

def 							__lowercase     (	UpperCAmelCase__   ):

					"""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.'   )

					__lowerCAmelCase       =				{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.'   )

					__lowerCAmelCase       =				len(UpperCAmelCase__   )

					__lowerCAmelCase       =				0
					__lowerCAmelCase       =				[]
					# put the x and x to the power values in a matrix
					while count_of_line < x:
										__lowerCAmelCase       =				0
										__lowerCAmelCase       =				coordinates[count_of_line][0]
										__lowerCAmelCase       =				[]
										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

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

					__lowerCAmelCase       =				0

					while count < x:
										__lowerCAmelCase       =				0
										while zahlen < x:
															if count == zahlen:
																				zahlen += 1
															if zahlen == x:
																				break
															__lowerCAmelCase       =				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

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

					__lowerCAmelCase       =				0
					__lowerCAmelCase       =				'f(x)='

					while count < x:
										__lowerCAmelCase       =				solution[count].split('E'   )
										if len(UpperCAmelCase__   ) > 1:
															__lowerCAmelCase       =				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, 1_1]]))
		print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
		print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))









import os

from apache_beam.io.filesystems import FileSystems
from apache_beam.pipeline import Pipeline

from .logging import get_logger


lowerCamelCase   					=  2 << 2_0  # 2mb
lowerCamelCase   					=  get_logger(__name__)





class      snake_case_      (					_a				):


							"""simple docstring"""





							def 							A__	(			self				):
												__lowerCAmelCase       =				self._options.get_all_options().get('runner'				)
												return runner in [None, "DirectRunner", "PortableRunner"]


def 							__lowercase     (	UpperCAmelCase__		,       UpperCAmelCase__		,       UpperCAmelCase__=False   ):

					"""simple docstring"""






					__lowerCAmelCase       =				FileSystems
					if fs.exists(UpperCAmelCase__   ):
										if force_upload:
															logger.info(F"""Remote path already exist: {remote_file_path}. Overwriting it as force_upload=True."""   )
										else:
															logger.info(F"""Remote path already exist: {remote_file_path}. Skipping it as force_upload=False."""   )
															return
					with fs.create(UpperCAmelCase__   ) as remote_file:
										with open(UpperCAmelCase__		,       'rb'   ) as local_file:
															__lowerCAmelCase       =				local_file.read(UpperCAmelCase__   )
															while chunk:
																				remote_file.write(UpperCAmelCase__   )
																				__lowerCAmelCase       =				local_file.read(UpperCAmelCase__   )


def 							__lowercase     (	UpperCAmelCase__		,       UpperCAmelCase__		,       UpperCAmelCase__=False   ):

					"""simple docstring"""






					__lowerCAmelCase       =				FileSystems
					if os.path.exists(UpperCAmelCase__   ):
										if force_download:
															logger.info(F"""Local path already exist: {remote_file_path}. Overwriting it as force_upload=True."""   )
										else:
															logger.info(F"""Local path already exist: {remote_file_path}. Skipping it as force_upload=False."""   )
															return
					with fs.open(UpperCAmelCase__   ) as remote_file:
										with open(UpperCAmelCase__		,       'wb'   ) as local_file:
															__lowerCAmelCase       =				remote_file.read(UpperCAmelCase__   )
															while chunk:
																				local_file.write(UpperCAmelCase__   )
																				__lowerCAmelCase       =				remote_file.read(UpperCAmelCase__   )






