




from math import factorial





class     lowerCamelCase__:
      def __init__(				self: List[str]    ,       UpperCamelCase_: str    ,       UpperCamelCase_: int   ):
             __lowerCamelCase   =						real
             if isinstance(UpperCamelCase_    ,       UpperCamelCase_   ):
                    __lowerCamelCase   =						[1] * rank
             else:
                    __lowerCamelCase   =						rank
      def __repr__(				self: List[Any]   ):
             return (
                 F'{self.real}+'
                 F'{"+".join(str(UpperCamelCase_   )+"E"+str(n+1   )for n,dual in enumerate(self.duals   )   )}'
             )
      def 			lowerCAmelCase__						(				self: Tuple   ):
             __lowerCamelCase   =						self.duals.copy()
             while cur[-1] == 0:
                    cur.pop(-1   )
             return Dual(self.real    ,       UpperCamelCase_   )
      def __add__(				self: int    ,       UpperCamelCase_: Tuple   ):
             if not isinstance(UpperCamelCase_    ,       UpperCamelCase_   ):
                    return Dual(self.real + other    ,       self.duals   )
             __lowerCamelCase   =						self.duals.copy()
             __lowerCamelCase   =						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_   ))   )
             __lowerCamelCase   =						[]
             for i in range(len(UpperCamelCase_   )   ):
                    new_duals.append(s_dual[i] + o_dual[i]   )
             return Dual(self.real + other.real    ,       UpperCamelCase_   )

      UpperCAmelCase__      :	Dict    		=   __add__
      def __sub__(				self: List[str]    ,       UpperCamelCase_: Dict   ):
             return self + other * -1
      def __mul__(				self: Any    ,       UpperCamelCase_: Optional[Any]   ):
             if not isinstance(UpperCamelCase_    ,       UpperCamelCase_   ):
                    __lowerCamelCase   =						[]
                    for i in self.duals:
                           new_duals.append(i * other   )
                    return Dual(self.real * other    ,       UpperCamelCase_   )
             __lowerCamelCase   =						[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_   )

      UpperCAmelCase__      :	Optional[int]    		=   __mul__
      def __truediv__(				self: Dict    ,       UpperCamelCase_: Tuple   ):
             if not isinstance(UpperCamelCase_    ,       UpperCamelCase_   ):
                    __lowerCamelCase   =						[]
                    for i in self.duals:
                           new_duals.append(i / other   )
                    return Dual(self.real / other    ,       UpperCamelCase_   )
             raise ValueError
      def __floordiv__(				self: int    ,       UpperCamelCase_: int   ):
             if not isinstance(UpperCamelCase_    ,       UpperCamelCase_   ):
                    __lowerCamelCase   =						[]
                    for i in self.duals:
                           new_duals.append(i // other   )
                    return Dual(self.real // other    ,       UpperCamelCase_   )
             raise ValueError




      def __pow__(				self: Tuple    ,       UpperCamelCase_: str   ):
             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
             __lowerCamelCase   =						self
             for _ in range(n - 1   ):
                    x *= self
             return x




def 					lowerCamelCase__       (						A__       :					List[str]  ,			A__       :					str  ,			A__       :					str	):





       '''simple docstring'''



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


if __name__ == "__main__":
       import doctest

       doctest.testmod()




       def 					lowerCamelCase__       (						A__       :					int	):





              '''simple docstring'''



              return y**2 * y**4




       print(differentiate(f, 9, 2))





from typing import Dict, List, Optional, Union

import numpy as np

from ...image_processing_utils import BaseImageProcessor, BatchFeature, get_size_dict
from ...image_transforms import (
    center_crop,
    convert_to_rgb,
    get_resize_output_image_size,
    normalize,
    rescale,
    resize,
    to_channel_dimension_format,
)
from ...image_utils import (
    OPENAI_CLIP_MEAN,
    OPENAI_CLIP_STD,
    ChannelDimension,
    ImageInput,
    PILImageResampling,
    make_list_of_images,
    to_numpy_array,
    valid_images,
)
from ...utils import TensorType, is_vision_available, logging


_A		       =						logging.get_logger(__name__)


if is_vision_available():
   import PIL






class     UpperCAmelCase__       (					A_		):



    """simple docstring"""






    UpperCAmelCase__						:		Union[str, Any]      =  ["pixel_values"]






    def __init__(			self					,   A_ = True					,   A_ = None					,   A_ = PILImageResampling.BICUBIC					,   A_ = True					,   A_ = None					,   A_ = True					,   A_ = 1 / 255					,   A_ = True					,   A_ = None					,   A_ = None					,   A_ = True					,   **A_					,   )  ->       None:
         super().__init__(**A_    )
         __UpperCamelCase    							=size if size is not None else {'shortest_edge': 224}
         __UpperCamelCase    							=get_size_dict(A_					,   default_to_square=A_    )
         __UpperCamelCase    							=crop_size if crop_size is not None else {'height': 224, 'width': 224}
         __UpperCamelCase    							=get_size_dict(A_					,   default_to_square=A_					,   param_name='crop_size'    )

