Runtime on this is around 30 hours. I think the program can get one more approximation before dying, but I've gone for 40 hours without anything new, so I'm calling it quits.
Here's the source.
pi = 3.14159265358979323846264338327950288419716939937510
Dig. = Digits of accuracy (not rounded.. just # of equal
digits past decimal point)
Num. = Number of similar fractions
Num. Den. = Result (Accuracy ) Dig. Num.
------------- ----------- = -------------------------------- (---------------------------------) ---- -----
1/ 1 = 1.000000000000000000000000000000 ( 2.141592653589793238462643383279) [ 0]
2/ 1 = 2.000000000000000000000000000000 ( 1.141592653589793238462643383279) [ 0]
3/ 1 = 3.000000000000000000000000000000 ( 0.141592653589793238462643383279) [ 0] (x 4)
13/ 4 = 3.250000000000000000000000000000 (-0.108407346410206761537356616721) [ 0]
16/ 5 = 3.200000000000000000000000000000 (-0.058407346410206761537356616721) [ 1]
19/ 6 = 3.166666666666666666666666666666 (-0.025074013076873428204023283387) [ 1]
22/ 7 = 3.142857142857142857142857142857 (-0.001264489267349618680213759578) [ 2] (x 8)
179/ 57 = 3.140350877192982456140350877192 ( 0.001241776396810782322292506087) [ 2]
201/ 64 = 3.140625000000000000000000000000 ( 0.000967653589793238462643383279) [ 3]
223/ 71 = 3.140845070422535211267605633802 ( 0.000747583167258027195037749477) [ 3]
245/ 78 = 3.141025641025641025641025641025 ( 0.000567012564152212821617742254) [ 3]
267/ 85 = 3.141176470588235294117647058823 ( 0.000416183001557944344996324456) [ 3]
289/ 92 = 3.141304347826086956521739130434 ( 0.000288305763706281940904252845) [ 3]
311/ 99 = 3.141414141414141414141414141414 ( 0.000178512175651824321229241865) [ 3]
333/ 106 = 3.141509433962264150943396226415 ( 0.000083219627529087519247156864) [ 4]
355/ 113 = 3.141592920353982300884955752212 (-0.000000266764189062422312368933) [ 6] (x 146)
52163/ 16604 = 3.141592387376535774512165743194 ( 0.000000266213257463950477640085) [ 6]
52518/ 16717 = 3.141592390979242687085003290064 ( 0.000000262610550551377640093215) [ 6]
52873/ 16830 = 3.141592394533571004159239453357 ( 0.000000259056222234303403929922) [ 6]
53228/ 16943 = 3.141592398040488697397155167325 ( 0.000000255549304541065488215954) [ 6]
53583/ 17056 = 3.141592401500938086303939962476 ( 0.000000252088855152158703420803) [ 6]
53938/ 17169 = 3.141592404915836682392684489486 ( 0.000000248673956556069958893793) [ 6]
54293/ 17282 = 3.141592408286078000231454692743 ( 0.000000245303715238231188690536) [ 6]
54648/ 17395 = 3.141592411612532336878413337165 ( 0.000000241977260901584230046114) [ 6]
55003/ 17508 = 3.141592414896047521133196253141 ( 0.000000238693745717329447130138) [ 6]
55358/ 17621 = 3.141592418137449633959480165711 ( 0.000000235452343604503163217568) [ 6]
55713/ 17734 = 3.141592421337543701364610352994 ( 0.000000232252249537098033030285) [ 6]
56068/ 17847 = 3.141592424497114360957023589398 ( 0.000000229092678877505619793881) [ 6]
56423/ 17960 = 3.141592427616926503340757238307 ( 0.000000225972866735121886144972) [ 6]
56778/ 18073 = 3.141592430697725889448348364964 ( 0.000000222892067349014295018315) [ 6]
57133/ 18186 = 3.141592433740239744858682503024 ( 0.000000219849553493603960880255) [ 6]
57488/ 18299 = 3.141592436745177332094649980873 ( 0.000000216844615906367993402406) [ 6]
57843/ 18412 = 3.141592439713230501846621768411 ( 0.000000213876562736616021614868) [ 6]
