






def 							_a	(  UpperCamelCase_       :					int )			-> str:
						"""simple docstring"""







						if bit_count < 0:
												raise ValueError("The given input must be positive" )

						# get the generated string sequence
						lowerCAmelCase__			   =    gray_code_sequence_string(A__ )
						#
						# convert them to integers
						for i in range(len(A__ ) ):
												lowerCAmelCase__			   =    int(sequence[i]     , 2 )

						return sequence


def 							_a	(  UpperCamelCase_       :					List[Any] )			-> List[str]:
						"""simple docstring"""







						if bit_count == 0:
												return ["0"]

						if bit_count == 1:
												return ["0", "1"]

						lowerCAmelCase__			   =    1 << bit_count  # defines the length of the sequence
						# 1<< n is equivalent to 2^n

						# recursive answer will generate answer for n-1 bits
						lowerCAmelCase__			   =    gray_code_sequence_string(bit_count - 1 )

						lowerCAmelCase__			   =    []

						# append 0 to first half of the smaller sequence generated
						for i in range(seq_len // 2 ):
												lowerCAmelCase__			   =    '0' + smaller_sequence[i]
												sequence.append(A__ )

						# append 1 to second half ... start from the end of the list
						for i in reversed(range(seq_len // 2 ) ):
												lowerCAmelCase__			   =    '1' + smaller_sequence[i]
												sequence.append(A__ )

						return sequence


if __name__ == "__main__":
	import doctest

	doctest.testmod()










def 						a__			(				A__,      A__				):

			def get_matched_characters(A__,      A__				) -> str:
						SCREAMING_SNAKE_CASE_		:     Dict					=					[]
						SCREAMING_SNAKE_CASE_		:     Any					=					min(len(_stra				),      len(_stra				)				) // 2
						for i, l in enumerate(_stra				):
									SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					int(max(0,      i - limit				)				)
									SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					int(min(i + limit + 1,      len(_stra				)				)				)
									if l in _stra[left:right]:
												matched.append(A__				)
												SCREAMING_SNAKE_CASE_		:     List[str]					=					F'''{_stra[0:_stra.index(A__				)]} {_stra[_stra.index(A__				) + 1:]}'''

						return "".join(A__				)

			# matching characters
			SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					get_matched_characters(A__,      A__				)
			SCREAMING_SNAKE_CASE_		:     int					=					get_matched_characters(A__,      A__				)
			SCREAMING_SNAKE_CASE_		:     Any					=					len(A__				)

			# transposition
			SCREAMING_SNAKE_CASE_		:     Optional[int]					=					(
			    len([(ca, ca) for ca, ca in zip(A__,      A__				) if ca != ca]				) // 2
			)

			if not match_count:
						SCREAMING_SNAKE_CASE_		:     Dict					=					0.0
			else:
						SCREAMING_SNAKE_CASE_		:     Optional[Any]					=					(
						    1
						    / 3
						    * (
						        match_count / len(A__				)
						        + match_count / len(A__				)
						        + (match_count - transpositions) / match_count
						    )
						)

			# common prefix up to 4 characters
			SCREAMING_SNAKE_CASE_		:     List[Any]					=					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'))





