{"abstract":"Cosine-pi omits the negative half-period sign.","category":"Floating-point arithmetic","checks":11,"contract":"Evaluate cos(pi*x) for finite x after modulo-two reduction. Half integers return positive zero; integers are exact; near half integers use a sine of the distance to the half integer. Finite results are rendered to eleven significant decimal digits; modeled domain violations and arithmetic errors are explicit strings.","evaluation_group":"s3-float-cospi","failed_approach":"The attempted local correction result=-abs(math.cos(math.pi*r)) still violates the explicit regression fixtures.","family":"s3-floating_point_arithmetic-cospi-branch-sign","id":"FA-16801","implementations":{"attempt":{"sha256":"b1c3074b4e1c8763e1c2e983061c32bfca2a97a6e1d28ee931650f20f027188d","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):\n    try:\n        r=abs(math.remainder(x,2.0))\n        if r==0: return '1'\n        if r==1: return '-1'\n        if r==0.5: return '0'\n        if r<0.5:\n            result=math.sin(math.pi*(0.5-r))\n        else:\n            result=-abs(math.cos(math.pi*r))\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('even integer', solve(2.0*N), \"1\")\ncheck('odd integer', solve(2.0*N+1), \"-1\")\ncheck('negative integer', solve(-2.0*N-1), \"-1\")\ncheck('half integer', solve(2.0*N+0.5), \"0\")\ncheck('quarter', solve(0.25), render(math.sqrt(0.5)))\ncheck('three quarters', solve(0.75), render(-math.sqrt(0.5)))\ncheck('near half below', solve(0.5-N*2.0**-50), render(math.sin(math.pi*N*2.0**-50)))\ncheck('near half above', solve(0.5+N*2.0**-50), render(-math.sin(math.pi*N*2.0**-50)))\ncheck('negative half', solve(-0.5), \"0\")\ncheck('negative quarter', solve(-0.25), render(math.sqrt(0.5)))\ncheck('large integer', solve(2.0**52+2*N), \"1\")\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":"3dbbf5778fa6648aca0296154a0b039fa8053f8a9f5b2f36aaa0b878656093ee","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):\n    try:\n        r=abs(math.remainder(x,2.0))\n        if r==0: return '1'\n        if r==1: return '-1'\n        if r==0.5: return '0'\n        if r<0.5:\n            result=math.sin(math.pi*(0.5-r))\n        else:\n            result=math.sin(math.pi*(r-0.5))\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('even integer', solve(2.0*N), \"1\")\ncheck('odd integer', solve(2.0*N+1), \"-1\")\ncheck('negative integer', solve(-2.0*N-1), \"-1\")\ncheck('half integer', solve(2.0*N+0.5), \"0\")\ncheck('quarter', solve(0.25), render(math.sqrt(0.5)))\ncheck('three quarters', solve(0.75), render(-math.sqrt(0.5)))\ncheck('near half below', solve(0.5-N*2.0**-50), render(math.sin(math.pi*N*2.0**-50)))\ncheck('near half above', solve(0.5+N*2.0**-50), render(-math.sin(math.pi*N*2.0**-50)))\ncheck('negative half', solve(-0.5), \"0\")\ncheck('negative quarter', solve(-0.25), render(math.sqrt(0.5)))\ncheck('large integer', solve(2.0**52+2*N), \"1\")\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":"41416c11b6aa6861cfaa3b93f83ae6b9d6dd75bceb32e3074e0a4dba029dbad1","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):\n    try:\n        r=abs(math.remainder(x,2.0))\n        if r==0: return '1'\n        if r==1: return '-1'\n        if r==0.5: return '0'\n        if r<0.5:\n            result=math.sin(math.pi*(0.5-r))\n        else:\n            result=-math.sin(math.pi*(r-0.5))\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('even integer', solve(2.0*N), \"1\")\ncheck('odd integer', solve(2.0*N+1), \"-1\")\ncheck('negative integer', solve(-2.0*N-1), \"-1\")\ncheck('half integer', solve(2.0*N+0.5), \"0\")\ncheck('quarter', solve(0.25), render(math.sqrt(0.5)))\ncheck('three quarters', solve(0.75), render(-math.sqrt(0.5)))\ncheck('near half below', solve(0.5-N*2.0**-50), render(math.sin(math.pi*N*2.0**-50)))\ncheck('near half above', solve(0.5+N*2.0**-50), render(-math.sin(math.pi*N*2.0**-50)))\ncheck('negative half', solve(-0.5), \"0\")\ncheck('negative quarter', solve(-0.25), render(math.sqrt(0.5)))\ncheck('large integer', solve(2.0**52+2*N), \"1\")\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-cospi-branch-sign","generated_at":"2026-09-29T14:39:40.135289+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 result=-math.sin(math.pi*(r-0.5)).","root_cause":"Cosine-pi omits the negative half-period sign. The faulty expression is result=math.sin(math.pi*(r-0.5)).","sha256":"bac3ee66d453fc1d228677acf6aac16048129cbb11e660942a12282bd542e03a","title":"Cosine-pi omits the negative half-period sign · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":43.832,"exit_code":1,"observations":[{"actual":"1","check":"even integer","expected":"1","passed":true},{"actual":"-1","check":"odd integer","expected":"-1","passed":true},{"actual":"-1","check":"negative integer","expected":"-1","passed":true},{"actual":"0","check":"half integer","expected":"0","passed":true},{"actual":"0.70710678119","check":"quarter","expected":"0.70710678119","passed":true},{"actual":"-0.70710678119","check":"three quarters","expected":"-0.70710678119","passed":true},{"actual":"2.7902947984e-15","check":"near half below","expected":"2.7902947984e-15","passed":true},{"actual":"-2.8253475241e-15","check":"near half