"""simple docstring"""







from math import isclose, sqrt



def     lowercase__( __SCREAMING_SNAKE_CASE							:  float  ,    __SCREAMING_SNAKE_CASE							:  float  ,    __SCREAMING_SNAKE_CASE							:  float   ):
     lowercase_			:       List[Any]									=			point_y / 4 / point_x
     lowercase_			:       int									=			2 * normal_gradient / (1 + normal_gradient * normal_gradient)
     lowercase_			:       Any									=			(1 - normal_gradient * normal_gradient) / (
         1 + normal_gradient * normal_gradient
     )
     lowercase_			:       List[str]									=			(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
     lowercase_			:       Union[str, Any]									=			outgoing_gradient**2 + 4
     lowercase_			:       Dict									=			2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
     lowercase_			:       Tuple									=			(point_y - outgoing_gradient * point_x) ** 2 - 1_00

     lowercase_			:       Tuple									=			(
         -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term   )
     ) / (2 * quadratic_term)
     lowercase_			:       str									=			(
         -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term   )
     ) / (2 * quadratic_term)

     # two solutions, one of which is our input point
     lowercase_			:       List[str]									=			x_minus if isclose(__SCREAMING_SNAKE_CASE  ,    __SCREAMING_SNAKE_CASE   ) else x_plus
     lowercase_			:       Optional[int]									=			point_y + outgoing_gradient * (next_x - point_x)

     return next_x, next_y, outgoing_gradient



def     lowercase__( __SCREAMING_SNAKE_CASE							:  float = 1.4  ,    __SCREAMING_SNAKE_CASE							:  float = -9.6   ):
     lowercase_			:       int									=			0
     lowercase_			:       float									=			first_x_coord
     lowercase_			:       float									=			first_y_coord
     lowercase_			:       float									=			(10.1 - point_y) / (0.0 - point_x)

     while not (-0.01 <= point_x <= 0.01 and point_y > 0):
          lowercase_			:       Optional[int]									=			next_point(__SCREAMING_SNAKE_CASE  ,    __SCREAMING_SNAKE_CASE  ,    __SCREAMING_SNAKE_CASE   )
          num_reflections += 1

     return num_reflections


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


"""simple docstring"""







from diffusers.utils.testing_utils import require_onnxruntime







@require_onnxruntime
class  UpperCamelCase	:
   pass

