
"""simple docstring"""




def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE							):
	if len(__SCREAMING_SNAKE_CASE							) == 0 or not all(len(__SCREAMING_SNAKE_CASE							) == 2 for pair in coordinates							):
		raise ValueError('The program cannot work out a fitting polynomial.'							)

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

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

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

	lowercase					  =   len(__SCREAMING_SNAKE_CASE							)

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

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

	lowercase					  =   0

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

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

	lowercase					  =   0
	lowercase					  =   'f(x)='

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



# This test is meant to be run in on an instance with TPUs like this:
#
#   python examples/pytorch/xla_spawn.py --num_cores=8 tests/test_trainer_tpu.py
#
# Replace 8 with the number of TPU cores you have.
#

import sys
from typing import Dict

from transformers import EvalPrediction, HfArgumentParser, TrainingArguments, is_torch_available
from transformers.utils import logging


UpperCAmelCase     				=			logging.get_logger(__name__)


if is_torch_available():
	import torch
	from torch import nn
	from torch.utils.data import Dataset

	from transformers import Trainer







	class 	A_       (      __lowerCamelCase       ):




		'''simple docstring'''


		def __init__(	self		,   snake_case = 101	):
			lowercase					  =   length





		def __len__(	self	):
			return self.length





		def __getitem__(	self		,   snake_case	):
			return i







	class 	A_       :




		'''simple docstring'''


		def __call__(	self		,   snake_case	):
			return {"input_ids": torch.tensor(snake_case	), "labels": torch.tensor(snake_case	)}







	class 	A_       (      nn.Module       ):




		'''simple docstring'''


		def __init__(	self	):
			super().__init__()
			# Add some (unused) params otherwise DDP will complain.
			lowercase					  =   nn.Linear(120		,   80	)





		def      SCREAMING_SNAKE_CASE__						(	self		,   snake_case		,   snake_case=None	):
			if labels is not None:
				return torch.tensor(0.0		,   device=input_ids.device	), input_ids
			else:
				return input_ids






def 							UpperCAmelCase_		(							):
	lowercase					  =   HfArgumentParser((TrainingArguments,)							)
	sys.argv += ["--output_dir", "./examples"]
	lowercase					  =   parser.parse_args_into_dataclasses()[0]

	logger.warning(
	    F'''Process rank: {training_args.local_rank}, device: {training_args.device}, '''
	    F'''tpu_num_cores: {training_args.tpu_num_cores}'''			,							)

	# Essentially, what we want to verify in the distributed case is
	# that we get all samples back, in the right order.
	# (this is crucial for prediction for instance)
	for dataset_length in [1001, 256, 15]:
		lowercase					  =   DummyDataset(__SCREAMING_SNAKE_CASE							)

		def compute_metrics(__SCREAMING_SNAKE_CASE							) -> Dict:
			lowercase					  =   list(range(len(__SCREAMING_SNAKE_CASE							)							)							)
			lowercase					  =   p.predictions.tolist() == sequential and p.label_ids.tolist() == sequential
			return {"success": success}

		lowercase					  =   Trainer(
		    model=DummyModel()			,							args=__SCREAMING_SNAKE_CASE			,							data_collator=DummyDataCollator()			,							eval_dataset=__SCREAMING_SNAKE_CASE			,							compute_metrics=__SCREAMING_SNAKE_CASE			,							)
		lowercase					  =   trainer.evaluate()
		logger.info(__SCREAMING_SNAKE_CASE							)
		if metrics["eval_success"] is not True:
			logger.error(__SCREAMING_SNAKE_CASE							)
			exit(1							)

		lowercase					  =   trainer.predict(__SCREAMING_SNAKE_CASE							)
		logger.info(p.metrics							)
		if p.metrics["test_success"] is not True:
			logger.error(p.metrics							)
			exit(1							)

		lowercase					  =   2

		lowercase					  =   trainer.evaluate()
		logger.info(__SCREAMING_SNAKE_CASE							)
		if metrics["eval_success"] is not True:
			logger.error(__SCREAMING_SNAKE_CASE							)
			exit(1							)

		lowercase					  =   trainer.predict(__SCREAMING_SNAKE_CASE							)
		logger.info(p.metrics							)
		if p.metrics["test_success"] is not True:
			logger.error(p.metrics							)
			exit(1							)

		lowercase					  =   None

	logger.info('🔥 All distributed tests successful'							)






def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE							):
	# For xla_spawn (TPUs)
	main()


if __name__ == "__main__":
	main()

