I don't actually have a fantastic example handy.
In retrospect, the \ example is not actually more performant as doing it directly -- it is equally performant, and avoids duplication of the fiddly part of the code.
We've long had the opinion that there were intermediate values that were useful in defining the pullback.
But maybe there are not or at least they are kinda rare.
I guess one obvious one though would be for various kinda of triple product f(a,b,c) = a*b*c which I think shows up in various forms of vectors, matrixes and transposes there of.
There is various efficient ways of doing this all at once,
But if you know you are going to want the pullback for a you really want to compute and store b*c for later use.