def find_root(a, b, tol, f):
    """prec:  the function f is continuous and 
    changes sign on [a,b].
    postc: returns a value that is within tol of 
    a root of f."""
    while b - a > tol:
        mid = (a + b)/2
        if f(a)*f(mid) < 0:  # change sign on [a, mid]
            b = mid
        elif f(mid)*f(b) <= 0: # change signon [mid, b]
            a = mid
    return mid

approximate = find_root(1, 2, .00001, lambda x: x*x - 2)
gold_standard = 2**(.5)
print(approximate - gold_standard)
print(.00001)