Approximating Pi

This is a really pointless post about approximating $\pi$, inspired by this post.

current_min_error <- abs(pi - 22 / 7)
current_best_numerator <- 22
current_best_denominator <- 7

for(denominator in 1:1e7) {
  numerator <- round(denominator * pi)
  pi_approx <- numerator / denominator
  pi_error <- abs(pi - pi_approx)
  if(pi_error < current_min_error) {
    current_min_error <- pi_error
    current_best_numerator <- numerator
    current_best_denominator <- denominator
    cat(paste(c('numerator:', numerator, 
                  ', denominator:', denominator, 
                  ', pi_approx: ', numerator / denominator, 
                  ', absolute error', pi_error, '\n')))
  }
}

numerator: 179 , denominator: 57 , pi_approx:  3.14035087719298 , absolute error 0.00124177639681067 
numerator: 201 , denominator: 64 , pi_approx:  3.140625 , absolute error 0.000967653589793116 
numerator: 223 , denominator: 71 , pi_approx:  3.14084507042254 , absolute error 0.000747583167258092 
numerator: 245 , denominator: 78 , pi_approx:  3.14102564102564 , absolute error 0.000567012564152147 
numerator: 267 , denominator: 85 , pi_approx:  3.14117647058824 , absolute error 0.000416183001557879 
numerator: 289 , denominator: 92 , pi_approx:  3.14130434782609 , absolute error 0.000288305763706198 
numerator: 311 , denominator: 99 , pi_approx:  3.14141414141414 , absolute error 0.000178512175651679 
numerator: 333 , denominator: 106 , pi_approx:  3.14150943396226 , absolute error 8.32196275291075e-05 
numerator: 355 , denominator: 113 , pi_approx:  3.14159292035398 , absolute error 2.66764189404967e-07 
numerator: 52163 , denominator: 16604 , pi_approx:  3.14159238737654 , absolute error 2.66213257216208e-07 
numerator: 52518 , denominator: 16717 , pi_approx:  3.14159239097924 , absolute error 2.62610550638698e-07 
numerator: 52873 , denominator: 16830 , pi_approx:  3.14159239453357 , absolute error 2.5905622225153e-07 
numerator: 53228 , denominator: 16943 , pi_approx:  3.14159239804049 , absolute error 2.55549304384317e-07 
numerator: 53583 , denominator: 17056 , pi_approx:  3.14159240150094 , absolute error 2.52088855123844e-07 
numerator: 53938 , denominator: 17169 , pi_approx:  3.14159240491584 , absolute error 2.48673956537715e-07 
numerator: 54293 , denominator: 17282 , pi_approx:  3.14159240828608 , absolute error 2.45303715118439e-07 
numerator: 54648 , denominator: 17395 , pi_approx:  3.14159241161253 , absolute error 2.41977260895254e-07 
numerator: 55003 , denominator: 17508 , pi_approx:  3.14159241489605 , absolute error 2.3869374565777e-07 
numerator: 55358 , denominator: 17621 , pi_approx:  3.14159241813745 , absolute error 2.35452343400055e-07 
numerator: 55713 , denominator: 17734 , pi_approx:  3.14159242133754 , absolute error 2.32252249432463e-07 
numerator: 56068 , denominator: 17847 , pi_approx:  3.14159242449711 , absolute error 2.29092678605269e-07 
numerator: 56423 , denominator: 17960 , pi_approx:  3.14159242761693 , absolute error 2.25972866640944e-07 
numerator: 56778 , denominator: 18073 , pi_approx:  3.14159243069773 , absolute error 2.22892067025526e-07 
numerator: 57133 , denominator: 18186 , pi_approx:  3.14159243374024 , absolute error 2.19849553229068e-07 
numerator: 57488 , denominator: 18299 , pi_approx:  3.14159243674518 , absolute error 2.16844615597012e-07 
numerator: 57843 , denominator: 18412 , pi_approx:  3.14159243971323 , absolute error 2.13876562682458e-07 
numerator: 58198 , denominator: 18525 , pi_approx:  3.14159244264507 , absolute error 2.10944719025719e-07 
numerator: 58553 , denominator: 18638 , pi_approx:  3.14159244554137 , absolute error 2.08048426042495e-07 