58198/ 18525 = 3.141592442645074224021592442645 ( 0.000000210944719014441050940634) [ 6]
58553/ 18638 = 3.141592445541367099474192509925 ( 0.000000208048426138988450873354) [ 6]
58908/ 18751 = 3.141592448402751853234494160311 ( 0.000000205187041385228149222968) [ 6]
59263/ 18864 = 3.141592451229855810008481764206 ( 0.000000202359937428454161619073) [ 6]
59618/ 18977 = 3.141592454023291352690098540338 ( 0.000000199566501885772544842941) [ 6]
59973/ 19090 = 3.141592456783656364588789942378 ( 0.000000196806136873873853440901) [ 6]
60328/ 19203 = 3.141592459511534656043326563557 ( 0.000000194078258582419316819722) [ 6]
60683/ 19316 = 3.141592462207496376061296334644 ( 0.000000191382296862401347048635) [ 6]
61038/ 19429 = 3.141592464872098409593906016779 ( 0.000000188717694828868737366500) [ 6]
61393/ 19542 = 3.141592467505884761027530447241 ( 0.000000186083908477435112936038) [ 6]
61748/ 19655 = 3.141592470109386924446705672856 ( 0.000000183480406314015937710423) [ 6]
62103/ 19768 = 3.141592472683124241197895588830 ( 0.000000180906668997264747794449) [ 6]
62458/ 19881 = 3.141592475227604245259292792113 ( 0.000000178362188993203350591166) [ 6]
62813/ 19994 = 3.141592477743322996899069720916 ( 0.000000175846470241563573662363) [ 6]
63168/ 20107 = 3.141592480230765405082806982642 ( 0.000000173359027833379836400637) [ 6]
63523/ 20220 = 3.141592482690405539070227497527 ( 0.000000170899387699392415885752) [ 6]
63878/ 20333 = 3.141592485122706929621797078640 ( 0.000000168467086308840846304639) [ 6]
64233/ 20446 = 3.141592487528122860217157390198 ( 0.000000166061670378245485993081) [ 6]
64588/ 20559 = 3.141592489907096648669682377547 ( 0.000000163682696589792961005732) [ 6]
64943/ 20672 = 3.141592492260061919504643962848 ( 0.000000161329731318957999420431) [ 6]
65298/ 20785 = 3.141592494587442867452489776280 ( 0.000000159002350371010153606999) [ 6]
65653/ 20898 = 3.141592496889654512393530481385 ( 0.000000156700138726069112901894) [ 6]
66008/ 21011 = 3.141592499167102946075865023083 ( 0.000000154422690292386778360196) [ 6]
66363/ 21124 = 3.141592501420185570914599507669 ( 0.000000152169607667548043875610) [ 6]
66718/ 21237 = 3.141592503649291331167302349672 ( 0.000000149940501907295341033607) [ 6]
67073/ 21350 = 3.141592505854800936768149882903 ( 0.000000147734992301694493500376) [ 6]
67428/ 21463 = 3.141592508037087080091319945953 ( 0.000000145552706158371323437326) [ 6]
67783/ 21576 = 3.141592510196514645902855024100 ( 0.000000143393278592559788359179) [ 6]
68138/ 21689 = 3.141592512333440914749412144404 ( 0.000000141256352323713231238875) [ 6]
68493/ 21802 = 3.141592514448215760022016328777 ( 0.000000139141577478440627054502) [ 6]
68848/ 21915 = 3.141592516541181838923112023728 ( 0.000000137048611399539531359551) [ 6]
69203/ 22028 = 3.141592518612674777555838024332 ( 0.000000134977118460906805358947) [ 6]
69558/ 22141 = 3.141592520663023350345512849464 ( 0.000000132926769888117130533815) [ 6]
69913/ 22254 = 3.141592522692549653994787453940 ( 0.000000130897243584467855929339) [ 6]
70268/ 22367 = 3.141592524701569276165779943667 ( 0.000000128888223962296863439612) [ 6]
70623/ 22480 = 3.141592526690391459074733096085 ( 0.000000126899401779387910287194) [ 6]
70978/ 22593 = 3.141592528659319258177311556676 ( 0.000000124930473980285331826603) [ 6]
71333/ 22706 = 3.141592530608649696115564168061 ( 0.000000122981143542347079215218) [ 6]
