from typing import Callable


def bisection(function: Callable[[float], float], a: float, b: float) -> float:
 """
 finds where function becomes 0 in [a,b] using bolzano
 >>> bisection(lambda x: x ** 3 - 1, -5, 5)
 1.0000000149011612
 >>> bisection(lambda x: x ** 3 - 1, 2, 1000)
 Traceback (most recent call last):
 ...
 ValueError: could not find root in given interval.
 >>> bisection(lambda x: x ** 2 - 4 * x + 3, 0, 2)
 1.0
 >>> bisection(lambda x: x ** 2 - 4 * x + 3, 2, 4)
 3.0
 >>> bisection(lambda x: x ** 2 - 4 * x + 3, 4, 1000)
 Traceback (most recent call last):
 ...
 ValueError: could not find root in given interval.
 """
 start: float = a
 end: float = b
 if function(a) == 0: # one of the a or b is a root for the function
 return a
 elif function(b) == 0:
 return b
 elif (
 function(a) * function(b) > 0
 ): # if none of these are root and they are both positive or negative,
 # then this algorithm can't find the root
 raise ValueError("could not find root in given interval.")
 else:
 mid: float = start + (end - start) / 2.0
 while abs(start - mid) > 10 ** -7: # until precisely equals to 10^-7
 if function(mid) == 0:
 return mid
 elif function(mid) * function(start) < 0:
 end = mid
 else:
 start = mid
 mid = start + (end - start) / 2.0
 return mid


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


if __name__ == "__main__":
 print(bisection(f, 1, 1000))

 import doctest

 doctest.testmod()