First attempt

In the example of the logistic map, we can reliably use rational arithmetic, because the logistic map is rational. But we have

1.

bad performance, and

2.

unfeasible resource consumption.

For n > 20, the computation of fⁿ(x₀) runs out of memory in a standard package for rational arithmetic--the reason is that rather large, relatively prime numerators and denominators arise. But floating-point computation in simple precision already seems to give a fairly accurate answer in this case. Nothing is gained in practice.