





from decimal import Decimal, getcontext
from math import ceil, factorial



def        __lowercase       (				_SCREAMING_SNAKE_CASE     )			->       str:


      '''simple docstring'''


      if not isinstance(lowercase__			,						lowercase__     ):
            raise TypeError("""Undefined for non-integers"""     )
      elif precision < 1:
            raise ValueError("""Undefined for non-natural numbers"""     )

      SCREAMING_SNAKE_CASE		=   precision
      SCREAMING_SNAKE_CASE		=   ceil(precision / 14     )
      SCREAMING_SNAKE_CASE		=   42_68_80 * Decimal(1_00_05     ).sqrt()
      SCREAMING_SNAKE_CASE		=   1
      SCREAMING_SNAKE_CASE		=   13_59_14_09
      SCREAMING_SNAKE_CASE		=   Decimal(lowercase__     )
      for k in range(1			,						lowercase__     ):
            SCREAMING_SNAKE_CASE		=   factorial(6 * k     ) // (factorial(3 * k     ) * factorial(lowercase__     ) ** 3)
            linear_term += 5_45_14_01_34
            exponential_term *= -26_25_37_41_26_40_76_80_00
            partial_sum += Decimal(multinomial_term * linear_term     ) / exponential_term
      return str(constant_term / partial_sum     )[:-1]


if __name__ == "__main__":
      SCREAMING_SNAKE_CASE_						       =	5_0
      print(F'''The first {n} digits of pi is: {pi(n)}''')









from __future__ import annotations



def        __lowercase       (				_SCREAMING_SNAKE_CASE			,						_SCREAMING_SNAKE_CASE     )			->       list[int]:


      '''simple docstring'''


      SCREAMING_SNAKE_CASE		=						0
      SCREAMING_SNAKE_CASE		=						len(_SCREAMING_SNAKE_CASE     ) - 1

      while i < j:
            if nums[i] + nums[j] == target:
                  return [i, j]
            elif nums[i] + nums[j] < target:
                  SCREAMING_SNAKE_CASE		=						i + 1
            else:
                  SCREAMING_SNAKE_CASE		=						j - 1

      return []


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      print(F'''{two_pointer([2, 7, 1_1, 1_5], 9) = }''')


