



'''simple docstring'''

from math import factorial


class 	a__     :


	def __init__(self : Optional[Any],					__UpperCAmelCase : Optional[Any],					__UpperCAmelCase : int      )     ->			Tuple:

					"""simple docstring"""

					SCREAMING_SNAKE_CASE  :						Union[str, Any]      					= real
					if isinstance(__UpperCAmelCase,					__UpperCAmelCase      ):
									SCREAMING_SNAKE_CASE  :						int      					= [1] * rank
					else:
									SCREAMING_SNAKE_CASE  :						Union[str, Any]      					= rank


	def __repr__(self : Optional[Any]      )     ->			Tuple:

					"""simple docstring"""

					return (
					    F'''{self.real}+'''
					    F'''{"+".join(str(__UpperCAmelCase      )+"E"+str(n+1      )for n,dual in enumerate(self.duals      )      )}'''
					)


	def 				lowercase__  (self : List[str]      )     ->			int:

					"""simple docstring"""

					SCREAMING_SNAKE_CASE  :						str      					= self.duals.copy()
					while cur[-1] == 0:
									cur.pop(-1      )
					return Dual(self.real,					__UpperCAmelCase      )


	def __add__(self : List[str],					__UpperCAmelCase : str      )     ->			List[str]:

					"""simple docstring"""

					if not isinstance(__UpperCAmelCase,					__UpperCAmelCase      ):
									return Dual(self.real + other,					self.duals      )
					SCREAMING_SNAKE_CASE  :						Optional[Any]      					= self.duals.copy()
					SCREAMING_SNAKE_CASE  :						Optional[Any]      					= 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      ))      )
					SCREAMING_SNAKE_CASE  :						List[str]      					= []
					for i in range(len(__UpperCAmelCase      )      ):
									new_duals.append(s_dual[i] + o_dual[i]      )
					return Dual(self.real + other.real,					__UpperCAmelCase      )

	__magic_name__		:    List[str]	    =     __add__


	def __sub__(self : int,					__UpperCAmelCase : int      )     ->			List[str]:

					"""simple docstring"""

					return self + other * -1


	def __mul__(self : Any,					__UpperCAmelCase : Dict      )     ->			List[str]:

					"""simple docstring"""

					if not isinstance(__UpperCAmelCase,					__UpperCAmelCase      ):
									SCREAMING_SNAKE_CASE  :						str      					= []
									for i in self.duals:
													new_duals.append(i * other      )
									return Dual(self.real * other,					__UpperCAmelCase      )
					SCREAMING_SNAKE_CASE  :						List[str]      					= [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      )

	__magic_name__		:    Any	    =     __mul__


	def __truediv__(self : List[Any],					__UpperCAmelCase : List[Any]      )     ->			List[Any]:

					"""simple docstring"""

					if not isinstance(__UpperCAmelCase,					__UpperCAmelCase      ):
									SCREAMING_SNAKE_CASE  :						Optional[Any]      					= []
									for i in self.duals:
													new_duals.append(i / other      )
									return Dual(self.real / other,					__UpperCAmelCase      )
					raise ValueError


	def __floordiv__(self : str,					__UpperCAmelCase : Tuple      )     ->			Dict:

					"""simple docstring"""

					if not isinstance(__UpperCAmelCase,					__UpperCAmelCase      ):
									SCREAMING_SNAKE_CASE  :						Tuple      					= []
									for i in self.duals:
													new_duals.append(i // other      )
									return Dual(self.real // other,					__UpperCAmelCase      )
					raise ValueError






	def __pow__(self : Tuple,					__UpperCAmelCase : List[str]      )     ->			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
					SCREAMING_SNAKE_CASE  :						str      					= self
					for _ in range(n - 1      ):
									x *= self
					return x






def       __lowercase  (_SCREAMING_SNAKE_CASE      :Optional[Any]    ,	_SCREAMING_SNAKE_CASE      :int    ,	_SCREAMING_SNAKE_CASE      :Optional[Any]       ):
				if not callable(_SCREAMING_SNAKE_CASE       ):
								raise ValueError('''differentiate() requires a function as input for func'''       )
				if not isinstance(_SCREAMING_SNAKE_CASE    ,	(float, int)       ):
								raise ValueError('''differentiate() requires a float as input for position'''       )
				if not isinstance(_SCREAMING_SNAKE_CASE    ,	_SCREAMING_SNAKE_CASE       ):
								raise ValueError('''differentiate() requires an int as input for order'''       )
				SCREAMING_SNAKE_CASE  :						List[str]      					= Dual(_SCREAMING_SNAKE_CASE    ,	1       )
				SCREAMING_SNAKE_CASE  :						int      					= func(_SCREAMING_SNAKE_CASE       )
				if order == 0:
								return result.real
				return result.duals[order - 1] * factorial(_SCREAMING_SNAKE_CASE       )


if __name__ == "__main__":
	import doctest

	doctest.testmod()






	def       __lowercase  (_SCREAMING_SNAKE_CASE      :str       ):
					return y**2 * y**4




	print(differentiate(f, 9, 2))









