This repository contains programs which can be used to compute binary or hex digits from pi without calculating the previous digits and programs to calculate pi using them.
They use two formulae which were discovered by Euler in 1779 and compare them to the original formula used by Bailey, Borwein and Plouffe in 1997.
A paper about this was published in Eulerania Journal, Vol 3, Issue 1 in March 2023.
- euler_bbp.py - simple program to calculate hex digits of pi
- euler_bbp.go - faster (100x) program to do the same
- check_bbp_equations.py - program to check the equations in the paper work
Source for the paper:
$ python euler_bbp.py
bbp_original hex digits of pi from digit 10 5a308d31319 in 0.000s
bbp_euler hex digits of pi from digit 10 5a308d31319 in 0.000s
bbp_euler2 hex digits of pi from digit 10 5a308d313198 in 0.000s
bbp_original hex digits of pi from digit 100 c29b7c97c50 in 0.000s
bbp_euler hex digits of pi from digit 100 c29b7c97c50 in 0.001s
bbp_euler2 hex digits of pi from digit 100 c29b7c97c50 in 0.000s
bbp_original hex digits of pi from digit 1000 349f1c09b0 in 0.004s
bbp_euler hex digits of pi from digit 1000 349f1c09b0 in 0.005s
bbp_euler2 hex digits of pi from digit 1000 349f1c09b0 in 0.002s
bbp_original hex digits of pi from digit 10000 68ac8fcfb in 0.030s
bbp_euler hex digits of pi from digit 10000 68ac8fcfb in 0.046s
bbp_euler2 hex digits of pi from digit 10000 68ac8fcfb in 0.019s
bbp_original hex digits of pi from digit 100000 535ea16c in 0.340s
bbp_euler hex digits of pi from digit 100000 535ea16c in 0.536s
bbp_euler2 hex digits of pi from digit 100000 535ea16c in 0.222s
bbp_original hex digits of pi from digit 1000000 26c65e52 in 3.885s
bbp_euler hex digits of pi from digit 1000000 26c65e5 in 6.115s
bbp_euler2 hex digits of pi from digit 1000000 26c65e52 in 2.627s
bbp_original hex digits of pi from digit 10000000 17af586 in 42.293s
bbp_euler hex digits of pi from digit 10000000 17af586 in 66.215s
bbp_euler2 hex digits of pi from digit 10000000 17af586 in 29.003s
bbp_original hex digits of pi from digit 100000000 ecb840 in 463.304s
bbp_euler hex digits of pi from digit 100000000 ecb840 in 752.472s
bbp_euler2 hex digits of pi from digit 100000000 ecb840 in 327.756s
bbp_original hex digits of pi from digit 1000000000 85895 in 9379.175s
$ go run euler_bbp.go
original hex digits of pi from digit 10 5a308d31319 in 26.87µs
euler hex digits of pi from digit 10 5a308d31319 in 9.498µs
euler2 hex digits of pi from digit 10 5a308d313198 in 18.084µs
original hex digits of pi from digit 100 c29b7c97c50 in 19.587µs
euler hex digits of pi from digit 100 c29b7c97c50 in 32.26µs
euler2 hex digits of pi from digit 100 c29b7c97c50 in 27.682µs
original hex digits of pi from digit 1000 349f1c09b0 in 248.003µs
euler hex digits of pi from digit 1000 349f1c09b0 in 369.24µs
euler2 hex digits of pi from digit 1000 349f1c09b0 in 131.676µs
original hex digits of pi from digit 10000 68ac8fcfb in 2.174754ms
euler hex digits of pi from digit 10000 68ac8fcfb in 3.45151ms
euler2 hex digits of pi from digit 10000 68ac8fcfb in 1.37503ms
original hex digits of pi from digit 100000 535ea16c in 26.643839ms
euler hex digits of pi from digit 100000 535ea16c in 20.853328ms
euler2 hex digits of pi from digit 100000 535ea16c in 16.504332ms
original hex digits of pi from digit 1000000 26c65e52 in 55.597885ms
euler hex digits of pi from digit 1000000 26c65e5 in 75.05774ms
euler2 hex digits of pi from digit 1000000 26c65e52 in 160.25139ms
original hex digits of pi from digit 10000000 17af586 in 480.289261ms
euler hex digits of pi from digit 10000000 17af586 in 698.296118ms
euler2 hex digits of pi from digit 10000000 17af586 in 362.285909ms
original hex digits of pi from digit 100000000 ecb840 in 5.287894989s
euler hex digits of pi from digit 100000000 ecb840 in 8.886897248s
euler2 hex digits of pi from digit 100000000 ecb840 in 4.047852407s
original hex digits of pi from digit 1000000000 85895 in 1m6.804017733s
euler hex digits of pi from digit 1000000000 85895 in 1m41.747295359s
euler2 hex digits of pi from digit 1000000000 85895 in 44.864847315s
original hex digits of pi from digit 10000000000 921c in 13m16.405684459s
euler hex digits of pi from digit 10000000000 921c in 19m46.698153514s
euler2 hex digits of pi from digit 10000000000 921c in 8m36.587514529s
original hex digits of pi from digit 100000000000 c9c in 2h28m30.473717759s
euler hex digits of pi from digit 100000000000 c9c in 3h43m8.795869344s
euler2 hex digits of pi from digit 100000000000 c9c in 1h36m58.833018519s
