Jul 032019
 

(best my Marchant from Robert Mařík)

Pi= 3.141592653589793238462643383279502884197169399375105820974…

22/7 = 3,14…

22/7-Pi = 0.0012644892673496186802137595776400046…

 

355/113 = 3.141592…

355/113-Pi = 2,667641e-7

 

103993/33102 = 3.141592653…

103993/33102 -Pi = -5,77890… e-10

 

 

pi

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http://qin.laya.com/tech_projects_approxpi.html

Fractional Approximations of Pi

After reading the American Scientist article, On the Teeth of Wheels, which describes the intricate interplay between pure and applied mathematics, and how clock makers independently developed mathematical methods to approximate gear ratios that were not feasibly made (such as representing gear ratios that were two primes with other gear ratios that were close in value), I decided to write a program to find fractional approximations of Pi. A couple of minutes searching google brought me to an existing fractional approximations of pi page, which attempted to find decimal approximations using an iterative approach, which turned out to be very slow. However since the code was open source, I decided to modify for my own uses. The stern-brocot method pretty much changes the O(n2) to about O(1) for the calculation of successive values.

You can get several versions of the source:
The original modification to use the stern-brocot method
A 64 bit version

Fractional Approximation Table

pi = 3.14159265358979323846264338327950288419716939937510

               Num.                Den. = Result                                   (Accuracy                                 )
      -------------         ----------- = --------------------------------         (---------------------------------        )

