{"abstract":"Negative updates round differently from positive ones, producing an asymmetric limit cycle.","category":"Digital signal filters","checks":7,"contract":"Input [alpha, samples]; alpha is Q15 in [0, 32768] (\"bad-alpha\" otherwise). Inputs are saturated to int16; y += ((x - y) * alpha + 2**14) >> 15 (round half up on the Q15 product via arithmetic shift), starting at y = 0. Return the integer outputs.","evaluation_group":"w2-digital_signal_filters-q15-one-pole-smoother","failed_approach":"The attempted repair uses round(product / 32768), whose banker ties differ from round-half-up.","family":"w2-digital_signal_filters-q15-one-pole-smoother-division-rounding-direction","id":"FA-91481","implementations":{"attempt":{"sha256":"24f95721ccdba06ffb214368811c0ea3edf4005b85d0aa6b3ce9ba6822169c2e","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(x):\n    alpha, xs = x\n    if not 0 <= alpha <= 32768:\n        return 'bad-alpha'\n    y = 0\n    out = []\n    for v in xs:\n        v = max(-32768, min(32767, v))\n        y = y + round((v - y) * alpha / 32768)\n        out.append(y)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: input beyond range', [16384, [40000, 40000, -40000]], [16384, 24576, -4096]], ['repair check: small negative tie', [16384, [-3, 0, 0]], [-1, 0, 0]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: alpha zero', [0, [5, 5]], [0, 0]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]]], [['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['regression: random smoother 3', [32768, [8, 9547, 10, -20745, -49, 4]], [8, 9547, 10, -20745, -49, 4]], ['regression: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]], ['control: random smoother 49', [24576, [-17286, -45, 28008]], [-12964, -3275, 20187]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]]], [['regression: random smoother 6', [8192, [16294, 2, -779, 14009]], [4074, 3056, 2097, 5075]], ['regression: random smoother 7', [1000, [-8425, 39]], [-257, -248]], ['regression: random smoother 20', [16384, [6, -12155, -29083, 31]], [3, -6076, -17579, -8774]], ['control: alpha zero', [0, [5, 5]], [0, 0]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]]], [['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['regression: random smoother 10', [32768, [-12, -3127, -26, -45, -25]], [-12, -3127, -26, -45, -25]], ['regression: random smoother 36', [16384, [19291, 32, 32408, 5]], [9646, 4839, 18624, 9315]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]], ['control: random smoother 49', [24576, [-17286, -45, 28008]], [-12964, -3275, 20187]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: alpha zero', [0, [5, 5]], [0, 0]]], [['regression: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]], ['regression: random smoother 13', [8192, [-8, -27561]], [-2, -6892]], ['regression: random smoother 40', [16384, [32955, 23, 14565, 33, 25313, 28, 7111]], [16384, 8204, 11385, 5709, 15511, 7770, 7441]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\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":"ca59439fe1694e4edab02de8724d5eda645e93600ab79d83782c30b0eb1becab","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(x):\n    alpha, xs = x\n    if not 0 <= alpha <= 32768:\n        return 'bad-alpha'\n    y = 0\n    out = []\n    for v in xs:\n        v = max(-32768, min(32767, v))\n        y = y + int(((v - y) * alpha + (1 << 14)) / 32768)\n        out.append(y)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: input beyond range', [16384, [40000, 40000, -40000]], [16384, 24576, -4096]], ['repair check: small negative tie', [16384, [-3, 0, 0]], [-1, 0, 0]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: alpha zero', [0, [5, 5]], [0, 0]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]]], [['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['regression: random smoother 3', [32768, [8, 9547, 10, -20745, -49, 4]], [8, 9547, 10, -20745, -49, 4]], ['regression: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]], ['control: random smoother 49', [24576, [-17286, -45, 28008]], [-12964, -3275, 20187]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]]], [['regression: random smoother 6', [8192, [16294, 2, -779, 14009]], [4074, 3056, 2097, 5075]], ['regression: random smoother 7', [1000, [-8425, 39]], [-257, -248]], ['regression: random smoother 20', [16384, [6, -12155, -29083, 31]], [3, -6076, -17579, -8774]], ['control: alpha zero', [0, [5, 5]], [0, 0]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]]], [['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['regression: random smoother 