I'm working on a multiple-precision library. I'd like to make it possible for users to ask for higher precision answers for results already computed at a fixed precision. My $\mathrm{sqrt}(x)$ can pick up where it left off, even if $x$ changes a bit, because it uses Newton-Raphson. But $\exp(x)$ computed using a Maclaurin series or continued fraction has to be computed from scratch.
Is there an iterative refinement (i.e. Newton-Raphson, gradient descent) method for computing $\exp(x)$ that uses only arithmetic and integer roots?
(I know Newton-Raphson can solve $\log(y)-x=0$ to compute $\exp(x)$. I am specifically not asking for that. Newton-Raphson can also solve $\exp(y)-x=0$ to compute $\log(x)$. Note that each requires the other. I have neither right now as an arbitrary-precision function. I have arithmetic, integer roots, and equality/inequality tests.)
Best Answer
There is an algorithm for computing $\log_2(x)$ that might suit you. Combine that with the spigot algorithm for $e$, and you can get $\ln(x)$. From there, you can use Newton-Raphson to get $\exp(x)$.
I don't know if this roundabout way ends up doing any better than just recomputing.