  We analyse the accuracy of the approximate WKB quantization for the case of
general one-dimensional quartic potential. In particular, we are interested in
the validity of semiclassically predicted energy eigenvalues when approaching
the limit $E\to \infty$, and in the accuracy of low lying energy levels below
the potential barrier in the case of generally asymmetric double-well quartic
potential. In the latter case, using the standard WKB quantization an unnatural
localization of eigenstates due to the negligence of tunneling is implied and
thus the validity of semiclassics is uncertain. In all computations the higher
order corrections to the leading semiclassical approximation are included using
the complex contour integration technique. We show that these corrections can
improve accuracy of semiclassical approximation greatly by many orders of
magnitude.
