FA-16636 / Floating-point arithmetic / Open access

Exact ratio reduction moves exponent in the wrong direction · case 01

Exact ratio reduction moves exponent in the wrong direction.

Verified by executionVariant 1 · 11 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Exact ratio reduction moves exponent in the wrong direction. The faulty expression is power+=2.

VERIFIED REPAIR

Apply the contract at this fault site using power+=1.

Unsuccessful approach: The attempted local correction power+=0 still violates the explicit regression fixtures.

Case contract

Decode a finite binary64 value into its exact reduced signed numerator and positive power-of-two denominator, using (0,1) for either zero. Infinity and NaN return null.

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(bits):
    sign=-1 if bits>>63 else 1
    e=(bits>>52)&2047
    f=bits&((1<<52)-1)
    if e==2047: return None
    if e:
        mantissa=f|(1<<52)
        power=e-1023-52
    else:
        mantissa=f
        power=-1074
    if mantissa==0: return [0,1]
    while power<0 and mantissa%2==0:
        mantissa//=2
        power+=2
    if power>=0: return [sign*(mantissa<<power),1]
    return [sign*mantissa,1<<(-power)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('large integer', solve((1023+60+N)<<52), [1<<(60+N),1])
check('positive integer', solve((1023+N)<<52), [1<<N,1])
check('negative integer', solve((1<<63)|((1023+N)<<52)), [-(1<<N),1])
check('fraction', solve(0x3fe0000000000000), [1,2])
check('nontrivial fraction', solve(0x3fe8000000000000), [3,4])
check('minimum subnormal', solve(N), list(math.ldexp(float(N),-1074).as_integer_ratio()))
check('negative subnormal', solve((1<<63)|N), list((-math.ldexp(float(N),-1074)).as_integer_ratio()))
check('zero', solve(0), [0,1])
check('negative zero', solve(1<<63), [0,1])
check('infinity', solve(0x7ff0000000000000), None)
check('nan', solve(0x7ff8000000000000|N), None)
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
large integer	`[2305843009213693952, 1]`	`[2305843009213693952, 1]`	Passed
positive integer	`[134217728, 1]`	`[2, 1]`	Failed
negative integer	`[-134217728, 1]`	`[-2, 1]`	Failed
fraction	`[67108864, 1]`	`[1, 2]`	Failed
nontrivial fraction	`[100663296, 1]`	`[3, 4]`	Failed
minimum subnormal	`[1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	`[1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	Passed
negative subnormal	`[-1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	`[-1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	Passed
zero	`[0, 1]`	`[0, 1]`	Passed
negative zero	`[0, 1]`	`[0, 1]`	Passed
infinity	`None`	`None`	Passed
nan	`None`	`None`	Passed

SHA-256 / c6827410a2a94caf44cc5f597fdf17d6f41efd892eed174a57226ae84429a7df

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(bits):
    sign=-1 if bits>>63 else 1
    e=(bits>>52)&2047
    f=bits&((1<<52)-1)
    if e==2047: return None
    if e:
        mantissa=f|(1<<52)
        power=e-1023-52
    else:
        mantissa=f
        power=-1074
    if mantissa==0: return [0,1]
    while power<0 and mantissa%2==0:
        mantissa//=2
        power+=0
    if power>=0: return [sign*(mantissa<<power),1]
    return [sign*mantissa,1<<(-power)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('large integer', solve((1023+60+N)<<52), [1<<(60+N),1])
check('positive integer', solve((1023+N)<<52), [1<<N,1])
check('negative integer', solve((1<<63)|((1023+N)<<52)), [-(1<<N),1])
check('fraction', solve(0x3fe0000000000000), [1,2])
check('nontrivial fraction', solve(0x3fe8000000000000), [3,4])
check('minimum subnormal', solve(N), list(math.ldexp(float(N),-1074).as_integer_ratio()))
check('negative subnormal', solve((1<<63)|N), list((-math.ldexp(float(N),-1074)).as_integer_ratio()))
check('zero', solve(0), [0,1])
check('negative zero', solve(1<<63), [0,1])
check('infinity', solve(0x7ff0000000000000), None)
check('nan', solve(0x7ff8000000000000|N), None)
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
large integer	`[2305843009213693952, 1]`	`[2305843009213693952, 1]`	Passed
positive integer	`[1, 2251799813685248]`	`[2, 1]`	Failed
negative integer	`[-1, 2251799813685248]`	`[-2, 1]`	Failed
fraction	`[1, 9007199254740992]`	`[1, 2]`	Failed
nontrivial fraction	`[3, 9007199254740992]`	`[3, 4]`	Failed
minimum subnormal	`[1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	`[1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	Passed
negative subnormal	`[-1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	`[-1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	Passed
zero	`[0, 1]`	`[0, 1]`	Passed
negative zero	`[0, 1]`	`[0, 1]`	Passed
infinity	`None`	`None`	Passed
nan	`None`	`None`	Passed