                  7/                  2 = 3.50000000000000000000000000000000000000 (-0.35840734641020676153735661672049711581) [ 0]
                 10/                  3 = 3.33333333333333333333333333333333333333 (-0.19174067974354009487068995005383044914) [ 0]
                 13/                  4 = 3.25000000000000000000000000000000000000 (-0.10840734641020676153735661672049711581) [ 0]
                 16/                  5 = 3.20000000000000000000000000000000000000 (-0.05840734641020676153735661672049711581) [ 1]
                 19/                  6 = 3.16666666666666666666666666666666666666 (-0.02507401307687342820402328338716378247) [ 1]
                 22/                  7 = 3.14285714285714285714285714285714285714 (-0.00126448926734961868021375957763997295) [ 2]
                179/                 57 = 3.14035087719298245614035087719298245614 ( 0.00124177639681078232229250608652042805) [ 2]
                201/                 64 = 3.14062500000000000000000000000000000000 ( 0.00096765358979323846264338327950288419) [ 3]
                223/                 71 = 3.14084507042253521126760563380281690140 ( 0.00074758316725802719503774947668598279) [ 3]
                245/                 78 = 3.14102564102564102564102564102564102564 ( 0.00056701256415221282161774225386185855) [ 3]
                267/                 85 = 3.14117647058823529411764705882352941176 ( 0.00041618300155794434499632445597347243) [ 3]
                289/                 92 = 3.14130434782608695652173913043478260869 ( 0.00028830576370628194090425284472027550) [ 3]
                311/                 99 = 3.14141414141414141414141414141414141414 ( 0.00017851217565182432122924186536147005) [ 3]
                333/                106 = 3.14150943396226415094339622641509433962 ( 0.00008321962752908751924715686440854457) [ 4]
                355/                113 = 3.14159292035398230088495575221238938053 (-0.00000026676418906242231236893288649634) [ 6]
              52163/              16604 = 3.14159238737653577451216574319441098530 ( 0.00000026621325746395047764008509189889) [ 6]
              52518/              16717 = 3.14159239097924268708500329006400669976 ( 0.00000026261055055137764009321549618443) [ 6]
              52873/              16830 = 3.14159239453357100415923945335710041592 ( 0.00000025905622223430340392992240246827) [ 6]
              53228/              16943 = 3.14159239804048869739715516732573924334 ( 0.00000025554930454106548821595376364085) [ 6]
              53583/              17056 = 3.14159240150093808630393996247654784240 ( 0.00000025208885515215870342080295504179) [ 6]
              53938/              17169 = 3.14159240491583668239268448948686586289 ( 0.00000024867395655606995889379263702130) [ 6]
              54293/              17282 = 3.14159240828607800023145469274389538247 ( 0.00000024530371523823118869053560750172) [ 6]
              54648/              17395 = 3.14159241161253233687841333716585225639 ( 0.00000024197726090158423004611365062780) [ 6]
              55003/              17508 = 3.14159241489604752113319625314142106465 ( 0.00000023869374571732944713013808181954) [ 6]
              55358/              17621 = 3.14159241813744963395948016571136711877 ( 0.00000023545234360450316321756813576542) [ 6]
              55713/              17734 = 3.14159242133754370136461035299424833652 ( 0.00000023225224953709803303028525454767) [ 6]
              56068/              17847 = 3.14159242449711436095702358939877850619 ( 0.00000022909267887750561979388072437800) [ 6]
              56423/              17960 = 3.14159242761692650334075723830734966592 ( 0.00000022597286673512188614497215321827) [ 6]
              56778/              18073 = 3.14159243069772588944834836496431140375 ( 0.00000022289206734901429501831519148044) [ 6]
              57133/              18186 = 3.14159243374023974485868250302430440998 ( 0.00000021984955349360396088025519847421) [ 6]
              57488/              18299 = 3.14159243674517733209464998087327176348 ( 0.00000021684461590636799340240623112071) [ 6]
              57843/              18412 = 3.14159243971323050184662176841190527916 ( 0.00000021387656273661602161486759760503) [ 6]
              58198/              18525 = 3.14159244264507422402159244264507422402 ( 0.00000021094471901444105094063442866017) [ 6]
              58553/              18638 = 3.14159244554136709947419250992595772078 ( 0.00000020804842613898845087335354516341) [ 6]
              58908/              18751 = 3.14159244840275185323449416031145005599 ( 0.00000020518704138522814922296805282820) [ 6]
              59263/              18864 = 3.14159245122985581000848176420695504664 ( 0.00000020235993742845416161907254783755) [ 6]
              59618/              18977 = 3.14159245402329135269009854033830426305 ( 0.00000019956650188577254484294119862114) [ 6]
              59973/              19090 = 3.14159245678365636458878994237820848611 ( 0.00000019680613687387385344090129439808) [ 6]
