 2026-09-29T14:39:36.015734+00:00.

Case digest / 1292da6d01ae550fb691d0cb10ffbd410020f0faa8979ab3d932237ffd8e9365