円周率に近い分数
1/1 からスタートして、分数が円周率より小さければ分子を、大きければ分母を増やしていく、それまでで最も円周率に近い分数が出たら表示、といった感じで分母100万まで計算してみた。
分数 p/q、円周率との誤差、近似の優秀さとしてでっち上げた式 *1の値を並べたら下の表になった。円周率≒3 はもっと評価されていいと思った。
1/1 -2.141593e+00 -inf 2/1 -1.141593e+00 -0.191048 3/1 -1.415927e-01 1.779337 13/4 1.084073e-01 0.866239 16/5 5.840735e-02 1.024427 19/6 2.507401e-02 1.251825 22/7 1.264489e-03 2.158847 179/57 -1.241776e-03 1.289901 201/64 -9.676536e-04 1.308738 223/71 -7.475832e-04 1.331318 245/78 -5.670126e-04 1.358804 267/85 -4.161830e-04 1.393241 289/92 -2.883058e-04 1.438559 311/99 -1.785122e-04 1.503687 333/106 -8.321963e-05 1.617389 355/113 2.667642e-07 2.577758 52163/16604 -2.662133e-07 1.393739 52518/16717 -2.626106e-07 1.394122 52873/16830 -2.590562e-07 1.394512 53228/16943 -2.555493e-07 1.394907 53583/17056 -2.520889e-07 1.395307 53938/17169 -2.486740e-07 1.395714 54293/17282 -2.453037e-07 1.396125 54648/17395 -2.419773e-07 1.396543 55003/17508 -2.386937e-07 1.396966 55358/17621 -2.354523e-07 1.397395 55713/17734 -2.322522e-07 1.397830 56068/17847 -2.290927e-07 1.398271 56423/17960 -2.259729e-07 1.398717 56778/18073 -2.228921e-07 1.399170 57133/18186 -2.198496e-07 1.399628 57488/18299 -2.168446e-07 1.400093 57843/18412 -2.138766e-07 1.400564 58198/18525 -2.109447e-07 1.401041 58553/18638 -2.080484e-07 1.401524 58908/18751 -2.051870e-07 1.402014 59263/18864 -2.023599e-07 1.402510 59618/18977 -1.995665e-07 1.403012 59973/19090 -1.968061e-07 1.403521 60328/19203 -1.940783e-07 1.404037 60683/19316 -1.913823e-07 1.404559 61038/19429 -1.887177e-07 1.405088 61393/19542 -1.860839e-07 1.405623 61748/19655 -1.834804e-07 1.406166 62103/19768 -1.809067e-07 1.406715 62458/19881 -1.783622e-07 1.407272 62813/19994 -1.758465e-07 1.407836 63168/20107 -1.733590e-07 1.408407 63523/20220 -1.708994e-07 1.408985 63878/20333 -1.684671e-07 1.409571 64233/20446 -1.660617e-07 1.410165 64588/20559 -1.636827e-07 1.410766 64943/20672 -1.613297e-07 1.411375 65298/20785 -1.590024e-07 1.411991 65653/20898 -1.567001e-07 1.412616 66008/21011 -1.544227e-07 1.413249 66363/21124 -1.521696e-07 1.413890 66718/21237 -1.499405e-07 1.414539 67073/21350 -1.477350e-07 1.415197 67428/21463 -1.455527e-07 1.415864 67783/21576 -1.433933e-07 1.416539 68138/21689 -1.412564e-07 1.417223 68493/21802 -1.391416e-07 1.417917 68848/21915 -1.370486e-07 1.418619 69203/22028 -1.349771e-07 1.419331 69558/22141 -1.329268e-07 1.420053 69913/22254 -1.308972e-07 1.420784 70268/22367 -1.288882e-07 1.421525 70623/22480 -1.268994e-07 1.422276 70978/22593 -1.249305e-07 1.423038 71333/22706 -1.229811e-07 1.423810 71688/22819 -1.210511e-07 1.424592 72043/22932 -1.191401e-07 1.425386 72398/23045 -1.172479e-07 1.426190 72753/23158 -1.153741e-07 1.427006 73108/23271 -1.135185e-07 1.427834 73463/23384 -1.116808e-07 1.428673 73818/23497 -1.098608e-07 1.429524 74173/23610 -1.080582e-07 1.430388 74528/23723 -1.062728e-07 1.431264 74883/23836 -1.045044e-07 1.432154 75238/23949 -1.027526e-07 1.433056 75593/24062 -1.010173e-07 1.433972 75948/24175 -9.929817e-08 1.434901 76303/24288 -9.759506e-08 