For some functions you can turn the Taylor series into a recurrence relation, which makes it blazing fast to calculate. For instance for the family of function

For some functions you can turn the Taylor series into a recurrence relation, which makes it blazing fast to calculate. For instance for the family of functions f(x0,x1,...) = exp(poly(x0,x1,...)), where poly is a multivariate polynomial of order m, you can compute the Taylor coefficients with a recurrence relation of order m (that needs to look back m steps). This shows up in quantum optics, for example.