numerator: 58908 , denominator: 18751 , pi_approx:  3.14159244840275 , absolute error 2.05187041135702e-07 
numerator: 59263 , denominator: 18864 , pi_approx:  3.14159245122986 , absolute error 2.02359937251373e-07 
numerator: 59618 , denominator: 18977 , pi_approx:  3.14159245402329 , absolute error 1.995665015464e-07 
numerator: 59973 , denominator: 19090 , pi_approx:  3.14159245678366 , absolute error 1.96806136720795e-07 
numerator: 60328 , denominator: 19203 , pi_approx:  3.14159245951153 , absolute error 1.94078258353159e-07 
numerator: 60683 , denominator: 19316 , pi_approx:  3.1415924622075 , absolute error 1.91382296677034e-07 
numerator: 61038 , denominator: 19429 , pi_approx:  3.1415924648721 , absolute error 1.88717694804552e-07 
numerator: 61393 , denominator: 19542 , pi_approx:  3.14159246750588 , absolute error 1.86083908282342e-07 
numerator: 61748 , denominator: 19655 , pi_approx:  3.14159247010939 , absolute error 1.83480405979708e-07 
numerator: 62103 , denominator: 19768 , pi_approx:  3.14159247268312 , absolute error 1.80906668756364e-07 
numerator: 62458 , denominator: 19881 , pi_approx:  3.1415924752276 , absolute error 1.78362189018344e-07 
numerator: 62813 , denominator: 19994 , pi_approx:  3.14159247774332 , absolute error 1.7584647027391e-07 
numerator: 63168 , denominator: 20107 , pi_approx:  3.14159248023077 , absolute error 1.73359027577646e-07 
numerator: 63523 , denominator: 20220 , pi_approx:  3.14159248269041 , absolute error 1.70899387530454e-07 
numerator: 63878 , denominator: 20333 , pi_approx:  3.14159248512271 , absolute error 1.6846708605911e-07 
numerator: 64233 , denominator: 20446 , pi_approx:  3.14159248752812 , absolute error 1.66061670192619e-07 
numerator: 64588 , denominator: 20559 , pi_approx:  3.1415924899071 , absolute error 1.63682696285861e-07 
numerator: 64943 , denominator: 20672 , pi_approx:  3.14159249226006 , absolute error 1.61329731351856e-07 
numerator: 65298 , denominator: 20785 , pi_approx:  3.14159249458744 , absolute error 1.5900235039723e-07 
numerator: 65653 , denominator: 20898 , pi_approx:  3.14159249688965 , absolute error 1.56700138642663e-07 
numerator: 66008 , denominator: 21011 , pi_approx:  3.1415924991671 , absolute error 1.54422690190614e-07 
numerator: 66363 , denominator: 21124 , pi_approx:  3.14159250142019 , absolute error 1.52169607581243e-07 
numerator: 66718 , denominator: 21237 , pi_approx:  3.14159250364929 , absolute error 1.499405017924e-07 
numerator: 67073 , denominator: 21350 , pi_approx:  3.1415925058548 , absolute error 1.47734992239634e-07 
numerator: 67428 , denominator: 21463 , pi_approx:  3.14159250803709 , absolute error 1.45552705888008e-07 
numerator: 67783 , denominator: 21576 , pi_approx:  3.14159251019651 , absolute error 1.43393278584369e-07 
numerator: 68138 , denominator: 21689 , pi_approx:  3.14159251233344 , absolute error 1.41256352392816e-07 
numerator: 68493 , denominator: 21802 , pi_approx:  3.14159251444822 , absolute error 1.39141577371049e-07 
numerator: 68848 , denominator: 21915 , pi_approx:  3.14159251654118 , absolute error 1.37048611126289e-07 
numerator: 69203 , denominator: 22028 , pi_approx:  3.14159251861267 , absolute error 1.34977118371182e-07 
numerator: 69558 , denominator: 22141 , pi_approx:  3.14159252066302 , absolute error 1.32926769591535e-07 
numerator: 69913 , denominator: 22254 , pi_approx:  3.14159252269255 , absolute error 1.30897243266759e-07 
numerator: 70268 , denominator: 22367 , pi_approx:  3.14159252470157 , absolute error 1.28888223649426e-07 
numerator: 70623 , denominator: 22480 , pi_approx:  3.14159252669039 , absolute error 1.26899401653446e-07 
