





'''simple docstring'''







def   _lowerCAmelCase		(		lowercase   ,      lowercase = False      )   ->   List[str]:
			if n == 2:
						return True
			if not n % 2 or n < 2:
						return False
			if n > 5 and n % 10 not in (1, 3, 7, 9):  # can quickly check last digit
						return False
			if n > 3_3170_4406_4679_8873_8596_1981 and not allow_probable:
						raise ValueError(
						    """Warning: upper bound of deterministic test is exceeded. """
						    """Pass allow_probable=True to allow probabilistic test. """
						    """A return value of True indicates a probable prime."""      )
			# array bounds provided by analysis
			__lowerCAmelCase   		=							[
			    2047,
			    137_3653,
			    2532_6001,
			    32_1503_1751,
			    2_1523_0289_8747,
			    3_4747_4966_0383,
			    341_5500_7172_8321,
			    1,
			    382_5123_0565_4641_3051,
			    1,
			    1,
			    3186_6585_7834_0311_5116_7461,
			    3_3170_4406_4679_8873_8596_1981,
			]

			__lowerCAmelCase   		=							[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
			for idx, _p in enumerate(UpperCamelCase_   ,      1      ):
						if n < _p:
									# then we have our last prime to check
									__lowerCAmelCase   		=							primes[:idx]
									break
			__lowerCAmelCase		,			__lowerCAmelCase   		=							n - 1, 0
			# break up n -1 into a power of 2 (s) and
			# remaining odd component
			# essentially, solve for d * 2 ** s == n - 1
			while d % 2 == 0:
						d //= 2
						s += 1
			for prime in plist:
						__lowerCAmelCase   		=							False
						for r in range(UpperCamelCase_      ):
									__lowerCAmelCase   		=							pow(UpperCamelCase_   ,      d * 2**r   ,      UpperCamelCase_      )
									# see article for analysis explanation for m
									if (r == 0 and m == 1) or ((m + 1) % n == 0):
												__lowerCAmelCase   		=							True
												# this loop will not determine compositeness
												break
						if pr:
									continue
						# if pr is False, then the above loop never evaluated to true,
						# and the n MUST be composite
						return False
			return True





def   _lowerCAmelCase		(		)   ->   List[Any]:
			assert not miller_rabin(561      )
			assert miller_rabin(563      )
			# 2047

			assert not miller_rabin(83_8201      )
			assert miller_rabin(83_8207      )
			# 1_373_653

			assert not miller_rabin(1731_6001      )
			assert miller_rabin(1731_6017      )
			# 25_326_001

			assert not miller_rabin(30_7838_6641      )
			assert miller_rabin(30_7838_6653      )
			# 3_215_031_751

			assert not miller_rabin(1_7130_4557_4801      )
			assert miller_rabin(1_7130_4557_4819      )
			# 2_152_302_898_747

			assert not miller_rabin(2_7797_9972_8307      )
			assert miller_rabin(2_7797_9972_8327      )
			# 3_474_749_660_383

			assert not miller_rabin(113_8500_2390_9441      )
			assert miller_rabin(113_8500_2390_9527      )
			# 341_550_071_728_321

			assert not miller_rabin(127_5041_0188_4880_4351      )
			assert miller_rabin(127_5041_0188_4880_4391      )
			# 3_825_123_056_546_413_051

			assert not miller_rabin(796_6646_4458_5077_8779_1867      )
			assert miller_rabin(796_6646_4458_5077_8779_1951      )
			# 318_665_857_834_031_151_167_461

			assert not miller_rabin(5528_4067_7446_6478_9766_0333      )
			assert miller_rabin(5528_4067_7446_6478_9766_0359      )
			# 3_317_044_064_679_887_385_961_981
			# upper limit for probabilistic test


if __name__ == "__main__":
	test_miller_rabin()





'''simple docstring'''
import string
from math import logaa







def       a_				(	UpperCamelCase_						,  UpperCamelCase_ ):
							A_			    =		document.translate(
							    str.maketrans(""						,  ""						,  string.punctuation ) ).replace("\n"						,  "" )
							A_			    =		document_without_punctuation.split(" " )  # word tokenization
							return len([word for word in tokenize_document if word.lower() == term.lower()] )







def       a_				(	UpperCamelCase_						,  UpperCamelCase_ ):
							A_			    =		corpus.lower().translate(
							    str.maketrans(""						,  ""						,  string.punctuation ) )  # strip all punctuation and replace it with ''
							A_			    =		corpus_without_punctuation.split("\n" )
							A_			    =		term.lower()
							return (len([doc for doc in docs if term in doc] ), len(UpperCamelCase_ ))







def       a_				(	UpperCamelCase_						,  UpperCamelCase_						,  UpperCamelCase_=False ):
							if smoothing:
														if n == 0:
																					raise ValueError("log10(0) is undefined." )
														return round(1 + logaa(n / (1 + df) )						,  3 )

							if df == 0:
														raise ZeroDivisionError("df must be > 0" )
							elif n == 0:
														raise ValueError("log10(0) is undefined." )
							return round(logaa(n / df )						,  3 )







def       a_				(	UpperCamelCase_						,  UpperCamelCase_ ):
							return round(tf * idf						,  3 )





