

'''simple docstring'''


def     SCREAMING_SNAKE_CASE(	UpperCamelCase ,UpperCamelCase = 1_0_0_0	)     ->     str:
	if (not isinstance(UpperCamelCase ,UpperCamelCase	)) or (digit_position <= 0):
		raise ValueError('Digit position must be a positive integer'	)
	elif (not isinstance(UpperCamelCase ,UpperCamelCase	)) 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(UpperCamelCase ,1 ,UpperCamelCase	)
	    - 2 * _subsum(UpperCamelCase ,4 ,UpperCamelCase	)
	    - _subsum(UpperCamelCase ,5 ,UpperCamelCase	)
	    - _subsum(UpperCamelCase ,6 ,UpperCamelCase	)
	)

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







def     SCREAMING_SNAKE_CASE(	UpperCamelCase ,UpperCamelCase ,UpperCamelCase	)     ->     float:
	UpperCAmelCase_		:   Optional[int]													=       0.0
	for sum_index in range(digit_pos_to_extract + precision	):
		UpperCAmelCase_		:   Any													=       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[Any]													=       pow(
			    1_6 ,digit_pos_to_extract - 1 - sum_index ,UpperCamelCase	)
		else:
			UpperCAmelCase_		:   Dict													=       pow(1_6 ,digit_pos_to_extract - 1 - sum_index	)
		total += exponential_term / denominator
	return total


if __name__ == "__main__":
						import doctest

						doctest.testmod()









'''simple docstring'''


from __future__ import annotations







def     SCREAMING_SNAKE_CASE(	UpperCamelCase ,UpperCamelCase ,UpperCamelCase ,UpperCamelCase	)     ->     int:

	# Base Case
	if counter == 0 or capacity == 0:
		return 0

	# If weight of the nth item is more than Knapsack of capacity,
	#   then this item cannot be included in the optimal solution,
	# else return the maximum of two cases:
	#   (1) nth item included
	#   (2) not included
	if weights[counter - 1] > capacity:
		return knapsack(UpperCamelCase ,UpperCamelCase ,UpperCamelCase ,counter - 1	)
	else:
		UpperCAmelCase_		:   int													=       capacity - weights[counter - 1]
		UpperCAmelCase_		:   Optional[int]													=       values[counter - 1] + knapsack(
		    UpperCamelCase ,UpperCamelCase ,UpperCamelCase ,counter - 1	)
		UpperCAmelCase_		:   Any													=       knapsack(UpperCamelCase ,UpperCamelCase ,UpperCamelCase ,counter - 1	)
		return max(UpperCamelCase ,UpperCamelCase	)


if __name__ == "__main__":
						import doctest

						doctest.testmod()






