Piecewise Linear AD via Source Transformation

y m i = vl i for i = 0, . . . ,m 1 Extraction . Here, x 2 X ✓ R denotes a vector of input variables, y = y(x) 2 R the corresponding output variables, and ' some smooth intermediate assignments from a library ⌘ +, , ⇤, /, p ·, exp, sin, cos, . . . . However, the smoothness assumption is violated in most real applications. For example, the evaluation routines of many physical applications contain nonsmooth expressions such as the absolute value, the maximum, and/or the minimum function, in order to avoid unrealistic quantities. In this case, the standard di↵erentiation rules do not necessarily apply any more. Thus, the derivatives provided by standard AD tools become unreliable since they are based on the chain rule. Moreover, the simpler models are questionable even if the derivatives are evaluated at points x 2 R where the function is di↵erentiable, since they do not take nearby kinks or nonsmoothness into account. A remedy for this situation was recently proposed in [1], where the author presents a method to compute (directional) piecewise linear models of the original abs-factorable function instead of a simple linearization. The piecewise linearization y = F (x; x), F : R ⇥R ! R, at a point x 2 R for a directional increment x 2 R represents the function in a more appropriate way and can be derived by a minor modification of the original code. Similar to the standard forward mode, the idea is based on defining an extended evaluation routine using the propagation rules