              60328/              19203 = 3.14159245951153465604332656355777743060 ( 0.00000019407825858241931681972172545359) [ 6]
              60683/              19316 = 3.14159246220749637606129633464485400704 ( 0.00000019138229686240134704863464887715) [ 6]
              61038/              19429 = 3.14159246487209840959390601677904163878 ( 0.00000018871769482886873736650046124541) [ 6]
              61393/              19542 = 3.14159246750588476102753044724183809231 ( 0.00000018608390847743511293603766479188) [ 6]
              61748/              19655 = 3.14159247010938692444670567285677944543 ( 0.00000018348040631401593771042272343876) [ 6]
              62103/              19768 = 3.14159247268312424119789558883043302306 ( 0.00000018090666899726474779444906986113) [ 6]
              62458/              19881 = 3.14159247522760424525929279211307278305 ( 0.00000017836218899320335059116643010114) [ 6]
              62813/              19994 = 3.14159247774332299689906972091627488246 ( 0.00000017584647024156357366236322800173) [ 6]
              63168/              20107 = 3.14159248023076540508280698264286069528 ( 0.00000017335902783337983640063664218891) [ 6]
              63523/              20220 = 3.14159248269040553907022749752720079129 ( 0.00000017089938769939241588575230209290) [ 6]
              63878/              20333 = 3.14159248512270692962179707864063345300 ( 0.00000016846708630884084630463886943119) [ 6]
              64233/              20446 = 3.14159248752812286021715739019857184779 ( 0.00000016606167037824548599308093103640) [ 6]
              64588/              20559 = 3.14159248990709664866968237754754608687 ( 0.00000016368269658979296100573195679732) [ 6]
              64943/              20672 = 3.14159249226006191950464396284829721362 ( 0.00000016132973131895799942043120567057) [ 6]
              65298/              20785 = 3.14159249458744286745248977628097185470 ( 0.00000015900235037101015360699853102949) [ 6]
              65653/              20898 = 3.14159249688965451239353048138577854340 ( 0.00000015670013872606911290189372434079) [ 6]
              66008/              21011 = 3.14159249916710294607586502308314692304 ( 0.00000015442269029238677836019635596115) [ 6]
              66363/              21124 = 3.14159250142018557091459950766900208293 ( 0.00000015216960766754804387561050080126) [ 6]
              66718/              21237 = 3.14159250364929133116730234967274097094 ( 0.00000014994050190729534103360676191325) [ 6]
              67073/              21350 = 3.14159250585480093676814988290398126463 ( 0.00000014773499230169449350037552161956) [ 6]
              67428/              21463 = 3.14159250803708708009131994595350137445 ( 0.00000014555270615837132343732600150974) [ 6]
              67783/              21576 = 3.14159251019651464590285502410085279940 ( 0.00000014339327859255978835917865008479) [ 6]
              68138/              21689 = 3.14159251233344091474941214440499792521 ( 0.00000014125635232371323123887450495898) [ 6]
              68493/              21802 = 3.14159251444821576002201632877717640583 ( 0.00000013914157747844062705450232647836) [ 6]
              68848/              21915 = 3.14159251654118183892311202372804015514 ( 0.00000013704861139953953135955146272905) [ 6]
              69203/              22028 = 3.14159251861267477755583802433266751407 ( 0.00000013497711846090680535894683537012) [ 6]
              69558/              22141 = 3.14159252066302335034551284946479382141 ( 0.00000013292676988811713053381470906278) [ 6]
              69913/              22254 = 3.14159252269254965399478745394086456367 ( 0.00000013089724358446785592933863832052) [ 6]
              70268/              22367 = 3.14159252470156927616577994366700943354 ( 0.00000012888822396229686343961249345065) [ 6]
              70623/              22480 = 3.14159252669039145907473309608540925266 ( 0.00000012689940177938791028719409363153) [ 6]
              70978/              22593 = 3.14159252865931925817731155667684681095 ( 0.00000012493047398028533182660265607324) [ 6]
              71333/              22706 = 3.14159253060864969611556416806130538183 ( 0.00000012298114354234707921521819750236) [ 6]
              71688/              22819 = 3.14159253253867391209080152504491870809 ( 0.00000012105111932637184185823458417610) [ 6]
              72043/              22932 = 3.14159253444967730682016396302110587824 ( 0.00000011914011593164247942025839700595) [ 6]
              72398/              23045 = 3.14159253634193968322846604469516164026 ( 0.00000011724785355523417733858434124393) [ 6]
              72753/              23158 = 3.14159253821573538302098626824423525347 ( 0.00000011537405785544165711503526763072) [ 6]
              73108/              23271 = 3.14159254007133341927721198057668342572 ( 0.00000011351845981918543140270281945847) [ 6]
              73463/              23384 = 3.14159254190899760520013684570646595963 ( 0.00000011168079563326250653757303692456) [ 6]
