Calculating Pi with fractions

\(\pi = \frac{4}{1+\frac{1^2}{2 + recursion(n)}}\)

import numpy as np


def frac_one(n: int, i: int = 0):
    i += 1
    n -= 1
    nominator = 1 + 2 * i
    if n == 0:
        return 1
    else:
        return 2 + (1 + 2 * i) ** 2 / frac_one(n, i)


def frac_inf(n: int, i: int = 0):
    i += 1
    n -= 1
    nominator = 1 + 2 * i
    if n == 0:
        return np.inf
    else:
        return 2 + (1 + 2 * i) ** 2 / frac_inf(n, i)

def pi_approx(func, n_iters):
    return 4 / (1 + 1 / func(n_iters))

def pi_mixed(n_iters):
    a = pi_approx(frac_one, n_iters)
    b = pi_approx(frac_inf, n_iters)
    return (a + b) / 2

# let's use our laptop's recursion limit to get maximum accuracy
import sys

max_recursion = int(sys.getrecursionlimit() * 0.98)
max_recursion

2940

pi_approx(frac_one, max_recursion)

3.141932789654053

pi_approx(frac_inf, max_recursion)

3.141252517545209

print(
    f"""
At recursion depth of {max_recursion} the approximation error is:
 * {(np.pi - pi_approx(frac_one, max_recursion))/np.pi:.3%} for frac_one
 * {(np.pi - pi_approx(frac_inf, max_recursion))/np.pi:.3%} for frac_inf
 * {(np.pi - pi_mixed(max_recursion))/np.pi:.12%} for pi_mixed
"""
)

At recursion depth of 2940 the approximation error is:
 * -0.011% for frac_one
 * 0.011% for frac_inf
 * -0.000000000313% for pi_mixed