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How do you know that the sets of samples were identical? There was at least a half-second difference in time of execution, although it's difficult to ascribe that to more samples taken.

I put code like [1] [STO + 0] [Rdown] inside the function being integrated. This gives me the number of function evaluations (assuming you know what was in R0 at the beginning). The set of points at which the function is evaluated is known from the non-uniform transformation. For a given number of evals, the set of points is always the same (before the transformation).

I've noticed the slight time difference, though I didn't measure it to 0.01 seconds :-) What it's doing in that time, I don't know.

The best I can figure is that in FIX 3 it was satisfied with the initial result and presented that. In FIX 4 it wasn't satisfied, but rather than doubling the number of points it first tried some other tweak, which did satisfy it. In FIX 5 the tweak wasn't enough so it does go ahead and double the number of points.

But just what that "tweak" on top of the Romberg extrapolation is, I don't know. And if that tweak only takes half a second, why doesn't it always apply it?

BTW, have you ever looked at all ten digits of a typical error estimate? Where is that other non-zero digit coming from, and why isn't it always in the same place relative to the first one?