SHA-256 / b7143a188cfa1de1549db5c423083ccc7f180b6c6cadb8ad96e13a1fda5da04a

3 / The verified repair

Exit 0

"""Failure Map reference implementation. Python standard library only."""
import json
import math
import struct
N = 1
observations = []
def solve(bits):
    sign=-1 if bits>>63 else 1
    e=(bits>>52)&2047
    f=bits&((1<<52)-1)
    if e==2047: return None
    if e:
        mantissa=f|(1<<52)
        power=e-1023-52
    else:
        mantissa=f
        power=-1074
    if mantissa==0: return [0,1]
    while power<0 and mantissa%2==0:
        mantissa//=2
        power+=1
    if power>=0: return [sign*(mantissa<<power),1]
    return [sign*mantissa,1<<(-power)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('large integer', solve((1023+60+N)<<52), [1<<(60+N),1])
check('positive integer', solve((1023+N)<<52), [1<<N,1])
check('negative integer', solve((1<<63)|((1023+N)<<52)), [-(1<<N),1])
check('fraction', solve(0x3fe0000000000000), [1,2])
check('nontrivial fraction', solve(0x3fe8000000000000), [3,4])
check('minimum subnormal', solve(N), list(math.ldexp(float(N),-1074).as_integer_ratio()))
check('negative subnormal', solve((1<<63)|N), list((-math.ldexp(float(N),-1074)).as_integer_ratio()))
check('zero', solve(0), [0,1])
check('negative zero', solve(1<<63), [0,1])
check('infinity', solve(0x7ff0000000000000), None)
check('nan', solve(0x7ff8000000000000|N), None)
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
large integer	`[2305843009213693952, 1]`	`[2305843009213693952, 1]`	Passed
positive integer	`[2, 1]`	`[2, 1]`	Passed
negative integer	`[-2, 1]`	`[-2, 1]`	Passed
fraction	`[1, 2]`	`[1, 2]`	Passed
nontrivial fraction	`[3, 4]`	`[3, 4]`	Passed
minimum subnormal	`[1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	`[1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	Passed
negative subnormal	`[-1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	`[-1, 202402253307310618352495346718917307049556649764142118356901358027430339567995346891960383701437124495187077864316811911389808737385793476867013399940738509921517424276566361364466907742093216341239767678472745068562007483424692698618103355649159556340810056512358769552333414615230502532186327508646006263307707741093494784]`	Passed
zero	`[0, 1]`	`[0, 1]`	Passed
negative zero	`[0, 1]`	`[0, 1]`	Passed
infinity	`None`	`None`	Passed
nan	`None`	`None`	Passed

SHA-256 / 232c5cccac878943ca1770bff8d7a21afb5af3860d3181aefa1b46f6590a6293

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 2026-09-29T14:39:38.073764+00:00.

Case digest / 6e3618056ccb8ee93d458520daa7f90ea16d52fd811a2b9fc13797e4d6eaef29