Re: Motion 4: P1788 on non-754?
> Date: Mon, 13 Apr 2009 15:00:37 +0200
> From: Vincent Lefevre <vincent@xxxxxxxxxx>
> To: "stds-1788@xxxxxxxx" <stds-1788@xxxxxxxx>
> Subject: Re: Motion 4: P1788 on non-754?
>
> On 2009-04-12 16:10:18 -0500, Corliss, George wrote:
> > At one level, how something may be implemented is not the concern of the
> > standard. At that level, the underlying hardware is irrelevant. At another
> > level, if something is impossible to implement (efficiently, depending on
> > one's point of view), there is no point putting it into a standard. Hence,
> > discussions leading to a standard SHOULD sometimes include discussions about
> > implementations. That is messy enough if the discussions are about possible
> > implementations in 754. The discussion complexity is probably unbounded if
> > we admit discussions of possible implementations on ANY architecture, known
> > or imagined.
>
> Not necessarily. IMHO, an arithmetic (not necessarily floating-point)
> with exceptions and directed rounding (possibly not correct rounding,
> but an implementation should provide error bounds, so that the user
> can know how tight his intervals will be) should be sufficient to
> define an interval arithmetic.
>
> Let's take an example: the double-double arithmetic, which can be
> implemented thanks to IEEE 754 BTW (so, on most architectures), and
> which is often used to extend the precision. I wonder if directed
> rounding is also used in practice with it, but anyway, this is
> possible. Should P1788 ignore such arithmetics? Is there any reason
> why such arithmetics would make IA definition significantly more
> complicated?
>
> --
> Vincent Lefèvre <vincent@xxxxxxxxxx> - Web: <http://www.vinc17.org/>
There is much here worthy of comment.
First, George is quite correct that, while we may prefer to
discuss our important interval issues in the abstract, it is
useful to know how those abstractions will be made manifest.
Or at least to have in the mind's eye how such a manifestation
might appear without working out the gritty details while
thinking of the abstraction.
Thus the focus on 754 which is the subject of Motion 4.
Next, Vincent is also correct to point out that all the
features of 754 are not, strictly speaking, necessary for our
interval work. Correctly rounded arithmetic is sufficient but
'incorrect' or approximate arithmetic is also sufficient so
long as you know of & can apply strict error bounds.
He uses double-double as an example.
Here is where we part company.
While double-double can be used as a starting point for a
larger & more general arithmetic, as a basis for an arithmetic
about which we wish to prove theorems it leaves much to be
desired. Let me point out just one problem: the density of
double-double numbers varies with the value of the number in
an inconsistent way. So while 1/3 may be approached from
either side by numbers near 1/3 +/- eps^2 (where eps ~ 2^-53),
we have that 1/2 is surrounded by a dense group of numbers
down to the limit of the exponent range (1/2 +/- 2^-1074).
Thus double-double is, in fact, an excellent example of an
arithmetic we do NOT want to consider as a basis for our
interval standard. Not because its bad. Its not. Not
because its inaccurate. It isn't. But because it has very
unusual properties. Properties that we would have to take
into account every time we made a decision about: containment,
acceptable roundoff error, exceptions (one part NaN & the
other not or one part over/underflowed & the other not), &
on & on.
It is best if we don't have to consider such things. And in
this case, by 'best' I mean 'simplest'.
Or at least as simple as we can make it & no simpler.
As Vincent is no doubt aware, this will cause difficulties
for MPFR as a basis for intervals. It is not because it is
a variable precision arithmetic. We made sure to account for
such things in 754-2008. It is because it is an arithmetic
that has made decisions to back off on things like correct
rounding for very good & practical reasons.
Still, if Motion 4 passes, it does not automatically rule
out MPFR. After all, even though MPFR does not meet 754-2008
there is nothing YET in 1788 that MPFR cannot do. Those
decisions are in the future. We are not yet ready to make
them.
And things can happen between now & then. We could decide
that very few requirements of 754 are necessary for 1788.
Or the MPFR folks could decide to make a 754-2008 compliant
version. They have the incentive, at least now they do.
We have the time.
If we use it wisely, that is.
Dan