  We propose an algorithm to compute the $C^\infty$-ring structure of arbitrary
Weil algebra. It allows us to do some analysis with higher infinitesimals
numerically and symbolically. To that end, we first give a brief description of
the (Forward-mode) automatic differentiation (AD) in terms of $C^\infty$-rings.
The notion of a $C^\infty$-ring was introduced by Lawvere and used as the
fundamental building block of smooth infinitesimal analysis and synthetic
differential geometry. We argue that interpreting AD in terms of
$C^\infty$-rings gives us a unifying theoretical framework and modular ways to
express multivariate partial derivatives. In particular, we can "package"
higher-order Forward-mode AD as a Weil algebra, and take tensor products to
compose them to achieve multivariate higher-order AD. The algorithms in the
present paper can also be used for a pedagogical purpose in learning and
studying smooth infinitesimal analysis as well.
