{
  "generated_at_utc": "2026-10-19T08:20:10.637453+00:00",
  "working_precision_dps": 80,
  "gamma_ratio": {
    "problem": "Gamma(n+0.5)/Gamma(n)",
    "reference_dps": 80,
    "python_version": "3.13.5",
    "mpmath_version": "1.3.0",
    "training_n": [
      200,
      400,
      600,
      800,
      1000,
      1200,
      1400,
      1600,
      1800
    ],
    "holdout_n": [
      25,
      75,
      150,
      250,
      750,
      1250,
      1750,
      2500,
      5000
    ],
    "fitted_log_coefficients": [
      "-1.5678154191903256595291274883185455e-22",
      "-0.12499999999999999922517013844510912",
      "-1.4581134527810487137544999793742465e-15",
      "0.0052083333346474074609256341910370822",
      "-0.00000000057654165277162720943687689477718449",
      "-0.0015623971346884110413482145227906944"
    ],
    "max_training_log_error": "8.5317094656143379372e-25",
    "source_sha256": "63511a5146bd181d7c1ef6bf67428f1f6220603025907aa06b53815a4abfbd6c",
    "mvdc_utils_sha256": "5ae44bf91bde396c2acb34230ac4e0a8cea3527468aea143e7790c26e9fcbf31",
    "notes": [
      "The fitted branch uses an analytic sqrt(n) main term and scaled high-precision QR.",
      "Product evaluation uses n-1 factors and is reported separately from fixed-order approximations.",
      "Stirling-2, Stirling-5 and log-Stirling through n^-5 are distinct declared baselines.",
      "Binary64 columns evaluate each formula using native float arithmetic; fitting and error references use 80 digits.",
      "No universal or equal-cost superiority follows from this finite accuracy comparison."
    ],
    "rows": [
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        "binary64_relative_errors": {
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  ],
  "factorial": [
    {
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      "log_stirling_through_n7_error": "8.2318871678644015726e-13",
      "ramanujan_sixth_root_error": "8.58720807515167285e-8"
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    {
      "n": 20,
      "log_stirling_through_n7_error": "1.6347590502249681837e-15",
      "ramanujan_sixth_root_error": "5.666064011483194711e-9"
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    {
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      "ramanujan_sixth_root_error": "1.4968322616070647388e-10"
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    {
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      "ramanujan_sixth_root_error": "9.4518895633892862111e-12"
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    {
      "n": 500,
      "log_stirling_through_n7_error": "4.3097250393381573352e-28",
      "ramanujan_sixth_root_error": "1.5246825382194052551e-14"
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  ],
  "scope": "Finite accuracy measurements with declared orders and training points; no website or universal claim is changed.",
  "source_sha256": {
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    "wallis_mvdc.py": "414b9388b0002b9bef12a7913a9548a331a6699acd79569ead2a41e0528005ec",
    "mvdc_factorial_analytic.py": "28178c9010fb14cefc85178d5c6495475861eed497b4d5639c9ef3c6449b8f3f"
  }
}