



"""simple docstring"""






def 			snake_case		(		lowerCAmelCase_  ,						lowerCAmelCase_	)						->				float:

  def get_matched_characters(lowerCAmelCase_  ,						lowerCAmelCase_	) -> str:
    _snake_case   	=      []
    _snake_case   	=      min(len(_stra	)  ,						len(_stra	)	) // 2
    for i, l in enumerate(_stra	):
      _snake_case   	=      int(max(0  ,						i - limit	)	)
      _snake_case   	=      int(min(i + limit + 1  ,						len(_stra	)	)	)
      if l in _stra[left:right]:
        matched.append(lowerCAmelCase_	)
        _snake_case   	=      f"""{_stra[0:_stra.index(lowerCAmelCase_	)]} {_stra[_stra.index(lowerCAmelCase_	) + 1:]}"""

    return "".join(lowerCAmelCase_	)

  # matching characters
  _snake_case   	=      get_matched_characters(lowerCAmelCase_  ,						lowerCAmelCase_	)
  _snake_case   	=      get_matched_characters(lowerCAmelCase_  ,						lowerCAmelCase_	)
  _snake_case   	=      len(lowerCAmelCase_	)

  # transposition
  _snake_case   	=      (
      len([(ca, ca) for ca, ca in zip(lowerCAmelCase_  ,						lowerCAmelCase_	) if ca != ca]	) // 2
  )

  if not match_count:
    _snake_case   	=      0.0
  else:
    _snake_case   	=      (
        1
        / 3
        * (
            match_count / len(lowerCAmelCase_	)
            + match_count / len(lowerCAmelCase_	)
            + (match_count - transpositions) / match_count
        )
    )

  # common prefix up to 4 characters
  _snake_case   	=      0
  for ca, ca in zip(stra[:4]  ,						stra[:4]	):
    if ca == ca:
      prefix_len += 1
    else:
      break

  return jaro + 0.1 * prefix_len * (1 - jaro)


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     print(jaro_winkler('''hello''', '''world'''))








'''simple docstring'''

def 							_UpperCamelCase				(   lowerCAmelCase__:					int		,lowerCAmelCase__:					bool = False   )      -> bool:
							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
							SCREAMING_SNAKE_CASE_				=				[
							    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,
							]

							SCREAMING_SNAKE_CASE_				=				[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41]
							for idx, _p in enumerate(lowerCAmelCase__		,1   ):
														if n < _p:
																					# then we have our last prime to check
																					SCREAMING_SNAKE_CASE_				=				primes[:idx]
																					break
							SCREAMING_SNAKE_CASE_				,    SCREAMING_SNAKE_CASE_				=				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:
														SCREAMING_SNAKE_CASE_				=				False
														for r in range(lowerCAmelCase__   ):
																					SCREAMING_SNAKE_CASE_				=				pow(lowerCAmelCase__		,d * 2**r		,lowerCAmelCase__   )
																					# see article for analysis explanation for m
																					if (r == 0 and m == 1) or ((m + 1) % n == 0):
																												SCREAMING_SNAKE_CASE_				=				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 							_UpperCamelCase				(   )      -> None:
							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()





