text: def double_factorial(n: int) -> int:
      """
      Compute double factorial using recursive method.
      Recursion can be costly for large numbers.
  
      To learn about the theory behind this algorithm:
      https://en.wikipedia.org/wiki/Double_factorial
  
      >>> import math
      >>> all(double_factorial(i) == math.prod(range(i, 0, -2)) for i in range(20))
      True
      >>> double_factorial(0.1)
      Traceback (most recent call last):
          ...
      ValueError: double_factorial() only accepts integral values
      >>> double_factorial(-1)
      Traceback (most recent call last):
          ...
      ValueError: double_factorial() not defined for negative values
      """
      if not isinstance(n, int):
          raise ValueError("double_factorial() only accepts integral values")
      if n < 0:
          raise ValueError("double_factorial() not defined for negative values")
      return 1 if n <= 1 else n * double_factorial(n - 2)
  
  
  if __name__ == "__main__":
      import doctest
  
      doctest.testmod()
  