Either use roots to solve for the roots of the quartic, or be forced to use symbolic form for the solutions. Note that roots will be blazingly fast, and it will give you virtually full double precision accuracy. The only case where roots will not be full double precision accuracy is when there are roots with higher multiplicity than 1, but that is not the fault of roots.
Note, there is absolutely no need to use the Cardano formula, as roots will be just as accurate and probably as fast.
If you find the numerical roots from roots are insufficiently accurate, then you have no choice but to use higher precision, and accept that if you insist on super high accuracy, then you must sacrifice speed.
There is no solution that will be both immensely accurate AND blazingly fast. So take your pick. There are no "tricks".