FA-15961 / Floating-point arithmetic / Open access

Nearest rounding loses sticky residue after a guard · case 01

Nearest rounding loses sticky residue after a guard.

Verified by executionVariant 1 · 12 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Nearest rounding loses sticky residue after a guard. The faulty expression is bool(g and (r or q%2)).

VERIFIED REPAIR

Apply the contract at this fault site using bool(g and (r or s or q%2)).

Unsuccessful approach: The attempted local correction bool(g and q%2) still violates the explicit regression fixtures.

Case contract

Round a nonnegative integer significand q with guard, round and sticky bits. Modes are nearest-even, nearest-away, zero, up, down; sign is +1 or -1. Return rounded signed integer, inexact flag, and whether rounding incremented the magnitude.

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(q,g,r,s,sign,mode):
    inexact=bool(g or r or s)
    if mode == 'nearest-even': increment=bool(g and (r or q%2))
    elif mode == 'nearest-away': increment=bool(g)
    elif mode == 'up': increment=sign>0 and inexact
    elif mode == 'down': increment=sign<0 and inexact
    else: increment=False
    value=sign*(q+int(increment))
    return [value,inexact,bool(increment)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('tie even', solve(2*N,1,0,0,1,"nearest-even"), [2*N,True,False])
check('tie odd', solve(2*N+1,1,0,0,1,"nearest-even"), [2*N+2,True,True])
check('above tie sticky', solve(2*N,1,0,1,1,"nearest-even"), [2*N+1,True,True])
check('above tie round', solve(2*N,1,1,0,1,"nearest-even"), [2*N+1,True,True])
check('below half', solve(2*N,0,1,1,1,"nearest-even"), [2*N,True,False])
check('nearest away', solve(2*N,1,0,0,-1,"nearest-away"), [-2*N-1,True,True])
check('up positive', solve(N,0,0,1,1,"up"), [N+1,True,True])
check('up negative', solve(N,1,1,1,-1,"up"), [-N,True,False])
check('down negative', solve(N,0,1,0,-1,"down"), [-N-1,True,True])
check('down positive', solve(N,1,1,1,1,"down"), [N,True,False])
check('truncate', solve(N,1,1,1,-1,"zero"), [-N,True,False])
check('exact', solve(N,0,0,0,1,"up"), [N,False,False])
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
tie even	`[2, True, False]`	`[2, True, False]`	Passed
tie odd	`[4, True, True]`	`[4, True, True]`	Passed
above tie sticky	`[2, True, False]`	`[3, True, True]`	Failed
above tie round	`[3, True, True]`	`[3, True, True]`	Passed
below half	`[2, True, False]`	`[2, True, False]`	Passed
nearest away	`[-3, True, True]`	`[-3, True, True]`	Passed
up positive	`[2, True, True]`	`[2, True, True]`	Passed
up negative	`[-1, True, False]`	`[-1, True, False]`	Passed
down negative	`[-2, True, True]`	`[-2, True, True]`	Passed
down positive	`[1, True, False]`	`[1, True, False]`	Passed
truncate	`[-1, True, False]`	`[-1, True, False]`	Passed
exact	`[1, False, False]`	`[1, False, False]`	Passed

