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Re: Discussion paper: what are the level 2 datums?



Dan, P1788

Thanks for the comments on this paper.

On 5 Oct 2010, at 19:41, Dan Zuras Intervals wrote:
> ...	While most of what you say about the meaning of 754
> 	Table 3.1 is correct, I have difficulty with the term
> 	'tagged'.  The phrase "for a particular format" is not
> 	so much to tag a floating-point value as to define
> 	which finite subset of the Reals lives at level 2 &,
> 	therefore, which set of representations need to live
> 	at level 3 to accomodate them.  That is, there is a
> 	different Table 3.1 for each floating-point datatype.
> 
> 	As far as 754 is concerned, the Binary32 value 2.0 is
> 	the same as the Binary64 value 2.0 as well as the
> 	Decimal128 value 2.0
I think you are 100% agreeing with what I MEAN, even if I'm not saying it as clearly as I would like.

> (although this last is further
> 	'tagged', if you will, with a particular element chosen
> 	from the cohort of values that are all equal to 2).

NO! The first Note in 1.1 explicitly says this is not the case. There, there is a "cohort" of values -0 and +0 equal to 0. But the interval [0,2] is NOT tagged by the particular one chosen.

> ...	The issue of whether two values chosen from two different
> 	datatypes may be compared or not is a language issue.
Precisely. WITHIN 754, they can't, not between different radices. 754§5.11 para 1:
"it shall be possible to compare one floating-point datum to another in that format (see 5.6.1). Additionally, floating-point data represented in different formats shall be comparable as long as the operands’ formats have the same radix."

> ...	(We toyed with the idea of providing inter-radix comparisons
> 	just for this purpose but ultimately rejected the idea as
> 	both too hard to do & too silly. :-)
Precisely.

> ...	Still, issues of radix aside, a level 2 value is a level
> 	2 value.  That is, it is an element of the Reals.  If two
> 	such values exist in two different level 2 sets, they are
> 	still considered equal without any consideration of the
> 	type they came from.

Yes. IN CONTEXT. You are still 100% agreeing with me but failing to see it!

> 	Similarly, I think it should be true that the interval
> 	[0,2] or [-0,2] or [+0,2] all be considered equal no
> 	matter which level 2 set they are chosen from.

YES, when asking what the mathematical value of sin([0,2]) should be. 
NO, when discussing to what pairs of values a "compareEqual" predicate may be applied.

John