rational_approx.py

import math
from operator import itemgetter

t = math.pi
N = 1000

def find_bounds(d):
    p = d*t
    m = int(p)
    n = m + 1
    assert n/d > t > m/d
    return (m,n)

rL = list()
for d in range(3,N):
    m,n = find_bounds(d)
    if math.gcd(m,d) == 1:
        e = m/d - t
        rL.append((m,d,e,abs(e)))
    if math.gcd(n,d) == 1:
        e = n/d-t
        rL.append((n,d,e,abs(e)))
        
def show(rL, j=len(rL)):
    i = 0
    for n,d,e,a in rL[:j]:
        i += 1
        print('%5d:  %5d %5d  %11.8f' % (i,n,d,e))
    
rL.sort(key=itemgetter(3))
show(rL,j=48)

'''
> p3 rational_approx.py
    1:    355   113   0.00000027
    2:   2818   897  -0.00000960
    3:   2862   911   0.00000998
    4:   2463   784  -0.00001102
    5:   2507   798   0.00001136
    6:   2108   671  -0.00001292
    7:   2152   685   0.00001319
    8:   1753   558  -0.00001559
    9:   1797   572   0.00001574
   10:   1442   459   0.00001955
   11:   1398   445  -0.00001962
   12:   2529   805   0.00002225
   13:   2441   777  -0.00002251
   14:   1087   346   0.00002584
   15:   1043   332  -0.00002639
   16:   2906   925   0.00002897
   17:   2774   883  -0.00002980
   18:   1819   579   0.00003084
   19:   1731   551  -0.00003186
   20:   2551   812   0.00003296
   21:   2419   770  -0.00003421
   22:   3107   989  -0.00003553
   23:    732   233   0.00003825
   24:    688   219  -0.00004014
   25:   2573   819   0.00004349
   26:   3085   982  -0.00004479
   27:   1841   586   0.00004557
   28:   2397   763  -0.00004613
   29:   2950   939   0.00004739
   30:   1709   544  -0.00004854
   31:   1109   353   0.00005041
   32:   2730   869  -0.00005065
   33:   2595   826   0.00005384
   34:   1021   325  -0.00005419
   35:   1486   473   0.00005640
   36:   2375   756  -0.00005826
   37:   1863   593   0.00005996
   38:   1354   431  -0.00006133
   39:   2240   713   0.00006233
   40:   3041   968  -0.00006373
   41:   2617   833   0.00006401
   42:   2994   953   0.00006527
   43:   1687   537  -0.00006565
   44:   2020   643  -0.00006855
   45:   2353   749  -0.00007062
   46:   2686   855  -0.00007219
   47:   3019   961  -0.00007340
   48:    377   120   0.00007401
>
'''