Several independent groups contribute to one total. Group i has group_sizes[i] people, each varying independently with standard deviation group_stds[i]. Return the standard deviation of the grand total.

Variances add, standard deviations do not:

$$\mathrm{Var}(\text{total}) = \sum_i n_i \sigma_i^2, \qquad \mathrm{SD} = \sqrt{\mathrm{Var}}$$

Input

group_sizes = [100, 200]
group_stds = [2.0, 3.0]

Output

46.9041575982343