'''simple docstring'''

def __UpperCamelCase(   _A					:				int  ,			_A					:				list    ):
   '''simple docstring'''
   _enforce_args(_A  ,			_A    )
   if n == 0:
      return 0
   UpperCAmelCase__							:				Any   				=  float('''-inf'''    )
   for i in range(1  ,			n + 1    ):
      UpperCAmelCase__							:				List[str]   				=  max(
          _A  ,			prices[i - 1] + naive_cut_rod_recursive(n - i  ,			_A    )    )

   return max_revue







def __UpperCamelCase(   _A					:				int  ,			_A					:				list    ):
   '''simple docstring'''
   _enforce_args(_A  ,			_A    )
   UpperCAmelCase__							:				int   				=  [float('''-inf'''    ) for _ in range(n + 1    )]
   return _top_down_cut_rod_recursive(_A  ,			_A  ,			_A    )







def __UpperCamelCase(   _A					:				int  ,			_A					:				list  ,			_A					:				list    ):
   '''simple docstring'''




   if max_rev[n] >= 0:
      return max_rev[n]
   elif n == 0:
      return 0
   else:
      UpperCAmelCase__							:				Optional[Any]   				=  float('''-inf'''    )
      for i in range(1  ,			n + 1    ):
         UpperCAmelCase__							:				int   				=  max(
             _A  ,			prices[i - 1] + _top_down_cut_rod_recursive(n - i  ,			_A  ,			_A    )  ,			)

      UpperCAmelCase__							:				List[Any]   				=  max_revenue

   return max_rev[n]







def __UpperCamelCase(   _A					:				int  ,			_A					:				list    ):
   '''simple docstring'''
   _enforce_args(_A  ,			_A    )

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

   for i in range(1  ,			n + 1    ):
      UpperCAmelCase__							:				Tuple   				=  max_rev[i]
      for j in range(1  ,			i + 1    ):
         UpperCAmelCase__							:				List[str]   				=  max(_A  ,			prices[j - 1] + max_rev[i - j]    )

      UpperCAmelCase__							:				Any   				=  max_revenue_i

   return max_rev[n]







def __UpperCamelCase(   _A					:				int  ,			_A					:				list    ):
   '''simple docstring'''




   if n < 0:
      UpperCAmelCase__							:				str   				=  F'''n must be greater than or equal to 0. Got n = {n}'''
      raise ValueError(_A    )

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







def __UpperCamelCase(   ):
   '''simple docstring'''
   UpperCAmelCase__							:				Optional[Any]   				=  [6, 10, 12, 15, 20, 23]
   UpperCAmelCase__							:				Tuple   				=  len(_A    )

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

   UpperCAmelCase__							:				Optional[Any]   				=  top_down_cut_rod(_A  ,			_A    )
   UpperCAmelCase__							:				int   				=  bottom_up_cut_rod(_A  ,			_A    )
   UpperCAmelCase__							:				Union[str, Any]   				=  naive_cut_rod_recursive(_A  ,			_A    )

   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()





'''simple docstring'''

import math







def __UpperCamelCase(   _A					:				float  ,			_A					:				float    ):
   '''simple docstring'''
   return math.pow(_A  ,			2    ) - a







def __UpperCamelCase(   _A					:				float    ):
   '''simple docstring'''
   return 2 * x







def __UpperCamelCase(   _A					:				float    ):
   '''simple docstring'''
   UpperCAmelCase__							:				int   				=  2.0

   while start <= a:
      UpperCAmelCase__							:				int   				=  math.pow(_A  ,			2    )

   return start







def __UpperCamelCase(   _A					:				float  ,			_A					:				int = 99_99  ,			_A					:				float = 0.0_0_0_0_0_0_0_0_0_0_0_0_0_1    ):
   '''simple docstring'''

   if a < 0:
      raise ValueError('''math domain error'''    )

   UpperCAmelCase__							:				Any   				=  get_initial_point(_A    )

   for _ in range(_A    ):
      UpperCAmelCase__							:				Optional[Any]   				=  value
      UpperCAmelCase__							:				Tuple   				=  value - fx(_A  ,			_A    ) / fx_derivative(_A    )
      if abs(prev_value - value    ) < tolerance:
         return value

   return value


if __name__ == "__main__":
  from doctest import testmod

  testmod()




