Back in the day, military folks used whichever nav computer to run the sums (either Dalton for the Allies or Knemeyer for the Axis). As a sidenote, the massive penetration into the market of these two basically blew all the others out of the water. The Knemeyer, for whatever reason, subsequently fell into disuse (why is totally beyond me as it was a pretty neat bit of kit) although Lahr's computer (subsequently the Jepp CR) can trace its lineage back to Knemeyer's device.
The TAS calculation is/was simple and just used the air density (from altitude and OAT) to correct EAS (which you can dumb down to CAS or IAS with minor errors - one just needs to keep in mind that CAS gets to be a bit average once the mach number ventures much above about 0.25 - the reason the techo folks invented EAS)
A typical link is
Airspeed Conversions (CAS/EAS/TAS/Mach) | AeroToolbox
The equation is the usual TAS = EAS / (1/√σ) where σ is the density ratio ρ/ρo (apologies, I can't find a convenient way to make the o a subscript). When setting pressure height and OAT in the relevant box (or the density height in the adjacent box), all the device is doing is acting as a look-up table and setting the sigma factor on the outer C/D slide rule scale in preparation for the above multiplication in the usual slide rule way of doing things. Easy peasy.
However, running a multivariate regression is neither necessary nor useful for understanding.
The various rules of thumb calculations one sees are really not necessary when the slide rule (nav computer) calculation is quite a straightforward exercise.
If my recollections be correct, RAS is/was the Brit term for CAS.
PS I've located the thread Jim refers to and linked to it in his first post.
Jim, if you like to email me the scan, I can put it into your post on your behalf.