Re: a draft motion on midpoint and radius
Nate
I agree with what you say about Dan's motion 32. But the contradiction you see between motion 37 and the current Level 1 text, is it between "has no value" and "has a value"?
Namely §7.6.8 says for an interval xx=[xlo,xhi]:
> midpoint (xlo + xhi)/2 if xx is nonempty bounded; no value if xx is empty or unbounded.
The current Level 2 §8.11.8 says "the closest F number to the Level 1 value if it exists, NaN if it doesn't exist".
The NaNs are intended as "safe" values (though Dan may disagree) -- the Level 2 version of Level 1's "no value". But I agree it is reasonable that Level 2 should be allowed to assign a value in some cases where no good Level 1 value exists. So the Level 2 function *approximates" the Level 1 one and may also *extend* it. (Maybe I didn't agree earlier, but I've changed my mind.)
Hence I see no contradiction.
However I think it's important to state properties, or invariants, that the Level 2 function shall satisfy.
Example 1: for an inf-sup type, so xx represented as [xlo,xhi] or equivalent
m = mid(xx) shall be monotone increasing in both xlo and xhi,
if one omits the case of [-oo,+oo].
I think that is true of Vladik's definition, and of any other sensible one I can think of.
Example 2: for an arbitrary type
r = rad(xx) shall be increasing under set inclusion.
That is xx \subseteq yy implies rad(xx) <= rad(yy).
That seems less obvious, even with Vladik's specification for inf-sup interval types. Vladik: is it true there?
754 arithmetic has many such invariants that have been of great value to numerical algorithm writers. E.g. round(x) is monotone in x, in all rounding modes; and 0.5 <= a/b <= 2 implies a-b is computed exactly; etc. We should aim to maximise the set of such (useful) invariants of our Level 2.
I think invariants were an important contribution of Dan's motion and should appear in the final Level 2 text in some form.
John Pryce
On 28 Sep 2012, at 17:06, Nate Hayes wrote:
> I agree. Personally I think the Level 2 definitions are good and it would be nice to find some way to provide at Level 1 a consistent definition. How to do this would be the question in my mind. Dan’s motion was a valiant effort.
> Nate
>
> From: Ralph Baker Kearfott
> Sent: Friday, September 28, 2012 10:58 AM
> To: Nate Hayes
> Cc: stds-1788 ; Michel Hack
> Subject: Re: a draft motion on midpoint and radius
>
> With motion 32 failing, I suppose it's not relevant to whether or not
> Motion 37 passes, that is, we wouldn't have an inconsistent set of
> guidance if Motion 37 passes. However, perhaps we should all understand the differences
> between the two motions.
>
> Baker
>
> On 09/28/2012 10:49 AM, Nate Hayes wrote:
> > IMO Motion 32 also gave a Level 1 definition that contradicted the Level 2 definition.
On 28 Sep 2012, at 17:48, Vincent Lefevre wrote:
> I think that Level 1 should define the midpoint on non-empty, bounded
> intervals only, and leave the remaining cases entirely to Level 2.