FA-17231 / Floating-point arithmetic / Open access

Scaled square root rounds the exponent quotient upward · case 01

Scaled square root rounds the exponent quotient upward.

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

ROOT CAUSE

Scaled square root rounds the exponent quotient upward. The faulty expression is q,r=(e+1)//2,e%2.

VERIFIED REPAIR

Apply the contract at this fault site using q,r=divmod(e,2).

Unsuccessful approach: The attempted local correction q,r=abs(e)//2,e%2 still violates the explicit regression fixtures.

Case contract

Nonnegative square root of finite binary64 x via exponent normalization. Negative input returns domain; negative zero is preserved. 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 'domain'
        if x==0: return render(x)
        m,e=math.frexp(x)
        q,r=(e+1)//2,e%2
        root=math.sqrt(math.ldexp(m,r))
        return render(math.ldexp(root,q))
    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('square', solve(float(N*N)), render(float(N)))
check('normal', solve(float(N)), render(math.sqrt(N)))
check('huge', solve(1e300), "1e+150")
check('tiny', solve(1e-300), "1e-150")
check('odd exponent', solve(2.0), render(math.sqrt(2)))
check('positive root zero', solve(0.0), "0")
check('negative domain', solve(-float(N)), "domain")
check('negative zero', solve(-0.0), "-0")
check('subnormal', solve(math.ldexp(1.0,-1074)), render(math.ldexp(1.0,-537)))
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
square	`2`	`1`	Failed
normal	`2`	`1`	Failed
huge	`2e+150`	`1e+150`	Failed
tiny	`1e-150`	`1e-150`	Passed
odd exponent	`1.4142135624`	`1.4142135624`	Passed
positive root zero	`0`	`0`	Passed
negative domain	`domain`	`domain`	Passed
negative zero	`-0`	`-0`	Passed
subnormal	`4.445517499e-162`	`2.2227587495e-162`	Failed

SHA-256 / f4f16258cd54f05dc6ca40c5fb625bcef67af19a573c192531cc5dff53b88423

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 'domain'
        if x==0: return render(x)
        m,e=math.frexp(x)
        q,r=abs(e)//2,e%2
        root=math.sqrt(math.ldexp(m,r))
        return render(math.ldexp(root,q))
    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('square', solve(float(N*N)), render(float(N)))
check('normal', solve(float(N)), render(math.sqrt(N)))
check('huge', solve(1e300), "1e+150")
check('tiny', solve(1e-300), "1e-150")
check('odd exponent', solve(2.0), render(math.sqrt(2)))
check('positive root zero', solve(0.0), "0")
check('negative domain', solve(-float(N)), "domain")
check('negative zero', solve(-0.0), "-0")
check('subnormal', solve(math.ldexp(1.0,-1074)), render(math.ldexp(1.0,-537)))
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
square	`1`	`1`	Passed
normal	`1`	`1`	Passed
huge	`1e+150`	`1e+150`	Passed
tiny	`6.6969287949e+149`	`1e-150`	Failed
odd exponent	`1.4142135624`	`1.4142135624`	Passed
positive root zero	`0`	`0`	Passed
negative domain	`domain`	`domain`	Passed
negative zero	`-0`	`-0`	Passed
subnormal	`2.2494568973e+161`	`2.2227587495e-162`	Failed

SHA-256 / 597c822b568b0678cc44d4f74875b003ed9ff63b7fe5c72bfbe16595ce428709

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 'domain'
        if x==0: return render(x)
        m,e=math.frexp(x)
        q,r=divmod(e,2)
        root=math.sqrt(math.ldexp(m,r))
        return render(math.ldexp(root,q))
    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('square', solve(float(N*N)), render(float(N)))
check('normal', solve(float(N)), render(math.sqrt(N)))
check('huge', solve(1e300), "1e+150")
check('tiny', solve(1e-300), "1e-150")
check('odd exponent', solve(2.0), render(math.sqrt(2)))
check('positive root zero', solve(0.0), "0")
check('negative domain', solve(-float(N)), "domain")
check('negative zero', solve(-0.0), "-0")
check('subnormal', solve(math.ldexp(1.0,-1074)), render(math.ldexp(1.0,-537)))
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
square	`1`	`1`	Passed
normal	`1`	`1`	Passed
huge	`1e+150`	`1e+150`	Passed
tiny	`1e-150`	`1e-150`	Passed
odd exponent	`1.4142135624`	`1.4142135624`	Passed
positive root zero	`0`	`0`	Passed
negative domain	`domain`	`domain`	Passed
negative zero	`-0`	`-0`	Passed
subnormal	`2.2227587495e-162`	`2.2227587495e-162`	Passed

SHA-256 / a2be440e40877a8ca6d5cbe5e4d7abbf5c9581a86035cddc86c6c1bf1b225c3e

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 2026-09-29T14:39:44.363901+00:00.

Case digest / 27ef4f5d153795dad7ca59db00780ee1f4388869d9f1ee34d4d965ceb8142b07