              73818/              23497 = 3.14159254372898667915052985487509043707 ( 0.00000010986080655931211352840441244712) [ 6]
              74173/              23610 = 3.14159254553155442609063955950868276154 ( 0.00000010805823881237200382377082012265) [ 6]
              74528/              23723 = 3.14159254731694979555705433545504362854 ( 0.00000010627284344290558904782445925565) [ 6]
              74883/              23836 = 3.14159254908541701627789897633831179728 ( 0.00000010450437622218474440694119108691) [ 6]
              75238/              23949 = 3.14159255083719570754520021712806380224 ( 0.00000010275259753091744316615143908195) [ 6]
              75593/              24062 = 3.14159255257252098744908985121768764026 ( 0.00000010101727225101355353206181524393) [ 6]
              75948/              24175 = 3.14159255429162357807652533609100310237 ( 0.00000009929816966038611804718849978182) [ 7]
              76303/              24288 = 3.14159255599472990777338603425559947299 ( 0.00000009759506333068925734902390341120) [ 7]
              76658/              24401 = 3.14159255768206221056514077291914265808 ( 0.00000009590773102789750261036036022611) [ 7]
              77013/              24514 = 3.14159255935383862282777188545321041037 ( 0.00000009423595461563487149782629247382) [ 7]
              77368/              24627 = 3.14159256101027327729727534819507045113 ( 0.00000009257951996116536803508443243306) [ 7]
              77723/              24740 = 3.14159256265157639450282942603071948261 ( 0.00000009093821684395981395724878340158) [ 7]
              78078/              24853 = 3.14159256427795437170562909910272401722 ( 0.00000008931183886675701428417677886697) [ 7]
              78433/              24966 = 3.14159256588960986942241448369782904750 ( 0.00000008770018336904022889958167383669) [ 7]
              78788/              25079 = 3.14159256748674189560987280194585111049 ( 0.00000008610305134285277058133365177370) [ 7]
              79143/              25192 = 3.14159256906954588758335979676087646872 ( 0.00000008452024735087928358651862641547) [ 7]
              79498/              25305 = 3.14159257063821379174076269511954159257 ( 0.00000008295157944672188068815996129162) [ 7]
              79853/              25418 = 3.14159257219293414115980801007160280116 ( 0.00000008139685909730283537320790008303) [ 7]
              80208/              25531 = 3.14159257373389213113469899338059613802 ( 0.00000007985590110732794438989890674617) [ 7]
              80563/              25644 = 3.14159257526126969271564498518171892060 ( 0.00000007832852354574699839809778396359) [ 7]
              80918/              25757 = 3.14159257677524556431261404666692549598 ( 0.00000007681454767415002933661257738821) [ 7]
              81273/              25870 = 3.14159257827599536142249710088906068805 ( 0.00000007531379787704014628239044219614) [ 7]
              81628/              25983 = 3.14159257976369164453681253127044606088 ( 0.00000007382610159392583085200905682331) [ 7]
              81983/              26096 = 3.14159258123850398528510116492949110974 ( 0.00000007235128925317754221835001177445) [ 7]
              82338/              26209 = 3.14159258270059903086725933839520775306 ( 0.00000007088919420759538404488429513113) [ 7]
              82693/              26322 = 3.14159258415014056682622900995365093837 ( 0.00000006943965267163641437332585194582) [ 7]
              83048/              26435 = 3.14159258558728957821070550406657839984 ( 0.00000006800250366025193787921292448435) [ 7]
              83403/              26548 = 3.14159258701220430917583245442217869519 ( 0.00000006657758892928681092885732418900) [ 7]
              83758/              26661 = 3.14159258842504032106822699823712538914 ( 0.00000006516475291739441638504237749505) [ 7]
              84113/              26774 = 3.14159258982595054904011354298946739374 ( 0.00000006376384268942252984029003549045) [ 7]
              84468/              26887 = 3.14159259121508535723583888124372373265 ( 0.00000006237470788122680450203577915154) [ 7]
              84823/              27000 = 3.14159259259259259259259259259259259259 ( 0.00000006099720064587005079068691029160) [ 7]
              85178/              27113 = 3.14159259395861763729576218050381735698 ( 0.00000005963117560116688120277568552721) [ 7]
              85533/              27226 = 3.14159259531330345992800999045030485565 ( 0.00000005827648977853463339282919802854) [ 7]
              85888/              27339 = 3.14159259665679066534986649109330992355 ( 0.00000005693300257311277689218619296064) [ 7]
              86243/              27452 = 3.14159259798921754334838991694594200786 ( 0.00000005560057569511425346633356087633) [ 7]
              86598/              27565 = 3.14159259931072011608924360602212951206 ( 0.00000005427907312237339977725737337213) [ 7]
              86953/              27678 = 3.14159260062143218440638774477924705542 ( 0.00000005296836105405625563850025582877) [ 7]
