






'''simple docstring'''





from __future__ import annotations

from collections.abc import MutableSequence
class  lowerCAmelCase_       :




							def __init__(   self     :   str ,							SCREAMING_SNAKE_CASE_     :   int ,							SCREAMING_SNAKE_CASE_     :   MutableSequence[float]	):
											if len(SCREAMING_SNAKE_CASE_	) != degree + 1:
															raise ValueError(
															    '''The number of coefficients should be equal to the degree + 1.'''	)

											lowerCAmelCase__      			=						list(SCREAMING_SNAKE_CASE_	)
											lowerCAmelCase__      			=						degree




							def __add__(   self     :   Optional[int] ,							SCREAMING_SNAKE_CASE_     :   Polynomial	):

											if self.degree > polynomial_a.degree:
															lowerCAmelCase__      			=						self.coefficients[:]
															for i in range(polynomial_a.degree + 1	):
																			coefficients[i] += polynomial_a.coefficients[i]
															return Polynomial(self.degree ,							SCREAMING_SNAKE_CASE_	)
											else:
															lowerCAmelCase__      			=						polynomial_a.coefficients[:]
															for i in range(self.degree + 1	):
																			coefficients[i] += self.coefficients[i]
															return Polynomial(polynomial_a.degree ,							SCREAMING_SNAKE_CASE_	)




							def __sub__(   self     :   Tuple ,							SCREAMING_SNAKE_CASE_     :   Polynomial	):
											return self + polynomial_a * Polynomial(0 ,							[-1]	)




							def __neg__(   self     :   Optional[Any]	):
											return Polynomial(self.degree ,							[-c for c in self.coefficients]	)




							def __mul__(   self     :   Tuple ,							SCREAMING_SNAKE_CASE_     :   Polynomial	):
											lowerCAmelCase__      			=						[0] * (self.degree + polynomial_a.degree + 1)
											for i in range(self.degree + 1	):
															for j in range(polynomial_a.degree + 1	):
																			coefficients[i + j] += (
																			    self.coefficients[i] * polynomial_a.coefficients[j]
																			)

											return Polynomial(self.degree + polynomial_a.degree ,							SCREAMING_SNAKE_CASE_	)




							def 						__snake_case					(   self     :   Dict ,							SCREAMING_SNAKE_CASE_     :   int | float	):
											lowerCAmelCase__      			=						0
											for i in range(self.degree + 1	):
															result += self.coefficients[i] * (substitution**i)
											return result




							def __str__(   self     :   Dict	):
											lowerCAmelCase__      			=						''''''
											for i in range(self.degree ,							-1 ,							-1	):
															if self.coefficients[i] == 0:
																			continue
															elif self.coefficients[i] > 0:
																			if polynomial:
																							polynomial += " + "
															else:
																			polynomial += " - "

															if i == 0:
																			polynomial += str(abs(self.coefficients[i]	)	)
															elif i == 1:
																			polynomial += str(abs(self.coefficients[i]	)	) + "x"
															else:
																			polynomial += str(abs(self.coefficients[i]	)	) + "x^" + str(SCREAMING_SNAKE_CASE_	)

											return polynomial




							def __repr__(   self     :   Union[str, Any]	):
											return self.__str__()




							def 						__snake_case					(   self     :   List[Any]	):
											lowerCAmelCase__      			=						[0] * self.degree
											for i in range(self.degree	):
															lowerCAmelCase__      			=						self.coefficients[i + 1] * (i + 1)
											return Polynomial(self.degree - 1 ,							SCREAMING_SNAKE_CASE_	)




							def 						__snake_case					(   self     :   List[str] ,							SCREAMING_SNAKE_CASE_     :   int | float = 0	):
											lowerCAmelCase__      			=						[0] * (self.degree + 2)
											lowerCAmelCase__      			=						constant
											for i in range(self.degree + 1	):
															lowerCAmelCase__      			=						self.coefficients[i] / (i + 1)
											return Polynomial(self.degree + 1 ,							SCREAMING_SNAKE_CASE_	)




							def __eq__(   self     :   List[str] ,							SCREAMING_SNAKE_CASE_     :   object	):
											if not isinstance(SCREAMING_SNAKE_CASE_ ,							SCREAMING_SNAKE_CASE_	):
															return False

											if self.degree != polynomial_a.degree:
															return False

											for i in range(self.degree + 1	):
															if self.coefficients[i] != polynomial_a.coefficients[i]:
																			return False

											return True




							def __ne__(   self     :   List[Any] ,							SCREAMING_SNAKE_CASE_     :   object	):
											return not self.__eq__(SCREAMING_SNAKE_CASE_	)













from math import isclose, sqrt





def 	lowerCAmelCase_		(lowercase__						:       float     ,					lowercase__						:       float     ,					lowercase__						:       float       )		->    tuple[float, float, float]:


				'''simple docstring'''




				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 - 1_00

				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(lowercase__     ,					lowercase__       ) else x_plus
				lowerCAmelCase__      			=						point_y + outgoing_gradient * (next_x - point_x)

				return next_x, next_y, outgoing_gradient





def 	lowerCAmelCase_		(lowercase__						:       float = 1.4     ,					lowercase__						:       float = -9.6       )		->    int:


				'''simple docstring'''




				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(lowercase__     ,					lowercase__     ,					lowercase__       )
								num_reflections += 1

				return num_reflections


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





