





import mpmath  # for roots of unity
import numpy as np

class   A							:



      def __init__(						self:    int		,	_lowerCAmelCase:    Optional[int]=None		,	_lowerCAmelCase:    str=None							)    ->     Tuple:
         '''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 						lowerCAmelCase__     (						self:    str		,	_lowerCAmelCase:    str							)    ->     Any:
         '''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 						lowerCAmelCase__     (						self:    Tuple							)    ->     Optional[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:    List[str]							)    ->     Optional[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()












def  a__ (__lowercase							:str					,  __lowercase							:str   )		-> float:

				def get_matched_characters(__lowercase							:str					,  __lowercase							:str   ) -> str:
								_A	:				Union[str, Any]   =      []
								_A	:				Dict   =      min(len(_stra   )					,  len(_stra   )   ) // 2
								for i, l in enumerate(_stra   ):
												_A	:				Tuple   =      int(max(0					,  i - limit   )   )
												_A	:				Union[str, Any]   =      int(min(i + limit + 1					,  len(_stra   )   )   )
												if l in _stra[left:right]:
																matched.append(__lowercase   )
																_A	:				List[Any]   =      f"""{_stra[0:_stra.index(__lowercase   )]} {_stra[_stra.index(__lowercase   ) + 1:]}"""

								return "".join(__lowercase   )

				# matching characters
				_A	:				str   =      get_matched_characters(__lowercase					,  __lowercase   )
				_A	:				Optional[Any]   =      get_matched_characters(__lowercase					,  __lowercase   )
				_A	:				Any   =      len(__lowercase   )

				# transposition
				_A	:				str   =      (
				    len([(ca, ca) for ca, ca in zip(__lowercase					,  __lowercase   ) if ca != ca]   ) // 2
				)

				if not match_count:
								_A	:				Optional[int]   =      0.0
				else:
								_A	:				str   =      (
								    1
								    / 3
								    * (
								        match_count / len(__lowercase   )
								        + match_count / len(__lowercase   )
								        + (match_count - transpositions) / match_count
								    )
								)

				# common prefix up to 4 characters
				_A	:				Any   =      0
				for ca, ca in zip(stra[:4]					,  stra[:4]   ):
								if ca == ca:
												prefix_len += 1
								else:
												break

				return jaro + 0.1 * prefix_len * (1 - jaro)


if __name__ == "__main__":
	import doctest

	doctest.testmod()
	print(jaro_winkler('hello', 'world'))



