def find_root(a, b, tol, f):
    """prec: [a,b] is an interval and f is continuous on [a,b]
    tol is a positive tolerance for error.
    f changes sign on [a,b]
    returns a value in the interval that is within tol of the actual root."""
    while abs(a - b) > 2*tol:
        mid = (b + a)/2
        if f(mid)*f(a) < 0:
            b = mid
        else:
            a = mid
    return (b + a)/2
        
a = find_root(1, 2, .000001, lambda x: x*x - 2)
b = 2**.5
print(f"error = {abs(a - b)}")