{
  "tid": "T509",
  "all_passed": true,
  "controls": [
    "pi",
    "displaced_value_rejected",
    "nan_rejected",
    "wide_ball_rejected",
    "integral_t_equals_half",
    "Q0_logarithm",
    "F0_free_sine",
    "G0_free_cosine"
  ],
  "source_sha256": {
    "balanced_checks.py": "7d29b572b3d3edcbb0ff9861c8eb1116777d72f0fdaa1282274fdb93adfd5c62",
    "balanced_generate.py": "1bac17812c98ca64baf6d54ba58e3bbdd947796f28c27cef6fab748461aee6ed"
  },
  "selection_sha256": "fbb87ff7eb0ec8c1579678a3e6fb86a6cfaef333398d4186163341110d76ada2",
  "count": 21,
  "independently_checked_additions": 14,
  "minimum_accuracy_bits": 731,
  "records": [
    {
      "params": {
        "nu": "0",
        "x": "0"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 999
    },
    {
      "params": {
        "nu": "0",
        "x": "critical-1"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 787,
      "root_certificate": {
        "lower": "9346590901445000620742286111649403288423403813751638971237722716083787373034269229009761862643338247983837589550073177042847069871388451642257129261837249292356844089954662525519697489674852340272764829652590681316865400485619535761449167017804478671535042212212107786356504466653027893264756102853148673371027016574177194315206134963998167450092741376065815163048288155579515433997087445986211691672341488702009306459035808394218770541670977077732044279466357805930657/1895163686890513836851781602127078314522565938863155042894350632356019229353487410071873057652156345154403968136146329597377417046038591786431014740040504320317938203150451544861163678749509820343338622062642173768179744694599680000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "upper": "9346590901445000620742286111649403288423403813751638971237722716083787373034269229009761862643338247983837589550073177042847069871388451642257129261837249292356844089954662525519697489674852340272764829652590681316865400485619535761449167018183511408913144979582464106781920129557541081037387111432018799842230862444874676329580746494429436480973535003295081082523771564787233791283290393994312555735929129332099615431268544144120734610338701490260479033102306744850593/1895163686890513836851781602127078314522565938863155042894350632356019229353487410071873057652156345154403968136146329597377417046038591786431014740040504320317938203150451544861163678749509820343338622062642173768179744694599680000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "endpoint_signs": [
          1,
          -1
        ],
        "second_derivative_sign": -1,
        "defining_interval": [
          "2525/512",
          "5051/1024"
        ]
      }
    },
    {
      "params": {
        "nu": "0",
        "x": "critical-2"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 775,
      "root_certificate": {
        "lower": "8911111378268551388128341205067209229639632602870886403988552407516613386219057914380170417853686104640263902710564485673549771821021117342471148314917856231948717822571177487546495762976119184825775820887135522687683226650634412535664372443944489513193490894719573755568332424716505134234842763737589605302915327741912271760297649890029383332915339971670662700816565056548588736058895887310402193962220942186215668015319282933336986744465022295914215505068643380504391/947581843445256918425890801063539157261282969431577521447175316178009614676743705035936528826078172577201984068073164798688708523019295893215507370020252160158969101575225772430581839374754910171669311031321086884089872347299840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "upper": "8911111378268551388128341205067209229639632602870886403988552407516613386219057914380170417853686104640263902710564485673549771821021117342471148314917856231948717822571177487546495762976119184825775820887135522687683226650634412535664372444134005881882542278404751915781040256168761728121158268027024668538517250677261012767484955655245017848355736785285295660554306761152447914701997361314452625994014762501260822501435650808287968778798884502178432881886617849964359/947581843445256918425890801063539157261282969431577521447175316178009614676743705035936528826078172577201984068073164798688708523019295893215507370020252160158969101575225772430581839374754910171669311031321086884089872347299840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "endpoint_signs": [
          -1,
          1
        ],
        "second_derivative_sign": 1,
        "defining_interval": [
          "9629/1024",
          "4815/512"
        ]
      }
    },
    {
      "params": {
        "nu": "0",
        "x": "critical-3"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 764,
      "root_certificate": {
