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On 27/12/2013 03:30, Vincent Lefevre wrote:
§12.10.1: Concerning Guillaume's remark about the "tightest" mode with implicit types, I would not disagree to relax the spec. Similarly, I also made a remark in quite old discussions, wondering why a particular minimal enclosure had to be chosen for the T-hull, and I don't remember if there was any good reason. One should notice that with some particular implicit types, a minimal enclosure may be much wider than other ones, but I see such issues as QoI (just like the choice of the precision for the interval types). However, later in Guillaume's remark: "As a consequence, I believe it is impossible to design a conforming implementation.", I don't think this is true; any example?
Imagine you have two mathematical functions f1 and f2 and you are able to implement the interval versions so that they always return minimal enclosures. If you tell that "f1 is tightest", then you have a conforming library. If you tell that "f2 is tightest", then you also have a conforming library. But if you tell that "both f1 and f2 are tightest", do you still have a conforming library? The lack of modularity cannot be a good sign.
As for an actual example, I guess it will have to be a bit farfetched. Consider cospi([0;0.25]) and sqrt([0.75;1]). (My trigonometry is a bit rusty, so bear with me if I got the inputs wrong.) What are the chances that the implementations return the exact same result for an implicit type? Any of these two functions could be tightest taken in isolation, but no longer when shipped together in a library.
Perhaps a bit less contrived: on one side you perform (1,1) + (1+e,1) in mid-rad; one the other side you downgrade (2+e,2) from a larger format (where 2+e is representable). Even if both the addition and the conversion return a minimal enclosure, how do you guarantee that both functions are tightest?
Best regards, Guillaume