Pi-Pourri

Several formulae for calculating 100 million digits of Pi in 3 minutes, using python and GMPY2

I wanted to see how long it would take to calulate pi to a million places to answer a kid's question

I found a page https://medium.com/@cosinekitty/how-to-calculate-a-million-digits-of-pi-d62ce3db8f58 that had a program using Machin's formula from 1706:

${\frac {\pi }{4}}=4\arctan {\frac {1}{5}}-\arctan {\frac {1}{239}}$

The article also included a link to several Machin-like formulae: https://en.wikipedia.org/wiki/Machin-like_formula
I added several to the code I also used GMPY's mpfr() and mpz types to speed things along instead of just big integer support in python The GMPY2 library has a wide range of high precision functions.

I added 7 Machin formulae like:

${\begin{aligned}{\frac {\pi }{4}}=&\;183\arctan {\frac {1}{239}}+32\arctan {\frac {1}{1023}}-68\arctan {\frac {1}{5832}}\\&+12\arctan {\frac {1}{110443}}-12\arctan {\frac {1}{4841182}}-100\arctan {\frac {1}{6826318}}\\\end{aligned}}$

Then I added Chudnovsky:

$\begin{align} a &= \sum^\infty_{k=0} \frac{(-1)^k (6k)!}{(3k)!(k!)^3 640320^{3k}} \\ &= 1 - \frac{6\cdot5\cdot4}{(1)^3 640320^3} + \frac{12\cdot11\cdot10\cdot9\cdot8\cdot7}{(2\cdot1)^3 640320^6} - \frac{18\cdot17\cdots13}{(3\cdot2\cdot1)^3 640320^{9}} + \cdots \\ b &= \sum^\infty_{k=0} \frac{(-1)^k (6k)!k}{(3k)!(k!)^3 640320^{3k}} \\ \frac{1}{\pi} &= \frac{13591409a + 545140134b}{426880 \sqrt{10005}} \\ \pi &= \frac{426880 \sqrt{10005}}{13591409a + 545140134b} \end{align}$

Finaly I added Arithmatic Geometric Mean

Here is the help for the program type python3 pi-pourri.py -h to see it

usage: pi-pourri.py [-h] [-f [FILENAME]] [-d [1 to 100,000,000]] [-a [1 to 10]]

 pi-pourri.py runs an algoritym from a list to calulate Pi to a number of decimal places
      Default: pi-pourri.py --digits 100000 --file pi.txt --alog 4

      So -d 100,000,000 will take a while to finish, -d 1,000,000 very quickly
      A last 5 digit check is done on powers of ten (10,...10000000,100000000)

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Pi-Pourri