        "lower": "13131844237227061619098820920978161361937632018337913786746729005692801611859974720190947186310132215703787458725665556580426511463421663857699034774960349620275334718597974955576554409737354615163317722345151368772875085490231415281720474887435152594344174438606836280087353945665620798210968244326727060609042438201713610692954795097493629739550459775575627871426794450650479838784588967974654288682174546651003980662421818449894817519251475116904450118487222482066891/947581843445256918425890801063539157261282969431577521447175316178009614676743705035936528826078172577201984068073164798688708523019295893215507370020252160158969101575225772430581839374754910171669311031321086884089872347299840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "upper": "13131844237227061619098820920978161361937632018337913786746729005692801611859974720190947186310132215703787458725665556580426511463421663857699034774960349620275334718597974955576554409737354615163317722345151368772875085490231415281720474887624668963033225822292014440300061777117877392097283748616162123844644361137062351700142100862709264254990856589190260831164536155254339017427690441978704720713968366966049135148538186324845799553585337323168667495305196951526859/947581843445256918425890801063539157261282969431577521447175316178009614676743705035936528826078172577201984068073164798688708523019295893215507370020252160158969101575225772430581839374754910171669311031321086884089872347299840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "endpoint_signs": [
          1,
          -1
        ],
        "second_derivative_sign": -1,
        "defining_interval": [
          "7095/512",
          "14191/1024"
        ]
      }
    },
    {
      "params": {
        "nu": "0",
        "x": "critical-4"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 753,
      "root_certificate": {
        "lower": "4336886169646522022900887075553386994635560659657850663200550564000441546734962625370109903245020611493662120470475036906890933356123693593237353317297131611095937276884741384195808577838540411347616785768638958368363798965327461049302267371607491715591285445509941980013816993077077728206680424240497739927474379917724101375465427836611634274724183483150485193851310302851720523332864952463386871997118615798845681671910831799913671603516132023031962454305376482333129/236895460861314229606472700265884789315320742357894380361793829044502403669185926258984132206519543144300496017018291199672177130754823973303876842505063040039742275393806443107645459843688727542917327757830271721022468086824960000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "upper": "4336886169646522022900887075553386994635560659657850663200550564000441546734962625370109903245020611493662120470475036906890933356123693593237353317297131611095937276884741384195808577838540411347616785768638958368363798965327461049302267371654870807763548291431236520066993950940141876678259300312856505736374860651561286627262254277915542903584282686554143433785745729002685317993640320964399480005067070877606970293439923768651417112099597574598016798509870099698121/236895460861314229606472700265884789315320742357894380361793829044502403669185926258984132206519543144300496017018291199672177130754823973303876842505063040039742275393806443107645459843688727542917327757830271721022468086824960000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "endpoint_signs": [
          -1,
          1
        ],
        "second_derivative_sign": 1,
        "defining_interval": [
          "9373/512",
          "18747/1024"
        ]
      }
    },
    {
      "params": {
        "nu": "0",
        "x": "critical-5"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 742,
      "root_certificate": {
        "lower": "10780540358622500420141846144160983611442599800729537593656101196934943707806207963464332634066114392133708137567612203486505796274063373789286473349119534085552114527605037816915494883512044316159562824519621870167631372488679009920462920267297504612448796366562672111842537980397242533692302590913619551084681344056185676349367440401642826695952697431656117480124858717832923067876143950281621030881413000368187767460190828415151509657981669817162024557563237095525633/473790921722628459212945400531769578630641484715788760723587658089004807338371852517968264413039086288600992034036582399344354261509647946607753685010126080079484550787612886215290919687377455085834655515660543442044936173649920000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "upper": "10780540358622500420141846144160983611442599800729537593656101196934943707806207963464332634066114392133708137567612203486505796274063373789286473349119534085552114527605037816915494883512044316159562824519621870167631372488679009920462920267392262796793322058405261191948891896123370830635460343058337082702482305523860046852961093284250643953672895838463433959993729570134852657197694687283646246897309910525710344703249012352627000675148600920294133245972224330255617/473790921722628459212945400531769578630641484715788760723587658089004807338371852517968264413039086288600992034036582399344354261509647946607753685010126080079484550787612886215290919687377455085834655515660543442044936173649920000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "endpoint_signs": [
