

'''simple docstring'''







from math import factorial


class 					A__      :



    def __init__(					self		,				UpperCamelCase__		,				UpperCamelCase__       )						->      List[str]:





           '''simple docstring'''
           A_						=      real
           if isinstance(UpperCamelCase__		,				UpperCamelCase__       ):
                  A_						=      [1] * rank
           else:
                  A_						=      rank



    def __repr__(					self       )						->      str:





           '''simple docstring'''
           return (
               f'''{self.real}+'''
               f'''{"+".join(str(UpperCamelCase__       )+"E"+str(n+1       )for n,dual in enumerate(self.duals       )       )}'''
           )



    def 					snake_case_			(					self       )						->      Optional[int]:





           '''simple docstring'''
           A_						=      self.duals.copy()
           while cur[-1] == 0:
                  cur.pop(-1       )
           return Dual(self.real		,				UpperCamelCase__       )



    def __add__(					self		,				UpperCamelCase__       )						->      Dict:





           '''simple docstring'''
           if not isinstance(UpperCamelCase__		,				UpperCamelCase__       ):
                  return Dual(self.real + other		,				self.duals       )
           A_						=      self.duals.copy()
           A_						=      other.duals.copy()
           if len(UpperCamelCase__       ) > len(UpperCamelCase__       ):
                  o_dual.extend([1] * (len(UpperCamelCase__       ) - len(UpperCamelCase__       ))       )
           elif len(UpperCamelCase__       ) < len(UpperCamelCase__       ):
                  s_dual.extend([1] * (len(UpperCamelCase__       ) - len(UpperCamelCase__       ))       )
           A_						=      []
           for i in range(len(UpperCamelCase__       )       ):
                  new_duals.append(s_dual[i] + o_dual[i]       )
           return Dual(self.real + other.real		,				UpperCamelCase__       )

    lowercase						=		__add__



    def __sub__(					self		,				UpperCamelCase__       )						->      Tuple:





           '''simple docstring'''
           return self + other * -1



    def __mul__(					self		,				UpperCamelCase__       )						->      int:





           '''simple docstring'''
           if not isinstance(UpperCamelCase__		,				UpperCamelCase__       ):
                  A_						=      []
                  for i in self.duals:
                         new_duals.append(i * other       )
                  return Dual(self.real * other		,				UpperCamelCase__       )
           A_						=      [0] * (len(self.duals       ) + len(other.duals       ) + 1)
           for i, item in enumerate(self.duals       ):
                  for j, jtem in enumerate(other.duals       ):
                         new_duals[i + j + 1] += item * jtem
           for k in range(len(self.duals       )       ):
                  new_duals[k] += self.duals[k] * other.real
           for index in range(len(other.duals       )       ):
                  new_duals[index] += other.duals[index] * self.real
           return Dual(self.real * other.real		,				UpperCamelCase__       )

    lowercase						=		__mul__



    def __truediv__(					self		,				UpperCamelCase__       )						->      Dict:





           '''simple docstring'''
           if not isinstance(UpperCamelCase__		,				UpperCamelCase__       ):
                  A_						=      []
                  for i in self.duals:
                         new_duals.append(i / other       )
                  return Dual(self.real / other		,				UpperCamelCase__       )
           raise ValueError



    def __floordiv__(					self		,				UpperCamelCase__       )						->      List[Any]:





           '''simple docstring'''
           if not isinstance(UpperCamelCase__		,				UpperCamelCase__       ):
                  A_						=      []
                  for i in self.duals:
                         new_duals.append(i // other       )
                  return Dual(self.real // other		,				UpperCamelCase__       )
           raise ValueError







    def __pow__(					self		,				UpperCamelCase__       )						->      Optional[Any]:





           '''simple docstring'''
           if n < 0 or isinstance(UpperCamelCase__		,				UpperCamelCase__       ):
                  raise ValueError("""power must be a positive integer"""       )
           if n == 0:
                  return 1
           if n == 1:
                  return self
           A_						=      self
           for _ in range(n - 1       ):
                  x *= self
           return x

def       UpperCAmelCase__    (      UpperCAmelCase__,       UpperCAmelCase__,       UpperCAmelCase__				)			->			Any:







       if not callable(UpperCAmelCase__				):
              raise ValueError("""differentiate() requires a function as input for func"""				)
       if not isinstance(UpperCAmelCase__,       (float, int)				):
              raise ValueError("""differentiate() requires a float as input for position"""				)
       if not isinstance(UpperCAmelCase__,       UpperCAmelCase__				):
              raise ValueError("""differentiate() requires an int as input for order"""				)
       A_						=      Dual(UpperCAmelCase__,       1				)
       A_						=      func(UpperCAmelCase__				)
       if order == 0:
              return result.real
       return result.duals[order - 1] * factorial(UpperCAmelCase__				)


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      def       UpperCAmelCase__    (      UpperCAmelCase__				)			->			Optional[Any]:
             return y**2 * y**4







      print(differentiate(f, 9, 2))



'''simple docstring'''







def       UpperCAmelCase__    (      UpperCAmelCase__				)			->			int:
       assert (
           isinstance(UpperCAmelCase__,       UpperCAmelCase__				) and number_of_steps > 0
       ), F'''number_of_steps needs to be positive integer, your input {number_of_steps}'''
       if number_of_steps == 1:
              return 1
       A_					,			A_						=      1, 1
       for _ in range(number_of_steps - 1				):
              A_					,			A_						=      current + previous, current
       return current


if __name__ == "__main__":
      import doctest

      doctest.testmod()
