On Wed, Nov 20, 2013 at 09:13:14PM +0100, FX wrote:
> > There is a missed optimization in
> >
> > + if (x < 12)
> > + {
> > + /* Compute directly as ERFC_SCALED(x) = ERFC(x) * EXP(X**2).
> > + This is not perfect, but much better than netlib. */
> > + return erfcq(x) * expq(x*x);
> > + }
> >
> > If x is less than approximately -8192, then erfc(x)*exp(x*x)
> > overflows.
>
> Hum, I get roughly -106 where you have -8192, so I?ll not
> commit immediately with the value I have, and let you check it first.
>
> OK to commit?
>
The optimization can come later. Your current version is fine,
because it fixes the ULP issue.
As to the optimization, for large negative x, one has
efrc(x)*exp(x*x) --> 2 * exp(x*x) = 2**emax
exp(x*x) = 2**(emax-1)
x*x = (emax-1)*log(2)
x = - sqrt((emax-1)*log(2)) ! Choosing minus branch
x = -106.56... ! emax=16384 (could be off-by-1, did not check)
So, yeah, you're correct. My suggestion was based on the not
so careful mistake of replacing x*x by x+x and dropping log(2).
That is, I had x+x = -emax --> x = - emax / 2.
--
Steve
