Use of C99 extra long double math functions after r236148

[Steve Kargl]

On Sun, Jul 08, 2012 at 07:29:30PM -0500, Stephen Montgomery-Smith wrote:
> On 07/08/2012 06:58 PM, Steve Kargl wrote:
> >On Sun, Jul 08, 2012 at 02:06:46PM -0500, Stephen Montgomery-Smith wrote:
> 
> >>So do people really work hard to get that last drop of ulp out of their
> >>calculations?
> >
> >I know very few scientist who work hard to reduce the ULP.  Most
> >have little understanding of floating point.
> >
> >>  Would a ulp=10 be considered unacceptable?
> >
> >Yes, it is unacceptable for the math library.  Remember ULPs
> >propagate and can quickly grow if the initial ulp for a
> >result is large.
> 
> OK.  But suppose I wanted ld80 precision.  What would be the advantage 
> of using an ld80 expl function with a ulp of 1 over an ld128 expl 
> function with a ulp of 10?  The error propagation in the latter case 
> could not be worse than the error propagation in the latter case.

Well, on the most popular hardware (that being i386/amd64), 
ld80 will use hardware fp instruction while ld128 must be
done completely in software.  The speed difference is 
significant.

> In other words, if I were asked to write a super-fantastic expl 
> function, where run time was not a problem, I would use mpfr, use 
> Taylor's series with a floating point precision that had way more digits 
> than I needed, and then just truncate away the last digits when 
> returning the answer.  And this would be guaranteed to give the correct 
> answer to just above 0.5 ulp (which I assume is best possible).

It's more like 1 ULP after truncation (, which isn't truncation 
but rounding).

The problem is run-time that 'run-time is the problem'.  Try
writing a expl() and simply use mpfr_exp() with 64-bit
precision.  If you're doing any serious simulation where 
exp() will be evaluate millions or billions of time, you'll
notice the difference.

> The only way I can see ULP errors greatly propagating is if one is 
> performing iteration type calculations (like f(f(f(f(x))))).

Have you read Goldberg's paper?

Not to mention, I've seen way too many examples of 'x - y'
where cancellation of significant digits causes
problems.  Throw in rather poor estimates of function
results with real poor ULP and you have problems.

-- 
Steve