From: Steven Jenkins <steven.jenkins@...>
Date: 2004-10-29T03:22:51+09:00
Subject: Re: Rounding error

trans. (T. Onoma) wrote:
> On Wednesday 27 October 2004 06:14 pm, Steven Jenkins wrote:
> | It's been done. Google for "matula floating slash". As a graduate
> | student long ago, I did a little Fortran programming on the topic for
> | David Matula.
> 
> Nice. I'm impressed. Unfortunately I couldn't find anything very detailed on 
> that method. (Have a good link?) But I did discover SLI arithmetic --not 
> perfect but it is more precise then f.p. and apparently is poised to have 
> great effects in supercomputing.

I was never involved in the analysis; I just wrote some code. Matula and 
Kornerup published a few papers, but I don't know whether they're on the 
web or not.

> But what was more interesting were the very real and practical solutions 
> coming into play. Take one part GMP and add:
> 
>   MPFR: http://www.mpfr.org/
> 
> or
> 
>   ARITHMOS: http://www.win.ua.ac.be/~cant/arithmos/
> 
> and presto! That was easy ;) Looks like the standards guys are way behind.

It was never the intent of IEEE 754 to prescribe optimal numerical 
properties. Like all industry standards, its purpose was to balance 
competing interests: numerical properties, performance, 
manufacturability, etc. Some of the companies that participate in the 
standards process are implementing these algorithms in proprietary code 
or silicon. They're not about to adopt a 'new' approach that has 
marginally better numerical properties if it puts them at a significant 
competitive disadvantage in terms of performance or cost.

Also, bear in mind that for many types of problems, the error in the 
mathematical approximations used to pose the problem and/or the 
algorithms used to solve it are considerably larger than the errors in 
the FP implementation. Additional precision there may cost you a lot and 
do you no good.

Steve