Programming F(n) for benchmarking purposes is interesting. However, the fastest way of all is probably:

F(n)= integer nearest (1/sqrt 5)[(1+sqrt 5)/2]^n
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Yes, but I'd like to understand for myself why various approaches are slower. Which means I have to find a number of independent approaches.

If n is extremely large, the exponentiation might become inaccurate, though.

That's why you use arbitrary precision, like one of the Math::Big* modules.