{"abstract":"Exponential divided difference divides by an unsigned separation.","category":"Floating-point arithmetic","checks":7,"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.","contract_signature":"a,b","evaluation_group":"s3-float-exp-divided-difference","failed_approach":"The attempted local correction ratio=abs(math.expm1(d))/d still violates the explicit regression fixtures.","family":"s3-floating_point_arithmetic-exp-divided-difference-orientation","id":"FA-16996","implementations":{"attempt":{"sha256":"e2e14fbb0be42c4aec3a186027e70a46a5c573347ca3a87e0223f19015ade737","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport struct\ndef render(x):\n    if math.isnan(x): return 'nan'\n    if math.isinf(x): return '-infinity' if x<0 else '+infinity'\n    return format(x,'.11g')\n\nN = 1\nobservations = []\ndef solve(a,b):\n    try:\n        if a==b: return render(math.exp(a))\n        d=b-a\n        if abs(d)<1e-8:\n            ratio=1+d/2+d*d/6\n        else:\n            ratio=abs(math.expm1(d))/d\n        result=math.exp(a)*ratio\n        return render(result)\n    except (ValueError, OverflowError, ZeroDivisionError, TypeError):\n        return \"arithmetic-error\"\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('coincident', solve(float(N),float(N)), render(math.exp(N)))\ncheck('near endpoints', solve(0.0,N*2.0**-60), \"1\")\ncheck('shifted near', solve(1.0,1.0+N*2.0**-50), render(math.exp(1)*(1+N*2.0**-50/2)))\ncheck('forward', solve(0.0,float(N)), render(math.expm1(N)/N))\ncheck('reverse', solve(float(N),0.0), render(math.expm1(N)/N))\ncheck('negative interval', solve(-float(N),0.0), render(-math.expm1(-N)/N))\ncheck('separated', solve(-1.0,1.0), render((math.exp(1)-math.exp(-1))/2))\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"e897f8c42a671a9108d507eb2ed9ad1565e24c665ab98275b1891a28419bb614","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport struct\ndef render(x):\n    if math.isnan(x): return 'nan'\n    if math.isinf(x): return '-infinity' if x<0 else '+infinity'\n    return format(x,'.11g')\n\nN = 1\nobservations = []\ndef solve(a,b):\n    try:\n        if a==b: return render(math.exp(a))\n        d=b-a\n        if abs(d)<1e-8:\n            ratio=1+d/2+d*d/6\n        else:\n            ratio=math.expm1(d)/abs(d)\n        result=math.exp(a)*ratio\n        return render(result)\n    except (ValueError, OverflowError, ZeroDivisionError, TypeError):\n        return \"arithmetic-error\"\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('coincident', solve(float(N),float(N)), render(math.exp(N)))\ncheck('near endpoints', solve(0.0,N*2.0**-60), \"1\")\ncheck('shifted near', solve(1.0,1.0+N*2.0**-50), render(math.exp(1)*(1+N*2.0**-50/2)))\ncheck('forward', solve(0.0,float(N)), render(math.expm1(N)/N))\ncheck('reverse', solve(float(N),0.0), render(math.expm1(N)/N))\ncheck('negative interval', solve(-float(N),0.0), render(-math.expm1(-N)/N))\ncheck('separated', solve(-1.0,1.0), render((math.exp(1)-math.exp(-1))/2))\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"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.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"s3-floating_point_arithmetic-exp-divided-difference-orientation","generated_at":"2026-09-29T14:39:41.953326+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"An offline floating representation model isolates a reproducible arithmetic fault.","root_cause":"Exponential divided difference divides by an unsigned separation. The faulty expression is ratio=math.expm1(d)/abs(d).","sha256":"67cf798cb638927ca5d66bf90f58a0c186f62b72f7628ff436681469b23ca20f","title":"Exponential divided difference divides by an unsigned separation · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":41.195,"exit_code":1,"observations":[{"actual":"2.7182818285","check":"coincident","expected":"2.7182818285","passed":true},{"actual":"1","check":"near endpoints","expected":"1","passed":true},{"actual":"2.7182818285","check":"shifted near","expected":"2.7182818285","passed":true},{"actual":"1.7182818285","check":"forward","expected":"1.7182818285","passed":true},{"actual":"-1.7182818285","check":"reverse","expected":"1.7182818285","passed":false},{"actual":"0.63212055883","check":"negative interval","expected":"0.63212055883","passed":true},{"actual":"1.1752011936","check":"separated","expected":"1.1752011936","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"coincident\", \"actual\": \"2.7182818285\", \"expected\": \"2.7182818285\", \"passed\": true}, {\"check\": \"near endpoints\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"shifted near\", \"actual\": \"2.7182818285\", \"expected\": \"2.7182818285\", \"passed\": true}, {\"check\": \"forward\", \"actual\": \"1.7182818285\", \"expected\": \"1.7182818285\", \"passed\": true}, {\"check\": \"reverse\", \"actual\": \"-1.7182818285\", \"expected\": \"1.7182818285\", \"passed\": false}, {\"check\": \"negative interval\", \"actual\": \"0.63212055883\", \"expected\": \"0.63212055883\", \"passed\": true}, {\"check\": \"separated\", \"actual\": \"1.1752011936\", \"expected\": \"1.1752011936\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.236,"exit_code":1,"observations":[{"actual":"2.7182818285","check":"coincident","expected":"2.7182818285","passed":true},{"actual":"1","check":"near endpoints","expected":"1","passed":true},{"actual":"2.7182818285","check":"shifted near","expected":"2.7182818285","passed":true},{"actual":"1.7182818285","check":"forward","expected":"1.7182818285","passed":true},{"actual":"-1.7182818285","check":"reverse","expected":"1.7182818285","passed":false},{"actual":"0.63212055883","check":"negative interval","expected":"0.63212055883","passed":true},{"actual":"1.1752011936","check":"separated","expected":"1.1752011936","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"coincident\", \"actual\": \"2.7182818285\", \"expected\": \"2.7182818285\", \"passed\": true}, {\"check\": \"near endpoints\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"shifted near\", \"actual\": \"2.7182818285\", \"expected\": \"2.7182818285\", \"passed\": true}, {\"check\": \"forward\", \"actual\": \"1.7182818285\", \"expected\": \"1.7182818285\", \"passed\": true}, {\"check\": \"reverse\", \"actual\": \"-1.7182818285\", \"expected\": \"1.7182818285\", \"passed\": false}, {\"check\": \"negative interval\", \"actual\": \"0.63212055883\", \"expected\": \"0.63212055883\", \"passed\": true}, {\"check\": \"separated\", \"actual\": \"1.1752011936\", \"expected\": \"1.1752011936\", \"passed\": true}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}