Originally Posted by latecoere240
TAS : CAS +/- density error +/- compressibility error or EAS +/- density error
TAS cannot be obtained from CAS or EAS by using independent linear terms.
Originally Posted by oggers
... the proper formula I gave ...
Originally Posted by oggers
The formula is: TAS = CAS/√Relative Density ...
This formula requires a loss of distinction between EAS and CAS which may be an acceptable assumption at low speed and low altitude. As either speed or altitude increase above some threshold - see Wittingham's post here on 2016/06/03 - the assumption leads to an increasingly erroneous TAS value and it is the difference from the correct value which is referred to as, rightly or wrongly, a "compressibility error." Latecoere240 therefore probably wanted to write your formula but with the addition of a corrective term. That is, I suspect the intention was TAS = CAS/sqrt(sigma) + compressibility_error; sigma being the ratio of ambient air density to the ISA MSL value as you've given.
It is incorrect to state TAS = EAS/sqrt(sigma) + compressibility_error; EAS might as well be defined by the expression TAS * sqrt(sigma), i.e. no corrective term is needed when mapping EAS to TAS or vice versa.
It's crucial to note that this compressibility_error is a function of CAS and altitude and that the effort needed to compute TAS this way is not less than working it out directly in the first instance. The function needed to do this mapping which is mechanically effected in modern airspeed indicators, and has been since the Tiger Moth, may be found in William Aiken Jr's NACA report 837 (1946),
Standard nomenclature for airspeeds with tables and charts for use in calculation of airspeed, available from the Aerade Reports Archive hosted by Cranfield University here:
NACA UK Mirror report description page. Eqn 3 can be rearranged to set TAS (V) or CAS (V_c) as the subject as required.