FA-16676 / Floating-point arithmetic / Open access

Stochastic rounding accepts an out-of-range remainder · case 01

Stochastic rounding accepts an out-of-range remainder.

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

ROOT CAUSE

Stochastic rounding accepts an out-of-range remainder. The faulty expression is if k<0 or rem<0 or rem>(1<<k) or draw<0 or draw>(1<<k): return "invalid".

THE FAILURE

Stochastic rounding accepts an out-of-range remainder. The faulty expression is if k<0 or rem<0 or rem>(1<<k) or draw<0 or draw>(1<<k): return "invalid".

Unsuccessful approach: The attempted local correction if k<0 or draw<0: return "invalid" still violates the explicit regression fixtures.

Case contract

Stipulated stochastic rounding of nonnegative significand with exact dyadic remainder rem/2**k. Draw is uniform integer in [0,2**k); increment magnitude iff draw<rem, then apply sign. Exact results never increment. Return rounded value and whether magnitude incremented.

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,rem,k,draw,sign):
    if k<0 or rem<0 or rem>(1<<k) or draw<0 or draw>(1<<k): return "invalid"
    if rem==0: return [sign*q,False]
    inc=draw<rem
    return [sign*(q+int(inc)),inc]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('draw below', solve(N,3,3,2,1), [N+1,True])
check('draw equality', solve(N,3,3,3,1), [N,False])
check('draw above', solve(N,3,3,7,1), [N,False])
check('negative increment', solve(N,3,3,1,-1), [-N-1,True])
check('negative no increment', solve(N,3,3,5,-1), [-N,False])
check('exact', solve(N,0,3,0,1), [N,False])
check('invalid remainder', solve(N,8,3,0,1), "invalid")
check('invalid draw', solve(N,3,3,8,1), "invalid")
check('large resolution', solve(N,1,54,0,1), [N+1,True])
check('resolution boundary', solve(N,1,54,1,1), [N,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
draw below	`[2, True]`	`[2, True]`	Passed
draw equality	`[1, False]`	`[1, False]`	Passed
draw above	`[1, False]`	`[1, False]`	Passed
negative increment	`[-2, True]`	`[-2, True]`	Passed
negative no increment	`[-1, False]`	`[-1, False]`	Passed
exact	`[1, False]`	`[1, False]`	Passed
invalid remainder	`[2, True]`	`invalid`	Failed
invalid draw	`[1, False]`	`invalid`	Failed
large resolution	`[2, True]`	`[2, True]`	Passed
resolution boundary	`[1, False]`	`[1, False]`	Passed

SHA-256 / 530828dcd61e90da4f240638498b7af86723524ff6d94eec676dd93214af5df4

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,rem,k,draw,sign):
    if k<0 or draw<0: return "invalid"
    if rem==0: return [sign*q,False]
    inc=draw<rem
    return [sign*(q+int(inc)),inc]
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('draw below', solve(N,3,3,2,1), [N+1,True])
check('draw equality', solve(N,3,3,3,1), [N,False])
check('draw above', solve(N,3,3,7,1), [N,False])
check('negative increment', solve(N,3,3,1,-1), [-N-1,True])
check('negative no increment', solve(N,3,3,5,-1), [-N,False])
check('exact', solve(N,0,3,0,1), [N,False])
check('invalid remainder', solve(N,8,3,0,1), "invalid")
check('invalid draw', solve(N,3,3,8,1), "invalid")
check('large resolution', solve(N,1,54,0,1), [N+1,True])
check('resolution boundary', solve(N,1,54,1,1), [N,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
draw below	`[2, True]`	`[2, True]`	Passed
draw equality	`[1, False]`	`[1, False]`	Passed
draw above	`[1, False]`	`[1, False]`	Passed
negative increment	`[-2, True]`	`[-2, True]`	Passed
negative no increment	`[-1, False]`	`[-1, False]`	Passed
exact	`[1, False]`	`[1, False]`	Passed
invalid remainder	`[2, True]`	`invalid`	Failed
invalid draw	`[1, False]`	`invalid`	Failed
large resolution	`[2, True]`	`[2, True]`	Passed
resolution boundary	`[1, False]`	`[1, False]`	Passed

SHA-256 / cf0393a49a268d261752c25da9dbd375fcd36dcd64e94956dfe6fdb3348b4cf1

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 10 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 2026-09-29T14:39:38.704297+00:00.

Case digest / 5940562d5fec45e2bfc50046c459aa496fd48ceeac8d0ed01531fb516212407b