         __UpperCamelCase    							=do_resize
         __UpperCamelCase    							=size
         __UpperCamelCase    							=resample
         __UpperCamelCase    							=do_center_crop
         __UpperCamelCase    							=crop_size
         __UpperCamelCase    							=do_rescale
         __UpperCamelCase    							=rescale_factor
         __UpperCamelCase    							=do_normalize
         __UpperCamelCase    							=image_mean if image_mean is not None else OPENAI_CLIP_MEAN
         __UpperCamelCase    							=image_std if image_std is not None else OPENAI_CLIP_STD
         __UpperCamelCase    							=do_convert_rgb






    def _a					(			self					,   A_					,   A_					,   A_ = PILImageResampling.BICUBIC					,   A_ = None					,   **A_					,   )  ->       np.ndarray:
         __UpperCamelCase    							=get_size_dict(A_					,   default_to_square=A_    )
         if "shortest_edge" not in size:
              raise ValueError(f'The `size` parameter must contain the key `shortest_edge`. Got {size.keys()}'    )
         __UpperCamelCase    							=get_resize_output_image_size(A_					,   size=size['shortest_edge']					,   default_to_square=A_    )
         return resize(A_					,   size=A_					,   resample=A_					,   data_format=A_					,   **A_    )






    def _a					(			self					,   A_					,   A_					,   A_ = None					,   **A_					,   )  ->       np.ndarray:
         __UpperCamelCase    							=get_size_dict(A_    )
         if "height" not in size or "width" not in size:
              raise ValueError(f'The `size` parameter must contain the keys (height, width). Got {size.keys()}'    )
         return center_crop(A_					,   size=(size['height'], size['width'])					,   data_format=A_					,   **A_    )






    def _a					(			self					,   A_					,   A_					,   A_ = None					,   **A_					,   )  ->       Union[str, Any]:
         return rescale(A_					,   scale=A_					,   data_format=A_					,   **A_    )






    def _a					(			self					,   A_					,   A_					,   A_					,   A_ = None					,   **A_					,   )  ->       np.ndarray:
         return normalize(A_					,   mean=A_					,   std=A_					,   data_format=A_					,   **A_    )






    def _a					(			self					,   A_					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = None					,   A_ = ChannelDimension.FIRST					,   **A_					,   )  ->       PIL.Image.Image:
         __UpperCamelCase    							=do_resize if do_resize is not None else self.do_resize
         __UpperCamelCase    							=size if size is not None else self.size
         __UpperCamelCase    							=get_size_dict(A_					,   param_name='size'					,   default_to_square=A_    )
         __UpperCamelCase    							=resample if resample is not None else self.resample
         __UpperCamelCase    							=do_center_crop if do_center_crop is not None else self.do_center_crop
         __UpperCamelCase    							=crop_size if crop_size is not None else self.crop_size
         __UpperCamelCase    							=get_size_dict(A_					,   param_name='crop_size'					,   default_to_square=A_    )
         __UpperCamelCase    							=do_rescale if do_rescale is not None else self.do_rescale
         __UpperCamelCase    							=rescale_factor if rescale_factor is not None else self.rescale_factor
         __UpperCamelCase    							=do_normalize if do_normalize is not None else self.do_normalize
         __UpperCamelCase    							=image_mean if image_mean is not None else self.image_mean
         __UpperCamelCase    							=image_std if image_std is not None else self.image_std
         __UpperCamelCase    							=do_convert_rgb if do_convert_rgb is not None else self.do_convert_rgb

         __UpperCamelCase    							=make_list_of_images(A_    )

         if not valid_images(A_    ):
              raise ValueError(
                  'Invalid image type. Must be of type PIL.Image.Image, numpy.ndarray, '
                  'torch.Tensor, tf.Tensor or jax.ndarray.'    )

         if do_resize and size is None:
              raise ValueError('Size must be specified if do_resize is True.'    )

         if do_center_crop and crop_size is None:
              raise ValueError('Crop size must be specified if do_center_crop is True.'    )

         if do_rescale and rescale_factor is None:
              raise ValueError('Rescale factor must be specified if do_rescale is True.'    )

         if do_normalize and (image_mean is None or image_std is None):
              raise ValueError('Image mean and std must be specified if do_normalize is True.'    )

         # PIL RGBA images are converted to RGB
         if do_convert_rgb:
              __UpperCamelCase    							=[convert_to_rgb(A_    ) for image in images]

         # All transformations expect numpy arrays.
         __UpperCamelCase    							=[to_numpy_array(A_    ) for image in images]

         if do_resize:
              __UpperCamelCase    							=[self.resize(image=A_					,   size=A_					,   resample=A_    ) for image in images]

         if do_center_crop:
              __UpperCamelCase    							=[self.center_crop(image=A_					,   size=A_    ) for image in images]

         if do_rescale:
              __UpperCamelCase    							=[self.rescale(image=A_					,   scale=A_    ) for image in images]

         if do_normalize:
              __UpperCamelCase    							=[self.normalize(image=A_					,   mean=A_					,   std=A_    ) for image in images]

         __UpperCamelCase    							=[to_channel_dimension_format(A_					,   A_    ) for image in images]

         __UpperCamelCase    							={'pixel_values': images}
         return BatchFeature(data=A_					,   tensor_type=A_    )