10', [32768, [-12, -3127, -26, -45, -25]], [-12, -3127, -26, -45, -25]], ['regression: random smoother 36', [16384, [19291, 32, 32408, 5]], [9646, 4839, 18624, 9315]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]], ['control: random smoother 49', [24576, [-17286, -45, 28008]], [-12964, -3275, 20187]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: alpha zero', [0, [5, 5]], [0, 0]]], [['regression: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]], ['regression: random smoother 13', [8192, [-8, -27561]], [-2, -6892]], ['regression: random smoother 40', [16384, [32955, 23, 14565, 33, 25313, 28, 7111]], [16384, 8204, 11385, 5709, 15511, 7770, 7441]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\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":"d345f70b1b6295c583329f8a7b4557e0aeec8e540cf3717a5911a4fe0cbb3eee","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(x):\n    alpha, xs = x\n    if not 0 <= alpha <= 32768:\n        return 'bad-alpha'\n    y = 0\n    out = []\n    for v in xs:\n        v = max(-32768, min(32767, v))\n        y = y + (((v - y) * alpha + (1 << 14)) >> 15)\n        out.append(y)\n    return out\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression: alpha one', [32768, [100, -5, 7]], [100, -5, 7]], ['regression: input beyond range', [16384, [40000, 40000, -40000]], [16384, 24576, -4096]], ['repair check: small negative tie', [16384, [-3, 0, 0]], [-1, 0, 0]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: alpha zero', [0, [5, 5]], [0, 0]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]]], [['regression: random smoother 2', [32768, [-33, 20874, -43, 6, -28]], [-33, 20874, -43, 6, -28]], ['regression: random smoother 3', [32768, [8, 9547, 10, -20745, -49, 4]], [8, 9547, 10, -20745, -49, 4]], ['regression: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]], ['control: random smoother 49', [24576, [-17286, -45, 28008]], [-12964, -3275, 20187]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]]], [['regression: random smoother 6', [8192, [16294, 2, -779, 14009]], [4074, 3056, 2097, 5075]], ['regression: random smoother 7', [1000, [-8425, 39]], [-257, -248]], ['regression: random smoother 20', [16384, [6, -12155, -29083, 31]], [3, -6076, -17579, -8774]], ['control: alpha zero', [0, [5, 5]], [0, 0]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]]], [['regression: random smoother 9', [32768, [-37759, -22144, -45]], [-32768, -22144, -45]], ['regression: random smoother 10', [32768, [-12, -3127, -26, -45, -25]], [-12, -3127, -26, -45, -25]], ['regression: random smoother 36', [16384, [19291, 32, 32408, 5]], [9646, 4839, 18624, 9315]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]], ['control: random smoother 49', [24576, [-17286, -45, 28008]], [-12964, -3275, 20187]], ['control: negative step', [16384, [-1, -1, -1]], [0, 0, 0]], ['control: alpha zero', [0, [5, 5]], [0, 0]]], [['regression: random smoother 12', [16384, [-15862, 2530, 35915]], [-7931, -2700, 15034]], ['regression: random smoother 13', [8192, [-8, -27561]], [-2, -6892]], ['regression: random smoother 40', [16384, [32955, 23, 14565, 33, 25313, 28, 7111]], [16384, 8204, 11385, 5709, 15511, 7770, 7441]], ['control: bad alpha', [40000, [1]], 'bad-alpha'], ['control: random smoother 1', [8192, [38, 13765]], [10, 3449]], ['control: random smoother 5', [32768, [5, 7014, 39535]], [5, 7014, 32767]], ['control: random smoother 39', [8192, [14896, 30856, 13380, 32942]], [3724, 10507, 11225, 16611]]]]\nfor label, args, expected in fixtures[N-1]:\n    check(label, solve(args), expected)\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":"A deterministic bounded teaching model with a stipulated toy contract; exact rational arithmetic or fixed-decimal rounding keeps outputs strict JSON. It is not a production DSP library and claims no standards conformance. 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":"w2-digital_signal_filters-q15-one-pole-smoother-division-rounding-direction","generated_at":"2026-09-29T14:51:36.309721+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Fixed-point exponential smoothers run on microcontrollers; rounding direction and saturation slips cause bias and limit cycles.","repair":"Use an arithmetic right shift (floor) after adding the half-LSB offset.","root_cause":"The shift is replaced with int(... / 32768), which truncates toward zero instead of flooring.","sha256":"3cfd57df5c719df69903863bc928972069da59cf74e48adfcb45b48b8b3c8b63","title":"Q15 smoother divides with truncation toward zero · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":37.365,"exit_code":1,"observations":[{"actual":[100,-5,7],"check":"regression: alpha one","expected":[100,-5,7],"passed":true},{"actual":[16384,24576,-4096],"check":"regression: input beyond range","expected":[16384,24576,-4096],"passed":true},{"actual":[-2,-1,-1],"check":"repair check: small negative tie","expected":[-1,0,0],"passed":false},{"actual":[0,0,0],"check":"control: negative step","expected":[0,0,0],"passed":true},{"actual":[0,0],"check":"control: alpha zero","expected":[0,0],"passed":true},{"actual":"bad-alpha","check":"control: bad alpha","expected":"bad-alpha","passed":true},{"actual":[10,3449],"check":"control: random smoother 1","expected":[10,3449],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: alpha one\", \"actual\": [100, -5, 7], \"expected\": [100, -5, 7], \"passed\": true}, {\"check\": \"regression: input beyond range\", \"actual\": [16384, 24576, -4096], \"expected\": [16384, 24576, -4096], \"passed\": true}, {\"check\": \"repair check: small negative tie\", \"actual\": [-2, -1, -1], \"expected\": [-1, 0, 0], \"passed\": false}, {\"check\": \"control: negative step\", \"actual\": [0, 0, 0], \"expected\": [0, 0, 0], \"passed\": true}, {\"check\": \"control: alpha zero\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"control: bad alpha\", \"actual\": \"bad-alpha\", \"expected\": \"bad-alpha\", \"passed\": true}, {\"check\": \"control: random smoother 1\", \"actual\": [10, 3449], \"expected\": [10, 3449], \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":40.232,"exit_code":1,"observations":[{"actual":[100,-4,7],"check":"regression: alpha one","expected":[100,-5,7],"passed":false},{"actual":[16384,24576,-4095],"check":"regression: input beyond range","expected":[16384,24576,-4096],"passed":false},{"actual":[-1,0,0],"check":"repair check: small negative tie","expected":[-1,0,0],"passed":true},{"actual":[0,0,0],"check":"control: negative step","expected":[0,0,0],"passed":true},{"actual":[0,0],"check":"control: alpha zero","expected":[0,0],"passed":true},{"actual":"bad-alpha","check":"control: bad alpha","expected":"bad-alpha","passed":true},{"actual":[10,3449],"check":"control: random smoother 1","expected":[10,3449],"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: alpha one\", \"actual\": [100, -4, 7], \"expected\": [100, -5, 7], \"passed\": false}, {\"check\": \"regression: input beyond range\", \"actual\": [16384, 24576, -4095], \"expected\": [16384, 24576, -4096], \"passed\": false}, {\"check\": \"repair check: small negative tie\", \"actual\": [-1, 0, 0], \"expected\": [-1, 0, 0], \"passed\": true}, {\"check\": \"control: negative step\", \"actual\": [0, 0, 0], \"expected\": [0, 0, 0], \"passed\": true}, {\"check\": \"control: alpha zero\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"control: bad alpha\", \"actual\": \"bad-alpha\", \"expected\": \"bad-alpha\", \"passed\": true}, {\"check\": \"control: random smoother 1\", \"actual\": [10, 3449], \"expected\": [10, 3449], \"passed\": true}], \"passed\": false}\n"},"fixed":{"elapsed_ms":39.817,"exit_code":0,"observations":[{"actual":[100,-5,7],"check":"regression: alpha one","expected":[100,-5,7],"passed":true},{"actual":[16384,24576,-4096],"check":"regression: input beyond range","expected":[16384,24576,-4096],"passed":true},{"actual":[-1,0,0],"check":"repair check: small negative tie","expected":[-1,0,0],"passed":true},{"actual":[0,0,0],"check":"control: negative step","expected":[0,0,0],"passed":true},{"actual":[0,0],"check":"control: alpha zero","expected":[0,0],"passed":true},{"actual":"bad-alpha","check":"control: bad alpha","expected":"bad-alpha","passed":true},{"actual":[10,3449],"check":"control: random smoother 1","expected":[10,3449],"passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression: alpha one\", \"actual\": [100, -5, 7], \"expected\": [100, -5, 7], \"passed\": true}, {\"check\": \"regression: input beyond range\", \"actual\": [16384, 24576, -4096], \"expected\": [16384, 24576, -4096], \"passed\": true}, {\"check\": \"repair check: small negative tie\", \"actual\": [-1, 0, 0], \"expected\": [-1, 0, 0], \"passed\": true}, {\"check\": \"control: negative step\", \"actual\": [0, 0, 0], \"expected\": [0, 0, 0], \"passed\": true}, {\"check\": \"control: alpha zero\", \"actual\": [0, 0], \"expected\": [0, 0], \"passed\": true}, {\"check\": \"control: bad alpha\", \"actual\": \"bad-alpha\", \"expected\": \"bad-alpha\", \"passed\": true}, {\"check\": \"control: random smoother 1\", \"actual\": [10, 3449], \"expected\": [10, 3449], \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}