Compile-Time Symbolic Differentiation Using C++ Expression Templates

Template metaprogramming is a popular technique for implementing compile time mechanisms for numerical computing. We demonstrate how expression templates can be used for compile time symbolic differentiation of algebraic expressions in C++ computer programs. Given a positive integer $N$ and an algebraic function of multiple variables, the compiler generates executable code for the $N$th partial derivatives of the function. Compile-time simplification of the derivative expressions is achieved using recursive templates. A detailed analysis indicates that current C++ compiler technology is already sufficient for practical use of our results, and highlights a number of issues where further improvements may be desirable.