


'''simple docstring'''
def 				lowerCamelCase_   (   __UpperCamelCase				:					list[list[int]]		)				->  str:







     """simple docstring"""
     if len(__UpperCamelCase		) == 0 or not all(len(__UpperCamelCase		) == 2 for pair in coordinates		):
          raise ValueError('The program cannot work out a fitting polynomial.'		)

     if len({tuple(__UpperCamelCase		) for pair in coordinates}		) != len(__UpperCamelCase		):
          raise ValueError('The program cannot work out a fitting polynomial.'		)

     _A				=       {x for x, _ in coordinates}
     if len(__UpperCamelCase		) == 1:
          return F'x={coordinates[0][0]}'

     if len(__UpperCamelCase		) != len(__UpperCamelCase		):
          raise ValueError('The program cannot work out a fitting polynomial.'		)

     _A				=       len(__UpperCamelCase		)

     _A				=       0
     _A				=       []
     # put the x and x to the power values in a matrix
     while count_of_line < x:
          _A				=       0
          _A				=       coordinates[count_of_line][0]
          _A				=       []
          while count_in_line < x:
               count_line.append(a ** (x - (count_in_line + 1))		)
               count_in_line += 1
          matrix.append(__UpperCamelCase		)
          count_of_line += 1

     _A				=       0
     # put the y values into a vector
     _A				=       []
     while count_of_line < x:
          vector.append(coordinates[count_of_line][1]		)
          count_of_line += 1

     _A				=       0

     while count < x:
          _A				=       0
          while zahlen < x:
               if count == zahlen:
                    zahlen += 1
               if zahlen == x:
                    break
               _A				=       matrix[zahlen][count] / matrix[count][count]
               for counting_columns, item in enumerate(matrix[count]		):
                    # manipulating all the values in the matrix
                    matrix[zahlen][counting_columns] -= item * bruch
               # manipulating the values in the vector
               vector[zahlen] -= vector[count] * bruch
               zahlen += 1
          count += 1

     _A				=       0
     # make solutions
     _A				=       []
     while count < x:
          solution.append(str(vector[count] / matrix[count][count]		)		)
          count += 1

     _A				=       0
     _A				=       'f(x)='

     while count < x:
          _A				=       solution[count].split('E'		)
          if len(__UpperCamelCase		) > 1:
               _A				=       F'{remove_e[0]}*10^{remove_e[1]}'
          solved += F'x^{x - (count + 1)}*{solution[count]}'
          if count + 1 != x:
               solved += "+"
          count += 1

     return solved


if __name__ == "__main__":
    print(points_to_polynomial([]))
    print(points_to_polynomial([[]]))
    print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
    print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
    print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
    print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
    print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
    print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
    print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
    print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))









'''simple docstring'''
from typing import List, Optional, Tuple, Union

from . import logging
from .import_utils import is_torch_available, is_torch_version


if is_torch_available():
    import torch

lowerCAmelCase  =				logging.get_logger(__name__)  # pylint: disable=invalid-name

try:
    from torch._dynamo import allow_in_graph as maybe_allow_in_graph
except (ImportError, ModuleNotFoundError):

    def 				lowerCamelCase_   (   cls				:					Tuple		)				->  Dict:







         """simple docstring"""
         return cls

def 				lowerCamelCase_   (   __UpperCamelCase				:					Union[Tuple, List]       ,						__UpperCamelCase				:					Optional[Union[List["torch.Generator"], "torch.Generator"]] = None       ,						__UpperCamelCase				:					Optional["torch.device"] = None       ,						__UpperCamelCase				:					Optional["torch.dtype"] = None       ,						__UpperCamelCase				:					Optional["torch.layout"] = None       ,						)				->  Tuple:







     """simple docstring"""
     _A				=       device
     _A				=       shape[0]

     _A				=       layout or torch.strided
     _A				=       device or torch.device('cpu'		)

     if generator is not None:
          _A				=       generator.device.type if not isinstance(__UpperCamelCase       ,						__UpperCamelCase		) else generator[0].device.type
          if gen_device_type != device.type and gen_device_type == "cpu":
               _A				=       'cpu'
               if device != "mps":
                    logger.info(
                        F'The passed generator was created on \'cpu\' even though a tensor on {device} was expected.'
                        F' Tensors will be created on \'cpu\' and then moved to {device}. Note that one can probably'
                        F' slighly speed up this function by passing a generator that was created on the {device} device.'		)
          elif gen_device_type != device.type and gen_device_type == "cuda":
               raise ValueError(F'Cannot generate a {device} tensor from a generator of type {gen_device_type}.'		)

     if isinstance(__UpperCamelCase       ,						__UpperCamelCase		):
          _A				=       (1,) + shape[1:]
          _A				=       [
              torch.randn(__UpperCamelCase       ,						generator=generator[i]       ,						device=__UpperCamelCase       ,						dtype=__UpperCamelCase       ,						layout=__UpperCamelCase		)
              for i in range(__UpperCamelCase		)
          ]
          _A				=       torch.cat(__UpperCamelCase       ,						dim=0		).to(__UpperCamelCase		)
     else:
          _A				=       torch.randn(__UpperCamelCase       ,						generator=__UpperCamelCase       ,						device=__UpperCamelCase       ,						dtype=__UpperCamelCase       ,						layout=__UpperCamelCase		).to(__UpperCamelCase		)

     return latents

def 				lowerCamelCase_   (   __UpperCamelCase				:					Any		)				->  Optional[int]:







     """simple docstring"""
     if is_torch_version('<'       ,						'2.0.0'		) or not hasattr(__UpperCamelCase       ,						'_dynamo'		):
          return False
     return isinstance(__UpperCamelCase       ,						torch._dynamo.eval_frame.OptimizedModule		)





