Several algorithms for arbitrary precision calculation of pi. Plus, how could I pass up that name? (Edit: Old name was taken by the package index. It doesn't apply anymore, anyway.) Based on Nick Craig-Wood's python implementations, but with my own gmpy2 and C++ changes. Includes an experimental timing program to verify certain functions. Rewritten in C++ and with gmp for speed.

To do:

• Better documentation
• Multi-threading: a possibility?
• Code profiling

Details in the code.

Machin: Uses the formula pi/4 = 4*arctan(1/5) - arctan(1/239). Features gmpy2's hardware level algorithm and an accelerated formula by Euler for arctan.

Chudnovsky: Uses a very fast rapidly convergent algorithm. Includes a regular implementation with gmpy2 and a binary-splitting method with gmpy2.

Gauss-Legendre: Uses a second order convergence algorithm based on the arithmatic-geometric mean. Only log2(n) iterations are needed.

Borwein: Just for fun, PyPi includes Borwein's 1984 algorithm with quartic convergence. Not as fast as Gauss-Legendre though, probably because of the 4 divisions per iteration.

Python

Digits >>>          10,000        100,000        1,000,000    10,000,000    100,000,000
---------------------------------------------------------------------------------------
Machin gmpy2        0.06 s        1.7 s          30.5 s       542.9 s       -
Machin Taylor       0.4 s         38.9 s         -            -             -
Machin Euler        0.3 s         34.4 s         -            -             -
Chudnovksy regular  0.01 s        0.7 s          79.7 s       -             -
Chudnovsky BS       0.004 s       0.1 s          1.6 s        28.9 s        432.3 s
Gauss-Legendre      0.01 s        0.4 s          5.8 s        101.5 s       -
Borwein             0.05 s        1.4 s          22.6 s       327.8 s       -

C++

Digits >>>          10,000        100,000        1,000,000    10,000,000    100,000,000
---------------------------------------------------------------------------------------
Chudnovsky BS       0.03 s        0.09 s         1.3 s        21.2 s        331.1 s

###Speed Optimizations (Python)

• Replaced gmpy2.log2 with math.log2 (less precision)
• Replaced gmpy2 integer square root multiplied by scale**2 with mpfr square root
  • 33% less time
• Use mpz instead of int exponentiation for scale
  • 50% less time