numerator: 70978 , denominator: 22593 , pi_approx:  3.14159252865932 , absolute error 1.2493047396589e-07 
numerator: 71333 , denominator: 22706 , pi_approx:  3.14159253060865 , absolute error 1.22981143491074e-07 
numerator: 71688 , denominator: 22819 , pi_approx:  3.14159253253867 , absolute error 1.21051119350568e-07 
numerator: 72043 , denominator: 22932 , pi_approx:  3.14159253444968 , absolute error 1.19140115995009e-07 
numerator: 72398 , denominator: 23045 , pi_approx:  3.14159253634194 , absolute error 1.17247853648195e-07 
numerator: 72753 , denominator: 23158 , pi_approx:  3.14159253821574 , absolute error 1.15374057862994e-07 
numerator: 73108 , denominator: 23271 , pi_approx:  3.14159254007133 , absolute error 1.13518459521345e-07 
numerator: 73463 , denominator: 23384 , pi_approx:  3.141592541909 , absolute error 1.11680795722435e-07 
numerator: 73818 , denominator: 23497 , pi_approx:  3.14159254372899 , absolute error 1.09860806229989e-07 
numerator: 74173 , denominator: 23610 , pi_approx:  3.14159254553155 , absolute error 1.08058238801334e-07 
numerator: 74528 , denominator: 23723 , pi_approx:  3.14159254731695 , absolute error 1.06272843414246e-07 
numerator: 74883 , denominator: 23836 , pi_approx:  3.14159254908542 , absolute error 1.04504376263748e-07 
numerator: 75238 , denominator: 23949 , pi_approx:  3.1415925508372 , absolute error 1.02752597541667e-07 
numerator: 75593 , denominator: 24062 , pi_approx:  3.14159255257252 , absolute error 1.0101727232481e-07 
numerator: 75948 , denominator: 24175 , pi_approx:  3.14159255429162 , absolute error 9.92981696867901e-08 
numerator: 76303 , denominator: 24288 , pi_approx:  3.14159255599473 , absolute error 9.75950631421085e-08 
numerator: 76658 , denominator: 24401 , pi_approx:  3.14159255768206 , absolute error 9.59077310902501e-08 
numerator: 77013 , denominator: 24514 , pi_approx:  3.14159255935384 , absolute error 9.42359545952343e-08 
numerator: 77368 , denominator: 24627 , pi_approx:  3.14159256101027 , absolute error 9.25795200501511e-08 
numerator: 77723 , denominator: 24740 , pi_approx:  3.14159256265158 , absolute error 9.0938216512626e-08 
numerator: 78078 , denominator: 24853 , pi_approx:  3.14159256427795 , absolute error 8.9311838813444e-08 
numerator: 78433 , denominator: 24966 , pi_approx:  3.14159256588961 , absolute error 8.77001831156576e-08 
numerator: 78788 , denominator: 25079 , pi_approx:  3.14159256748674 , absolute error 8.61030513554795e-08 
numerator: 79143 , denominator: 25192 , pi_approx:  3.14159256906955 , absolute error 8.45202472454787e-08 
numerator: 79498 , denominator: 25305 , pi_approx:  3.14159257063821 , absolute error 8.29515793832059e-08 
numerator: 79853 , denominator: 25418 , pi_approx:  3.14159257219293 , absolute error 8.13968590307468e-08 
numerator: 80208 , denominator: 25531 , pi_approx:  3.14159257373389 , absolute error 7.9855901002901e-08 
numerator: 80563 , denominator: 25644 , pi_approx:  3.14159257526127 , absolute error 7.83285232230924e-08 
numerator: 80918 , denominator: 25757 , pi_approx:  3.14159257677525 , absolute error 7.68145476115478e-08 
numerator: 81273 , denominator: 25870 , pi_approx:  3.141592578276 , absolute error 7.5313797864851e-08 
numerator: 81628 , denominator: 25983 , pi_approx:  3.14159257976369 , absolute error 7.38261016763886e-08 
numerator: 81983 , denominator: 26096 , pi_approx:  3.1415925812385 , absolute error 7.23512889599931e-08 
numerator: 82338 , denominator: 26209 , pi_approx:  3.1415925827006 , absolute error 7.08891940703893e-08 
numerator: 82693 , denominator: 26322 , pi_approx:  3.14159258415014 , absolute error 6.94396526945695e-08 
