




import colorsys

from PIL import Image  # type: ignore







def lowerCamelCase__  (     _lowercase					,		_lowercase					,		_lowercase ):

     '''simple docstring'''


     UpperCAmelCase_     :				Union[str, Any]  =     x
     UpperCAmelCase_     :				List[str]  =     y
     for step in range(_lowercase ):  # noqa: B007
          UpperCAmelCase_     :				Union[str, Any]  =     a * a - b * b + x
          UpperCAmelCase_     :				List[Any]  =     2 * a * b + y
          UpperCAmelCase_     :				List[Any]  =     a_new

          # divergence happens for all complex number with an absolute value
          # greater than 4
          if a * a + b * b > 4:
               break
     return step / (max_step - 1)







def lowerCamelCase__  (     _lowercase ):

     '''simple docstring'''


     if distance == 1:
          return (0, 0, 0)
     else:
          return (255, 255, 255)







def lowerCamelCase__  (     _lowercase ):

     '''simple docstring'''


     if distance == 1:
          return (0, 0, 0)
     else:
          return tuple(round(i * 255 ) for i in colorsys.hsv_to_rgb(_lowercase					,		1					,		1 ) )







def lowerCamelCase__  (     _lowercase = 800					,		_lowercase = 600					,		_lowercase = -0.6					,		_lowercase = 0					,		_lowercase = 3.2					,		_lowercase = 50					,		_lowercase = True					,		):

     '''simple docstring'''


     UpperCAmelCase_     :				int  =     Image.new('''RGB'''					,		(image_width, image_height) )
     UpperCAmelCase_     :				List[str]  =     img.load()

     # loop through the image-coordinates
     for image_x in range(_lowercase ):
          for image_y in range(_lowercase ):
               # determine the figure-coordinates based on the image-coordinates
               UpperCAmelCase_     :				str  =     figure_width / image_width * image_height
               UpperCAmelCase_     :				Dict  =     figure_center_x + (image_x / image_width - 0.5) * figure_width
               UpperCAmelCase_     :				Any  =     figure_center_y + (image_y / image_height - 0.5) * figure_height

               UpperCAmelCase_     :				int  =     get_distance(_lowercase					,		_lowercase					,		_lowercase )

               # color the corresponding pixel based on the selected coloring-function
               if use_distance_color_coding:
                    UpperCAmelCase_     :				Dict  =     get_color_coded_rgb(_lowercase )
               else:
                    UpperCAmelCase_     :				int  =     get_black_and_white_rgb(_lowercase )

     return img


if __name__ == "__main__":
     import doctest

     doctest.testmod()

     # colored version, full figure
     __a							     =					get_image()

     # uncomment for colored version, different section, zoomed in
     # img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
     # figure_width = 0.8)

     # uncomment for black and white version, full figure
     # img = get_image(use_distance_color_coding = False)

     # uncomment to save the image
     # img.save("mandelbrot.png")

     img.show()






'''simple docstring'''




import mpmath  # for roots of unity
import numpy as np


class 					SCREAMING_SNAKE_CASE       :


     def __init__(		self		,      _UpperCAmelCase=None		,      _UpperCAmelCase=None):

          '''simple docstring'''






          __A		:					List[Any]						=			list(poly_a or [0])[:]
          __A		:					Optional[int]						=			list(poly_b or [0])[:]

          # Remove leading zero coefficients
          while self.polyA[-1] == 0:
               self.polyA.pop()
          __A		:					Union[str, Any]						=			len(self.polyA)

          while self.polyB[-1] == 0:
               self.polyB.pop()
          __A		:					Optional[int]						=			len(self.polyB)

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

          # The product
          __A		:					Tuple						=			self.__multiply()


     def 				SCREAMING_SNAKE_CASE    (		self		,      _UpperCAmelCase):

          '''simple docstring'''






          __A		:					Optional[int]						=			[[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]
          #
          __A		:					Dict						=			self.c_max_length // 2
          while next_ncol > 0:
               __A		:					Optional[Any]						=			[[] for i in range(_UpperCAmelCase)]
               __A		:					Tuple						=			self.root**next_ncol

               # First half of next step
               __A		:					Optional[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
               __A		:					List[str]						=			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
               __A		:					Optional[int]						=			new_dft
               __A		:					Tuple						=			next_ncol // 2
          return dft[0]


     def 				SCREAMING_SNAKE_CASE    (		self):

          '''simple docstring'''






          __A		:					int						=			self.__dft('A')
          __A		:					Optional[Any]						=			self.__dft('B')
          __A		:					str						=			[[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
          __A		:					Dict						=			2
          while next_ncol <= self.c_max_length:
               __A		:					Optional[int]						=			[[] for i in range(_UpperCAmelCase)]
               __A		:					Any						=			self.root ** (next_ncol // 2)
               __A		:					Tuple						=			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
               __A		:					int						=			new_inverse_c
               next_ncol *= 2
          # Unpack
          __A		:					Optional[int]						=			[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):

          '''simple docstring'''






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