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Re: Comparisons and decorations, part 2

Dan Zuras Intervals wrote:

Date: Tue, 28 Sep 2010 14:26:23 +0200
From: Arnold Neumaier <Arnold.Neumaier@xxxxxxxxxxxx>
To: Dan Zuras Intervals <intervals08@xxxxxxxxxxxxxx>, 1788 <stds-1788@xxxxxxxxxxxxxxxxx> 
Subject: Re: Comparisons and decorations, part 2

Dan Zuras Intervals wrote:


	Then let me suggest that there exist a
	decoration for empty.

	Or, at least, some combination of the
	existing decorations that implies empty.

	That way we do not NEED a representation
	for empty within the interval part of a
	1788 interval.

	And we will have anything + empty = empty
	fall out to be both fast & easy no matter
	what we use for empty so long as the
	decoration makes it so.

Since bare[1,1]/bare[0,0] or sqrt(bare[-2,-1]) must be empty,
this conflicts with the need of a fast mode for arithmetic
operations for bare intervals.


	On the contrary, it is consistent with that.

	I was discussing fast & easy intervals for
	which the interpretation of empty is to be
	found in the decoration.

	That you might need some further restriction,
	say your favorite [NaN,NaN], for bare intervals
	is still consistent.

	And, should we feel that it is the appropriate
	way to go within that context, I have no
	objection.


It seems that you suggest the following:

(i) Constructors always create decorated intervals, never bare ones.
Bare intervals are created by stripping decorated ones using the
operator Bare.

(ii) The empty interval Empty is represented by [NaN,NaN] when bare,
and by ([NaN,NaN],IsEmpty) when created by a constructor.
However, _every_ decorated interval of the form ([any,any'],IsEmpty)
is interpreted as Empty.

(iii) The stripping of a decorated interval may modify its bare part
iff the semantics necessitates it. In particular,
      Bare(([any,any'],IsEmpty)) = [NaN,NaN].

(iv) Arithmetic operations on decorated intervals always preserve
an IsEmpty decoration.

This indeed captures the necessary behavior of Empty, and improves
the efficiency of propagating decorated intervals in general, while
not harming fast expression evaluation for bare intervals.

However, it still means that
    bare[a,b] interior bare[c,d]
needs a branch to handle bareEmpty.