





from math import loga






def 					lowerCamelCase_						(    _UpperCamelCase			)		->						int:






				"""simple docstring"""


				if a < 0:
								raise ValueError('''Input value must be a positive integer'''			)
				elif isinstance(_UpperCamelCase			,    _UpperCamelCase			):
								raise TypeError('''Input value must be a \'int\' type'''			)
				return 0 if (a == 0) else int(loga(a & -a			)			)


if __name__ == "__main__":
	import doctest

	doctest.testmod()







from math import isclose, sqrt






def 					lowerCamelCase_						(    _UpperCamelCase			,    _UpperCamelCase			,    _UpperCamelCase			)		->						tuple[float, float, float]:






				"""simple docstring"""


				snake_case_						:   Dict           =  point_y / 4 / point_x
				snake_case_						:   List[str]           =  2 * normal_gradient / (1 + normal_gradient * normal_gradient)
				snake_case_						:   Union[str, Any]           =  (1 - normal_gradient * normal_gradient) / (
				    1 + normal_gradient * normal_gradient
				)
				snake_case_						:   Tuple           =  (sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

				# to find the next point, solve the simultaeneous equations:
				# y^2 + 4x^2 = 100
				# y - b = m * (x - a)
				# ==> A x^2 + B x + C = 0
				snake_case_						:   Union[str, Any]           =  outgoing_gradient**2 + 4
				snake_case_						:   Tuple           =  2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
				snake_case_						:   Optional[Any]           =  (point_y - outgoing_gradient * point_x) ** 2 - 100

				snake_case_						:   Dict           =  (
				    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term			)
				) / (2 * quadratic_term)
				snake_case_						:   Optional[int]           =  (
				    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term			)
				) / (2 * quadratic_term)

				# two solutions, one of which is our input point
				snake_case_						:   Any           =  x_minus if isclose(_UpperCamelCase			,    _UpperCamelCase			) else x_plus
				snake_case_						:   int           =  point_y + outgoing_gradient * (next_x - point_x)

				return next_x, next_y, outgoing_gradient






def 					lowerCamelCase_						(    _UpperCamelCase = 1.4			,    _UpperCamelCase = -9.6			)		->						int:






				"""simple docstring"""


				snake_case_						:   int           =  0
				snake_case_						:   float           =  first_x_coord
				snake_case_						:   float           =  first_y_coord
				snake_case_						:   float           =  (10.1 - point_y) / (0.0 - point_x)

				while not (-0.01 <= point_x <= 0.01 and point_y > 0):
								snake_case_      ,							snake_case_      ,							snake_case_						:   str           =  next_point(_UpperCamelCase			,    _UpperCamelCase			,    _UpperCamelCase			)
								num_reflections += 1

				return num_reflections


if __name__ == "__main__":
	print(F'''{solution() = }''')
