





'''simple docstring'''

import mpmath  # for roots of unity
import numpy as np
class     __lowerCAmelCase					:






       """simple docstring"""





       def __init__(							self       :		str	,			lowerCAmelCase__       :		int=None	,			lowerCAmelCase__       :		str=None					)   ->					Any:

            '''simple docstring'''




            _UpperCamelCase       =	list(poly_a or [0]					)[:]
            _UpperCamelCase       =	list(poly_b or [0]					)[:]

            # Remove leading zero coefficients
            while self.polyA[-1] == 0:
                 self.polyA.pop()
            _UpperCamelCase       =	len(self.polyA					)

            while self.polyB[-1] == 0:
                 self.polyB.pop()
            _UpperCamelCase       =	len(self.polyB					)

            # Add 0 to make lengths equal a power of 2
            _UpperCamelCase       =	int(
                2 ** np.ceil(np.loga(len(self.polyA					) + len(self.polyB					) - 1					)					)					)

            while len(self.polyA					) < self.c_max_length:
                 self.polyA.append(0					)
            while len(self.polyB					) < self.c_max_length:
                 self.polyB.append(0					)
            # A complex root used for the fourier transform
            _UpperCamelCase       =	complex(mpmath.root(x=1	,			n=self.c_max_length	,			k=1					)					)

            # The product
            _UpperCamelCase       =	self.__multiply()





       def snake_case__						(							self       :		Tuple	,			lowerCAmelCase__       :		Dict					)   ->					int:

            '''simple docstring'''




            _UpperCamelCase       =	[[x] for x in self.polyA] if which == '''A''' else [[x] for x in self.polyB]
            # Corner case
            if len(lowerCAmelCase__					) <= 1:
                 return dft[0]
            #
            _UpperCamelCase       =	self.c_max_length // 2
            while next_ncol > 0:
                 _UpperCamelCase       =	[[] for i in range(lowerCAmelCase__					)]
                 _UpperCamelCase       =	self.root**next_ncol

                 # First half of next step
                 _UpperCamelCase       =	1
                 for j in range(self.c_max_length // (next_ncol * 2)					):
                      for i in range(lowerCAmelCase__					):
                           new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]					)
                      current_root *= root
                 # Second half of next step
                 _UpperCamelCase       =	1
                 for j in range(self.c_max_length // (next_ncol * 2)					):
                      for i in range(lowerCAmelCase__					):
                           new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]					)
                      current_root *= root
                 # Update
                 _UpperCamelCase       =	new_dft
                 _UpperCamelCase       =	next_ncol // 2
            return dft[0]





       def snake_case__						(							self       :		Optional[Any]					)   ->					Union[str, Any]:

            '''simple docstring'''




            _UpperCamelCase       =	self.__dft('''A'''					)
            _UpperCamelCase       =	self.__dft('''B'''					)
            _UpperCamelCase       =	[[dft_a[i] * dft_b[i] for i in range(self.c_max_length					)]]
            del dft_a
            del dft_b

            # Corner Case
            if len(inverce_c[0]					) <= 1:
                 return inverce_c[0]
            # Inverse DFT
            _UpperCamelCase       =	2
            while next_ncol <= self.c_max_length:
                 _UpperCamelCase       =	[[] for i in range(lowerCAmelCase__					)]
                 _UpperCamelCase       =	self.root ** (next_ncol // 2)
                 _UpperCamelCase       =	1
                 # First half of next step
                 for j in range(self.c_max_length // next_ncol					):
                      for i in range(next_ncol // 2					):
                           # Even positions
                           new_inverse_c[i].append(
                               (
                                   inverce_c[i][j]
                                   + inverce_c[i][j + self.c_max_length // next_ncol]
                               )
                               / 2					)
                           # Odd positions
                           new_inverse_c[i + next_ncol // 2].append(
                               (
                                   inverce_c[i][j]
                                   - inverce_c[i][j + self.c_max_length // next_ncol]
                               )
                               / (2 * current_root)					)
                      current_root *= root
                 # Update
                 _UpperCamelCase       =	new_inverse_c
                 next_ncol *= 2
            # Unpack
            _UpperCamelCase       =	[round(x[0].real	,			8					) + round(x[0].imag	,			8					) * 1J for x in inverce_c]

            # Remove leading 0's
            while inverce_c[-1] == 0:
                 inverce_c.pop()
            return inverce_c







       def __str__(							self       :		Optional[Any]					)   ->					List[Any]:

            '''simple docstring'''




            _UpperCamelCase       =	'''A = ''' + ''' + '''.join(
                f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyA[: self.len_A]					)					)
            _UpperCamelCase       =	'''B = ''' + ''' + '''.join(
                f"""{coef}*x^{i}""" for coef, i in enumerate(self.polyB[: self.len_B]					)					)
            _UpperCamelCase       =	'''A*B = ''' + ''' + '''.join(
                f"""{coef}*x^{i}""" for coef, i in enumerate(self.product					)					)

            return f"""{a}\n{b}\n{c}"""


# Unit tests
if __name__ == "__main__":
   import doctest

   doctest.testmod()










'''simple docstring'''

from math import isclose, sqrt







def  a__    (    lowercase :					float, lowercase :					float, lowercase :					float				)		->		tuple[float, float, float]:





     """simple docstring"""

     _UpperCamelCase       =	point_y / 4 / point_x
     _UpperCamelCase       =	2 * normal_gradient / (1 + normal_gradient * normal_gradient)
     _UpperCamelCase       =	(1 - normal_gradient * normal_gradient) / (
         1 + normal_gradient * normal_gradient
     )
     _UpperCamelCase       =	(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
     _UpperCamelCase       =	outgoing_gradient**2 + 4
     _UpperCamelCase       =	2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
     _UpperCamelCase       =	(point_y - outgoing_gradient * point_x) ** 2 - 100

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

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

     return next_x, next_y, outgoing_gradient







def  a__    (    lowercase :					float = 1.4, lowercase :					float = -9.6				)		->		int:





     """simple docstring"""

     _UpperCamelCase       =	0
     _UpperCamelCase       =	first_x_coord
     _UpperCamelCase       =	first_y_coord
     _UpperCamelCase       =	(1_0.1 - point_y) / (0.0 - point_x)

     while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
          _UpperCamelCase   ,    _UpperCamelCase   ,    _UpperCamelCase       =	next_point(lowercase, lowercase, lowercase				)
          num_reflections += 1

     return num_reflections


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



