Re: MidRad and reproducibility
Vincent Lefevre wrote:
On 2009-09-21 14:15:45 +0200, Arnold Neumaier wrote:
Can you please give an example of what beyond enclosure you want to
require from an implementation of midrad multiplication?
1. Something about the accuracy (i.e. an accuracy mode 'accurate',
which would have some additional requirements, and an accuracy
mode 'valid').
This would seem possible, although is is not trivial to come up with
an algorithm-free definition of accurate that is provably implementable
for +,-,*,and /. What would you suggest?
2. Something about the midpoint of an interval. If you evaluate
yy = f(xx), what's the relation between mid(yy) and f(mid(xx))?
Do we have correct rounding or some error bound?
Please make a specific suggestions of the relation that you'd want to
see. Forcing the two equal forces overestimation in the width of the
product by a factor of up to 1.5. Would that be compatible with an
accurate mode?
3. Possibly a specification of the exceptions (like this will be
done for infsup). Could be optional.
One could probably take over all annotations.
I think that since midrad is more targeted at performance-critical
applications [in multiple precision] (if performance doesn't matter,
infsup should be better), the representation should be left to the
implementation.
Please explain why performance could be critical. If you work with
N digits, an addition/subtraction takes O(N) work while the bounding
part costs only O(1), and a multiplication takes O(N^2) work while the
bounding part costs only O(N), whether done with lowprecision midrad or
with triplex. Division and elementary functions are surely not less complex.
Arnold Neumaier