I disagree. Both variants are more similar than it may seem at a first glance. In essence, the question here is which of the following calculations is more efficient for big n:

a) the power function for arbitrary precision floating point

b) the power function for 2x2 matrices of arbitrary precision integers

Assuming that we can ignore the initial effort to calculate the golden ratio (can we?), Binet's formula is based on a), while the article's best algorith is an application of b).

I'm absolutely undecided which one works better, mostly because both kinds of power functions are working essentially the same way, with a complexity of O(log(n)).

Also, the precisions required for both cases seem to be quite similar.

This trick is especially funny when performed using a hand calculator. For instance, just calculate the powers of the golden ratio and observe numbers like "xxxxx,0000yyy" "xxxxx,9999yyy", which become closer and closer to integers.

You can’t get an exact answer in general by using fixed-size floating-point numbers, as your pseudocode does. The number of bits of precision you need to get an exact answer is going to depend on the input value, presumably linearly.