              87308/              27791 = 3.14159260192148537296246986434457198373 ( 0.00000005166830786550017351893493090046) [ 7]
              87663/              27904 = 3.14159260321100917431192660550458715596 ( 0.00000005037878406415071677777491572823) [ 7]
              88018/              28017 = 3.14159260449013099189777635007316986115 ( 0.00000004909966224656486703320633302304) [ 7]
              88373/              28130 = 3.14159260575897618201208674013508709562 ( 0.00000004783081705645055664314441578857) [ 7]
              88728/              28243 = 3.14159260701766809474914138016499663633 ( 0.00000004657212514371350200311450624786) [ 7]
              89083/              28356 = 3.14159260826632811397940471152489772887 ( 0.00000004532346512448323867175460515532) [ 7]
              89438/              28469 = 3.14159260950507569637149179809617478660 ( 0.00000004408471754209115158518332809759) [ 7]
              89793/              28582 = 3.14159261073402840948848925897417955356 ( 0.00000004285576482897415412430532333063) [ 7]
              90148/              28695 = 3.14159261195330196898414357902073531974 ( 0.00000004163649126947849980425876756445) [ 7]
              90503/              28808 = 3.14159261316301027492363232435434601499 ( 0.00000004042678296353901105892515686920) [ 7]
              90858/              28921 = 3.14159261436326544725286124269561910030 ( 0.00000003922652779120978214058388378389) [ 7]
              91213/              29034 = 3.14159261555417786043948474202658951574 ( 0.00000003803561537802315864125291336845) [ 7]
              91568/              29147 = 3.14159261673585617730812776615089031461 ( 0.00000003685393706115451561712861256958) [ 7]
              91923/              29260 = 3.14159261790840738209159261790840738209 ( 0.00000003568138585637105076537109550210) [ 7]
              92278/              29373 = 3.14159261907193681271916385796479760324 ( 0.00000003451785642574347952531470528095) [ 7]
              92633/              29486 = 3.14159262022654819236247710777996337244 ( 0.00000003336324504610016627549953951175) [ 7]
              92988/              29599 = 3.14159262137234366025879252677455319436 ( 0.00000003221744957820385085650494968983) [ 7]
              93343/              29712 = 3.14159262250942380183091007000538502961 ( 0.00000003108036943663173331327411785458) [ 7]
              93698/              29825 = 3.14159262363788767812238055322715842414 ( 0.00000002995190556034026283005234446005) [ 7]
              94053/              29938 = 3.14159262475783285456610328011223194602 ( 0.00000002883196038389654010316727093817) [ 7]
              94408/              30051 = 3.14159262586935542910385677681275165551 ( 0.00000002772043780935878660646675122868) [ 7]
              94763/              30164 = 3.14159262697255005967378331786235247314 ( 0.00000002661724317878886006541715041105) [ 7]
              95118/              30277 = 3.14159262806750999108233972982792218515 ( 0.00000002552228324738030365345158069904) [ 7]
              95473/              30390 = 3.14159262915432708127673576834485027969 ( 0.00000002443546615718590761493465260450) [ 7]
              95828/              30503 = 3.14159263023309182703340655017539258433 ( 0.00000002335670141142923683310411029986) [ 7]
              96183/              30616 = 3.14159263130389338907760648027175333159 ( 0.00000002228589984938503690300774955260) [ 7]
              96538/              30729 = 3.14159263236681961664876826450584138761 ( 0.00000002122297362181387511877366149658) [ 7]
              96893/              30842 = 3.14159263342195707152584138512418131119 ( 0.00000002016783616693680199815532157300) [ 7]
              97248/              30955 = 3.14159263446939105152640930382813761912 ( 0.00000001912040218693623407945136526507) [ 7]
              97603/              31068 = 3.14159263550920561349298313377108278614 ( 0.00000001808058762496966024950842009805) [ 7]
              97958/              31181 = 3.14159263654148359577948109425611750745 ( 0.00000001704830964268316228902338537674) [ 7]
              98313/              31294 = 3.14159263756630664025052725762126925289 ( 0.00000001602348659821211612565823363130) [ 7]
              98668/              31407 = 3.14159263858375521380583946254019804502 ( 0.00000001500603802465680392073930483917) [ 7]
              99023/              31520 = 3.14159263959390862944162436548223350253 ( 0.00000001399588460902101901779726938166) [ 7]
              99378/              31633 = 3.14159264059684506686055701324566117661 ( 0.00000001299294817160208637003384170758) [ 7]
              99733/              31746 = 3.14159264159264159264159264159264159264 ( 0.00000001199715164582105074168686129155) [ 7]
             100088/              31859 = 3.14159264258137417998053925107504943657 ( 0.00000001100841905848210413220445344762) [ 7]
             100443/              31972 = 3.14159264356311772801201050919554610283 ( 0.00000001002667551045063287408395678136) [ 7]
