{"abstract":"Power of one plus a small increment rounds the base before amplification.","category":"Floating-point arithmetic","checks":7,"contract":"Evaluate (1+x)**y without first rounding a small increment into unity. Domain requires x>-1; y is finite, and fixtures have finite results. Finite results are rendered to eleven significant decimal digits; modeled domain violations and arithmetic errors are explicit strings.","evaluation_group":"s3-float-power1p","failed_approach":"The attempted local correction exponent=y*math.log(max(1+x,1e-300)) still violates the explicit regression fixtures.","family":"s3-floating_point_arithmetic-power1p-base-rounding","id":"FA-16956","implementations":{"attempt":{"sha256":"3c2cea1f30efa25274eee0b0c56456146f5cb07036595ce91e393a64e2eaedb5","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(x,y):\n    try:\n        if x<=-1: return 'domain'\n        if y==0 or x==0: return '1'\n        exponent=y*math.log(max(1+x,1e-300))\n        result=math.exp(exponent)\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('tiny amplified', solve(N*2.0**-60,2.0**60), render(math.exp(N)))\ncheck('negative amplified', solve(-N*2.0**-60,2.0**60), render(math.exp(-N)))\ncheck('normal', solve(1.0,float(N)), render(2.0**N))\ncheck('negative power', solve(1.0,-float(N)), render(2.0**(-N)))\ncheck('zero exponent', solve(0.5,0.0), \"1\")\ncheck('zero increment', solve(0.0,float(N)), \"1\")\ncheck('invalid base', solve(-2.0,float(N)), \"domain\")\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":"4f3c09e020c3551e24d12f589174b2c57915cc3c192e3e2c82cf73b67dc1e598","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(x,y):\n    try:\n        if x<=-1: return 'domain'\n        if y==0 or x==0: return '1'\n        exponent=y*math.log(1+x)\n        result=math.exp(exponent)\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('tiny amplified', solve(N*2.0**-60,2.0**60), render(math.exp(N)))\ncheck('negative amplified', solve(-N*2.0**-60,2.0**60), render(math.exp(-N)))\ncheck('normal', solve(1.0,float(N)), render(2.0**N))\ncheck('negative power', solve(1.0,-float(N)), render(2.0**(-N)))\ncheck('zero exponent', solve(0.5,0.0), \"1\")\ncheck('zero increment', solve(0.0,float(N)), \"1\")\ncheck('invalid base', solve(-2.0,float(N)), \"domain\")\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"},"fixed":{"sha256":"c4b47d9a689e667e8fe276dbfd388973a6af1c5fd37c142044212b14d63fbbf6","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(x,y):\n    try:\n        if x<=-1: return 'domain'\n        if y==0 or x==0: return '1'\n        exponent=y*math.log1p(x)\n        result=math.exp(exponent)\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('tiny amplified', solve(N*2.0**-60,2.0**60), render(math.exp(N)))\ncheck('negative amplified', solve(-N*2.0**-60,2.0**60), render(math.exp(-N)))\ncheck('normal', solve(1.0,float(N)), render(2.0**N))\ncheck('negative power', solve(1.0,-float(N)), render(2.0**(-N)))\ncheck('zero exponent', solve(0.5,0.0), \"1\")\ncheck('zero increment', solve(0.0,float(N)), \"1\")\ncheck('invalid base', solve(-2.0,float(N)), \"domain\")\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-power1p-base-rounding","generated_at":"2026-09-29T14:39:41.646897+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.","repair":"Apply the contract at this fault site using exponent=y*math.log1p(x).","root_cause":"Power of one plus a small increment rounds the base before amplification. The faulty expression is exponent=y*math.log(1+x).","sha256":"d708ec1e3f076730a7ac1813d8ef5a9e3e299e1567e70c53aedc8f3c1673781f","title":"Power of one plus a small increment rounds the base before amplification · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":43.111,"exit_code":1,"observations":[{"actual":"1","check":"tiny amplified","expected":"2.7182818285","passed":false},{"actual":"1","check":"negative amplified","expected":"0.36787944117","passed":false},{"actual":"2","check":"normal","expected":"2","passed":true},{"actual":"0.5","check":"negative power","expected":"0.5","passed":true},{"actual":"1","check":"zero exponent","expected":"1","passed":true},{"actual":"1","check":"zero increment","expected":"1","passed":true},{"actual":"domain","check":"invalid base","expected":"domain","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"tiny amplified\", \"actual\": \"1\", \"expected\": \"2.7182818285\", \"passed\": false}, {\"check\": \"negative amplified\", \"actual\": \"1\", \"expected\": \"0.36787944117\", \"passed\": false}, {\"check\": \"normal\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"negative power\", \"actual\": \"0.5\", \"expected\": \"0.5\", \"passed\": true}, {\"check\": \"zero exponent\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"zero increment\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"invalid base\", \"actual\": \"domain\", \"expected\": \"domain\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":43.695,"exit_code":1,"observations":[{"actual":"1","check":"tiny amplified","expected":"2.7182818285","passed":false},{"actual":"1","check":"negative amplified","expected":"0.36787944117","passed":false},{"actual":"2","check":"normal","expected":"2","passed":true},{"actual":"0.5","check":"negative power","expected":"0.5","passed":true},{"actual":"1","check":"zero exponent","expected":"1","passed":true},{"actual":"1","check":"zero increment","expected":"1","passed":true},{"actual":"domain","check":"invalid base","expected":"domain","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"tiny amplified\", \"actual\": \"1\", \"expected\": \"2.7182818285\", \"passed\": false}, {\"check\": \"negative amplified\", \"actual\": \"1\", \"expected\": \"0.36787944117\", \"passed\": false}, {\"check\": \"normal\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"negative power\", \"actual\": \"0.5\", \"expected\": \"0.5\", \"passed\": true}, {\"check\": \"zero exponent\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"zero increment\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"invalid base\", \"actual\": \"domain\", \"expected\": \"domain\", \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":44.859,"exit_code":0,"observations":[{"actual":"2.7182818285","check":"tiny amplified","expected":"2.7182818285","passed":true},{"actual":"0.36787944117","check":"negative amplified","expected":"0.36787944117","passed":true},{"actual":"2","check":"normal","expected":"2","passed":true},{"actual":"0.5","check":"negative power","expected":"0.5","passed":true},{"actual":"1","check":"zero exponent","expected":"1","passed":true},{"actual":"1","check":"zero increment","expected":"1","passed":true},{"actual":"domain","check":"invalid base","expected":"domain","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"tiny amplified\", \"actual\": \"2.7182818285\", \"expected\": \"2.7182818285\", \"passed\": true}, {\"check\": \"negative amplified\", \"actual\": \"0.36787944117\", \"expected\": \"0.36787944117\", \"passed\": true}, {\"check\": \"normal\", \"actual\": \"2\", \"expected\": \"2\", \"passed\": true}, {\"check\": \"negative power\", \"actual\": \"0.5\", \"expected\": \"0.5\", \"passed\": true}, {\"check\": \"zero exponent\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"zero increment\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"invalid base\", \"actual\": \"domain\", \"expected\": \"domain\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}