  Let ${\cal H}(x,\xi)$ be a holomorphic Hamiltonian of quadratic growth on $
R^{2n}$, $b$ a holomorphic exponentially localized observable, $H$, $B$ the
corresponding operators on $L^2(R^n)$ generated by Weyl quantization, and
$U(t)=\exp{iHt/\hbar}$. It is proved that the $L^2$ norm of the difference
between the Heisenberg observable $B_t=U(t)BU(-t)$ and its semiclassical
approximation of order ${N-1}$ is majorized by $K N^{(6n+1)N}(-\hbar
ln\hbar)^N$ for $t\in [0,T_N(\hbar)]$ where $T_N(\hbar)=-{2 ln\hbar\over
{N-1}}$. Choosing a suitable $N(\hbar)$ the error is majorized by
$C\hbar^{ln|ln\hbar|}$, $0\leq t\leq |ln\hbar|/ln|ln\hbar|$. (Here $K,C$ are
constants independent of $N,\hbar$).