71688/ 22819 = 3.141592532538673912090801525044 ( 0.000000121051119326371841858235) [ 6]
72043/ 22932 = 3.141592534449677306820163963021 ( 0.000000119140115931642479420258) [ 6]
72398/ 23045 = 3.141592536341939683228466044695 ( 0.000000117247853555234177338584) [ 6]
72753/ 23158 = 3.141592538215735383020986268244 ( 0.000000115374057855441657115035) [ 6]
73108/ 23271 = 3.141592540071333419277211980576 ( 0.000000113518459819185431402703) [ 6]
73463/ 23384 = 3.141592541908997605200136845706 ( 0.000000111680795633262506537573) [ 6]
73818/ 23497 = 3.141592543728986679150529854875 ( 0.000000109860806559312113528404) [ 6]
74173/ 23610 = 3.141592545531554426090639559508 ( 0.000000108058238812372003823771) [ 6]
74528/ 23723 = 3.141592547316949795557054335455 ( 0.000000106272843442905589047824) [ 6]
74883/ 23836 = 3.141592549085417016277898976338 ( 0.000000104504376222184744406941) [ 6]
75238/ 23949 = 3.141592550837195707545200217128 ( 0.000000102752597530917443166151) [ 6]
75593/ 24062 = 3.141592552572520987449089851217 ( 0.000000101017272251013553532062) [ 6]
75948/ 24175 = 3.141592554291623578076525336091 ( 0.000000099298169660386118047188) [ 7]
76303/ 24288 = 3.141592555994729907773386034255 ( 0.000000097595063330689257349024) [ 7]
76658/ 24401 = 3.141592557682062210565140772919 ( 0.000000095907731027897502610360) [ 7]
77013/ 24514 = 3.141592559353838622827771885453 ( 0.000000094235954615634871497826) [ 7]
77368/ 24627 = 3.141592561010273277297275348195 ( 0.000000092579519961165368035084) [ 7]
77723/ 24740 = 3.141592562651576394502829426030 ( 0.000000090938216843959813957249) [ 7]
78078/ 24853 = 3.141592564277954371705629099102 ( 0.000000089311838866757014284177) [ 7]
78433/ 24966 = 3.141592565889609869422414483697 ( 0.000000087700183369040228899582) [ 7]
78788/ 25079 = 3.141592567486741895609872801945 ( 0.000000086103051342852770581334) [ 7]
79143/ 25192 = 3.141592569069545887583359796760 ( 0.000000084520247350879283586519) [ 7]
79498/ 25305 = 3.141592570638213791740762695119 ( 0.000000082951579446721880688160) [ 7]
79853/ 25418 = 3.141592572192934141159808010071 ( 0.000000081396859097302835373208) [ 7]
80208/ 25531 = 3.141592573733892131134698993380 ( 0.000000079855901107327944389899) [ 7]
80563/ 25644 = 3.141592575261269692715644985181 ( 0.000000078328523545746998398098) [ 7]
80918/ 25757 = 3.141592576775245564312614046666 ( 0.000000076814547674150029336613) [ 7]
81273/ 25870 = 3.141592578275995361422497100889 ( 0.000000075313797877040146282390) [ 7]
81628/ 25983 = 3.141592579763691644536812531270 ( 0.000000073826101593925830852009) [ 7]
81983/ 26096 = 3.141592581238503985285101164929 ( 0.000000072351289253177542218350) [ 7]
82338/ 26209 = 3.141592582700599030867259338395 ( 0.000000070889194207595384044884) [ 7]
82693/ 26322 = 3.141592584150140566826229009953 ( 0.000000069439652671636414373326) [ 7]
83048/ 26435 = 3.141592585587289578210705504066 ( 0.000000068002503660251937879213) [ 7]
83403/ 26548 = 3.141592587012204309175832454422 ( 0.000000066577588929286810928857) [ 7]
83758/ 26661 = 3.141592588425040321068226998237 ( 0.000000065164752917394416385042) [ 7]
84113/ 26774 = 3.141592589825950549040113542989 ( 0.000000063763842689422529840290) [ 7]
84468/ 26887 = 3.141592591215085357235838881243 ( 0.000000062374707881226804502036) [ 7]
84823/ 27000 = 3.141592592592592592592592592592 ( 0.000000060997200645870050790687) [ 7]
85178/ 27113 = 3.141592593958617637295762180503 ( 0.000000059631175601166881202776) [ 7]
