




"""simple docstring"""


import colorsys

from PIL import Image  # type: ignore


def  UpperCAmelCase__	(snake_case__							: float		,       snake_case__							: float		,       snake_case__							: int   ):
		"""simple docstring"""

		_snake_case							:		Any								=  x
		_snake_case							:		List[str]								=  y
		for step in range(snake_case__   ):  # noqa: B007
				_snake_case							:		Union[str, Any]								=  a * a - b * b + x
				_snake_case							:		List[Any]								=  2 * a * b + y
				_snake_case							:		int								=  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  UpperCAmelCase__	(snake_case__							: float   ):
		"""simple docstring"""

		if distance == 1:
				return (0, 0, 0)
		else:
				return (2_55, 2_55, 2_55)


def  UpperCAmelCase__	(snake_case__							: float   ):
		"""simple docstring"""

		if distance == 1:
				return (0, 0, 0)
		else:
				return tuple(round(i * 2_55   ) for i in colorsys.hsv_to_rgb(snake_case__		,       1		,       1   )   )


def  UpperCAmelCase__	(snake_case__							: int = 8_00		,       snake_case__							: int = 6_00		,       snake_case__							: float = -0.6		,       snake_case__							: float = 0		,       snake_case__							: float = 3.2		,       snake_case__							: int = 50		,       snake_case__							: bool = True		,       ):
		"""simple docstring"""

		_snake_case							:		str								=  Image.new("""RGB"""		,       (image_width, image_height)   )
		_snake_case							:		Optional[int]								=  img.load()

		# loop through the image-coordinates
		for image_x in range(snake_case__   ):
				for image_y in range(snake_case__   ):
						# determine the figure-coordinates based on the image-coordinates
						_snake_case							:		Any								=  figure_width / image_width * image_height
						_snake_case							:		Optional[int]								=  figure_center_x + (image_x / image_width - 0.5) * figure_width
						_snake_case							:		Any								=  figure_center_y + (image_y / image_height - 0.5) * figure_height

						_snake_case							:		str								=  get_distance(snake_case__		,       snake_case__		,       snake_case__   )

						# color the corresponding pixel based on the selected coloring-function
						if use_distance_color_coding:
								_snake_case							:		Optional[Any]								=  get_color_coded_rgb(snake_case__   )
						else:
								_snake_case							:		str								=  get_black_and_white_rgb(snake_case__   )

		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"""


def  UpperCAmelCase__	(snake_case__							: int = 1_00_00_00   ):
  """simple docstring"""

  _snake_case							:		Dict								=  [i - 1 for i in range(limit + 1   )]

  for i in range(2		,       limit + 1   ):
    if phi[i] == i - 1:
      for j in range(2 * i		,       limit + 1		,       snake_case__   ):
        phi[j] -= phi[j] // i

  return sum(phi[2 : limit + 1]   )


if __name__ == "__main__":
     print(solution())


