How does one analyze the accuracy of the following method to evaluate the summation?
Also, what is the difference between analyzing the accuracy and proving that it is backward stable? I think I have to prove it is backward stable, since the accuracy of solution depends on the stability of the algorithm and conditioning of the problem.
I would avoid this method.
For one thing, if one of the $x_i$ is very large and negative, $e^{x_i}$ will be evaluated as zero, so the product will be zero no matter what the other terms are.
Also, for large positive $x_i$, the error in $e^{x_i}$ will be much larger than the error in $x_i$, since $e^{x+c} =e^x e^c $, and $c$ can be fairly large if $x$ is quite large.