HP Prime Miscalculating

[parisse]

Actually, there seems to be a confusion. Inside the CAS, you should use exact representation of objects (i.e. here rationals) if you want to perform exact arithmetic, and approx representation of objects (i.e. floating point) if you want to perform approx computations.
Floating points should not be used to do exact computations unless you know exactly what you do, because they are not designed to do that, they will do exact computations for *some* rationals with denominator dividing a power of 10 in HOME or a power of 2 in CAS.
In other words, you can perfectly do your floating point computations inside the CAS and some floating point computation inside HOME are done by conversion to CAS floats calling the CAS numeric stuff (for example numeric integration) or are faster in CAS (like matrix inversion).