numerator: 83048 , denominator: 26435 , pi_approx:  3.14159258558729 , absolute error 6.80025036281506e-08 
numerator: 83403 , denominator: 26548 , pi_approx:  3.1415925870122 , absolute error 6.65775887753739e-08 
numerator: 83758 , denominator: 26661 , pi_approx:  3.14159258842504 , absolute error 6.51647527050159e-08 
numerator: 84113 , denominator: 26774 , pi_approx:  3.14159258982595 , absolute error 6.37638426503884e-08 
numerator: 84468 , denominator: 26887 , pi_approx:  3.14159259121509 , absolute error 6.237470762116e-08 
numerator: 84823 , denominator: 27000 , pi_approx:  3.14159259259259 , absolute error 6.09972006238024e-08 
numerator: 85178 , denominator: 27113 , pi_approx:  3.14159259395862 , absolute error 5.96311755529655e-08 
numerator: 85533 , denominator: 27226 , pi_approx:  3.1415925953133 , absolute error 5.82764898560129e-08 
numerator: 85888 , denominator: 27339 , pi_approx:  3.14159259665679 , absolute error 5.69330023125758e-08 
numerator: 86243 , denominator: 27452 , pi_approx:  3.14159259798922 , absolute error 5.56005756990885e-08 
numerator: 86598 , denominator: 27565 , pi_approx:  3.14159259931072 , absolute error 5.42790727919851e-08 
numerator: 86953 , denominator: 27678 , pi_approx:  3.14159260062143 , absolute error 5.29683608085918e-08 
numerator: 87308 , denominator: 27791 , pi_approx:  3.14159260192149 , absolute error 5.16683078544133e-08 
numerator: 87663 , denominator: 27904 , pi_approx:  3.14159260321101 , absolute error 5.03787838113112e-08 
numerator: 88018 , denominator: 28017 , pi_approx:  3.14159260449013 , absolute error 4.90996621138606e-08 
numerator: 88373 , denominator: 28130 , pi_approx:  3.14159260575898 , absolute error 4.78308170848152e-08 
numerator: 88728 , denominator: 28243 , pi_approx:  3.14159260701767 , absolute error 4.65721248232853e-08 
numerator: 89083 , denominator: 28356 , pi_approx:  3.14159260826633 , absolute error 4.53234649810952e-08 
numerator: 89438 , denominator: 28469 , pi_approx:  3.14159260950508 , absolute error 4.40847172100689e-08 
numerator: 89793 , denominator: 28582 , pi_approx:  3.14159261073403 , absolute error 4.28557647147443e-08 
numerator: 90148 , denominator: 28695 , pi_approx:  3.1415926119533 , absolute error 4.16364911437483e-08 
numerator: 90503 , denominator: 28808 , pi_approx:  3.14159261316301 , absolute error 4.04267828102434e-08 
numerator: 90858 , denominator: 28921 , pi_approx:  3.14159261436327 , absolute error 3.92265278037485e-08 
numerator: 91213 , denominator: 29034 , pi_approx:  3.14159261555418 , absolute error 3.80356151019612e-08 
numerator: 91568 , denominator: 29147 , pi_approx:  3.14159261673586 , absolute error 3.68539367912035e-08 
numerator: 91923 , denominator: 29260 , pi_approx:  3.14159261790841 , absolute error 3.56813858459759e-08 
numerator: 92278 , denominator: 29373 , pi_approx:  3.14159261907194 , absolute error 3.45178561289572e-08 
numerator: 92633 , denominator: 29486 , pi_approx:  3.14159262022655 , absolute error 3.33632450555399e-08 
numerator: 92988 , denominator: 29599 , pi_approx:  3.14159262137234 , absolute error 3.22174495970273e-08 
numerator: 93343 , denominator: 29712 , pi_approx:  3.14159262250942 , absolute error 3.10803693892581e-08 
numerator: 93698 , denominator: 29825 , pi_approx:  3.14159262363789 , absolute error 2.99519054003383e-08 
numerator: 94053 , denominator: 29938 , pi_approx:  3.14159262475783 , absolute error 2.88319603747311e-08 
numerator: 94408 , denominator: 30051 , pi_approx:  3.14159262586936 , absolute error 2.77204375009887e-08 
numerator: 94763 , denominator: 30164 , pi_approx:  3.14159262697255 , absolute error 2.66172430762879e-08 
