How can I improve performance via a high-level approach when implementing long equations in C++

Question

I am developing some engineering simulations. This involves implementing some long equations such as this equation to calculate stress in a rubber like material:

Answer 1

This may be a little terse, but I've actually found good speedup for polynomials (interpolation of energy functions) by using Horner Form, which basically rewrites ax^3 + bx^2 + cx + d as d + x(c + x(b + x(a))). This will avoid a lot of repeated calls to pow() and stops you from doing silly things like separately calling pow(x,6) and pow(x,7) instead of just doing x*pow(x,6).

This is not directly applicable to your current problem, but if you have high order polynomials with integer powers it can help. You might have to watch out for numerical stability and overflow issues since the order of operations is important for that (although in general I actually think Horner Form helps for this, since x^20 and x are usually many orders of magnitude apart).

Also as a practical tip, if you haven't done so already, try to simplify the expression in maple first. You can probably get it to do most of the common subexpression elimination for you. I don't know how much it affects the code generator in that program in particular, but I know in Mathematica doing a FullSimplify before generating the code can result in a huge difference.