How to convert IAS/CAS to TAS up to Mach 3?
For higher airspeeds, TAS is always calculated based on the Mach number:
$$
\mathrm{TAS} = M \cdot a_0 \sqrt{\frac{T}{T_0}}
$$
where $ a_0 $ is the speed of sound at ground level ($ a_0 \approx 661 \, \mathrm{kt} $) and $ T_0 = 288.15 \, \mathrm{K} $ is the temperature at ground level. This formula is just based on the fact that the speed of sound in air is proportional to the square-root of the temperature.

How to calculate the Mach Number

Now, we need to distinguish between subsonic and supersonic flow. The pitot tube will give you the ratio of impact pressure $ q $ and static pressure $ p $. From this ratio, you calculate the the Mach number:


subsonic flow:
$$
M = \sqrt{5 \left( (q/p)^{5/7} - 1 \right)}
$$
supersonic flow:
$$
M \approx 0.8813 \sqrt{\frac{q}{p} \left( 1 - \frac{1}{7 M^2} \right)^{5/2}}
$$
This equation needs to be solved iteratively.


The following plot shows the Mach Number as a function of the $ p / q $ ratio (red for subsonic flow, blue for supersonic flow):



Calculate the True Air Speed

Now that we have the Mach number, we need to calculate the TAS using the formula from above. The last missing piece is the static air temperature $ T $, which can be calculated from the measured total air temperature $ T_\mathrm{tot} $:

$$
T = \frac{T_\mathrm{tot}}{1 + M^2 / 5}
$$

This gives the final answer:


Unfortunately, there is no easy way to calculate this based on KIAS as an input. You will need the ratio $ q / p $ and the temperature $ T_\mathrm{tot} $ as measured by the aircraft.