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made grammatical changes and simplified wording for clarity

\label{Formula_Response_Sampling}

As you may know, it is extremely difficult to determine whether a given expression
is exactly equal to another expression in general. For example, is $\sin 2x=2\sin x\cos x$?
Symbolically proving it one way or another is impossible in general. Therefore,
LON-CAPA uses a sampling system. If your answer and the student's answer
agree at the sampling points within your given tolerance factor, the student's
answer will be accepted. If the student's answer does not agree at the sampling points within your given tolerance factor, it will be rejected.

To specify where to sample the formulas for determining whether the student's
answer is correct, you need to put a sampling specification in the \textbf{Sample
Points}\index{Sample Points} field. The sampling specifications take the
following format:

\begin{enumerate}
\item A comma-separated list of the variables you wish to interpret,
\item followed by {}``@'' (not in quotes),
\item followed by any number of the following two things, separated by semi-colons:

\begin{enumerate}
\item a comma-separated list of as many numbers as there are variables, which specifies
one sampling point, OR
\item a comma-separated list of as many numbers as there are variables, followed
by a colon, followed by another list of as many numbers as there are variables,
followed by a \#, followed by an integer.
\end{enumerate}
\end{enumerate}
The first form specifies one point to sample. The second form specifies a
range for each variable, and the system will take as many random samples
from that range as the number after the \#.

For $2x^{2}+4$, with one variable {}``x'', one could specify:

\begin{itemize}
\item {}``x@2'', which will sample the answers only at 2. (This is generally
a bad idea, as the student could get lucky and match at that point)
\item {}``x@1:5\#4'' will takes 4 samples from somewhere between 1 and 5.
\item {}``x@1:5\#4;10'' will takes 4 samples from somewhere between 1 and 5,
and also sample at 10.
\end{itemize}
For $2x^{2}+3y^{3}$, which has two variables, one could specify:

\begin{itemize}
\item {}``x,y@4,5:10,12\#4;0,0'', which take four samples from the box determined
by the points (4, 5) and (10, 12), and also sample the point (0, 0).
\end{itemize}