"""simple docstring"""




from math import factorial
class 						__A							:






     def __init__(		self				,     a__				,     a__  ):
       _lowerCAmelCase						:       Dict  			=     real
       if isinstance(a__				,     a__  ):
         _lowerCAmelCase						:       int  			=     [1] * rank
       else:
         _lowerCAmelCase						:       int  			=     rank






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






     def     __A  (		self  ):
       _lowerCAmelCase						:       str  			=     self.duals.copy()
       while cur[-1] == 0:
         cur.pop(-1  )
       return Dual(self.real				,     a__  )






     def __add__(		self				,     a__  ):
       if not isinstance(a__				,     a__  ):
         return Dual(self.real + other				,     self.duals  )
       _lowerCAmelCase						:       List[str]  			=     self.duals.copy()
       _lowerCAmelCase						:       Optional[int]  			=     other.duals.copy()
       if len(a__  ) > len(a__  ):
         o_dual.extend([1] * (len(a__  ) - len(a__  ))  )
       elif len(a__  ) < len(a__  ):
         s_dual.extend([1] * (len(a__  ) - len(a__  ))  )
       _lowerCAmelCase						:       List[str]  			=     []
       for i in range(len(a__  )  ):
         new_duals.append(s_dual[i] + o_dual[i]  )
       return Dual(self.real + other.real				,     a__  )

     _UpperCamelCase							:			Optional[Any]							=		__add__






     def __sub__(		self				,     a__  ):
       return self + other * -1






     def __mul__(		self				,     a__  ):
       if not isinstance(a__				,     a__  ):
         _lowerCAmelCase						:       Any  			=     []
         for i in self.duals:
           new_duals.append(i * other  )
         return Dual(self.real * other				,     a__  )
       _lowerCAmelCase						:       Union[str, Any]  			=     [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				,     a__  )

     _UpperCamelCase							:			Optional[Any]							=		__mul__






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






     def __floordiv__(		self				,     a__  ):
       if not isinstance(a__				,     a__  ):
         _lowerCAmelCase						:       List[str]  			=     []
         for i in self.duals:
           new_duals.append(i // other  )
         return Dual(self.real // other				,     a__  )
       raise ValueError






     def __pow__(		self				,     a__  ):
       if n < 0 or isinstance(a__				,     a__  ):
         raise ValueError("""power must be a positive integer"""  )
       if n == 0:
         return 1
       if n == 1:
         return self
       _lowerCAmelCase						:       Union[str, Any]  			=     self
       for _ in range(n - 1  ):
         x *= self
       return x







def 							SCREAMING_SNAKE_CASE     (			_lowerCamelCase						:       Union[str, Any]			,_lowerCamelCase						:       int			,_lowerCamelCase						:       List[str]	)       ->      Tuple:
  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"""	)
  _lowerCAmelCase						:       int  			=     Dual(_lowerCamelCase			,1	)
  _lowerCAmelCase						:       str  			=     func(_lowerCamelCase	)
  if order == 0:
    return result.real
  return result.duals[order - 1] * factorial(_lowerCamelCase	)


if __name__ == "__main__":
     import doctest

     doctest.testmod()





     def 							SCREAMING_SNAKE_CASE     (			_lowerCamelCase						:       Dict	)       ->      Dict:
       return y**2 * y**4






     print(differentiate(f, 9, 2))





"""simple docstring"""




_a      :		List[str]           =   {
    'Pillow': 'Pillow',
    'accelerate': 'accelerate>=0.11.0',
    'compel': 'compel==0.1.8',
    'black': 'black~=23.1',
    'datasets': 'datasets',
    'filelock': 'filelock',
    'flax': 'flax>=0.4.1',
    'hf-doc-builder': 'hf-doc-builder>=0.3.0',
    'huggingface-hub': 'huggingface-hub>=0.13.2',
    'requests-mock': 'requests-mock==1.10.0',
    'importlib_metadata': 'importlib_metadata',
    'invisible-watermark': 'invisible-watermark',
    'isort': 'isort>=5.5.4',
    'jax': 'jax>=0.2.8,!=0.3.2',
    'jaxlib': 'jaxlib>=0.1.65',
    'Jinja2': 'Jinja2',
    'k-diffusion': 'k-diffusion>=0.0.12',
    'torchsde': 'torchsde',
    'note_seq': 'note_seq',
    'librosa': 'librosa',
    'numpy': 'numpy',
    'omegaconf': 'omegaconf',
    'parameterized': 'parameterized',
    'protobuf': 'protobuf>=3.20.3,<4',
    'pytest': 'pytest',
    'pytest-timeout': 'pytest-timeout',
    'pytest-xdist': 'pytest-xdist',
    'ruff': 'ruff>=0.0.241',
    'safetensors': 'safetensors',
    'sentencepiece': 'sentencepiece>=0.1.91,!=0.1.92',
    'scipy': 'scipy',
    'onnx': 'onnx',
    'regex': 'regex!=2019.12.17',
    'requests': 'requests',
    'tensorboard': 'tensorboard',
    'torch': 'torch>=1.4',
    'torchvision': 'torchvision',
    'transformers': 'transformers>=4.25.1',
    'urllib3': 'urllib3<=2.0.0',
}




