



'''simple docstring'''






def 			snake_case_					( _lowerCAmelCase  :						list[list[int]]  )				->       str:
       if len(_lowerCAmelCase  ) == 0 or not all(len(_lowerCAmelCase  ) == 2 for pair in coordinates  ):
              raise ValueError('''The program cannot work out a fitting polynomial.'''  )

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

       UpperCAmelCase			:	Optional[Any]										=		{x for x, _ in coordinates}
       if len(_lowerCAmelCase  ) == 1:
              return f"""x={coordinates[0][0]}"""

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

       UpperCAmelCase			:	List[Any]										=		len(_lowerCAmelCase  )

       UpperCAmelCase			:	Dict										=		0
       UpperCAmelCase			:	list[list[float]]										=		[]
       # put the x and x to the power values in a matrix
       while count_of_line < x:
              UpperCAmelCase			:	List[Any]										=		0
              UpperCAmelCase			:	Optional[int]										=		coordinates[count_of_line][0]
              UpperCAmelCase			:	list[float]										=		[]
              while count_in_line < x:
                     count_line.append(a ** (x - (count_in_line + 1))  )
                     count_in_line += 1
              matrix.append(_lowerCAmelCase  )
              count_of_line += 1

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

       UpperCAmelCase			:	List[str]										=		0

       while count < x:
              UpperCAmelCase			:	Tuple										=		0
              while zahlen < x:
                     if count == zahlen:
                            zahlen += 1
                     if zahlen == x:
                            break
                     UpperCAmelCase			:	Union[str, Any]										=		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

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

       UpperCAmelCase			:	Optional[Any]										=		0
       UpperCAmelCase			:	Optional[int]										=		'''f(x)='''

       while count < x:
              UpperCAmelCase			:	list[str]										=		solution[count].split('''E'''  )
              if len(_lowerCAmelCase  ) > 1:
                     UpperCAmelCase			:	int										=		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'''






import importlib.metadata
import subprocess
import sys


def 			snake_case_					( _lowerCAmelCase  :						bool = False  )				->       str:
							UpperCAmelCase			:	List[str]										=		False
							if "IPython" in sys.modules:
														UpperCAmelCase			:	Optional[Any]										=		'''google.colab''' in str(sys.modules['''IPython'''].get_ipython()  )

							if in_colab:
														if upgrade:
																					UpperCAmelCase			:	List[Any]										=		['''pip''', '''install''', '''-U''', '''torch''']
																					subprocess.run(_lowerCAmelCase       ,				check=_lowerCAmelCase  )
														# get the current version of torch
														UpperCAmelCase			:	Dict										=		importlib.metadata.version('''torch'''  )
														UpperCAmelCase			:	int										=		torch_version[: torch_version.rindex('''.'''  )]
														UpperCAmelCase			:	Any										=		f"""https://storage.googleapis.com/tpu-pytorch/wheels/colab/torch_xla-{torch_version_trunc}-cp37-cp37m-linux_x86_64.whl"""
														UpperCAmelCase			:	List[str]										=		['''pip''', '''install''', xla_wheel]
														subprocess.run(_lowerCAmelCase       ,				check=_lowerCAmelCase  )
							else:
														raise RuntimeError('''`install_xla` utility works only on google colab.'''  )





