





"""simple docstring"""




def        a_				(							_lowercase       ,						_lowercase = 1000   ):
						if (not isinstance(_lowercase       ,						_lowercase   )) or (digit_position <= 0):
												raise ValueError('''Digit position must be a positive integer'''   )
						elif (not isinstance(_lowercase       ,						_lowercase   )) or (precision < 0):
												raise ValueError('''Precision must be a nonnegative integer'''   )

						# compute an approximation of (16 ** (n - 1)) * pi whose fractional part is mostly
						# accurate
						_UpperCamelCase			:					List[str]     =    (
						    4 * _subsum(_lowercase       ,						1       ,						_lowercase   )
						    - 2 * _subsum(_lowercase       ,						4       ,						_lowercase   )
						    - _subsum(_lowercase       ,						5       ,						_lowercase   )
						    - _subsum(_lowercase       ,						6       ,						_lowercase   )
						)

						# return the first hex digit of the fractional part of the result
						return hex(int((sum_result % 1) * 16   )   )[2:]





def        a_				(							_lowercase       ,						_lowercase       ,						_lowercase   ):
						_UpperCamelCase			:					int     =    0.0
						for sum_index in range(digit_pos_to_extract + precision   ):
												_UpperCamelCase			:					str     =    8 * sum_index + denominator_addend
												if sum_index < digit_pos_to_extract:
																		# if the exponential term is an integer and we mod it by the denominator
																		# before dividing, only the integer part of the sum will change;
																		# the fractional part will not
																		_UpperCamelCase			:					Optional[int]     =    pow(
																		    16       ,						digit_pos_to_extract - 1 - sum_index       ,						_lowercase   )
												else:
																		_UpperCamelCase			:					Tuple     =    pow(16       ,						digit_pos_to_extract - 1 - sum_index   )
												total += exponential_term / denominator
						return total


if __name__ == "__main__":
						import doctest

						doctest.testmod()










"""simple docstring"""




from functools import lru_cache





@lru_cache
def        a_				(							_lowercase   ):
						if not isinstance(_lowercase       ,						_lowercase   ):
												raise TypeError('''k must be an integer.'''   )
						if k < 0:
												raise ValueError('''k integer must be greater or equal to zero.'''   )
						if k == 0:
												return 0
						elif k == 1:
												return 1
						else:
												return fibonacci(k - 1   ) + fibonacci(k - 2   )





def        a_				(							_lowercase       ,						_lowercase   ):
						_UpperCamelCase			:					Any     =    len(_lowercase   )
						# Find m such that F_m >= n where F_i is the i_th fibonacci number.
						_UpperCamelCase			:					Optional[int]     =    0
						while True:
												if fibonacci(_lowercase   ) >= len_list:
																		_UpperCamelCase			:					List[Any]     =    i
																		break
												i += 1
						_UpperCamelCase			:					Tuple     =    0
						while fibb_k > 0:
												_UpperCamelCase			:					Union[str, Any]     =    min(
												    offset + fibonacci(fibb_k - 1   )       ,						len_list - 1   )  # Prevent out of range
												_UpperCamelCase			:					Any     =    arr[index_k]
												if item_k_1 == val:
																		return index_k
												elif val < item_k_1:
																		fibb_k -= 1
												elif val > item_k_1:
																		offset += fibonacci(fibb_k - 1   )
																		fibb_k -= 2
						else:
												return -1


if __name__ == "__main__":
						import doctest

						doctest.testmod()



