FA-16446 / Floating-point arithmetic / Open access

Positive-real square root subtracts nearly equal radii · case 01

Positive-real square root subtracts nearly equal radii.

Verified by executionVariant 1 · 10 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Positive-real square root subtracts nearly equal radii. The faulty expression is imag=math.copysign(math.sqrt((r-x)/2),y).

VERIFIED REPAIR

Apply the contract at this fault site using imag=y/(2*t).

Unsuccessful approach: The attempted local correction imag=math.copysign(math.sqrt(max(0,(r-x)/2)),y) still violates the explicit regression fixtures.

Case contract

Principal complex square root for finite components, using stable branch reconstruction and preserving the imaginary signed-zero side of the negative real branch cut. Fixtures 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 ['0',render(y)]
        r=math.hypot(x,y)
        t=math.sqrt(r/2+abs(x)/2)
        if x>=0:
            real=t
            imag=math.copysign(math.sqrt((r-x)/2),y)
        else:
            imag=math.copysign(t,y)
            real=abs(y)/(2*t)
        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('positive axis', solve(float(N*N),0.0), [render(float(N)),"0"])
check('negative axis above', solve(-float(N*N),0.0), ["0",render(float(N))])
check('negative axis below', solve(-float(N*N),-0.0), ["0",render(-float(N))])
check('upper quadrant', solve(3.0,4.0), ["2","1"])
check('lower quadrant', solve(3.0,-4.0), ["2","-1"])
check('negative real quadrant', solve(-3.0,4.0), ["1","2"])
check('huge axis', solve(1e308,0.0), ["1e+154","0"])
check('tiny imaginary', solve(4.0,N*1e-200), ["2",render(N*1e-200/4)])
check('origin above', solve(0.0,0.0), ["0","0"])
check('origin below', solve(0.0,-0.0), ["0","-0"])
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 axis	`['1', '0']`	`['1', '0']`	Passed
negative axis above	`['0', '1']`	`['0', '1']`	Passed
negative axis below	`['0', '-1']`	`['0', '-1']`	Passed
upper quadrant	`['2', '1']`	`['2', '1']`	Passed
lower quadrant	`['2', '-1']`	`['2', '-1']`	Passed
negative real quadrant	`['1', '2']`	`['1', '2']`	Passed
huge axis	`['1e+154', '0']`	`['1e+154', '0']`	Passed
tiny imaginary	`['2', '0']`	`['2', '2.5e-201']`	Failed
origin above	`['0', '0']`	`['0', '0']`	Passed
origin below	`['0', '-0']`	`['0', '-0']`	Passed

SHA-256 / 7a87c80628adc3e80eeb10c45b8981412d40b342b82e72dc3cff38a1652122a9

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 ['0',render(y)]
        r=math.hypot(x,y)
        t=math.sqrt(r/2+abs(x)/2)
        if x>=0:
            real=t
            imag=math.copysign(math.sqrt(max(0,(r-x)/2)),y)
        else:
            imag=math.copysign(t,y)
            real=abs(y)/(2*t)
        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('positive axis', solve(float(N*N),0.0), [render(float(N)),"0"])
check('negative axis above', solve(-float(N*N),0.0), ["0",render(float(N))])
check('negative axis below', solve(-float(N*N),-0.0), ["0",render(-float(N))])
check('upper quadrant', solve(3.0,4.0), ["2","1"])
check('lower quadrant', solve(3.0,-4.0), ["2","-1"])
check('negative real quadrant', solve(-3.0,4.0), ["1","2"])
check('huge axis', solve(1e308,0.0), ["1e+154","0"])
check('tiny imaginary', solve(4.0,N*1e-200), ["2",render(N*1e-200/4)])
check('origin above', solve(0.0,0.0), ["0","0"])
check('origin below', solve(0.0,-0.0), ["0","-0"])
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 axis	`['1', '0']`	`['1', '0']`	Passed
negative axis above	`['0', '1']`	`['0', '1']`	Passed
negative axis below	`['0', '-1']`	`['0', '-1']`	Passed
upper quadrant	`['2', '1']`	`['2', '1']`	Passed
lower quadrant	`['2', '-1']`	`['2', '-1']`	Passed
negative real quadrant	`['1', '2']`	`['1', '2']`	Passed
huge axis	`['1e+154', '0']`	`['1e+154', '0']`	Passed
tiny imaginary	`['2', '0']`	`['2', '2.5e-201']`	Failed
origin above	`['0', '0']`	`['0', '0']`	Passed
origin below	`['0', '-0']`	`['0', '-0']`	Passed

SHA-256 / d02f5ee265ed7002b1a7f652b5a550c4f99919021ea401356a9f8ccc36948921

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,y):
    try:
        if x==0 and y==0: return ['0',render(y)]
        r=math.hypot(x,y)
        t=math.sqrt(r/2+abs(x)/2)
        if x>=0:
            real=t
            imag=y/(2*t)
        else:
            imag=math.copysign(t,y)
            real=abs(y)/(2*t)
        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('positive axis', solve(float(N*N),0.0), [render(float(N)),"0"])
check('negative axis above', solve(-float(N*N),0.0), ["0",render(float(N))])
check('negative axis below', solve(-float(N*N),-0.0), ["0",render(-float(N))])
check('upper quadrant', solve(3.0,4.0), ["2","1"])
check('lower quadrant', solve(3.0,-4.0), ["2","-1"])
check('negative real quadrant', solve(-3.0,4.0), ["1","2"])
check('huge axis', solve(1e308,0.0), ["1e+154","0"])
check('tiny imaginary', solve(4.0,N*1e-200), ["2",render(N*1e-200/4)])
check('origin above', solve(0.0,0.0), ["0","0"])
check('origin below', solve(0.0,-0.0), ["0","-0"])
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 axis	`['1', '0']`	`['1', '0']`	Passed
negative axis above	`['0', '1']`	`['0', '1']`	Passed
negative axis below	`['0', '-1']`	`['0', '-1']`	Passed
upper quadrant	`['2', '1']`	`['2', '1']`	Passed
lower quadrant	`['2', '-1']`	`['2', '-1']`	Passed
negative real quadrant	`['1', '2']`	`['1', '2']`	Passed
huge axis	`['1e+154', '0']`	`['1e+154', '0']`	Passed
tiny imaginary	`['2', '2.5e-201']`	`['2', '2.5e-201']`	Passed
origin above	`['0', '0']`	`['0', '0']`	Passed
origin below	`['0', '-0']`	`['0', '-0']`	Passed

SHA-256 / 9a44494b75975666e33f4c600a76b1f2e736a5bb390106b9c4809fc638a9569d

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 2026-09-29T14:39:36.377226+00:00.

Case digest / 579489d3631f1b5d8149763798a2083e98751f1b65556b5ca2d0f6aa37858d11