Algorithmic differentiation in Axiom

This paper describes the design and implementation of an algorithmic differentiation framework in the Axiom computer algebra system. Our implementation works by transformations on Spad programs at the level of the typed abstract syntax tree -- Spad is the language for extending Axiom with libraries. The framework illustrates an algebraic theory of algorithmic differentiation, here only for Spad programs, but we suggest that the theory is general. In particular, if it is possible to define a compositional semantics for programs, we define the exact requirements for when a program can be algorithmically differentiated. This leads to a general algorithmic differentiation system, and is not confined to functions which compute with basic data types, such as floating point numbers.