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FA-16041 / Floating-point arithmetic / Open access

Exponent and fraction are intersected instead of combined · case 01

Exponent and fraction are intersected instead of combined.

Verified by executionVariant 1 · 14 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Exponent and fraction are intersected instead of combined. The faulty expression is return sign|((exponent<<3)&fraction).

THE FAILURE

Exponent and fraction are intersected instead of combined. The faulty expression is return sign|((exponent<<3)&fraction).

Unsuccessful approach: The attempted local correction return sign|(exponent+fraction) still violates the explicit regression fixtures.

Case contract

Encode finite exactly represented dyadic x into sign/4-bit-exponent/3-bit-fraction toy float with bias 7, gradual underflow, round-to-nearest-even and infinity overflow. Return the unsigned byte. The miniature format is stipulated, not a hardware conformance claim.

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
N = 1
observations = []
def solve(x):
    sign=128 if math.copysign(1.0,x)<0 else 0
    x=abs(x)
    if x == 0: return sign
    if x >= 248: return sign|120
    if x < 2**-6:
        q=round(x*512)
        return sign|q
    m,e=math.frexp(x)
    e-=1
    q=round(m*16)
    if q == 16:
        q=8
        e+=1
    if e>7: return sign|120
    exponent=e+7
    fraction=q-8
    return sign|((exponent<<3)&fraction)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('zero', solve(0.0), 0)
check('negative zero', solve(-0.0), 128)
check('subnormal', solve(N/512), N)
check('negative subnormal', solve(-N/512), 128|N)
check('underflow tie', solve(1/1024), 0)
check('subnormal carry', solve(15/1024), 8)
check('normal varying', solve(2.0**N), (N+7)<<3)
check('normal fraction', solve((8+N)/8), 56+N)
check('binade carry', solve(1.9375), 64)
check('below midpoint', solve(1.90625), 63)
check('largest finite', solve(240.0), 119)
check('overflow tie', solve(248.0), 120)
check('negative overflow', solve(-256.0), 248)
check('minimum normal', solve(1/64), 8)
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
zero00Passed
negative zero128128Passed
subnormal11Passed
negative subnormal129129Passed
underflow tie00Passed
subnormal carry88Passed
normal varying064Failed
normal fraction057Failed
binade carry064Failed
below midpoint063Failed
largest finite0119Failed
overflow tie120120Passed
negative overflow248248Passed
minimum normal08Failed

SHA-256 / d58bd2722edef02c18982bd631344b8b6687b05e244825f70d9ef0eadf9b9ba2

2 / The unsuccessful fix

Exit 1
"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
N = 1
observations = []
def solve(x):
    sign=128 if math.copysign(1.0,x)<0 else 0
    x=abs(x)
    if x == 0: return sign
    if x >= 248: return sign|120
    if x < 2**-6:
        q=round(x*512)
        return sign|q
    m,e=math.frexp(x)
    e-=1
    q=round(m*16)
    if q == 16:
        q=8
        e+=1
    if e>7: return sign|120
    exponent=e+7
    fraction=q-8
    return sign|(exponent+fraction)
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('zero', solve(0.0), 0)
check('negative zero', solve(-0.0), 128)
check('subnormal', solve(N/512), N)
check('negative subnormal', solve(-N/512), 128|N)
check('underflow tie', solve(1/1024), 0)
check('subnormal carry', solve(15/1024), 8)
check('normal varying', solve(2.0**N), (N+7)<<3)
check('normal fraction', solve((8+N)/8), 56+N)
check('binade carry', solve(1.9375), 64)
check('below midpoint', solve(1.90625), 63)
check('largest finite', solve(240.0), 119)
check('overflow tie', solve(248.0), 120)
check('negative overflow', solve(-256.0), 248)
check('minimum normal', solve(1/64), 8)
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
zero00Passed
negative zero128128Passed
subnormal11Passed
negative subnormal129129Passed
underflow tie00Passed
subnormal carry88Passed
normal varying864Failed
normal fraction857Failed
binade carry864Failed
below midpoint1463Failed
largest finite21119Failed
overflow tie120120Passed
negative overflow248248Passed
minimum normal18Failed

SHA-256 / 584b829ab88599a65d887a0eac6f73685a08175dd9a71cbc3d7ffd543c92bb34

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 14 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.

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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 2026-09-29T14:39:32.729985+00:00.

Case digest / 7a891d8daa928883724f6dce560cd8882966e0a26974275b36845d58b6aa3426