def    __lowerCamelCase	(		lowerCAmelCase__			,						lowerCAmelCase__ = 1_0_0_0			):



							if (not isinstance(__A			,						__A			)) or (digit_position <= 0):
														raise ValueError('Digit position must be a positive integer'			)
							elif (not isinstance(__A			,						__A			)) 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
							lowerCAmelCase__  			=      (
							    4 * _subsum(__A			,						1			,						__A			)
							    - 2 * _subsum(__A			,						4			,						__A			)
							    - _subsum(__A			,						5			,						__A			)
							    - _subsum(__A			,						6			,						__A			)
							)

							# return the first hex digit of the fractional part of the result
							return hex(int((sum_result % 1) * 1_6			)			)[2:]






def    __lowerCamelCase	(		lowerCAmelCase__			,						lowerCAmelCase__			,						lowerCAmelCase__			):
							lowerCAmelCase__  			=      0.0
							for sum_index in range(digit_pos_to_extract + precision			):
														lowerCAmelCase__  			=      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
																					lowerCAmelCase__  			=      pow(
																					    1_6			,						digit_pos_to_extract - 1 - sum_index			,						__A			)
														else:
																					lowerCAmelCase__  			=      pow(1_6			,						digit_pos_to_extract - 1 - sum_index			)
														total += exponential_term / denominator
							return total


if __name__ == "__main__":
				import doctest

				doctest.testmod()

from ..utils import DummyObject, requires_backends
class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						Any    ,					**lowercase__      :						List[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						List[str]    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Dict    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Dict    ,					*lowercase__      :						int    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[Any]    ,					*lowercase__      :						Dict    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						int    ,					**lowercase__      :						str):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Dict    ,					**lowercase__      :						int):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						Tuple    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[Any]    ,					*lowercase__      :						int    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						int    ,					*lowercase__      :						Optional[Any]    ,					**lowercase__      :						Union[str, Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Tuple    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Any):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Tuple    ,					*lowercase__      :						Optional[Any]    ,					**lowercase__      :						Any):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						int    ,					*lowercase__      :						Tuple    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[Any]    ,					*lowercase__      :						List[Any]    ,					**lowercase__      :						Union[str, Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						List[Any]    ,					**lowercase__      :						Any):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Any    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						int    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						List[str]    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Tuple    ,					**lowercase__      :						List[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						int    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						List[Any]    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						Any    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						int    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Tuple    ,					*lowercase__      :						Any    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Dict    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[Any]    ,					*lowercase__      :						str    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[int]    ,					*lowercase__      :						Any    ,					**lowercase__      :						Union[str, Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[int]    ,					*lowercase__      :						List[str]    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Dict    ,					*lowercase__      :						List[Any]    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[Any]    ,					*lowercase__      :						Tuple    ,					**lowercase__      :						Optional[int]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						int    ,					*lowercase__      :						Any    ,					**lowercase__      :						str):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Dict    ,					*lowercase__      :						Dict    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[Any]    ,					*lowercase__      :						str    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[int]    ,					*lowercase__      :						str    ,					**lowercase__      :						List[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						Any    ,					**lowercase__      :						Union[str, Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[Any]    ,					*lowercase__      :						int    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[Any]    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						Optional[int]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						Optional[int]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Union[str, Any]    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[int]    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						int):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						Tuple    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Union[str, Any]    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[Any]    ,					*lowercase__      :						Tuple    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Union[str, Any]    ,					*lowercase__      :						Union[str, Any]    ,					**lowercase__      :						List[str]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[int]    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						str):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Optional[int]    ,					*lowercase__      :						Any    ,					**lowercase__      :						Any):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						str    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Optional[Any]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						List[str]    ,					*lowercase__      :						str    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Any    ,					*lowercase__      :						Dict    ,					**lowercase__      :						Tuple):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Tuple    ,					*lowercase__      :						Any    ,					**lowercase__      :						Dict):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Union[str, Any]    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						Optional[int]):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])






class        a_				(		metaclass=SCREAMING_SNAKE_CASE		):






 '''simple docstring'''






 UpperCAmelCase_							=      ['tokenizers']




 def __init__(    self      :						Dict    ,					*lowercase__      :						Optional[int]    ,					**lowercase__      :						str):







        '''simple docstring'''



        requires_backends(self    ,					['tokenizers'])