'''simple docstring'''

from ..utils import DummyObject, requires_backends


class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[Any]	    =     ["sentencepiece"]


	def __init__(self : Optional[Any],					*__UpperCAmelCase : List[Any],					**__UpperCAmelCase : List[Any]      )     ->			Optional[int]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Tuple	    =     ["sentencepiece"]


	def __init__(self : Optional[int],					*__UpperCAmelCase : int,					**__UpperCAmelCase : List[str]      )     ->			int:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : List[str],					*__UpperCAmelCase : str,					**__UpperCAmelCase : List[Any]      )     ->			str:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : Optional[int],					*__UpperCAmelCase : str,					**__UpperCAmelCase : Optional[Any]      )     ->			List[Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[str]	    =     ["sentencepiece"]


	def __init__(self : Tuple,					*__UpperCAmelCase : Union[str, Any],					**__UpperCAmelCase : Tuple      )     ->			Any:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Union[str, Any]	    =     ["sentencepiece"]


	def __init__(self : List[Any],					*__UpperCAmelCase : List[Any],					**__UpperCAmelCase : Optional[Any]      )     ->			str:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    int	    =     ["sentencepiece"]


	def __init__(self : Tuple,					*__UpperCAmelCase : int,					**__UpperCAmelCase : Any      )     ->			Union[str, Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    str	    =     ["sentencepiece"]


	def __init__(self : int,					*__UpperCAmelCase : Optional[Any],					**__UpperCAmelCase : Tuple      )     ->			Optional[Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    str	    =     ["sentencepiece"]


	def __init__(self : Tuple,					*__UpperCAmelCase : Optional[Any],					**__UpperCAmelCase : Tuple      )     ->			Dict:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[Any]	    =     ["sentencepiece"]


	def __init__(self : Tuple,					*__UpperCAmelCase : Optional[int],					**__UpperCAmelCase : Tuple      )     ->			int:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Tuple	    =     ["sentencepiece"]


	def __init__(self : List[Any],					*__UpperCAmelCase : Optional[int],					**__UpperCAmelCase : Optional[int]      )     ->			Union[str, Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    int	    =     ["sentencepiece"]


	def __init__(self : str,					*__UpperCAmelCase : str,					**__UpperCAmelCase : str      )     ->			Optional[int]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[str]	    =     ["sentencepiece"]


	def __init__(self : int,					*__UpperCAmelCase : List[str],					**__UpperCAmelCase : str      )     ->			Union[str, Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    str	    =     ["sentencepiece"]


	def __init__(self : Tuple,					*__UpperCAmelCase : Tuple,					**__UpperCAmelCase : str      )     ->			List[Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Any	    =     ["sentencepiece"]


	def __init__(self : Dict,					*__UpperCAmelCase : Optional[int],					**__UpperCAmelCase : List[str]      )     ->			Optional[Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : str,					*__UpperCAmelCase : Optional[int],					**__UpperCAmelCase : Tuple      )     ->			Dict:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[Any]	    =     ["sentencepiece"]


	def __init__(self : Union[str, Any],					*__UpperCAmelCase : int,					**__UpperCAmelCase : str      )     ->			Optional[int]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : Any,					*__UpperCAmelCase : str,					**__UpperCAmelCase : List[Any]      )     ->			Optional[int]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : List[Any],					*__UpperCAmelCase : Union[str, Any],					**__UpperCAmelCase : List[Any]      )     ->			List[Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : Dict,					*__UpperCAmelCase : Optional[Any],					**__UpperCAmelCase : str      )     ->			List[Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[Any]	    =     ["sentencepiece"]


	def __init__(self : List[Any],					*__UpperCAmelCase : Any,					**__UpperCAmelCase : str      )     ->			int:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    str	    =     ["sentencepiece"]


	def __init__(self : Optional[int],					*__UpperCAmelCase : str,					**__UpperCAmelCase : Tuple      )     ->			Any:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[int]	    =     ["sentencepiece"]


	def __init__(self : Union[str, Any],					*__UpperCAmelCase : Dict,					**__UpperCAmelCase : Optional[int]      )     ->			Tuple:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Tuple	    =     ["sentencepiece"]


	def __init__(self : List[str],					*__UpperCAmelCase : Union[str, Any],					**__UpperCAmelCase : Dict      )     ->			List[str]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Any	    =     ["sentencepiece"]


	def __init__(self : int,					*__UpperCAmelCase : Union[str, Any],					**__UpperCAmelCase : Optional[Any]      )     ->			Union[str, Any]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[Any]	    =     ["sentencepiece"]


	def __init__(self : Union[str, Any],					*__UpperCAmelCase : Tuple,					**__UpperCAmelCase : List[str]      )     ->			Optional[int]:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Any	    =     ["sentencepiece"]


	def __init__(self : Optional[int],					*__UpperCAmelCase : str,					**__UpperCAmelCase : str      )     ->			str:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    int	    =     ["sentencepiece"]


	def __init__(self : Dict,					*__UpperCAmelCase : Any,					**__UpperCAmelCase : List[str]      )     ->			Dict:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    Optional[Any]	    =     ["sentencepiece"]


	def __init__(self : List[Any],					*__UpperCAmelCase : Optional[Any],					**__UpperCAmelCase : int      )     ->			Any:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    int	    =     ["sentencepiece"]


	def __init__(self : Tuple,					*__UpperCAmelCase : Optional[Any],					**__UpperCAmelCase : int      )     ->			int:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )



class 	a__     (				metaclass=_lowercase	):
	__magic_name__		:    List[Any]	    =     ["sentencepiece"]


	def __init__(self : Optional[int],					*__UpperCAmelCase : str,					**__UpperCAmelCase : List[str]      )     ->			str:

					"""simple docstring"""

					requires_backends(self,					['''sentencepiece''']      )




