






'''simple docstring'''



SCREAMING_SNAKE_CASE						=	tuple[float, float, float]
SCREAMING_SNAKE_CASE						=	tuple[float, float, float]


def 							lowercase_				(    __A :       Pointad					,      __A :       Pointad			)       ->      Vectorad:



 """simple docstring"""




 lowercase					:   Optional[Any]				  =end_pointa[0] - end_pointa[0]
 lowercase					:   Dict				  =end_pointa[1] - end_pointa[1]
 lowercase					:   List[Any]				  =end_pointa[2] - end_pointa[2]
 return (x, y, z)


def 							lowercase_				(    __A :       Vectorad					,      __A :       Vectorad			)       ->      Vectorad:



 """simple docstring"""




 lowercase					:   List[str]				  =ab[1] * ac[2] - ab[2] * ac[1]  # *i
 lowercase					:   Optional[int]				  =(ab[0] * ac[2] - ab[2] * ac[0]) * -1  # *j
 lowercase					:   str				  =ab[0] * ac[1] - ab[1] * ac[0]  # *k
 return (x, y, z)


def 							lowercase_				(    __A :       Vectorad					,      __A :       int			)       ->      bool:



 """simple docstring"""




 return tuple(round(__A					,      __A			) for x in vector			) == (0, 0, 0)





def 							lowercase_				(    __A :       Pointad					,      __A :       Pointad					,      __A :       Pointad					,      __A :       int = 1_0			)       ->      bool:



 """simple docstring"""




 lowercase					:   Optional[int]				  =create_vector(__A					,      __A			)
 lowercase					:   int				  =create_vector(__A					,      __A			)
 return is_zero_vector(get_ad_vectors_cross(__A					,      __A			)					,      __A			)











'''simple docstring'''



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




      """simple docstring"""




      def __init__( self			:		List[Any]     ,      UpperCAmelCase			:		Dict=None     ,      UpperCAmelCase			:		Optional[int]=None		)  ->  Optional[Any]:


       '''simple docstring'''

       lowercase					:   Any				  =list(poly_a or [0]		)[:]
       lowercase					:   Dict				  =list(poly_b or [0]		)[:]

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

       while self.polyB[-1] == 0:
        self.polyB.pop()
       lowercase					:   List[str]				  =len(self.polyB		)

       # Add 0 to make lengths equal a power of 2
       lowercase					:   Tuple				  =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
       lowercase					:   Optional[int]				  =complex(mpmath.root(x=1     ,      n=self.c_max_length     ,      k=1		)		)

       # The product
       lowercase					:   str				  =self.__multiply()






      def 				A__      ( self			:		Optional[Any]     ,      UpperCAmelCase			:		Union[str, Any]		)  ->  Tuple:


       '''simple docstring'''

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

        # First half of next step
        lowercase					:   Union[str, Any]				  =1
        for j in range(self.c_max_length // (next_ncol * 2)		):
         for i in range(UpperCAmelCase		):
          new_dft[i].append(dft[i][j] + current_root * dft[i + next_ncol][j]		)
         current_root *= root
        # Second half of next step
        lowercase					:   Any				  =1
        for j in range(self.c_max_length // (next_ncol * 2)		):
         for i in range(UpperCAmelCase		):
          new_dft[i].append(dft[i][j] - current_root * dft[i + next_ncol][j]		)
         current_root *= root
        # Update
        lowercase					:   Tuple				  =new_dft
        lowercase					:   List[Any]				  =next_ncol // 2
       return dft[0]






      def 				A__      ( self			:		int		)  ->  str:


       '''simple docstring'''

       lowercase					:   List[Any]				  =self.__dft('''A'''		)
       lowercase					:   Union[str, Any]				  =self.__dft('''B'''		)
       lowercase					:   Any				  =[[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
       lowercase					:   Tuple				  =2
       while next_ncol <= self.c_max_length:
        lowercase					:   Tuple				  =[[] for i in range(UpperCAmelCase		)]
        lowercase					:   Tuple				  =self.root ** (next_ncol // 2)
        lowercase					:   Optional[int]				  =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
        lowercase					:   List[Any]				  =new_inverse_c
        next_ncol *= 2
       # Unpack
       lowercase					:   List[str]				  =[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			:		Dict		)  ->  Optional[Any]:


       '''simple docstring'''

       lowercase					:   Optional[Any]				  ='''A = ''' + ''' + '''.join(
           f'{coef}*x^{i}' for coef, i in enumerate(self.polyA[: self.len_A]		)		)
       lowercase					:   List[str]				  ='''B = ''' + ''' + '''.join(
           f'{coef}*x^{i}' for coef, i in enumerate(self.polyB[: self.len_B]		)		)
       lowercase					:   Optional[Any]				  ='''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()



