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That's probably faster, but it's not O(1). Calculating sqrt(5) to the needed precision, and raising phi to the n power, are not O(1).

> Calculating sqrt(5) to the needed precision...

I think this one could be optimized by precomputing and storing (1 + sqrt(5)) at the highest precision available.

If you want to find an arbitrarily high F(n), you will need an arbitrarily high precision of sqrt(5).