My question is pretty simple and to the point. Is there a known way to efficiently compute logistic maps to within a specified precision? In other words, the input is a value $x$ and integers $d,n$; the desired output is the result of $n$ iterations of the logistic map applied to $x$, to $d$ bits of precision.
I know of a way to do this using exact real arithmetic but the representations of real numbers that I know and the algorithms I know all take exponential time with respect to the requested number of bits of precision. Using fixed-point arithmetic doesn't work because each multiplication doubles the number of bits of precision needed, so the number of bits needed is exponential in the number of iterations. Is there a known efficient way to compute logistic maps to with specified precision?