Symmetric and asymmetric discretization

Figure 4: Different discretization schemes for 10 bins per decade. The last two lines are almost indistinguishable.
  

The discretization of the tables usually works as

$$i_{\textrm{e}}=\textrm{round}(log_{10}(E)*\textrm{icnde})+1$$

which is just the inverse of the above formula. In this case, the energy $E_{i}$ is the middle of a bin going from

$$E_{i}/c\le E<E_{i}\times c, c=10^{1/\textrm{icnde}}$$

Each bin will have a contribution of $w=\frac{E}{E_{i}}$ to ensure energy conservation in the discretization method.

For some applications an asymmetric binning method can be advantageous.In this case

$$i_{\textrm{e}}=\textrm{round}(log_{10}(E)*\textrm{icnde}+\textrm{asbin})+1$$

and

$$E_{i}\times a/c\le E<E_{i}\times a\times c, a=10^{\textrm{-asbin}/\textrm{icnde}}$$

i.e. the borders of the bin are shifted by a factor $a$ or by $\textrm{-asbin/icnde }$ on a $\log_{10}$-scale. Which value is suitable has to be found empirically by comparing MC and CE. E.g. for electromagnetic CE, a value of $\textrm{asbin$=-0.07$ }$proved useful for when $\textrm{icnde=10}$, see Fig. 4. But better is