          1,
          -1
        ],
        "second_derivative_sign": -1,
        "defining_interval": [
          "23299/1024",
          "5825/256"
        ]
      }
    },
    {
      "params": {
        "nu": "0",
        "x": "critical-6"
      },
      "origin": "retained website entry",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 731,
      "root_certificate": {
        "lower": "25773488103037886487806127288612187450712795655392176056463486551950138014989486853085878427100537987977631476770451782458735078637226394045810803974963669730567481191034664667044419160243594237099784927235395569027879271636277877958522235074844915278042192033433839051220631821402574307132719748235461207505318978785290336697732492356550480435611102350685395461757217660039851454960249346203248307157431423862678882057497269044751864712183006870761138856356485177379391/947581843445256918425890801063539157261282969431577521447175316178009614676743705035936528826078172577201984068073164798688708523019295893215507370020252160158969101575225772430581839374754910171669311031321086884089872347299840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "upper": "25773488103037886487806127288612187450712795655392176056463486551950138014989486853085878427100537987977631476770451782458735078637226394045810803974963669730567481191034664667044419160243594237099784927235395569027879271636277877958522235075034431646731243417119017211433339652854830901019035252524896270740920901720639077704919798121766114951051499164300028421494959364643710633603350820207298739189225244177724036543613636919702846746516869077025356233174459646839359/947581843445256918425890801063539157261282969431577521447175316178009614676743705035936528826078172577201984068073164798688708523019295893215507370020252160158969101575225772430581839374754910171669311031321086884089872347299840000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000",
        "endpoint_signs": [
          -1,
          1
        ],
        "second_derivative_sign": 1,
        "defining_interval": [
          "6963/256",
          "27853/1024"
        ]
      }
    },
    {
      "params": {
        "nu": "0",
        "x": "1/2"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 995,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "0",
        "x": "1"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 994,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": [],
      "integral_relative_error": "7.71456935621e-152"
    },
    {
      "params": {
        "nu": "0",
        "x": "2"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 991,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": [],
      "integral_relative_error": "9.43347128424e-152"
    },
    {
      "params": {
        "nu": "0",
        "x": "3"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 987,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "0",
        "x": "5"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 982,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "0",
        "x": "8"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 972,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "0",
        "x": "sqrt(2)"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 991,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "0",
        "x": "2*sqrt(2)"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 987,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "0",
        "x": "4*sqrt(2)"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 980,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "1",
        "x": "1"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 991,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": [],
      "integral_relative_error": "3.15597859808e-151"
    },
    {
      "params": {
        "nu": "2",
        "x": "1"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 993,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "3",
        "x": "1"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 993,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": []
    },
    {
      "params": {
        "nu": "1/2",
        "x": "1"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 993,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": [
        "elementary_half_order_rotation_and_phase"
      ]
    },
    {
      "params": {
        "nu": "1",
        "x": "2"
      },
      "origin": "balanced expansion",
      "stored_interval_contains_ball": true,
      "accuracy_bits": 989,
      "mpmath_160_digits": true,
      "mpmath_210_digits": true,
      "structural_checks": [],
      "integral_relative_error": "2.38519624069e-151"
    }
  ],
  "limitations": "Retained values are fully Arb-recomputed; new values additionally have independent mpmath and family-identity checks. Existing manual numbers and annotations are not claimed to originate in this run."
}
