{
  "precision_decimal_digits": 70,
  "status": "passed",
  "classification": {
    "numerical": "70-digit quadrature checks are evidence, not interval certificates",
    "analytic": "Remainder bounds and rational endpoints are justified in the accompanying derivations"
  },
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    "x": 8,
    "value_quadrature": "0.112279639253499311825517032554",
    "rows": [
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        "N": 0,
        "sum": "0.0",
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        "absolute_error": "0.112279639253499311825517032554",
        "analytic_bound": "0.125"
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        "N": 1,
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        "absolute_error": "0.0127203607465006881744829674456",
        "analytic_bound": "0.015625"
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        "N": 2,
        "sum": "0.109375",
        "signed_error": "0.00290463925349931182551703255441",
        "absolute_error": "0.00290463925349931182551703255441",
        "analytic_bound": "0.00390625"
      },
      {
        "N": 3,
        "sum": "0.11328125",
        "signed_error": "-0.00100161074650068817448296744559",
        "absolute_error": "0.00100161074650068817448296744559",
        "analytic_bound": "0.00146484375"
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        "N": 4,
        "sum": "0.11181640625",
        "signed_error": "0.000463233003499311825517032554413",
        "absolute_error": "0.000463233003499311825517032554413",
        "analytic_bound": "0.000732421875"
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        "N": 5,
        "sum": "0.112548828125",
        "signed_error": "-0.000269188871500688174482967445587",
        "absolute_error": "0.000269188871500688174482967445587",
        "analytic_bound": "0.000457763671875"
      },
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        "N": 6,
        "sum": "0.112091064453125",
        "signed_error": "0.000188574800374311825517032554413",
        "absolute_error": "0.000188574800374311825517032554413",
        "analytic_bound": "0.00034332275390625"
      },
      {
        "N": 7,
        "sum": "0.11243438720703125",
        "signed_error": "-0.000154747953531938174482967445587",
        "absolute_error": "0.000154747953531938174482967445587",
        "analytic_bound": "0.00030040740966796875"
      },
      {
        "N": 8,
        "sum": "0.11213397979736328125",
        "signed_error": "0.000145659456136030575517032554413",
        "absolute_error": "0.000145659456136030575517032554413",
        "analytic_bound": "0.00030040740966796875"
      },
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        "N": 9,
        "sum": "0.11243438720703125",
        "signed_error": "-0.000154747953531938174482967445587",
        "absolute_error": "0.000154747953531938174482967445587",
        "analytic_bound": "0.00033795833587646484375"
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      {
        "N": 10,
        "sum": "0.11209642887115478515625",
        "signed_error": "0.000183210382344526669267032554413",
        "absolute_error": "0.000183210382344526669267032554413",
        "analytic_bound": "0.0004224479198455810546875"
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        "N": 11,
        "sum": "0.1125188767910003662109375",
        "signed_error": "-0.000239237537501054385420467445587",
        "absolute_error": "0.000239237537501054385420467445587",
        "analytic_bound": "0.0005808658897876739501953125"
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        "N": 12,
        "sum": "0.1119380109012126922607421875",
        "signed_error": "0.000341628352286619564774845054413",
        "absolute_error": "0.000341628352286619564774845054413",
        "analytic_bound": "0.00087129883468151092529296875"
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      {
        "N": 13,
        "sum": "0.11280930973589420318603515625",
        "signed_error": "-0.000529670482394891360518123695587",
        "absolute_error": "0.000529670482394891360518123695587",
        "analytic_bound": "0.00141586060635745525360107421875"
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      {
        "N": 14,
        "sum": "0.111393449129536747932434082031",
        "signed_error": "0.000886190123962563893082950523163",
        "absolute_error": "0.000886190123962563893082950523163",
        "analytic_bound": "0.00247775606112554669380187988281"
      },
      {
        "N": 15,
        "sum": "0.113871205190662294626235961914",
        "signed_error": "-0.00159156593716298280071892935965",
        "absolute_error": "0.00159156593716298280071892935965",
        "analytic_bound": "0.00464579261461040005087852478027"
      },
      {
