
'''simple docstring'''
import torch

from accelerate import Accelerator, DistributedType


class        SCREAMING_SNAKE_CASE__	:







				def __enter__(    self			)->						str:







											'''simple docstring'''






											if self.enabled:
																		UpperCamelCase					       =			self.model.no_sync()
																		self.model_sync_obj.__enter__()

											return self







				def __exit__(    self		,							A_		,							A_		,							A_			)->						Tuple:







											'''simple docstring'''






											if self.enabled:
																		# Average all models on exit
																		self._sync_and_avg_model_params()
																		self.model_sync_obj.__exit__(A_		,							A_		,							A_			)







				def __init__(    self		,							A_		,							A_		,							A_		,							A_ = True			)->						Any:







											'''simple docstring'''






											if accelerator.distributed_type not in [
											    DistributedType.NO,
											    DistributedType.MULTI_CPU,
											    DistributedType.MULTI_GPU,
											]:
																		raise NotImplementedError('LocalSGD is supported only for CPUs and GPUs (no DeepSpeed or MegatronLM)'			)
											UpperCamelCase					       =			enabled and accelerator.distributed_type != DistributedType.NO
											UpperCamelCase					       =			0
											if self.enabled:
																		UpperCamelCase					       =			accelerator
																		UpperCamelCase					       =			model
																		UpperCamelCase					       =			local_sgd_steps







				def 		UpperCAmelCase_  (    self			)->						Any:







											'''simple docstring'''






											self.num_steps += 1
											if not self.enabled:
																		return

											if self.num_steps % self.local_sgd_steps == 0:
																		self._sync_and_avg_model_params()







				def 		UpperCAmelCase_  (    self			)->						List[Any]:







											'''simple docstring'''






											self.accelerator.wait_for_everyone()
											with self.accelerator.autocast():
																		for param in self.model.parameters():
																									UpperCamelCase					       =			self.accelerator.reduce(param.data		,							reduction='mean'			)






'''simple docstring'''
def 						A_(			A		:						list[list[int]]):
							if len(A) == 0 or not all(len(A) == 2 for pair in coordinates):
														raise ValueError('The program cannot work out a fitting polynomial.')

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

							UpperCamelCase					       =			{x for x, _ in coordinates}
							if len(A) == 1:
														return f'''x={coordinates[0][0]}'''

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

							UpperCamelCase					       =			len(A)

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

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

							UpperCamelCase					       =			0

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

							UpperCamelCase					       =			0
							UpperCamelCase					       =			'f(x)='

							while count < x:
														UpperCamelCase					       =			solution[count].split('E')
														if len(A) > 1:
																					UpperCamelCase					       =			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]]))




