Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
> From: "Nate Hayes" <nh@xxxxxxxxxxxxxxxxx>
> To: <rbk@xxxxxxxxxxxxx>
> Cc: "Dan Zuras Intervals" <intervals08@xxxxxxxxxxxxxx>,
> "P-1788" <stds-1788@xxxxxxxxxxxxxxxxx>
> Subject: Re: A question Re: Level 1 <---> level 2 mappings; arithmetic versus applications
> Date: Wed, 30 Jun 2010 19:56:53 -0500
>
> . . .
>
> (also, I know I said I was going to shut-up and listen, but now two people
> have addressed me specifically...)
Its OK. I understand. A friend asked me today
what I've been up to & I told her that I started
a bar fight last night but didn't get out of the
bar until I was trampled. It is now some 85 email
notes later & I'm still trying to get to the door.
>
> . . .
>
> Let's take a closer look:
>
> (for the sake of having a concrete discussion, I like to speak specifically
> again about Level 1 and Level 2 inf-sup and mid-rad intervals. This is just
> for illustration and discussion, ok? So Dan, please indulge me).
Again, its OK. I understand your meaning.
>
> If we have Level 1 inf-sup interval [a,b] and Level 1 mid-rad interval
> (m;r), with m=(a+b)/2 and r=(b-a)/2, then we also have Level 2 inf-sup
> interval [A,B] and Level 2 mid-rad interval (M;R), each the tightest
> possible superset such that [a,b] \subset [A,B] is true and (m;r) \subset
> (M;R) is true. Generally speaking, it may not be the case that A=M-R or
> B=M+R, however this is no problem, since at Level 1 [a,b] \subset [M-R,M+R]
> will always be true (please pay special attention to the case, and note that
> I'm assuming so far all arithmetic is calculated with infinite precision).
All true. And your subset observations are important.
>
> This agrees with the statement "we can think of mid-rad at level 2 as
> giving a different set of objects than inf-sup, just as we think of binary
> and decimal floating point data as different sets".
This is the part I am trying to change but more on that
below.
>
> So far so good. It all hangs together.
>
> However, if we require the endpoints of any Level 2 interval must be
> extractable in a lossless manner as a floating-point number, this implies
> A=M-R and B=M+R must both be true. But these equalities may not always hold.
> It may even be possible that there does not exist any such floating-point
> numbers M-R or M+R. These numbers M-R and M+R might only exist at Level 1
> when the arithmetic is performed in infinite precision.
This is close. What I am advocating is that we MAKE IT
TRUE. Again, more below.
>
> HOWEVER:
>
> As Dan explains in is e-mail:
>
> > It sounds complicated but its not, really. If I have
> > a Real level 1 interval for which the Real midpoint is
> > midR & the Real radius is radR, I can use the subset of
> > mid-rad elements defined by the assignments
> >
> > mid <-- roundToNearest(midR)
> > rad <-- roundAway(mid + radR) - mid.
> >
> > Not all mid-rad pairs have the property that they can
> > be summed to an element of F exactly, but THIS SUBSET
> > of the mid-rad pairs DOES have that property, by
> > construction.
> ...
> > And it doesn't solve everything.
> >
> > It will often return an interval that is slightly wider
> > than would be returned in an inf-sup form but only by
> > an ULP on one side or the other. This will piss off
> > the inf-sup guys who want narrowest interval uber alles
> > (no nationality dig intended :-) but it is the price of
> > freedom from concern about the nature of the format.
>
> My point is this: WHY do we even need to make these restrictions and
> compromises, when everything we already hope to achieve is already specified
> by simply saying something along the lines:
Why, you ask? We need not. But it offers us no
hope for compromise unless we do something.
>
> "The Level 2 result of an operation is the tightest possible superset of the
> true result within the set of floating-point intervals represented by the
> Level 2 type."
>
> This may potentially result in some widening when converting from mid-rad to
> inf-sup. But it doesn't require any restrictions on mid-rad Level 2 objects,
> and Dan's solution causes widening in this case anyways.
>
> Nate
Nate is quite correct in his observations above
that some abstract Real interval may be projected
onto level 2 in either a [A,B] or [M-R,M+R] manner
with both being supersets of the original Real
interval & either being a strict subset of the
other.
(I think it is also possible that NEITHER need be
a proper subset of the other but I am less sure of
this.)
There is an old joke.
Patient to doctor: "Doctor, it hurts when I do this."
Doctor to patient: "Then don't do that."
That is what I am advocating.
Let's not do that.
Let's not have a standard in which two level 2 subsets
that are associated with the same floating-point type
F have two different sets of elements hanging around.
Let's define it to admit only one subset for any given
associated floating-point type & then take steps to
MAKE IT THAT WAY.
I chose to define it by the exact extraction of bounds.
You could argue that it could also be defined by the
exact extraction of midpoint & radius.
But if we start that argument it will never end & we
will never come to an agreement.
Pick something.
And live with what you pick.
Can someone get the door for me?
I really need to get some fresh air & lick my wounds. :-)
Dan