
"""simple docstring"""


from math import isclose, sqrt






def  lowercase						(	lowerCAmelCase__	,lowerCAmelCase__	,lowerCAmelCase__						):
			lowerCamelCase_					=  point_y / 4 / point_x
			lowerCamelCase_					=  2 * normal_gradient / (1 + normal_gradient * normal_gradient)
			lowerCamelCase_					=  (1 - normal_gradient * normal_gradient) / (
			    1 + normal_gradient * normal_gradient
			)
			lowerCamelCase_					=  (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
			lowerCamelCase_					=  outgoing_gradient**2 + 4
			lowerCamelCase_					=  2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
			lowerCamelCase_					=  (point_y - outgoing_gradient * point_x) ** 2 - 100

			lowerCamelCase_					=  (
			    -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term						)
			) / (2 * quadratic_term)
			lowerCamelCase_					=  (
			    -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term						)
			) / (2 * quadratic_term)

			# two solutions, one of which is our input point
			lowerCamelCase_					=  x_minus if isclose(lowerCAmelCase__	,lowerCAmelCase__						) else x_plus
			lowerCamelCase_					=  point_y + outgoing_gradient * (next_x - point_x)

			return next_x, next_y, outgoing_gradient






def  lowercase						(	lowerCAmelCase__ = 1.4	,lowerCAmelCase__ = -9.6						):
			lowerCamelCase_					=  0
			lowerCamelCase_					=  first_x_coord
			lowerCamelCase_					=  first_y_coord
			lowerCamelCase_					=  (10.1 - point_y) / (0.0 - point_x)

			while not (-0.01 <= point_x <= 0.01 and point_y > 0):
						lowerCamelCase_		, lowerCamelCase_		, lowerCamelCase_					=  next_point(lowerCAmelCase__	,lowerCAmelCase__	,lowerCAmelCase__						)
						num_reflections += 1

			return num_reflections


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





"""simple docstring"""


def  lowercase						(	lowerCAmelCase__						):
			return " ".join(input_str.split()[::-1]						)


if __name__ == "__main__":
						import doctest

						doctest.testmod()