             100798/              32085 = 3.14159264453794608072307932055477637525 ( 0.00000000905184715773956406272472650894) [ 8]
             101153/              32198 = 3.14159264550593204546866264985402820050 ( 0.00000000808386119299398073342547468369) [ 8]
             101508/              32311 = 3.14159264646714741109838754603695335953 ( 0.00000000712264582736425583724254952466) [ 8]
             101863/              32424 = 3.14159264742166296570441648161855415741 ( 0.00000000616813027275822690166094872678) [ 8]
             102218/              32537 = 3.14159264836954851399944678366167747487 ( 0.00000000522024472446319659961782540932) [ 8]
             102573/              32650 = 3.14159264931087289433384379785604900459 ( 0.00000000427892034412879958542345387960) [ 8]
             102928/              32763 = 3.14159265024570399536062021182431401275 ( 0.00000000334408924310202317145518887144) [ 8]
             103283/              32876 = 3.14159265117410877235673439591191142474 ( 0.00000000241568446610590898736759145945) [ 8]
             103638/              32989 = 3.14159265209615326320894843735790718118 ( 0.00000000149363997525369494592159570301) [ 8]
             103993/              33102 = 3.14159265301190260407226149477372968400 ( 0.00000000057789063439038188850577320019) [ 9]
             104348/              33215 = 3.14159265392142104470871594159265392142 (-0.00000000033162780624607255831315103723) [ 9]
             208341/              66317 = 3.14159265346743670552045478534915632492 ( 0.00000000012235653294218859793034655927) [ 9]
             312689/              99532 = 3.14159265361893662339750030141060161556 (-0.00000000002914338493485691813109873137) [10]
             833719/             265381 = 3.14159265358107777120441930658185778183 ( 0.00000000000871546725822407669764510236) [11]
            1146408/             364913 = 3.14159265359140397848254241421927966391 (-0.00000000000161074001989903093977677972) [11]
            3126535/             995207 = 3.14159265358865040137378454934501063597 ( 0.00000000000114283708885883393449224822) [11]
            4272943/            1360120 = 3.14159265358938917154368732170690821398 ( 0.00000000000040406691895606157259467021) [12]
            5419351/            1725033 = 3.14159265358981538324194377730744861112 (-0.00000000000002214477930039402794572693) [13]
           42208400/           13435351 = 3.14159265358977223594679439338801048070 ( 0.00000000000002100251584898989149240349) [13]
           47627751/           15160384 = 3.14159265358977714548655231951908342163 ( 0.00000000000001609297609106376041946256) [13]
           53047102/           16885417 = 3.14159265358978105189821489158366654492 ( 0.00000000000001218656442849169583633927) [13]
           58466453/           18610450 = 3.14159265358978423412652568852445803298 ( 0.00000000000000900433611769475504485121) [14]
           63885804/           20335483 = 3.14159265358978687646612573696921779531 ( 0.00000000000000636199651764631028508888) [14]
           69305155/           22060516 = 3.14159265358978910556761228975786423128 ( 0.00000000000000413289503109352163865291) [14]
           74724506/           23785549 = 3.14159265358979101134054126730478241221 ( 0.00000000000000222712210211597472047198) [14]
           80143857/           25510582 = 3.14159265358979265937562694571217544154 ( 0.00000000000000057908701643756732744265) [15]
          165707065/           52746197 = 3.14159265358979340254615892023457160333 (-0.00000000000000016408351553695506871914) [15]
          245850922/           78256779 = 3.14159265358979316028327718420406748404 ( 0.00000000000000007817936619907543540015) [16]
          411557987/          131002976 = 3.14159265358979325782644815641440084536 (-0.00000000000000001936380477313489796117) [16]
          657408909/          209259755 = 3.14159265358979322134827119529027452029 ( 0.00000000000000001711437218798922836390) [16]
         1068966896/          340262731 = 3.14159265358979323539256492948091926059 ( 0.00000000000000000307007845379858362360) [17]
         2549491779/          811528438 = 3.14159265358979323901400975919959073572 (-0.00000000000000000055136637592008785153) [18]
         3618458675/         1151791169 = 3.14159265358979323794416069185889026381 ( 0.00000000000000000051848269142061262038) [18]
         6167950454/         1963319607 = 3.14159265358979323838637750639037956696 ( 0.00000000000000000007626587688912331723) [19]
        14885392687/         4738167652 = 3.14159265358979323849387505801156062596 (-0.00000000000000000003123167473205774177) [19]
        21053343141/         6701487259 = 3.14159265358979323846238174277486901359 ( 0.00000000000000000000026164050463387060) [21]
       899125804609/       286200632530 = 3.14159265358979323846290312739302875642 (-0.00000000000000000000025974411352587223) [21]
       920179147750/       292902119789 = 3.14159265358979323846289119831454290206 (-0.00000000000000000000024781503504001787) [21]
