It is the technique of backwards-error analysis that is naive, not the resulting bound. A “naive” method is not “simple” or “dumb” (or “aggressive” or whatever you think the opposite of “conservative” might be). It is simply a method that doesn’t use some a priori knowledge that’s available for the specific problem at hand. Backwards-error analysis is a naive technique in that it doesn’t benefit from specialized theorems of floating-point that allow one to establish tighter bounds, like the one I proved.
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I pointed part of this out:
It is the technique of backwards-error analysis that is naive, not the resulting bound. A “naive” method is not “simple” or “dumb” (or “aggressive” or whatever you think the opposite of “conservative” might be). It is simply a method that doesn’t use some a priori knowledge that’s available for the specific problem at hand. Backwards-error analysis is a naive technique in that it doesn’t benefit from specialized theorems of floating-point that allow one to establish tighter bounds, like the one I proved.