




def 		a_       (   __lowerCAmelCase      ,   __lowerCAmelCase = 10_00							):
  if (not isinstance(__lowerCAmelCase      ,   __lowerCAmelCase							)) or (digit_position <= 0):
    raise ValueError('''Digit position must be a positive integer'''							)
  elif (not isinstance(__lowerCAmelCase      ,   __lowerCAmelCase							)) 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(__lowerCAmelCase      ,   1      ,   __lowerCAmelCase							)
      - 2 * _subsum(__lowerCAmelCase      ,   4      ,   __lowerCAmelCase							)
      - _subsum(__lowerCAmelCase      ,   5      ,   __lowerCAmelCase							)
      - _subsum(__lowerCAmelCase      ,   6      ,   __lowerCAmelCase							)
  )

  # return the first hex digit of the fractional part of the result
  return hex(int((sum_result % 1) * 16							)							)[2:]
def 		a_       (   __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(
          16      ,   digit_pos_to_extract - 1 - sum_index      ,   __lowerCAmelCase							)
    else:
      lowerCAmelCase__								=			pow(16      ,   digit_pos_to_extract - 1 - sum_index							)
    total += exponential_term / denominator
  return total


if __name__ == "__main__":
      import doctest

      doctest.testmod()







def 		a_       (   __lowerCAmelCase      ,   __lowerCAmelCase							):
  lowerCAmelCase__								=			0
  for ele in arr:
    total_sum += product_sum(__lowerCAmelCase      ,   depth + 1							) if isinstance(__lowerCAmelCase      ,   __lowerCAmelCase							) else ele
  return total_sum * depth
def 		a_       (   __lowerCAmelCase							):
  return product_sum(__lowerCAmelCase      ,   1							)


if __name__ == "__main__":
      import doctest

      doctest.testmod()

