






from math import factorial
class       _UpperCAmelCase					:

				"""simple docstring"""




				def __init__(     self						:    int,  lowerCamelCase						:    Tuple,  lowerCamelCase						:    Tuple  ):

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




				def __repr__(     self						:    str  ):

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




				def 				lowercase__   (     self						:    Dict  ):

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




				def __add__(     self						:    int,  lowerCamelCase						:    List[str]  ):

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

				lowercase__											=       __add__




				def __sub__(     self						:    Dict,  lowerCamelCase						:    Union[str, Any]  ):

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




				def __mul__(     self						:    List[str],  lowerCamelCase						:    str  ):

								'''simple docstring'''
								if not isinstance(lowerCamelCase,  lowerCamelCase  ):
												lowercase__    =			[]
												for i in self.duals:
																new_duals.append(i * other  )
												return Dual(self.real * other,  lowerCamelCase  )
								lowercase__    =			[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,  lowerCamelCase  )

				lowercase__											=       __mul__




				def __truediv__(     self						:    Dict,  lowerCamelCase						:    Dict  ):

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




				def __floordiv__(     self						:    Tuple,  lowerCamelCase						:    int  ):

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




				def __pow__(     self						:    List[str],  lowerCamelCase						:    Tuple  ):

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






def   a			(						lowerCamelCase_		,   lowerCamelCase_		,   lowerCamelCase_				):






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


if __name__ == "__main__":
							import doctest

							doctest.testmod()






							def   a			(						lowerCamelCase_				):






											'''simple docstring'''
											return y**2 * y**4




							print(differentiate(f, 9, 2))












import warnings


warnings.warn(
    'memory_utils has been reorganized to utils.memory. Import `find_executable_batchsize` from the main `__init__`: '
    '`from accelerate import find_executable_batch_size` to avoid this warning.',
    FutureWarning,
)




