Is there a general / simple formula for Slater's rules?
Sadly, no. It wouldn't be too hard to program a "get Slater shielding" script. But it wouldn't be terribly useful.

First off, let's back up briefly. Slater's rules allow you to estimate the effective nuclear charge $Z_{eff}$ from the real number of protons in the nucleus and the effective shielding of electrons in each orbital "shell" (e.g., to compare the effective nuclear charge and shielding for say $3d$ vs. $4s$ in transition metals).

Slater wanted to use these for quantum calculations, such that the radial wavefunction would be something like:

$$\psi_{ns}(r) = r^{n-1}e^{-\frac{(Z-\sigma)r}{n}}$$

So the rules he devised were fairly simple and produced fairly accurate predictions of things like the electron configurations of transition metals and ionization potentials.

Later, others performed better optimizations of $\sigma$ and $Z_{eff}$ using variational Hartree-Fock methods. For example, Clementi and Raimondi published "Atomic Screening Constants from SCF Functions." J Chem Phys (1963) 38, 2686–2689. (That article covers elements up to Kr.)



It's pretty obvious that no simple formula will cover these curves.

These days, basis set optimization is done by computer, and I doubt many people even glance at the exponents.

In my mind, Slater's rules serve two main purposes at this point:


They're relatively simple to teach and learn and allow quick qualitative predictions, particularly for electron configurations in the transition metals.
They emphasize that exponents for multi-electron atoms must be empirically derived.


I still teach them, particularly, for the first reason. To a lesser degree, they also reinforce the idea of effective nuclear charge and provide some semi-quantitive basis for that concept.