





"""simple docstring"""





from __future__ import annotations

from collections.abc import Callable

_UpperCamelCase							=      list[list[float | int]]



def 				lowerCAmelCase_			(      SCREAMING_SNAKE_CASE       :							Matrix    ,						SCREAMING_SNAKE_CASE       :							Matrix   ):


							'''simple docstring'''







							__lowerCamelCase      :      int         =len(SCREAMING_SNAKE_CASE   )
							__lowerCamelCase      :      Matrix         =[[0 for _ in range(size + 1   )] for _ in range(SCREAMING_SNAKE_CASE   )]
							__lowerCamelCase      :      int
							__lowerCamelCase      :      int
							__lowerCamelCase      :      int
							__lowerCamelCase      :      int
							__lowerCamelCase      :      int
							__lowerCamelCase      :      float

							for row in range(SCREAMING_SNAKE_CASE   ):
														for col in range(SCREAMING_SNAKE_CASE   ):
																					__lowerCamelCase      :      Union[str, Any]         =matrix[row][col]

														__lowerCamelCase      :      Optional[int]         =vector[row][0]

							__lowerCamelCase      :      Any         =0
							__lowerCamelCase      :      Any         =0
							while row < size and col < size:
														# pivoting
														__lowerCamelCase      :      Any         =max((abs(augmented[rowa][col]   ), rowa) for rowa in range(SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE   )   )[
														    1
														]
														if augmented[pivot_row][col] == 0:
																					col += 1
																					continue
														else:
																					__lowerCamelCase    ,						__lowerCamelCase      :      int         =augmented[pivot_row], augmented[row]

														for rowa in range(row + 1    ,						SCREAMING_SNAKE_CASE   ):
																					__lowerCamelCase      :      Dict         =augmented[rowa][col] / augmented[row][col]
																					__lowerCamelCase      :      Dict         =0
																					for cola in range(col + 1    ,						size + 1   ):
																												augmented[rowa][cola] -= augmented[row][cola] * ratio

														row += 1
														col += 1

							# back substitution
							for col in range(1    ,						SCREAMING_SNAKE_CASE   ):
														for row in range(SCREAMING_SNAKE_CASE   ):
																					__lowerCamelCase      :      int         =augmented[row][col] / augmented[col][col]
																					for cola in range(SCREAMING_SNAKE_CASE    ,						size + 1   ):
																												augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
							return [
							    [round(augmented[row][size] / augmented[row][row]    ,						10   )] for row in range(SCREAMING_SNAKE_CASE   )
							]



def 				lowerCAmelCase_			(      SCREAMING_SNAKE_CASE       :							list[int]   ):


							'''simple docstring'''







							__lowerCamelCase      :      int         =len(SCREAMING_SNAKE_CASE   )
							__lowerCamelCase      :      Matrix         =[[0 for _ in range(SCREAMING_SNAKE_CASE   )] for _ in range(SCREAMING_SNAKE_CASE   )]
							__lowerCamelCase      :      Matrix         =[[0] for _ in range(SCREAMING_SNAKE_CASE   )]
							__lowerCamelCase      :      Matrix
							__lowerCamelCase      :      int
							__lowerCamelCase      :      int
							__lowerCamelCase      :      int

							for x_val, y_val in enumerate(SCREAMING_SNAKE_CASE   ):
														for col in range(SCREAMING_SNAKE_CASE   ):
																					__lowerCamelCase      :      Dict         =(x_val + 1) ** (size - col - 1)
														__lowerCamelCase      :      Tuple         =y_val

							__lowerCamelCase      :      Union[str, Any]         =solve(SCREAMING_SNAKE_CASE    ,						SCREAMING_SNAKE_CASE   )

							def interpolated_func(SCREAMING_SNAKE_CASE       :							int   ) -> int:
														return sum(
														    round(coeffs[x_val][0]   ) * (var ** (size - x_val - 1))
														    for x_val in range(SCREAMING_SNAKE_CASE   )   )

							return interpolated_func



def 				lowerCAmelCase_			(      SCREAMING_SNAKE_CASE       :							int   ):


