def 		__lowerCamelCase							(UpperCAmelCase__   :							list[list[int]]     ):
					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."     )

					SCREAMING_SNAKE_CASE				=				{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."     )

					SCREAMING_SNAKE_CASE				=				len(UpperCAmelCase__     )

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

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

					SCREAMING_SNAKE_CASE				=				0

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

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

					SCREAMING_SNAKE_CASE				=				0
					SCREAMING_SNAKE_CASE				=				"f(x)="

					while count < x:
										SCREAMING_SNAKE_CASE				=				solution[count].split("E"     )
										if len(UpperCAmelCase__     ) > 1:
															SCREAMING_SNAKE_CASE				=				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]]))





import json
import os
from dataclasses import dataclass

import numpy as np
import pyarrow as pa

import datasets
from utils import get_duration


_lowerCamelCase    :      int	 =						10_00_00_00_00_00
_lowerCamelCase    :      Union[str, Any]	 =						1_00_00

_lowerCamelCase ,  _lowerCamelCase    :      List[str]	 =						os.path.split(__file__)
_lowerCamelCase    :      Any	 =						os.path.join(RESULTS_BASEPATH, '''results''', RESULTS_FILENAME.replace('''.py''', '''.json'''))





def 		__lowerCamelCase							(UpperCAmelCase__   :							int					,       UpperCAmelCase__   :							int     ):
					SCREAMING_SNAKE_CASE				=				pa.Table.from_pydict({"col": [0] * chunk_size}     )
					SCREAMING_SNAKE_CASE				=				pa.concat_tables([table] * (num_examples // chunk_size)     )
					return datasets.Dataset(UpperCAmelCase__					,       fingerprint="table_100B"     )


@dataclass
class   lowercase			:
							lowercase__  :				int
							lowercase__  :				int
							lowercase__  :				int
							lowercase__  :				int




							def 		__snake_case(  self      :    Tuple					)    -> Dict:



												'''simple docstring'''

												SCREAMING_SNAKE_CASE				=				np.random.default_rng(self.seed					)
												SCREAMING_SNAKE_CASE				=				rng.integers(low=self.low		,							high=self.high		,							size=self.size					).tolist()




							def __iter__(  self      :    Union[str, Any]					)    -> Dict:



												'''simple docstring'''

												return iter(self._sampled_values					)





							def __len__(  self      :    Any					)    -> int:



												'''simple docstring'''

												return self.size





@get_duration
def 		__lowerCamelCase							(UpperCAmelCase__   :							datasets.Dataset     ):
					SCREAMING_SNAKE_CASE				=				dataset[0]





@get_duration
def 		__lowerCamelCase							(UpperCAmelCase__   :							datasets.Dataset     ):
					SCREAMING_SNAKE_CASE				=				dataset[-1]





@get_duration
def 		__lowerCamelCase							(UpperCAmelCase__   :							datasets.Dataset     ):
					SCREAMING_SNAKE_CASE				=				dataset[range(len(UpperCAmelCase__     ) // 2					,       len(UpperCAmelCase__     ) // 2 + 1_0_2_4     )]





@get_duration
def 		__lowerCamelCase							(UpperCAmelCase__   :							datasets.Dataset     ):
					SCREAMING_SNAKE_CASE				=				dataset[RandIter(0					,       len(UpperCAmelCase__     )					,       1_0_2_4					,       seed=4_2     )]





def 		__lowerCamelCase							():
					SCREAMING_SNAKE_CASE				=				{"num examples": SPEED_TEST_N_EXAMPLES}
					SCREAMING_SNAKE_CASE				=				(get_first_row, get_last_row, get_batch_of_aaaa_rows, get_batch_of_aaaa_random_rows)
					print("generating dataset"     )
					SCREAMING_SNAKE_CASE				=				generate_100B_dataset(num_examples=UpperCAmelCase__					,       chunk_size=UpperCAmelCase__     )
					print("Functions"     )
					for func in functions:
										print(func.__name__     )
										SCREAMING_SNAKE_CASE				=				func(UpperCAmelCase__     )

					with open(UpperCAmelCase__					,       "wb"     ) as f:
										f.write(json.dumps(UpperCAmelCase__     ).encode("utf-8"     )     )


if __name__ == "__main__":  # useful to run the profiler
							benchmark_table_100B()




