





'''simple docstring'''

import mpmath  # for roots of unity
import numpy as np


class 					a_     :

    '''simple docstring'''


    def __init__(		self ,		A=None ,		A=None       )		->   List[str]:
        _SCREAMING_SNAKE_CASE									=			list(poly_a or [0]       )[:]
        _SCREAMING_SNAKE_CASE									=			list(poly_b or [0]       )[:]

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

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

        # Add 0 to make lengths equal a power of 2
        _SCREAMING_SNAKE_CASE									=			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
        _SCREAMING_SNAKE_CASE									=			complex(mpmath.root(x=1 ,		n=self.c_max_length ,		k=1       )       )

        # The product
        _SCREAMING_SNAKE_CASE									=			self.__multiply()


    def 			snake_case_(		self ,		A       )		->   Dict:
        _SCREAMING_SNAKE_CASE									=			[[x] for x in self.polyA] if which == "A" else [[x] for x in self.polyB]
        # Corner case
        if len(_A       ) <= 1:
            return dft[0]
        #
        _SCREAMING_SNAKE_CASE									=			self.c_max_length // 2
        while next_ncol > 0:
            _SCREAMING_SNAKE_CASE									=			[[] for i in range(_A       )]
            _SCREAMING_SNAKE_CASE									=			self.root**next_ncol

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


    def 			snake_case_(		self       )		->   Optional[Any]:
        _SCREAMING_SNAKE_CASE									=			self.__dft("""A"""       )
        _SCREAMING_SNAKE_CASE									=			self.__dft("""B"""       )
        _SCREAMING_SNAKE_CASE									=			[[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
        _SCREAMING_SNAKE_CASE									=			2
        while next_ncol <= self.c_max_length:
            _SCREAMING_SNAKE_CASE									=			[[] for i in range(_A       )]
            _SCREAMING_SNAKE_CASE									=			self.root ** (next_ncol // 2)
            _SCREAMING_SNAKE_CASE									=			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
            _SCREAMING_SNAKE_CASE									=			new_inverse_c
            next_ncol *= 2
        # Unpack
        _SCREAMING_SNAKE_CASE									=			[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[Any]:
        _SCREAMING_SNAKE_CASE									=			"A = " + " + ".join(
            f'{coef}*x^{i}' for coef, i in enumerate(self.polyA[: self.len_A]       )       )
        _SCREAMING_SNAKE_CASE									=			"B = " + " + ".join(
            f'{coef}*x^{i}' for coef, i in enumerate(self.polyB[: self.len_B]       )       )
        _SCREAMING_SNAKE_CASE									=			"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()




from math import factorial, radians




def       _snake_case	(   lowerCAmelCase							:   float   , lowerCAmelCase							:   int = 1_8   , lowerCAmelCase							:   int = 1_0	):
		"""simple docstring"""







		SCREAMING_SNAKE_CASE_  :		List[str]				      =    angle_in_degrees - ((angle_in_degrees // 360.0) * 360.0)

		# Converting from degrees to radians
		SCREAMING_SNAKE_CASE_  :		Tuple				      =    radians(lowerCAmelCase	)

		SCREAMING_SNAKE_CASE_  :		Optional[Any]				      =    angle_in_radians
		SCREAMING_SNAKE_CASE_  :		List[str]				      =    3
		SCREAMING_SNAKE_CASE_  :		str				      =    -1

		for _ in range(lowerCAmelCase	):
				result += (b * (angle_in_radians**a)) / factorial(lowerCAmelCase	)

				SCREAMING_SNAKE_CASE_  :		Dict				      =    -b  # One positive term and the next will be negative and so on...
				a += 2  # Increased by 2 for every term.

		return round(lowerCAmelCase   , lowerCAmelCase	)


if __name__ == "__main__":
		__import__('''doctest''').testmod()