       941232490891/       299603607048 = 3.14159265358979323846287980289163130623 (-0.00000000000000000000023641961212842204) [21]
       962285834032/       306305094307 = 3.14159265358979323846286890609758924162 (-0.00000000000000000000022552281808635743) [21]
       983339177173/       313006581566 = 3.14159265358979323846285847590540629124 (-0.00000000000000000000021509262590340705) [21]
      1004392520314/       319708068825 = 3.14159265358979323846284848297337933163 (-0.00000000000000000000020509969387644744) [21]
      1025445863455/       326409556084 = 3.14159265358979323846283890036945343710 (-0.00000000000000000000019551708995055291) [21]
      1046499206596/       333111043343 = 3.14159265358979323846282970332883684002 (-0.00000000000000000000018632004933395583) [21]
      1067552549737/       339812530602 = 3.14159265358979323846282086904029653302 (-0.00000000000000000000017748576079364883) [21]
      1088605892878/       346514017861 = 3.14159265358979323846281237645725178231 (-0.00000000000000000000016899317774889812) [21]
      1109659236019/       353215505120 = 3.14159265358979323846280420613037215131 (-0.00000000000000000000016082285086926712) [21]
      1130712579160/       359916992379 = 3.14159265358979323846279634005887720665 (-0.00000000000000000000015295677937432246) [21]
      1151765922301/       366618479638 = 3.14159265358979323846278876155814494589 (-0.00000000000000000000014537827864206170) [21]
      1172819265442/       373319966897 = 3.14159265358979323846278145514157963557 (-0.00000000000000000000013807186207675138) [21]
      1193872608583/       380021454156 = 3.14159265358979323846277440641497885695 (-0.00000000000000000000013102313547597276) [21]
      1214925951724/       386722941415 = 3.14159265358979323846276760198188357586 (-0.00000000000000000000012421870238069167) [21]
      1235979294865/       393424428674 = 3.14159265358979323846276102935860166316 (-0.00000000000000000000011764607909877897) [21]
      1257032638006/       400125915933 = 3.14159265358979323846275467689777075912 (-0.00000000000000000000011129361826787493) [21]
      1278085981147/       406827403192 = 3.14159265358979323846274853371947582775 (-0.00000000000000000000010515043997294356) [21]
      1299139324288/       413528890451 = 3.14159265358979323846274258964906440289 (-0.00000000000000000000009920636956151870) [22]
      1320192667429/       420230377710 = 3.14159265358979323846273683516091186105 (-0.00000000000000000000009345188140897686) [22]
      1341246010570/       426931864969 = 3.14159265358979323846273126132748294414 (-0.00000000000000000000008787804798005995) [22]
      1362299353711/       433633352228 = 3.14159265358979323846272585977311658438 (-0.00000000000000000000008247649361370019) [22]
      1383352696852/       440334839487 = 3.14159265358979323846272062263203084137 (-0.00000000000000000000007723935252795718) [22]
      1404406039993/       447036326746 = 3.14159265358979323846271554251010510785 (-0.00000000000000000000007215923060222366) [22]
      1425459383134/       453737814005 = 3.14159265358979323846271061245004906498 (-0.00000000000000000000006722917054618079) [22]
      1446512726275/       460439301264 = 3.14159265358979323846270582589961333896 (-0.00000000000000000000006244262011045477) [22]
      1467566069416/       467140788523 = 3.14159265358979323846270117668253641040 (-0.00000000000000000000005779340303352621) [22]
      1488619412557/       473842275782 = 3.14159265358979323846269665897195688730 (-0.00000000000000000000005327569245400311) [22]
      1509672755698/       480543763041 = 3.14159265358979323846269226726605047424 (-0.00000000000000000000004888398654759005) [22]
      1530726098839/       487245250300 = 3.14159265358979323846268799636567745111 (-0.00000000000000000000004461308617456692) [22]
      1551779441980/       493946737559 = 3.14159265358979323846268384135384972222 (-0.00000000000000000000004045807434683803) [22]
      1572832785121/       500648224818 = 3.14159265358979323846267979757684694309 (-0.00000000000000000000003641429734405890) [22]
      1593886128262/       507349712077 = 3.14159265358979323846267586062682924856 (-0.00000000000000000000003247734732636437) [22]
      1614939471403/       514051199336 = 3.14159265358979323846267202632581000779 (-0.00000000000000000000002864304630712360) [22]
      1635992814544/       520752686595 = 3.14159265358979323846266829071086609244 (-0.00000000000000000000002490743136320825) [22]
      1657046157685/       527454173854 = 3.14159265358979323846266465002047559662 (-0.00000000000000000000002126674097271243) [22]
      1678099500826/       534155661113 = 3.14159265358979323846266110068188399415 (-0.00000000000000000000001771740238110996) [22]
      1699152843967/       540857148372 = 3.14159265358979323846265763929940953314 (-0.00000000000000000000001425601990664895) [22]
      1720206187108/       547558635631 = 3.14159265358979323846265426264360740155 (-0.00000000000000000000001087936410451736) [22]
