I would like to find an approximation for:
$$ \log \left(\sum_{i=1}^{N} a_i\exp(-b_i^2)\exp(-c_i^2)\right) $$
with
$$ a_i = \frac{1}{\sqrt{(e^2 + e_i^2)(g^2 + g_i^2)}} \\ b_i = \frac{b-d_i}{2(e^2 + e_i^2)} \\ c_i = \frac{c-f_i}{2(g^2 + g_i^2)} $$
where $b, e, c, g$ are floats independent of $i$ and $d_i, e_i, f_i, g_i$ are also floats which depend on $i$. These floats can safely be considered to take random values between $(0,100]$.
Finally $N>1$ and $a_i>0 \,\forall\, i$.