I once witnessed a programmer with a PhD in Maths find closed form formulas for a lot of questions where it was expected to write some code with loops building/accumulating a result. As a simple example, to explain what was going on, if the question would be "calculate the 100th fibonacci number", she would just use Binet's formula to do so (as opposed to using a loop). I was rather impressed how often that happened.
For evaluating the Fibonacci numbers (as with any other integer linear recurrence), I'd generally prefer the matrix-exponentiation-by-squaring approach, or one of the simplified formulas based on it. Those don't need anything more complicated than bigint multiplication. [And from there, taking the ratio between two values gives you a quick way to approximate the golden ratio!]
If you want to beat the O(n^2) runtime of the trivial iteration, you pretty much have to use Newton's method for φ, exponentiation by squaring on the matrix form, or another method with faster-than-linear convergence.
The astonishment doesn't get less, but it shifts from Binet's single formula to the exponential map, and maybe the fundamental theorem of algebra (or generalisations).