85533/ 27226 = 3.141592595313303459928009990450 ( 0.000000058276489778534633392829) [ 7]
85888/ 27339 = 3.141592596656790665349866491093 ( 0.000000056933002573112776892186) [ 7]
86243/ 27452 = 3.141592597989217543348389916945 ( 0.000000055600575695114253466334) [ 7]
86598/ 27565 = 3.141592599310720116089243606022 ( 0.000000054279073122373399777257) [ 7]
86953/ 27678 = 3.141592600621432184406387744779 ( 0.000000052968361054056255638500) [ 7]
87308/ 27791 = 3.141592601921485372962469864344 ( 0.000000051668307865500173518935) [ 7]
87663/ 27904 = 3.141592603211009174311926605504 ( 0.000000050378784064150716777775) [ 7]
88018/ 28017 = 3.141592604490130991897776350073 ( 0.000000049099662246564867033206) [ 7]
88373/ 28130 = 3.141592605758976182012086740135 ( 0.000000047830817056450556643144) [ 7]
88728/ 28243 = 3.141592607017668094749141380164 ( 0.000000046572125143713502003115) [ 7]
89083/ 28356 = 3.141592608266328113979404711524 ( 0.000000045323465124483238671755) [ 7]
89438/ 28469 = 3.141592609505075696371491798096 ( 0.000000044084717542091151585183) [ 7]
89793/ 28582 = 3.141592610734028409488489258974 ( 0.000000042855764828974154124305) [ 7]
90148/ 28695 = 3.141592611953301968984143579020 ( 0.000000041636491269478499804259) [ 7]
90503/ 28808 = 3.141592613163010274923632324354 ( 0.000000040426782963539011058925) [ 7]
90858/ 28921 = 3.141592614363265447252861242695 ( 0.000000039226527791209782140584) [ 7]
91213/ 29034 = 3.141592615554177860439484742026 ( 0.000000038035615378023158641253) [ 7]
91568/ 29147 = 3.141592616735856177308127766150 ( 0.000000036853937061154515617129) [ 7]
91923/ 29260 = 3.141592617908407382091592617908 ( 0.000000035681385856371050765371) [ 7]
92278/ 29373 = 3.141592619071936812719163857964 ( 0.000000034517856425743479525315) [ 7]
92633/ 29486 = 3.141592620226548192362477107779 ( 0.000000033363245046100166275500) [ 7]
92988/ 29599 = 3.141592621372343660258792526774 ( 0.000000032217449578203850856505) [ 7]
93343/ 29712 = 3.141592622509423801830910070005 ( 0.000000031080369436631733313274) [ 7]
93698/ 29825 = 3.141592623637887678122380553227 ( 0.000000029951905560340262830052) [ 7]
94053/ 29938 = 3.141592624757832854566103280112 ( 0.000000028831960383896540103167) [ 7]
94408/ 30051 = 3.141592625869355429103856776812 ( 0.000000027720437809358786606467) [ 7]
94763/ 30164 = 3.141592626972550059673783317862 ( 0.000000026617243178788860065417) [ 7]
95118/ 30277 = 3.141592628067509991082339729827 ( 0.000000025522283247380303653452) [ 7]
95473/ 30390 = 3.141592629154327081276735768344 ( 0.000000024435466157185907614935) [ 7]
95828/ 30503 = 3.141592630233091827033406550175 ( 0.000000023356701411429236833104) [ 7]
96183/ 30616 = 3.141592631303893389077606480271 ( 0.000000022285899849385036903008) [ 7]
96538/ 30729 = 3.141592632366819616648768264505 ( 0.000000021222973621813875118774) [ 7]
96893/ 30842 = 3.141592633421957071525841385124 ( 0.000000020167836166936801998155) [ 7]
97248/ 30955 = 3.141592634469391051526409303828 ( 0.000000019120402186936234079451) [ 7]
97603/ 31068 = 3.141592635509205613492983133771 ( 0.000000018080587624969660249508) [ 7]
97958/ 31181 = 3.141592636541483595779481094256 ( 0.000000017048309642683162289023) [ 7]
98313/ 31294 = 3.141592637566306640250527257621 ( 0.000000016023486598212116125658) [ 7]
98668/ 31407 = 3.141592638583755213805839462540 ( 0.000000015006038024656803920739) [ 7]