numerator: 95118 , denominator: 30277 , pi_approx:  3.14159262806751 , absolute error 2.55222829537161e-08 
numerator: 95473 , denominator: 30390 , pi_approx:  3.14159262915433 , absolute error 2.44354660949853e-08 
numerator: 95828 , denominator: 30503 , pi_approx:  3.14159263023309 , absolute error 2.33567014618075e-08 
numerator: 96183 , denominator: 30616 , pi_approx:  3.14159263130389 , absolute error 2.22858997922515e-08 
numerator: 96538 , denominator: 30729 , pi_approx:  3.14159263236682 , absolute error 2.1222973600743e-08 
numerator: 96893 , denominator: 30842 , pi_approx:  3.14159263342196 , absolute error 2.01678358457968e-08 
numerator: 97248 , denominator: 30955 , pi_approx:  3.14159263446939 , absolute error 1.91204021504632e-08 
numerator: 97603 , denominator: 31068 , pi_approx:  3.14159263550921 , absolute error 1.80805876937029e-08 
numerator: 97958 , denominator: 31181 , pi_approx:  3.14159263654148 , absolute error 1.70483094308338e-08 
numerator: 98313 , denominator: 31294 , pi_approx:  3.14159263756631 , absolute error 1.60234865376196e-08 
numerator: 98668 , denominator: 31407 , pi_approx:  3.14159263858376 , absolute error 1.50060377457351e-08 
numerator: 99023 , denominator: 31520 , pi_approx:  3.14159263959391 , absolute error 1.39958844513899e-08 
numerator: 99378 , denominator: 31633 , pi_approx:  3.14159264059685 , absolute error 1.2992948050794e-08 
numerator: 99733 , denominator: 31746 , pi_approx:  3.14159264159264 , absolute error 1.19971517165141e-08 
numerator: 100088 , denominator: 31859 , pi_approx:  3.14159264258137 , absolute error 1.1008419065206e-08 
numerator: 100443 , denominator: 31972 , pi_approx:  3.14159264356312 , absolute error 1.00266754898826e-08 
numerator: 100798 , denominator: 32085 , pi_approx:  3.14159264453795 , absolute error 9.0518468276457e-09 
numerator: 101153 , denominator: 32198 , pi_approx:  3.14159264550593 , absolute error 8.08386113604342e-09 
numerator: 101508 , denominator: 32311 , pi_approx:  3.14159264646715 , absolute error 7.12264558444531e-09 
numerator: 101863 , denominator: 32424 , pi_approx:  3.14159264742166 , absolute error 6.1681300067562e-09 
numerator: 102218 , denominator: 32537 , pi_approx:  3.14159264836955 , absolute error 5.22024468097015e-09 
numerator: 102573 , denominator: 32650 , pi_approx:  3.14159264931087 , absolute error 4.27892032917043e-09 
numerator: 102928 , denominator: 32763 , pi_approx:  3.1415926502457 , absolute error 3.34408900570793e-09 
numerator: 103283 , denominator: 32876 , pi_approx:  3.14159265117411 , absolute error 2.41568454129037e-09 
numerator: 103638 , denominator: 32989 , pi_approx:  3.14159265209615 , absolute error 1.49363987844708e-09 
numerator: 103993 , denominator: 33102 , pi_approx:  3.1415926530119 , absolute error 5.77890624242627e-10 
numerator: 104348 , denominator: 33215 , pi_approx:  3.14159265392142 , absolute error 3.31628058347633e-10 
numerator: 208341 , denominator: 66317 , pi_approx:  3.14159265346744 , absolute error 1.22356347276309e-10 
numerator: 312689 , denominator: 99532 , pi_approx:  3.14159265361894 , absolute error 2.91433543964104e-11 
numerator: 833719 , denominator: 265381 , pi_approx:  3.14159265358108 , absolute error 8.71525074330748e-12 
numerator: 1146408 , denominator: 364913 , pi_approx:  3.1415926535914 , absolute error 1.61071156412618e-12 
numerator: 3126535 , denominator: 995207 , pi_approx:  3.14159265358865 , absolute error 1.14264153694421e-12 
numerator: 4272943 , denominator: 1360120 , pi_approx:  3.14159265358939 , absolute error 4.04121180963557e-13 
numerator: 5419351 , denominator: 1725033 , pi_approx:  3.14159265358982 , absolute error 2.22044604925031e-14