THE IF-PROBLEM IN AUTOMATIC DIFFERENTIATION

We deal with a severe problem that arises in the automatic differentiation of functions. Many computer programs defining a function employ statements of the form if B(x) then S1 else S2, where B(x) is a Boolean expression and S1 and S2 denote subprograms. This often leads to a piecewise definition of the function under consideration. Automatic differentiation of the pieces may be hazardous, for instance in cases where the underlying function is differentiable but one or the other piece is not. In such cases available software often fails to produce correct results. To resolve this perplexity, we distinguish between a function and its representations. In particular, we introduce the notion derivative-consistent. Automatic differentiation applied to a derivative-consistent representation of a function yields correct results.