You will alsö find these equations in the AITC. They have been around for years. They are iterative in nature in that initially you have to assume a "d" to generate the "S" value. To work, the d you assume has to be greater than or equal to the d you compute.
Top formula is for a no-slab condition, and the bottom is for a slab present. The "S" values are different between the two cases, as are the formulas.
I have an old publication at work that has there equations too. I'll take a look tomorrow.
Mike McCann, PE, SE (WA)