I am informed that you are "counting lattice points inside of a polyhedron."

Here is a lecture on the subject - the picture on page six looks like the version of the problem you are interested in. To be honest, I found these notes by doing a google search. I am told that this is a huge field!

It might help if you could narrow your problem even further. For example, you say that the $x_i$ are bounded real numbers. Do you know these to some high precision? Or can you give some information on how the $x_i$ are given? And can you say the same for $S$?

I am informed that you are "counting lattice points inside of a polyhedron."

Here is a lecture on the subject - the picture on page six looks like the version of the problem you are interested in. To be honest, I found these notes by doing a google search. I am told that this is a huge field!

It might help if you could narrow your problem even further. For example, you say that the $x_i$ are bounded real numbers. Do you know these to some high precision? Or can you give some information on how the $x_i$ are given? And can you say the same for $S$?

EDIT: Here is a survey paper by the same author, Jesús De Loera, covering the same material in greater detail.