Compile-Time Symbolic Differentiation Using C++ Expression Templates

TL;DR

This paper presents a method using C++ expression templates to perform compile-time symbolic differentiation of algebraic functions, generating derivative code efficiently during compilation.

Contribution

It introduces a novel approach for compile-time symbolic differentiation in C++ using expression templates and recursive simplification techniques.

Findings

Current C++ compilers are capable of supporting practical implementation.

The method efficiently generates derivatives at compile time.

Further compiler improvements could enhance performance and capabilities.

Abstract

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.