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 -\frac{\log|p/q-\pi|}{\log p}*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