


from math import isclose, sqrt




def 	__lowercase		(  _UpperCamelCase,			_UpperCamelCase,			_UpperCamelCase   )			->Optional[Any]:







      """simple docstring"""

      lowercase	:      str          =						point_y / 4 / point_x
      lowercase	:      Any          =						2 * normal_gradient / (1 + normal_gradient * normal_gradient)
      lowercase	:      Dict          =						(1 - normal_gradient * normal_gradient) / (
          1 + normal_gradient * normal_gradient
      )
      lowercase	:      Optional[int]          =						(sa - ca * incoming_gradient) / (ca + sa * incoming_gradient)

      # to find the next point, solve the simultaeneous equations:
      # y^2 + 4x^2 = 100
      # y - b = m * (x - a)
      # ==> A x^2 + B x + C = 0
      lowercase	:      List[Any]          =						outgoing_gradient**2 + 4
      lowercase	:      Union[str, Any]          =						2 * outgoing_gradient * (point_y - outgoing_gradient * point_x)
      lowercase	:      List[str]          =						(point_y - outgoing_gradient * point_x) ** 2 - 100

      lowercase	:      str          =						(
          -linear_term - sqrt(linear_term**2 - 4 * quadratic_term * constant_term   )
      ) / (2 * quadratic_term)
      lowercase	:      int          =						(
          -linear_term + sqrt(linear_term**2 - 4 * quadratic_term * constant_term   )
      ) / (2 * quadratic_term)

      # two solutions, one of which is our input point
      lowercase	:      Optional[int]          =						x_minus if isclose(snake_case_,			snake_case_   ) else x_plus
      lowercase	:      str          =						point_y + outgoing_gradient * (next_x - point_x)

      return next_x, next_y, outgoing_gradient




def 	__lowercase		(  _UpperCamelCase = 1.4,			_UpperCamelCase = -9.6   )			->List[Any]:







      """simple docstring"""

      lowercase	:      int          =						0
      lowercase	:      float          =						first_x_coord
      lowercase	:      float          =						first_y_coord
      lowercase	:      float          =						(1_0.1 - point_y) / (0.0 - point_x)

      while not (-0.0_1 <= point_x <= 0.0_1 and point_y > 0):
            lowercase	:      Tuple          =						next_point(snake_case_,			snake_case_,			snake_case_   )
            num_reflections += 1

      return num_reflections


if __name__ == "__main__":
  print(F'''{solution() = }''')







def 				lowerCAmelCase_	(     snake_case_,snake_case_     ):
      _enforce_args(snake_case_,snake_case_     )
      if n == 0:
            return 0
      _A							:     Tuple			=						float("""-inf"""     )
      for i in range(1,n + 1     ):
            _A							:     str			=						max(
                snake_case_,prices[i - 1] + naive_cut_rod_recursive(n - i,snake_case_     )     )

      return max_revue







def 				lowerCAmelCase_	(     snake_case_,snake_case_     ):
      _enforce_args(snake_case_,snake_case_     )
      _A							:     Dict			=						[float("""-inf"""     ) for _ in range(n + 1     )]
      return _top_down_cut_rod_recursive(snake_case_,snake_case_,snake_case_     )







def 				lowerCAmelCase_	(     snake_case_,snake_case_,snake_case_     ):
      if max_rev[n] >= 0:
            return max_rev[n]
      elif n == 0:
            return 0
      else:
            _A							:     List[str]			=						float("""-inf"""     )
            for i in range(1,n + 1     ):
                  _A							:     Optional[Any]			=						max(
                      snake_case_,prices[i - 1] + _top_down_cut_rod_recursive(n - i,snake_case_,snake_case_     ),)

            _A							:     Tuple			=						max_revenue

      return max_rev[n]







def 				lowerCAmelCase_	(     snake_case_,snake_case_     ):
      _enforce_args(snake_case_,snake_case_     )

      # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
      # length 0.
      _A							:     List[Any]			=						[float("""-inf"""     ) for _ in range(n + 1     )]
      _A							:     Any			=						0

      for i in range(1,n + 1     ):
            _A							:     Optional[Any]			=						max_rev[i]
            for j in range(1,i + 1     ):
                  _A							:     int			=						max(snake_case_,prices[j - 1] + max_rev[i - j]     )

            _A							:     int			=						max_revenue_i

      return max_rev[n]







def 				lowerCAmelCase_	(     snake_case_,snake_case_     ):
      if n < 0:
            _A							:     Optional[Any]			=						f'''n must be greater than or equal to 0. Got n = {n}'''
            raise ValueError(snake_case_     )

      if n > len(snake_case_     ):
            _A							:     Any			=						(
                """Each integral piece of rod must have a corresponding price. """
                f'''Got n = {n} but length of prices = {len(snake_case_     )}'''
            )
            raise ValueError(snake_case_     )







def 				lowerCAmelCase_	(     ):
      _A							:     Tuple			=						[6, 10, 12, 15, 20, 23]
      _A							:     List[Any]			=						len(snake_case_     )

      # the best revenue comes from cutting the rod into 6 pieces, each
      # of length 1 resulting in a revenue of 6 * 6 = 36.
      _A							:     Any			=						36

      _A							:     List[Any]			=						top_down_cut_rod(snake_case_,snake_case_     )
      _A							:     List[Any]			=						bottom_up_cut_rod(snake_case_,snake_case_     )
      _A							:     Dict			=						naive_cut_rod_recursive(snake_case_,snake_case_     )

      assert expected_max_revenue == max_rev_top_down
      assert max_rev_top_down == max_rev_bottom_up
      assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
  main()

