Carlson arcsin demo. Calculates pi.

carlson_asin.py

""" pi from 10 * asin((phi-1)/2)

    Uses Carlson's accelerated version of
    Borchardt's modified AGM algorithm
    Written by PM 2Ring 2022.05.11
"""

from itertools import count
from decimal import Decimal as D, getcontext

def agm_asin(x):
    ctx = getcontext()
    eps = 16 / D(10) ** ctx.prec
    ctx.prec += 2
    b = one = D(1)
    a = (one - x*x).sqrt()
    g = one
    d = [a]
    for i in count(1):
        a = (a + g) / 2
        pd, d = d, [a]
        q = one
        for u in pd:
            q /= 4
            d.append(d[-1] - u * q)
        r = one - q
        b *= r
        y = b * x / d[-1]; print(f"{i:2}: {y}")
        if abs(pd[-1] * r - d[-1]) < eps:
            break
        g = (a * g).sqrt()
    #y = b * x / d[-1]
    ctx.prec -= 2
    return +y

ctx = getcontext()
ctx.prec = 50
x = (D(5).sqrt() - 1) / 4
y = agm_asin(x)
print(10 * y)

Live demo running on SageMathCell.