							'''simple docstring'''







							return (
							    1
							    - variable
							    + variable**2
							    - variable**3
							    + variable**4
							    - variable**5
							    + variable**6
							    - variable**7
							    + variable**8
							    - variable**9
							    + variable**10
							)



def 				lowerCAmelCase_			(      SCREAMING_SNAKE_CASE       :							Callable[[int], int] = question_function    ,						SCREAMING_SNAKE_CASE       :							int = 10   ):


							'''simple docstring'''







							__lowerCamelCase      :      list[int]         =[func(SCREAMING_SNAKE_CASE   ) for x_val in range(1    ,						order + 1   )]

							__lowerCamelCase      :      list[Callable[[int], int]]         =[
							    interpolate(data_points[:max_coeff]   ) for max_coeff in range(1    ,						order + 1   )
							]

							__lowerCamelCase      :      int         =0
							__lowerCamelCase      :      Callable[[int], int]
							__lowerCamelCase      :      int

							for poly in polynomials:
														__lowerCamelCase      :      List[str]         =1
														while func(SCREAMING_SNAKE_CASE   ) == poly(SCREAMING_SNAKE_CASE   ):
																					x_val += 1

														ret += poly(SCREAMING_SNAKE_CASE   )

							return ret


if __name__ == "__main__":
							print(f'''{solution() = }''')











"""simple docstring"""





from typing import Dict, List

from nltk.translate import gleu_score

import datasets
from datasets import MetricInfo


_UpperCamelCase							=      '\\n@misc{wu2016googles,\n      title={Google\'s Neural Machine Translation System: Bridging the Gap between Human and Machine Translation},\n      author={Yonghui Wu and Mike Schuster and Zhifeng Chen and Quoc V. Le and Mohammad Norouzi and Wolfgang Macherey\n              and Maxim Krikun and Yuan Cao and Qin Gao and Klaus Macherey and Jeff Klingner and Apurva Shah and Melvin\n              Johnson and Xiaobing Liu and Łukasz Kaiser and Stephan Gouws and Yoshikiyo Kato and Taku Kudo and Hideto\n              Kazawa and Keith Stevens and George Kurian and Nishant Patil and Wei Wang and Cliff Young and\n              Jason Smith and Jason Riesa and Alex Rudnick and Oriol Vinyals and Greg Corrado and Macduff Hughes\n              and Jeffrey Dean},\n      year={2016},\n      eprint={1609.08144},\n      archivePrefix={arXiv},\n      primaryClass={cs.CL}\n}\n'

_UpperCamelCase							=      '\\nThe BLEU score has some undesirable properties when used for single\nsentences, as it was designed to be a corpus measure. We therefore\nuse a slightly different score for our RL experiments which we call\nthe \'GLEU score\'. For the GLEU score, we record all sub-sequences of\n1, 2, 3 or 4 tokens in output and target sequence (n-grams). We then\ncompute a recall, which is the ratio of the number of matching n-grams\nto the number of total n-grams in the target (ground truth) sequence,\nand a precision, which is the ratio of the number of matching n-grams\nto the number of total n-grams in the generated output sequence. Then\nGLEU score is simply the minimum of recall and precision. This GLEU\nscore\'s range is always between 0 (no matches) and 1 (all match) and\nit is symmetrical when switching output and target. According to\nour experiments, GLEU score correlates quite well with the BLEU\nmetric on a corpus level but does not have its drawbacks for our per\nsentence reward objective.\n'

