A quick python function that can approximate pi "quickly" using the method outlined in https://www.youtube.com/watch?v=RZBhSi_PwHU

approxPi.py

import gmpy2 as g

def approxPi(randomState, numTrials=5000, randbits=1024, prec=200):
    """Compute one approximation of pi using pairs of random numbers."""
    with g.local_context(g.get_context()) as ctx:
        ctx.precision = prec
        coprime = g.mpz(0)
        for i in range(numTrials):
            a = g.mpz_urandomb(randomState, randbits)+1
            b = g.mpz_urandomb(randomState, randbits)+1
            if g.gcd(a, b) == 1:
                coprime = coprime + 1
        probabilityCoprime = g.mpfr(coprime)/numTrials
        return g.sqrt(6/probabilityCoprime)
rs = g.random_state()

[approxPi(rs, numTrials=50000, prec=80) for i in range(10)]

"""
In my very brief experimentation, 50k trials are required with numbers in range
0 to 2*1024-1 before you reliably get 3.13 or 3.14 out. With that many trials,
Python bogs down. I'm sure there's a way to speed this up, and I'm sure I don't
know it readily, given the logic involved.
"""