Minimum number of points for a good exponential curve fit We are currently trying to fit data from a process that generates data that decays exponentially.  We employ several techniques for fitting this exponential decay in including FFT analysis and a weighted least-squares algorithm.  We are attempting to fit a lot of these decays every second (in some cases 8000 decays/s) on varied computing platforms.  

We want to minimize our computational time so we now have the question as to how do we determine the minimum number of points required for a "good" fit (I know the term is ambiguous).  Just to be clear, we aren't interested in how long the sample required for a good fit is; we want to be able to determine how many points for a given portion of a curve would be required for a good fit.  Is there an explicit approach to determining this or will this require modeling?
If the data is clean, you only need two points because there are only two degrees of freedom-the original rate and the decay time.  You use more points when there is noise in the data to get a better estimate, so the number of points needed depends on how noisy the data is.  You can linearize your fit by taking the logairithm, getting $\log(\text{counts}(t))=\log (\text{counts at t=0})-\text{(decay rate*time)}$  Now a linear least squares fit will give you estimates of the errors in the parameters.  I would take a few curves and fit them each with lots of points.  You can then take a random sample of the points and see how much the fit changes.
As stated in the original post, the function is a single exponential.

Also, as stated in the original post, we are trying to DETERMINE the number of points needed.  As a rough guideline, we sample 100s to 1000s of points for each curve currently.  But maybe we only need 10 or 50 or 100.