FA-17001 / Floating-point arithmetic / Open access

Exponential divided difference reverses its exponent separation · case 01

Exponential divided difference reverses its exponent separation.

Verified by executionVariant 1 · 7 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

Exponential divided difference reverses its exponent separation. The faulty expression is d=a-b.

THE FAILURE

Exponential divided difference reverses its exponent separation. The faulty expression is d=a-b.

Unsuccessful approach: The attempted local correction d=abs(b-a) still violates the explicit regression fixtures.

Case contract

Evaluate (exp(b)-exp(a))/(b-a) with continuous value exp(a) when a=b. Fixture exponents are bounded; use expm1 for close endpoints. 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(a,b):
    try:
        if a==b: return render(math.exp(a))
        d=a-b
        if abs(d)<1e-8:
            ratio=1+d/2+d*d/6
        else:
            ratio=math.expm1(d)/d
        result=math.exp(a)*ratio
        return render(result)
    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('coincident', solve(float(N),float(N)), render(math.exp(N)))
check('near endpoints', solve(0.0,N*2.0**-60), "1")
check('shifted near', solve(1.0,1.0+N*2.0**-50), render(math.exp(1)*(1+N*2.0**-50/2)))
check('forward', solve(0.0,float(N)), render(math.expm1(N)/N))
check('reverse', solve(float(N),0.0), render(math.expm1(N)/N))
check('negative interval', solve(-float(N),0.0), render(-math.expm1(-N)/N))
check('separated', solve(-1.0,1.0), render((math.exp(1)-math.exp(-1))/2))
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
coincident	`2.7182818285`	`2.7182818285`	Passed
near endpoints	`1`	`1`	Passed
shifted near	`2.7182818285`	`2.7182818285`	Passed
forward	`0.63212055883`	`1.7182818285`	Failed
reverse	`4.6707742705`	`1.7182818285`	Failed
negative interval	`0.23254415793`	`0.63212055883`	Failed
separated	`0.1590461864`	`1.1752011936`	Failed

SHA-256 / 0812c0a79acd8d2193ed3094a1063567f32715520a44fe94b1a07ec65273961b

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(a,b):
    try:
        if a==b: return render(math.exp(a))
        d=abs(b-a)
        if abs(d)<1e-8:
            ratio=1+d/2+d*d/6
        else:
            ratio=math.expm1(d)/d
        result=math.exp(a)*ratio
        return render(result)
    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('coincident', solve(float(N),float(N)), render(math.exp(N)))
check('near endpoints', solve(0.0,N*2.0**-60), "1")
check('shifted near', solve(1.0,1.0+N*2.0**-50), render(math.exp(1)*(1+N*2.0**-50/2)))
check('forward', solve(0.0,float(N)), render(math.expm1(N)/N))
check('reverse', solve(float(N),0.0), render(math.expm1(N)/N))
check('negative interval', solve(-float(N),0.0), render(-math.expm1(-N)/N))
check('separated', solve(-1.0,1.0), render((math.exp(1)-math.exp(-1))/2))
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
coincident	`2.7182818285`	`2.7182818285`	Passed
near endpoints	`1`	`1`	Passed
shifted near	`2.7182818285`	`2.7182818285`	Passed
forward	`1.7182818285`	`1.7182818285`	Passed
reverse	`4.6707742705`	`1.7182818285`	Failed
negative interval	`0.63212055883`	`0.63212055883`	Passed
separated	`1.1752011936`	`1.1752011936`	Passed

SHA-256 / 6b56252c10d5a1011fa160a85753aa2c9e9aa1c33d31f223bbefe9460927cb94

HELD IN THE MEMBER ARCHIVE

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

This mechanism has 7 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.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

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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 2026-09-29T14:39:41.953326+00:00.

Case digest / 981abc4f69857ad23db9c948e435d6f17a3c54938a3f3338b9d967cac1dc392c