99023/ 31520 = 3.141592639593908629441624365482 ( 0.000000013995884609021019017797) [ 7]
99378/ 31633 = 3.141592640596845066860557013245 ( 0.000000012992948171602086370034) [ 7]
99733/ 31746 = 3.141592641592641592641592641592 ( 0.000000011997151645821050741687) [ 7]
100088/ 31859 = 3.141592642581374179980539251075 ( 0.000000011008419058482104132204) [ 7]
100443/ 31972 = 3.141592643563117728012010509195 ( 0.000000010026675510450632874084) [ 7]
100798/ 32085 = 3.141592644537946080723079320554 ( 0.000000009051847157739564062725) [ 8]
101153/ 32198 = 3.141592645505932045468662649854 ( 0.000000008083861192993980733425) [ 8]
101508/ 32311 = 3.141592646467147411098387546036 ( 0.000000007122645827364255837243) [ 8]
101863/ 32424 = 3.141592647421662965704416481618 ( 0.000000006168130272758226901661) [ 8]
102218/ 32537 = 3.141592648369548513999446783661 ( 0.000000005220244724463196599618) [ 8]
102573/ 32650 = 3.141592649310872894333843797856 ( 0.000000004278920344128799585423) [ 8]
102928/ 32763 = 3.141592650245703995360620211824 ( 0.000000003344089243102023171455) [ 8]
103283/ 32876 = 3.141592651174108772356734395911 ( 0.000000002415684466105908987368) [ 8]
103638/ 32989 = 3.141592652096153263208948437357 ( 0.000000001493639975253694945922) [ 8]
103993/ 33102 = 3.141592653011902604072261494773 ( 0.000000000577890634390381888506) [ 9]
104348/ 33215 = 3.141592653921421044708715941592 (-0.000000000331627806246072558313) [ 9]
208341/ 66317 = 3.141592653467436705520454785349 ( 0.000000000122356532942188597930) [ 9]
312689/ 99532 = 3.141592653618936623397500301410 (-0.000000000029143384934856918131) [10] (x 2)
833719/ 265381 = 3.141592653581077771204419306581 ( 0.000000000008715467258224076698) [11]
1146408/ 364913 = 3.141592653591403978482542414219 (-0.000000000001610740019899030940) [11] (x 2)
3126535/ 995207 = 3.141592653588650401373784549345 ( 0.000000000001142837088858833934) [11]
4272943/ 1360120 = 3.141592653589389171543687321706 ( 0.000000000000404066918956061573) [12]
5419351/ 1725033 = 3.141592653589815383241943777307 (-0.000000000000022144779300394028) [13] (x 7)
42208400/ 13435351 = 3.141592653589772235946794393388 ( 0.000000000000021002515848989891) [13]
47627751/ 15160384 = 3.141592653589777145486552319519 ( 0.000000000000016092976091063760) [13]
53047102/ 16885417 = 3.141592653589781051898214891583 ( 0.000000000000012186564428491696) [13]
58466453/ 18610450 = 3.141592653589784234126525688524 ( 0.000000000000009004336117694755) [14]
63885804/ 20335483 = 3.141592653589786876466125736969 ( 0.000000000000006361996517646310) [14]
69305155/ 22060516 = 3.141592653589789105567612289757 ( 0.000000000000004132895031093522) [14]
74724506/ 23785549 = 3.141592653589791011340541267304 ( 0.000000000000002227122102115975) [14]
80143857/ 25510582 = 3.141592653589792659375626945712 ( 0.000000000000000579087016437567) [15] (x 2)
165707065/ 52746197 = 3.141592653589793402546158920234 (-0.000000000000000164083515536955) [15]
245850922/ 78256779 = 3.141592653589793160283277184204 ( 0.000000000000000078179366199075) [16]
411557987/ 131002976 = 3.141592653589793257826448156414 (-0.000000000000000019363804773135) [16]
657408909/ 209259755 = 3.141592653589793221348271195290 ( 0.000000000000000017114372187989) [16]
1068966896/ 340262731 = 3.141592653589793235392564929480 ( 0.000000000000000003070078453799) [17]
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