  Weak lensing leads to the non-Gaussian magnification distribution of standard
candles at given redshift $z$, $p(\mu|z)$. In this paper, we give accurate and
simple empirical fitting formulae of the weak lensing numerical simulation
results with the generalized Dyer-Roeder prescription. The smoothness parameter
$\tilde{\alpha}$ essentially represents the amount of matter that can cause
magnification of a given source. Since matter distribution in our universe is
inhomogeneous, we can think of our universe as a mosaic of cones centered on
the observer, each with a different value of $\tilde{\alpha}$. We define the
{\it direction dependent} smoothness parameter $\tilde{\alpha}$ via the
Dyer-Roeder equation; there is a unique mapping between $\tilde{\alpha}$ and
the magnification of a source. We find that the distribution of
$\tilde{\alpha}$ at given $z$, $p(\tilde{\alpha}|z)$, is well described by a
modified Gaussian distribution. For the same matter distribution, i.e., the
same $p(\tilde{\alpha}|z)$, different values of $\Omega_m$ and
$\Omega_{\Lambda}$ can lead to very different magnification distributions.
  Our formulae can be conveniently used to calculate the weak lensing effects
for observed Type Ia supernovae at arbitrary redshifts.