      1741259530249/       554260122890 = 3.14159265358979323846265096764121998082 (-0.00000000000000000000000758436171709663) [23]
      1762312873390/       560961610149 = 3.14159265358979323846264775136584745085 (-0.00000000000000000000000436808634456666) [23]
      1783366216531/       567663097408 = 3.14159265358979323846264461102927921823 (-0.00000000000000000000000122774977633404) [23]
      3587785776203/      1142027682075 = 3.14159265358979323846264306850252143875 ( 0.00000000000000000000000031477698144544) [24]
      5371151992734/      1709690779483 = 3.14159265358979323846264358066268961876 (-0.00000000000000000000000019738318673457) [24]
      8958937768937/      2851718461558 = 3.14159265358979323846264337555790890412 ( 0.00000000000000000000000000772159398007) [26]
     77042654144230/     24523438471947 = 3.14159265358979323846264338985711710108 (-0.00000000000000000000000000657761421689) [26]
     86001591913167/     27375156933505 = 3.14159265358979323846264338836754332073 (-0.00000000000000000000000000508804043654) [26]
     94960529682104/     30226875395063 = 3.14159265358979323846264338715903365924 (-0.00000000000000000000000000387953077505) [26]
    103919467451041/     33078593856621 = 3.14159265358979323846264338615889617509 (-0.00000000000000000000000000287939329090) [26]
    112878405219978/     35930312318179 = 3.14159265358979323846264338531751659443 (-0.00000000000000000000000000203801371024) [26]
    121837342988915/     38782030779737 = 3.14159265358979323846264338459987358119 (-0.00000000000000000000000000132037069700) [26]
    130796280757852/     41633749241295 = 3.14159265358979323846264338398054099481 (-0.00000000000000000000000000070103811062) [27]
    139755218526789/     44485467702853 = 3.14159265358979323846264338344061241435 (-0.00000000000000000000000000016110953016) [27]
    288469374822515/     91822653867264 = 3.14159265358979323846264338319580081089 ( 0.00000000000000000000000000008370207330) [28]
    428224593349304/    136308121570117 = 3.14159265358979323846264338327569743446 ( 0.00000000000000000000000000000380544973) [29]
   3137327371971917/    998642318693672 = 3.14159265358979323846264338328304372840 (-0.00000000000000000000000000000354084421) [29]
   3565551965321221/   1134950440263789 = 3.14159265358979323846264338328216143479 (-0.00000000000000000000000000000265855060) [29]
   3993776558670525/   1271258561833906 = 3.14159265358979323846264338328146834546 (-0.00000000000000000000000000000196546127) [29]
   4422001152019829/   1407566683404023 = 3.14159265358979323846264338328090949305 (-0.00000000000000000000000000000140660886) [29]
   4850225745369133/   1543874804974140 = 3.14159265358979323846264338328044932237 (-0.00000000000000000000000000000094643818) [30]
   5278450338718437/   1680182926544257 = 3.14159265358979323846264338328006381618 (-0.00000000000000000000000000000056093199) [30]
   5706674932067741/   1816491048114374 = 3.14159265358979323846264338327973616618 (-0.00000000000000000000000000000023328199) [30]
   6134899525417045/   1952799169684491 = 3.14159265358979323846264338327945425704 ( 0.00000000000000000000000000000004862715) [31]
  17976473982901831/   5722089387483356 = 3.14159265358979323846264338327954374978 (-0.00000000000000000000000000000004086559) [31]
  24111373508318876/   7674888557167847 = 3.14159265358979323846264338327952097924 (-0.00000000000000000000000000000001809505) [31]
  30246273033735921/   9627687726852338 = 3.14159265358979323846264338327950744587 (-0.00000000000000000000000000000000456168) [32]
  36381172559152966/  11580486896536829 = 3.14159265358979323846264338327949847672 ( 0.00000000000000000000000000000000440747) [32]
  66627445592888887/  21208174623389167 = 3.14159265358979323846264338327950254837 ( 0.00000000000000000000000000000000033582) [33]
 230128609812402582/  73252211597019839 = 3.14159265358979323846264338327950319206 (-0.00000000000000000000000000000000030787) [33]
 296756055405291469/  94460386220409006 = 3.14159265358979323846264338327950304754 (-0.00000000000000000000000000000000016335) [33]
 363383500998180356/ 115668560843798173 = 3.14159265358979323846264338327950295601 (-0.00000000000000000000000000000000007182) [34]
 430010946591069243/ 136876735467187340 = 3.14159265358979323846264338327950289285 (-0.00000000000000000000000000000000000866) [35]
1356660285366096616/ 431838381024951187 = 3.14159265358979323846264338327950287593 ( 0.00000000000000000000000000000000000826) [35]
1786671231957165859/ 568715116492138527 = 3.14159265358979323846264338327950288000 ( 0.00000000000000000000000000000000000419) [35]
2216682178548235102/ 705591851959325867 = 3.14159265358979323846264338327950288250 ( 0.00000000000000000000000000000000000169) [35]
2646693125139304345/ 842468587426513207 = 3.14159265358979323846264338327950288418 ( 0.00000000000000000000000000000000000001) [37]

 

 

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