_UpperCamelCase							=      '\\nComputes corpus-level Google BLEU (GLEU) score of translated segments against one or more references.\nInstead of averaging the sentence level GLEU scores (i.e. macro-average precision), Wu et al. (2016) sum up the matching\ntokens and the max of hypothesis and reference tokens for each sentence, then compute using the aggregate values.\n\nArgs:\n    predictions (list of str): list of translations to score.\n        Each translation should be tokenized into a list of tokens.\n    references (list of list of str): list of lists of references for each translation.\n        Each reference should be tokenized into a list of tokens.\n    min_len (int): The minimum order of n-gram this function should extract. Defaults to 1.\n    max_len (int): The maximum order of n-gram this function should extract. Defaults to 4.\n\nReturns:\n    \'google_bleu\': google_bleu score\n\nExamples:\n    Example 1:\n        >>> hyp1 = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'which\',\n        ...         \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'always\',\n        ...         \'disobeys\', \'the\', \'commands\', \'of\', \'the\', \'cat\']\n        >>> ref1a = [\'It\', \'is\', \'the\', \'guiding\', \'principle\', \'which\',\n        ...          \'guarantees\', \'the\', \'rubber\', \'duck\', \'forces\', \'never\',\n        ...          \'being\', \'under\', \'the\', \'command\', \'of\', \'the\', \'cat\']\n\n        >>> hyp2 = [\'he\', \'read\', \'the\', \'book\', \'because\', \'he\', \'was\',\n        ...         \'interested\', \'in\', \'world\', \'history\']\n        >>> ref2a = [\'he\', \'was\', \'interested\', \'in\', \'world\', \'history\',\n        ...          \'because\', \'he\', \'read\', \'the\', \'book\']\n\n        >>> list_of_references = [[ref1a], [ref2a]]\n        >>> hypotheses = [hyp1, hyp2]\n        >>> google_bleu = datasets.load_metric("google_bleu")\n        >>> results = google_bleu.compute(predictions=hypotheses, references=list_of_references)\n        >>> print(round(results["google_bleu"], 2))\n        0.44\n\n    Example 2:\n        >>> hyp1 = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'which\',\n        ...         \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'always\',\n        ...         \'disobeys\', \'the\', \'commands\', \'of\', \'the\', \'cat\']\n        >>> ref1a = [\'It\', \'is\', \'the\', \'guiding\', \'principle\', \'which\',\n        ...          \'guarantees\', \'the\', \'rubber\', \'duck\', \'forces\', \'never\',\n        ...          \'being\', \'under\', \'the\', \'command\', \'of\', \'the\', \'cat\']\n        >>> ref1b = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'that\',\n        ...          \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'will\', \'never\',\n        ...          \'heed\', \'the\', \'cat\', \'commands\']\n        >>> ref1c = [\'It\', \'is\', \'the\', \'practical\', \'guide\', \'for\', \'the\',\n        ...          \'rubber\', \'duck\', \'army\', \'never\', \'to\', \'heed\', \'the\', \'directions\',\n        ...          \'of\', \'the\', \'cat\']\n\n        >>> hyp2 = [\'he\', \'read\', \'the\', \'book\', \'because\', \'he\', \'was\',\n        ...         \'interested\', \'in\', \'world\', \'history\']\n        >>> ref2a = [\'he\', \'was\', \'interested\', \'in\', \'world\', \'history\',\n        ...          \'because\', \'he\', \'read\', \'the\', \'book\']\n\n        >>> list_of_references = [[ref1a, ref1b, ref1c], [ref2a]]\n        >>> hypotheses = [hyp1, hyp2]\n        >>> google_bleu = datasets.load_metric("google_bleu")\n        >>> results = google_bleu.compute(predictions=hypotheses, references=list_of_references)\n        >>> print(round(results["google_bleu"], 2))\n        0.61\n\n    Example 3:\n        >>> hyp1 = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'which\',\n        ...         \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'always\',\n        ...         \'disobeys\', \'the\', \'commands\', \'of\', \'the\', \'cat\']\n        >>> ref1a = [\'It\', \'is\', \'the\', \'guiding\', \'principle\', \'which\',\n        ...          \'guarantees\', \'the\', \'rubber\', \'duck\', \'forces\', \'never\',\n        ...          \'being\', \'under\', \'the\', \'command\', \'of\', \'the\', \'cat\']\n        >>> ref1b = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'that\',\n        ...          \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'will\', \'never\',\n        ...          \'heed\', \'the\', \'cat\', \'commands\']\n        >>> ref1c = [\'It\', \'is\', \'the\', \'practical\', \'guide\', \'for\', \'the\',\n        ...          \'rubber\', \'duck\', \'army\', \'never\', \'to\', \'heed\', \'the\', \'directions\',\n        ...          \'of\', \'the\', \'cat\']\n\n        >>> hyp2 = [\'he\', \'read\', \'the\', \'book\', \'because\', \'he\', \'was\',\n        ...         \'interested\', \'in\', \'world\', \'history\']\n        >>> ref2a = [\'he\', \'was\', \'interested\', \'in\', \'world\', \'history\',\n        ...          \'because\', \'he\', \'read\', \'the\', \'book\']\n\n        >>> list_of_references = [[ref1a, ref1b, ref1c], [ref2a]]\n        >>> hypotheses = [hyp1, hyp2]\n        >>> google_bleu = datasets.load_metric("google_bleu")\n        >>> results = google_bleu.compute(predictions=hypotheses, references=list_of_references, min_len=2)\n        >>> print(round(results["google_bleu"], 2))\n        0.53\n\n    Example 4:\n        >>> hyp1 = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'which\',\n        ...         \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'always\',\n        ...         \'disobeys\', \'the\', \'commands\', \'of\', \'the\', \'cat\']\n        >>> ref1a = [\'It\', \'is\', \'the\', \'guiding\', \'principle\', \'which\',\n        ...          \'guarantees\', \'the\', \'rubber\', \'duck\', \'forces\', \'never\',\n        ...          \'being\', \'under\', \'the\', \'command\', \'of\', \'the\', \'cat\']\n        >>> ref1b = [\'It\', \'is\', \'a\', \'guide\', \'to\', \'action\', \'that\',\n        ...          \'ensures\', \'that\', \'the\', \'rubber\', \'duck\', \'will\', \'never\',\n        ...          \'heed\', \'the\', \'cat\', \'commands\']\n        >>> ref1c = [\'It\', \'is\', \'the\', \'practical\', \'guide\', \'for\', \'the\',\n        ...          \'rubber\', \'duck\', \'army\', \'never\', \'to\', \'heed\', \'the\', \'directions\',\n        ...          \'of\', \'the\', \'cat\']\n\n        >>> hyp2 = [\'he\', \'read\', \'the\', \'book\', \'because\', \'he\', \'was\',\n        ...         \'interested\', \'in\', \'world\', \'history\']\n        >>> ref2a = [\'he\', \'was\', \'interested\', \'in\', \'world\', \'history\',\n        ...          \'because\', \'he\', \'read\', \'the\', \'book\']\n\n        >>> list_of_references = [[ref1a, ref1b, ref1c], [ref2a]]\n        >>> hypotheses = [hyp1, hyp2]\n        >>> google_bleu = datasets.load_metric("google_bleu")\n        >>> results = google_bleu.compute(predictions=hypotheses,references=list_of_references, min_len=2, max_len=6)\n        >>> print(round(results["google_bleu"], 2))\n        0.4\n'
@datasets.utils.file_utils.add_start_docstrings(_DESCRIPTION       ,  _KWARGS_DESCRIPTION   )
class 		SCREAMING_SNAKE_CASE_							(    datasets.Metric   ):
				"""simple docstring"""




				def 					__lowercase  (     self  :Optional[Any]			):
											return datasets.MetricInfo(
											    description=_DESCRIPTION						,    citation=_CITATION						,    inputs_description=_KWARGS_DESCRIPTION						,    features=datasets.Features(
											        {
											            '''predictions''': datasets.Sequence(datasets.Value('''string'''						,    id='''token'''			)						,    id='''sequence'''			),
											            '''references''': datasets.Sequence(
											                datasets.Sequence(datasets.Value('''string'''						,    id='''token'''			)						,    id='''sequence'''			)						,    id='''references'''			),
											        }			)						,    )




				def 					__lowercase  (     self  :Optional[Any]						,    __lowercase  :List[List[List[str]]]						,    __lowercase  :List[List[str]]						,    __lowercase  :int = 1						,    __lowercase  :int = 4						,    ):
											return {
											    "google_bleu": gleu_score.corpus_gleu(
											        list_of_references=__lowercase						,    hypotheses=__lowercase						,    min_len=__lowercase						,    max_len=__lowercase			)
											}