1.435845 76658/24401 -9.590773e-08 1.436803 77013/24514 -9.423595e-08 1.437776 77368/24627 -9.257952e-08 1.438764 77723/24740 -9.093822e-08 1.439767 78078/24853 -8.931184e-08 1.440787 78433/24966 -8.770018e-08 1.441823 78788/25079 -8.610305e-08 1.442875 79143/25192 -8.452025e-08 1.443945 79498/25305 -8.295158e-08 1.445033 79853/25418 -8.139686e-08 1.446139 80208/25531 -7.985590e-08 1.447263 80563/25644 -7.832852e-08 1.448407 80918/25757 -7.681455e-08 1.449570 81273/25870 -7.531380e-08 1.450754 81628/25983 -7.382610e-08 1.451959 81983/26096 -7.235129e-08 1.453186 82338/26209 -7.088919e-08 1.454435 82693/26322 -6.943965e-08 1.455707 83048/26435 -6.800250e-08 1.457003 83403/26548 -6.657759e-08 1.458323 83758/26661 -6.516475e-08 1.459669 84113/26774 -6.376384e-08 1.461041 84468/26887 -6.237471e-08 1.462440 84823/27000 -6.099720e-08 1.463867 85178/27113 -5.963118e-08 1.465324 85533/27226 -5.827649e-08 1.466811 85888/27339 -5.693300e-08 1.468329 86243/27452 -5.560058e-08 1.469880 86598/27565 -5.427907e-08 1.471465 86953/27678 -5.296836e-08 1.473084 87308/27791 -5.166831e-08 1.474741 87663/27904 -5.037878e-08 1.476436 88018/28017 -4.909966e-08 1.478171 88373/28130 -4.783082e-08 1.479947 88728/28243 -4.657212e-08 1.481767 89083/28356 -4.532346e-08 1.483633 89438/28469 -4.408472e-08 1.485546 89793/28582 -4.285576e-08 1.487509 90148/28695 -4.163649e-08 1.489524 90503/28808 -4.042678e-08 1.491594 90858/28921 -3.922653e-08 1.493723 91213/29034 -3.803562e-08 1.495912 91568/29147 -3.685394e-08 1.498166 91923/29260 -3.568139e-08 1.500488 92278/29373 -3.451786e-08 1.502882 92633/29486 -3.336325e-08 1.505352 92988/29599 -3.221745e-08 1.507904 93343/29712 -3.108037e-08 1.510541 93698/29825 -2.995191e-08 1.513271 94053/29938 -2.883196e-08 1.516099 94408/30051 -2.772044e-08 1.519032 94763/30164 -2.661724e-08 1.522079 95118/30277 -2.552228e-08 1.525247 95473/30390 -2.443547e-08 1.528547 95828/30503 -2.335670e-08 1.531988 96183/30616 -2.228590e-08 1.535585 96538/30729 -2.122297e-08 1.539350 96893/30842 -2.016784e-08 1.543299 97248/30955 -1.912040e-08 1.547451 97603/31068 -1.808059e-08 1.551828 97958/31181 -1.704831e-08 1.556453 98313/31294 -1.602349e-08 1.561356 98668/31407 -1.500604e-08 1.566571 99023/31520 -1.399588e-08 1.572141 99378/31633 -1.299295e-08 1.578114 99733/31746 -1.199715e-08 1.584552 100088/31859 -1.100842e-08 1.591533 100443/31972 -1.002668e-08 1.599155 100798/32085 -9.051847e-09 1.607543 101153/32198 -8.083861e-09 1.616866 101508/32311 -7.122646e-09 1.627356 101863/32424 -6.168130e-09 1.639341 102218/32537 -5.220245e-09 1.653311 102573/32650 -4.278920e-09 1.670048 102928/32763 -3.344089e-09 1.690906 103283/32876 -2.415685e-09 1.718570 103638/32989 -1.493640e-09 1.759689 103993/33102 -5.778906e-10 1.841369 104348/33215 3.316281e-10 1.888887 208341/66317 -1.223563e-10 1.863657 312689/99532 2.914335e-11 1.917242 833719/265381 -8.715251e-12 1.867874 1146408/364913 1.610712e-12 1.946249 3126535/995207 -1.142642e-12 1.838641
注意: 倍精度浮動小数点数で計算したので、誤差と近似の優秀さの精度は下の方ではやや低くなってます。
*1:合ってる桁数/分母の桁数 的な何か