I need a way to find a rational approximation of any radical expression (say, $\sqrt{1+\sqrt[3]{2}}$) in such a way that I know a bound on the error. Given a rational number, I think I can always use Newton's method to approximate its nth root within a known bound, so I imagine the way to approximate the whole expression is to build it up from its components - approximate $\sqrt[3]{2}$, then use the result to approximate the whole thing, in this example. But how do I account for the errors in the earlier calculations while doing the later ones? I'll need to be able to handle complex numbers too, which seems like it might pose additional complications.
edit - more information about the problem context: by "radical expression," I mean any element of the smallest set of complex numbers that is closed under the arithmetic operations and rational exponentiation. I have such a radical expression, $n$, and a set $S$ of some number of its conjugates, also given as radical expressions. I have to determine whether $n$ is equal to some element of $S$, which will be true iff there is some element of $S$ that is closer to $n$ than any two elements of $S$ are to each other.
Compute the expression using interval arithmetic. The result will have lower and upper bounds on the value.
You may have to try different forms of the expression to get better bounds.
If your interval arithmetic package can use arbitrary precision, that helps.