above","expected":"-2.7902947984e-15","passed":false},{"actual":"0","check":"negative half","expected":"0","passed":true},{"actual":"0.70710678119","check":"negative quarter","expected":"0.70710678119","passed":true},{"actual":"1","check":"large integer","expected":"1","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"even integer\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"odd integer\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"negative integer\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"half integer\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"quarter\", \"actual\": \"0.70710678119\", \"expected\": \"0.70710678119\", \"passed\": true}, {\"check\": \"three quarters\", \"actual\": \"-0.70710678119\", \"expected\": \"-0.70710678119\", \"passed\": true}, {\"check\": \"near half below\", \"actual\": \"2.7902947984e-15\", \"expected\": \"2.7902947984e-15\", \"passed\": true}, {\"check\": \"near half above\", \"actual\": \"-2.8253475241e-15\", \"expected\": \"-2.7902947984e-15\", \"passed\": false}, {\"check\": \"negative half\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"negative quarter\", \"actual\": \"0.70710678119\", \"expected\": \"0.70710678119\", \"passed\": true}, {\"check\": \"large integer\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":75.749,"exit_code":1,"observations":[{"actual":"1","check":"even integer","expected":"1","passed":true},{"actual":"-1","check":"odd integer","expected":"-1","passed":true},{"actual":"-1","check":"negative integer","expected":"-1","passed":true},{"actual":"0","check":"half integer","expected":"0","passed":true},{"actual":"0.70710678119","check":"quarter","expected":"0.70710678119","passed":true},{"actual":"0.70710678119","check":"three quarters","expected":"-0.70710678119","passed":false},{"actual":"2.7902947984e-15","check":"near half below","expected":"2.7902947984e-15","passed":true},{"actual":"2.7902947984e-15","check":"near half above","expected":"-2.7902947984e-15","passed":false},{"actual":"0","check":"negative half","expected":"0","passed":true},{"actual":"0.70710678119","check":"negative quarter","expected":"0.70710678119","passed":true},{"actual":"1","check":"large integer","expected":"1","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"even integer\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"odd integer\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"negative integer\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"half integer\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"quarter\", \"actual\": \"0.70710678119\", \"expected\": \"0.70710678119\", \"passed\": true}, {\"check\": \"three quarters\", \"actual\": \"0.70710678119\", \"expected\": \"-0.70710678119\", \"passed\": false}, {\"check\": \"near half below\", \"actual\": \"2.7902947984e-15\", \"expected\": \"2.7902947984e-15\", \"passed\": true}, {\"check\": \"near half above\", \"actual\": \"2.7902947984e-15\", \"expected\": \"-2.7902947984e-15\", \"passed\": false}, {\"check\": \"negative half\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"negative quarter\", \"actual\": \"0.70710678119\", \"expected\": \"0.70710678119\", \"passed\": true}, {\"check\": \"large integer\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":43.332,"exit_code":0,"observations":[{"actual":"1","check":"even integer","expected":"1","passed":true},{"actual":"-1","check":"odd integer","expected":"-1","passed":true},{"actual":"-1","check":"negative integer","expected":"-1","passed":true},{"actual":"0","check":"half integer","expected":"0","passed":true},{"actual":"0.70710678119","check":"quarter","expected":"0.70710678119","passed":true},{"actual":"-0.70710678119","check":"three quarters","expected":"-0.70710678119","passed":true},{"actual":"2.7902947984e-15","check":"near half below","expected":"2.7902947984e-15","passed":true},{"actual":"-2.7902947984e-15","check":"near half above","expected":"-2.7902947984e-15","passed":true},{"actual":"0","check":"negative half","expected":"0","passed":true},{"actual":"0.70710678119","check":"negative quarter","expected":"0.70710678119","passed":true},{"actual":"1","check":"large integer","expected":"1","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"even integer\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"odd integer\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"negative integer\", \"actual\": \"-1\", \"expected\": \"-1\", \"passed\": true}, {\"check\": \"half integer\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"quarter\", \"actual\": \"0.70710678119\", \"expected\": \"0.70710678119\", \"passed\": true}, {\"check\": \"three quarters\", \"actual\": \"-0.70710678119\", \"expected\": \"-0.70710678119\", \"passed\": true}, {\"check\": \"near half below\", \"actual\": \"2.7902947984e-15\", \"expected\": \"2.7902947984e-15\", \"passed\": true}, {\"check\": \"near half above\", \"actual\": \"-2.7902947984e-15\", \"expected\": \"-2.7902947984e-15\", \"passed\": true}, {\"check\": \"negative half\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}, {\"check\": \"negative quarter\", \"actual\": \"0.70710678119\", \"expected\": \"0.70710678119\", \"passed\": true}, {\"check\": \"large integer\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}