        "N": 16,
        "sum": "0.109225412576051894575357437134",
        "signed_error": "0.00305422667744741725015959542062",
        "absolute_error": "0.00305422667744741725015959542062",
        "analytic_bound": "0.00929158522922080010175704956055"
      },
      {
        "N": 17,
        "sum": "0.118516997805272694677114486694",
        "signed_error": "-0.00623735855177338285159745413992",
        "absolute_error": "0.00623735855177338285159745413992",
        "analytic_bound": "0.0197446186120942002162337303162"
      },
      {
        "N": 18,
        "sum": "0.0987723791931784944608807563782",
        "signed_error": "0.0135072600603208173646362761762",
        "absolute_error": "0.0135072600603208173646362761762",
        "analytic_bound": "0.0444253918772119504865258932114"
      },
      {
        "N": 19,
        "sum": "0.14319777107039044494740664959",
        "signed_error": "-0.0309181318168911331218896170351",
        "absolute_error": "0.0309181318168911331218896170351",
        "analytic_bound": "0.105510305708378382405498996377"
      },
      {
        "N": 20,
        "sum": "0.0376874653620120625419076532125",
        "signed_error": "0.0745921738914872492836093793419",
        "absolute_error": "0.0745921738914872492836093793419",
        "analytic_bound": "0.263775764270945956013747490942"
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    "exact_rational_enclosure": {
      "lower": {
        "numerator": 117581,
        "denominator": 1048576,
        "decimal": "0.11213397979736328125"
      },
      "upper": {
        "numerator": 14737,
        "denominator": 131072,
        "decimal": "0.11243438720703125"
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    }
  },
  "watson": {
    "x": 16,
    "value": "0.430396489620679044930960072391",
    "three_terms": "0.430564350989010851357831212384",
    "absolute_error": "0.00016786136833180642687113999315",
    "analytic_bound": "0.000202841245754873202586917970658"
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  "laplace": [
    {
      "lambda": 8,
      "endpoint_exact_numeric": "0.105342302753714008422836467419",
      "endpoint_two_terms": "0.09375",
      "endpoint_error": "0.0115923027537140084228364674193",
      "interior_exact_numeric": "0.827652685949025441685127141584",
      "interior_two_terms": "0.803143151191561949869482140889",
      "interior_error": "0.0245095347574634918156450006953",
      "interior_analytic_bound": "0.0500153487712751424528991307458"
    },
    {
      "lambda": 32,
      "endpoint_exact_numeric": "0.0295815478641950838279982420528",
      "endpoint_two_terms": "0.029296875",
      "endpoint_error": "0.000284672864195083827998242052821",
      "interior_exact_numeric": "0.433876841531915413535638866609",
      "interior_two_terms": "0.432727990943729498852091670738",
      "interior_error": "0.00114885058818591468354719587178",
      "interior_analytic_bound": "0.00141989575373253237180308290556"
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      "lambda": 128,
      "endpoint_exact_numeric": "0.00769574770362076851763629646982",
      "endpoint_two_terms": "0.0076904296875",
      "endpoint_error": "0.00000531801612076851763629646982494",
      "interior_exact_numeric": "0.220300301539772985447923586395",
      "interior_two_terms": "0.220258547390358314915714660405",
      "interior_error": "0.0000417541494146705322089259898088",
      "interior_analytic_bound": "0.0000443715225088785130658883085874"
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  "stationary": [
    {
      "lambda": 8,
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      "exact_imag": "0.537191106585789107908652435008",
      "quadrature_difference": "5.84924137193459963488004253983e-65",
      "leading_relative_error": "0.124285764474586796495533390004"
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    {
      "lambda": 32,
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      "exact_imag": "0.303103175173299794830414487154",
      "quadrature_difference": "1.50466879949862605927339639725e-65",
      "leading_relative_error": "0.0312385742606912617978094311132"
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      "lambda": 128,
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      "exact_imag": "0.155426173810461406896373465035",
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      "leading_relative_error": "0.00781232120402567751435717378309"
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  "nonstationary": {
    "lambda": 7,
    "exact_formula_difference": "0.0"
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  "degenerate_quartic": {
    "constant_numeric": "1.81280495411095415596534257793",
    "constant_gamma": "1.81280495411095415596534257793",
    "scale": "lambda^(-1/4), not lambda^(-1/2)"
  }
}