
'''simple docstring'''





from math import pi, sqrt







def snake_case_     (					lowercase__		):
  if num <= 0:
    raise ValueError("math domain error"		)
  if num > 171.5:
    raise OverflowError("math range error"		)
  elif num - int(lowercase__		) not in (0, 0.5):
    raise NotImplementedError("num must be an integer or a half-integer"		)
  elif num == 0.5:
    return sqrt(lowercase__		)
  else:
    return 1.0 if num == 1 else (num - 1) * gamma(num - 1		)







def snake_case_     (					):
  assert gamma(0.5		) == sqrt(lowercase__		)
  assert gamma(1		) == 1.0
  assert gamma(2		) == 1.0


if __name__ == "__main__":
      from doctest import testmod

      testmod()
      SCREAMING_SNAKE_CASE     =       1.0
      while num:
            SCREAMING_SNAKE_CASE     =       float(input("""Gamma of: """))
            print(F'gamma({num}) = {gamma(num)}')
            print("""\nEnter 0 to exit...""")





'''simple docstring'''





import math







def snake_case_     (					lowercase__		):

  if 1 < number < 4:
    # 2 and 3 are primes
    return True
  elif number < 2 or number % 2 == 0 or number % 3 == 0:
    # Negatives, 0, 1, all even numbers, all multiples of 3 are not primes
    return False

  # All primes number are in format of 6k +/- 1
  for i in range(5   ,       int(math.sqrt(lowercase__		) + 1		)   ,       6		):
    if number % i == 0 or number % (i + 2) == 0:
      return False
  return True







def snake_case_     (					lowercase__ = 0.1		):
  UpperCAmelCase__      :					Dict				       =							3
  UpperCAmelCase__      :					Tuple				       =							3

  while primes / (2 * j - 1) >= ratio:
    for i in range(j * j + j + 1   ,       (j + 2) * (j + 2)   ,       j + 1		):
      primes += is_prime(lowercase__		)
    j += 2
  return j


if __name__ == "__main__":
      import doctest

      doctest.testmod()



