Are there numerical rounding issues in using the cubic formula to find roots of cubic equations? Similarly with the quartic formula?
I do know for the quadratic formula to solve $ax^2+bx+c = 0$ that you use the formulas $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ or $x = \frac{2c}{-b \mp \sqrt{b^2-4ac}}$ so that the numerator or denominator respectively are chosen so that they are the sum of two terms with the same sign.
It could certainly be.
You have
$$S = \sqrt [3] {R + \sqrt{Q^3 + R^2}}$$ and $$T = \sqrt [3] {R - \sqrt{Q^3 + R^2}}$$
so if $Q$ is small compared with $R$, there could be trouble.
The $\sqrt{}$s could be computed safely in that case, as in the quadratic case, by $a-\sqrt{a^2+b} =(a-\sqrt{a^2+b})\frac{a+\sqrt{a^2+b}}{a+\sqrt{a^2+b}} =\frac{-b}{a+\sqrt{a^2+b}} $.
I wouldn't be surprised if standard libraries took these kind of precautions.