Every mixed-precision kernel makes the same decision, and it is usually made by hand: pick a format, pick a limb count, pick an accumulation width, and check afterwards whether the answer is close enough. The check is always "how wrong was it", never "how wrong could it have been" — so nobody can say which target accuracies are reachable at all in a given format, and which ones no amount of engineering will deliver.

There is a reason the second question goes unasked: it needs a ground truth, and floating point rarely offers one. The paper How Many Bits Does a Trusted Result Need? builds one — exact integer and rational arithmetic over a corpus pinned by SHA-256 — and makes the accuracy budget itself the thing being measured.…