Hmm. isqrt_frac_rto is unnecessarily complicated. Here's a more streamlined version of the code.


import math

def isqrt_frac_rto(n, m):
    """
    Square root of n/m, rounded to the nearest integer using round-to-odd.
    """
    a = math.isqrt(n*m) // m
    return a | (a*a*m != n)

def sqrt_frac(n, m):
    """
    Square root of n/m as a float, correctly rounded.
    """
    q = (n.bit_length() - m.bit_length() - 109) // 2
    if q >= 0:
        return float(isqrt_frac_rto(n, m << 2 * q) << q)
    else:
        return isqrt_frac_rto(n << -2 * q, m) / (1 << -q)