FA-16466 / Floating-point arithmetic / Open access

Complex logarithm squares components before scaling · case 01

Complex logarithm squares components before scaling.

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

ROOT CAUSE

Complex logarithm squares components before scaling. The faulty expression is real=0.5math.log(xx+y*y).

THE FAILURE

Complex logarithm squares components before scaling. The faulty expression is real=0.5*math.log(x*x+y*y).

Unsuccessful approach: The attempted local correction real=0.5*math.log(max(x*x+y*y,1e-300)) still violates the explicit regression fixtures.

Case contract

Principal logarithm of nonzero finite complex x+iy, using hypot and atan2; zero returns a pole marker. Inputs keep hypot finite. 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,y):
    try:
        if x==0 and y==0: return 'pole'
        a=abs(x); b=abs(y)
        hi=max(a,b); lo=min(a,b)
        ratio=lo/hi
        real=0.5*math.log(x*x+y*y)
        imag=math.atan2(y,x)
        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('unequal components', solve(3.0,4.0), [render(math.log(5)),render(math.atan2(4,3))])
check('huge', solve(N*1e200,1e200), [render(math.log(math.hypot(N*1e200,1e200))),render(math.atan2(1,N))])
check('tiny', solve(N*1e-200,-1e-200), [render(math.log(math.hypot(N*1e-200,1e-200))),render(math.atan2(-1,N))])
check('negative real above', solve(-float(N),0.0), [render(math.log(N)),render(math.pi)])
check('negative real below', solve(-float(N),-0.0), [render(math.log(N)),render(-math.pi)])
check('imaginary axis', solve(0.0,float(N)), [render(math.log(N)),render(math.pi/2)])
check('real axis', solve(float(N),0.0), [render(math.log(N)),"0"])
check('origin', solve(0.0,0.0), "pole")
check('quadrant three', solve(-float(N),-float(N)), [render(math.log(math.hypot(N,N))),render(-3*math.pi/4)])
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
unequal components	`['1.6094379124', '0.927295218']`	`['1.6094379124', '0.927295218']`	Passed
huge	`['+infinity', '0.7853981634']`	`['460.86359219', '0.7853981634']`	Failed
tiny	`arithmetic-error`	`['-460.17044501', '-0.7853981634']`	Failed
negative real above	`['0', '3.1415926536']`	`['0', '3.1415926536']`	Passed
negative real below	`['0', '-3.1415926536']`	`['0', '-3.1415926536']`	Passed
imaginary axis	`['0', '1.5707963268']`	`['0', '1.5707963268']`	Passed
real axis	`['0', '0']`	`['0', '0']`	Passed
origin	`pole`	`pole`	Passed
quadrant three	`['0.34657359028', '-2.3561944902']`	`['0.34657359028', '-2.3561944902']`	Passed

SHA-256 / 0f3dd936436d00f4b4ff2024ce1ecdafa91d3097509c7d28560148003b49fe0d

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,y):
    try:
        if x==0 and y==0: return 'pole'
        a=abs(x); b=abs(y)
        hi=max(a,b); lo=min(a,b)
        ratio=lo/hi
        real=0.5*math.log(max(x*x+y*y,1e-300))
        imag=math.atan2(y,x)
        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('unequal components', solve(3.0,4.0), [render(math.log(5)),render(math.atan2(4,3))])
check('huge', solve(N*1e200,1e200), [render(math.log(math.hypot(N*1e200,1e200))),render(math.atan2(1,N))])
check('tiny', solve(N*1e-200,-1e-200), [render(math.log(math.hypot(N*1e-200,1e-200))),render(math.atan2(-1,N))])
check('negative real above', solve(-float(N),0.0), [render(math.log(N)),render(math.pi)])
check('negative real below', solve(-float(N),-0.0), [render(math.log(N)),render(-math.pi)])
check('imaginary axis', solve(0.0,float(N)), [render(math.log(N)),render(math.pi/2)])
check('real axis', solve(float(N),0.0), [render(math.log(N)),"0"])
check('origin', solve(0.0,0.0), "pole")
check('quadrant three', solve(-float(N),-float(N)), [render(math.log(math.hypot(N,N))),render(-3*math.pi/4)])
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
unequal components	`['1.6094379124', '0.927295218']`	`['1.6094379124', '0.927295218']`	Passed
huge	`['+infinity', '0.7853981634']`	`['460.86359219', '0.7853981634']`	Failed
tiny	`['-345.38776395', '-0.7853981634']`	`['-460.17044501', '-0.7853981634']`	Failed
negative real above	`['0', '3.1415926536']`	`['0', '3.1415926536']`	Passed
negative real below	`['0', '-3.1415926536']`	`['0', '-3.1415926536']`	Passed
imaginary axis	`['0', '1.5707963268']`	`['0', '1.5707963268']`	Passed
real axis	`['0', '0']`	`['0', '0']`	Passed
origin	`pole`	`pole`	Passed
quadrant three	`['0.34657359028', '-2.3561944902']`	`['0.34657359028', '-2.3561944902']`	Passed

SHA-256 / 42f8e86ec686fd3d2ad631d8c5c8815ae1614c9659c6b2524d9d43fc6a5f3625

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 2026-09-29T14:39:36.734838+00:00.

Case digest / 232be7a1b312e799ca6a17d41f2298d38be93ca12072fec0e3bf12639af331fd