The logistic map is a classic example where floating point numbers fail. It's also a great example of where error propagates very badly in general in numerical algorithms even when dealing with bignums. I was wondering if there are any known algorithms for taming this issue? Is there an efficient way to compute a logistic map that doesn't require naively computing it with huge precision?
It is a classic example because it is a chaotic system. The entire point of a chaotic system is that it shows unbelievable sensitivity to initial conditions. To get an answer within 5% of correct after n iterations requires starting with O(n) digits of the number. Not because your algorithm is bad, but because changing any of those digits changes what the answer should be.
So, no. While you can potentially speed up the calculation somewhat, you can't get away with starting with lower precision.