code: def fibonacci_approx(n):
      r
      sqrt_5 = math.sqrt(5)
      phi = (1 + sqrt_5) / 2
      return ((phi ** n) - (-phi) ** (-n)) / sqrt_5
docs: r"""
      approximate value (due to numerical errors) of fib(n) using closed form
      expression
  
      Args:
          n (int):
  
      Returns:
          int: the n-th fib number
  
      CommandLine:
          python -m utool.util_alg fibonacci_approx
  
      Example:
          >>> # DISABLE_DOCTEST
          >>> from utool.util_alg import *  # NOQA
          >>> import utool as ut
          >>> series = [int(fibonacci_approx(n)) for n in range(20)]
          >>> result = ('series = %s' % (str(series[0:10]),))
          >>> print(result)