SHA-256 / 43b4461303fb6a583c10c5cc3c3e3161775986ceeca035aebff812ae30041c6c

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(q,g,r,s,sign,mode):
    inexact=bool(g or r or s)
    if mode == 'nearest-even': increment=bool(g and q%2)
    elif mode == 'nearest-away': increment=bool(g)
    elif mode == 'up': increment=sign>0 and inexact
    elif mode == 'down': increment=sign<0 and inexact
    else: increment=False
    value=sign*(q+int(increment))
    return [value,inexact,bool(increment)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('tie even', solve(2*N,1,0,0,1,"nearest-even"), [2*N,True,False])
check('tie odd', solve(2*N+1,1,0,0,1,"nearest-even"), [2*N+2,True,True])
check('above tie sticky', solve(2*N,1,0,1,1,"nearest-even"), [2*N+1,True,True])
check('above tie round', solve(2*N,1,1,0,1,"nearest-even"), [2*N+1,True,True])
check('below half', solve(2*N,0,1,1,1,"nearest-even"), [2*N,True,False])
check('nearest away', solve(2*N,1,0,0,-1,"nearest-away"), [-2*N-1,True,True])
check('up positive', solve(N,0,0,1,1,"up"), [N+1,True,True])
check('up negative', solve(N,1,1,1,-1,"up"), [-N,True,False])
check('down negative', solve(N,0,1,0,-1,"down"), [-N-1,True,True])
check('down positive', solve(N,1,1,1,1,"down"), [N,True,False])
check('truncate', solve(N,1,1,1,-1,"zero"), [-N,True,False])
check('exact', solve(N,0,0,0,1,"up"), [N,False,False])
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
tie even	`[2, True, False]`	`[2, True, False]`	Passed
tie odd	`[4, True, True]`	`[4, True, True]`	Passed
above tie sticky	`[2, True, False]`	`[3, True, True]`	Failed
above tie round	`[2, True, False]`	`[3, True, True]`	Failed
below half	`[2, True, False]`	`[2, True, False]`	Passed
nearest away	`[-3, True, True]`	`[-3, True, True]`	Passed
up positive	`[2, True, True]`	`[2, True, True]`	Passed
up negative	`[-1, True, False]`	`[-1, True, False]`	Passed
down negative	`[-2, True, True]`	`[-2, True, True]`	Passed
down positive	`[1, True, False]`	`[1, True, False]`	Passed
truncate	`[-1, True, False]`	`[-1, True, False]`	Passed
exact	`[1, False, False]`	`[1, False, False]`	Passed

SHA-256 / 4852e2ecc96128fe716cea7446ae021a8b4ad7c7ba40f34db4706de06230ab63

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(q,g,r,s,sign,mode):
    inexact=bool(g or r or s)
    if mode == 'nearest-even': increment=bool(g and (r or s or q%2))
    elif mode == 'nearest-away': increment=bool(g)
    elif mode == 'up': increment=sign>0 and inexact
    elif mode == 'down': increment=sign<0 and inexact
    else: increment=False
    value=sign*(q+int(increment))
    return [value,inexact,bool(increment)]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('tie even', solve(2*N,1,0,0,1,"nearest-even"), [2*N,True,False])
check('tie odd', solve(2*N+1,1,0,0,1,"nearest-even"), [2*N+2,True,True])
check('above tie sticky', solve(2*N,1,0,1,1,"nearest-even"), [2*N+1,True,True])
check('above tie round', solve(2*N,1,1,0,1,"nearest-even"), [2*N+1,True,True])
check('below half', solve(2*N,0,1,1,1,"nearest-even"), [2*N,True,False])
check('nearest away', solve(2*N,1,0,0,-1,"nearest-away"), [-2*N-1,True,True])
check('up positive', solve(N,0,0,1,1,"up"), [N+1,True,True])
check('up negative', solve(N,1,1,1,-1,"up"), [-N,True,False])
check('down negative', solve(N,0,1,0,-1,"down"), [-N-1,True,True])
check('down positive', solve(N,1,1,1,1,"down"), [N,True,False])
check('truncate', solve(N,1,1,1,-1,"zero"), [-N,True,False])
check('exact', solve(N,0,0,0,1,"up"), [N,False,False])
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
tie even	`[2, True, False]`	`[2, True, False]`	Passed
tie odd	`[4, True, True]`	`[4, True, True]`	Passed
above tie sticky	`[3, True, True]`	`[3, True, True]`	Passed
above tie round	`[3, True, True]`	`[3, True, True]`	Passed
below half	`[2, True, False]`	`[2, True, False]`	Passed
nearest away	`[-3, True, True]`	`[-3, True, True]`	Passed
up positive	`[2, True, True]`	`[2, True, True]`	Passed
up negative	`[-1, True, False]`	`[-1, True, False]`	Passed
down negative	`[-2, True, True]`	`[-2, True, True]`	Passed
down positive	`[1, True, False]`	`[1, True, False]`	Passed
truncate	`[-1, True, False]`	`[-1, True, False]`	Passed
exact	`[1, False, False]`	`[1, False, False]`	Passed

SHA-256 / a5a73ab805edc4aec7482b1c8c458ae63df1e594277495dbf7311049afcbfc91

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 2026-09-29T14:39:31.976912+00:00.

Case digest / 8b4e2d174f4e99cb173980b1b6888e9d4894488734acd717546a9af2239b2550