Limitations of Taylor method-based elementary function implementation

Since last week's paragraph Jan has continued pondering ways to leverage the Taylor series methods implementation developed for the new IVP solver. An obvious idea is to use Taylor methods to compute polynomial approximations for elementary functions. There are however two possible limitations to this approach, both stemming from the same fundamental compositionality problem, namely that Taylor coefficients need to be available for all parameters of the computed functions. The first case is that of computing, say the cosine of the solution of a differential equation. If the solution is not itself computed using Taylor methods, but rather by a constant or affine approximation method, then at most the first two terms in the series for the solution are available as input to the Taylor method algorithm. The second case is that of approximating insufficiently smooth functions, such as the absolute value. In this case one has to make an approximation of the domain over which the function lacks a continuous derivative, in our case where the parameter is zero, and instead use a smoother approximation, again losing precision in the process. The resulting approximations may very well still be acceptable, but may lead to unpredictable degrees of over-approximation, causing confusion for the end-users. Still, the method seems to Jan as a simple way to implement elementary functions in a rigorous manner.