"""Newton's Method."""

# Newton's Method - https://en.wikipedia.org/wiki/Newton%27s_method
from typing import Callable

RealFunc = Callable[[float], float] # type alias for a real -> real function


# function is the f(x) and derivative is the f'(x)
def newton(
 function: RealFunc,
 derivative: RealFunc,
 starting_int: int,
) -> float:
 """
 >>> newton(lambda x: x ** 3 - 2 * x - 5, lambda x: 3 * x ** 2 - 2, 3)
 2.0945514815423474
 >>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -2)
 1.0
 >>> newton(lambda x: x ** 3 - 1, lambda x: 3 * x ** 2, -4)
 1.0000000000000102
 >>> import math
 >>> newton(math.sin, math.cos, 1)
 0.0
 >>> newton(math.sin, math.cos, 2)
 3.141592653589793
 >>> newton(math.cos, lambda x: -math.sin(x), 2)
 1.5707963267948966
 >>> newton(math.cos, lambda x: -math.sin(x), 0)
 Traceback (most recent call last):
 ...
 ZeroDivisionError: Could not find root
 """
 prev_guess = float(starting_int)
 while True:
 try:
 next_guess = prev_guess - function(prev_guess) / derivative(prev_guess)
 except ZeroDivisionError:
 raise ZeroDivisionError("Could not find root") from None
 if abs(prev_guess - next_guess) < 10 ** -5:
 return next_guess
 prev_guess = next_guess


def f(x: float) -> float:
 return (x ** 3) - (2 * x) - 5


def f1(x: float) -> float:
 return 3 * (x ** 2) - 2


if __name__ == "__main__":
 print(newton(f, f1, 3))