  Recent advances in the asymptotic analysis of energy levels of potentials
produce relative errors in eigenvalue sums of order $10^{-34}$, but few
non-trivial potentials have been solved numerically to such accuracy. We solve
the general quartic potential (arbitrary linear combination of $x^2$ and $x^4$
) beyond this level of accuracy using a basis of several hundred oscillator
states. We list the lowest 20 eigenvalues for 9 such potentials. We confirm the
known asymptotic expansion for the levels of the pure quartic oscillator, and
extract the next 2 terms in the asymptotic expansion. We give analytic formulas
for expansion in up to 3 even basis states. We confirm the virial theorem for
the various energy components to similar accuracy. The sextic oscillator levels
are also given. These benchmark results should be useful for extreme tests of
approximations in several areas of chemical physics and beyond.
