The Salamin / Brent / Gauss Arithmetic-Geometric Mean pi formula, using fixed point arithmetic

AGM_pi.py

""" Gauss / Salamin / Brent true AGM pi
    Binary fixed point   
    Written by PM 2Ring 2023.11.1
"""

from math import isqrt

def AGM_pi(loops, bits):
    a, b = 1 << bits, isqrt(1 << 2*bits-1)
    s = 1 << bits-2
    for i in range(loops):
        c = (a - b) >> 1
        s -= c * c >> bits-i
        a, b = (a + b) >> 1, isqrt(a * b)
    return a * a // s

# Tested upto 18 loops
loops = 11
print(f"{loops=}")
bits = ((161139 << loops) - 165914) // 35553
print(f"{bits=}")

digits = bits * 30103 // 100000
print(f"{digits=}")

prec = bits + 8 + loops
print(f"{prec=}")

mypi = AGM_pi(loops, prec) * 10**digits >> prec
#print(mypi)
   
mypi = str(mypi)[1:]
print("pi ~=\n3.")
for i in range(0, len(mypi), 5):
    ii = i + 5
    j = 0 if ii % 25 else 1 if ii % 100 else 2 if ii % 1000 else 3
    print(mypi[i:ii], end=j*"\n" or " ")

Version using the decimal module: https://stackoverflow.com/a/26478803/4014959
Live SageMathCell version

bits = (2**loops * pi - log(8*pi)) / log(2)
See (PDF) Jonathan Borwein, Pi and the AGM by Richard P. Brent.