One of Richardson's most celebrated achievements is his retroactive attempt to forecast the weather during a single day-20 May 1910-by direct computation.

A direct computation now gives:

This can be found either using by direct computation or using the method of supplementary variables.

However, this can be confirmed by direct computation of the optical scalars of the null congruence .

The direct computation of the numerator , involves two nested iterations, as characterized by the following pseudo-code:

Then this can be seen by a direct computation for n any integer, and the general case reduces to this case without much difficulty.

This allows more direct computation of terms.

The direct computation of this integral is quite difficult, but we can simplify the derivation of the result using the divergence theorem:

Assuming that (in the case , the direct computation is much faster) the binary expression of has digits, of which k are ones.

On the other hand if the saturation removes some components, the direct computation may be dramatically faster.