cexpl() (was: Re: Update ENTERI() macro)

[Steve Kargl]

On Sat, Mar 09, 2019 at 12:51:07AM +1100, Bruce Evans wrote:
> On Thu, 7 Mar 2019, Steve Kargl wrote:
> 
> > On Thu, Mar 07, 2019 at 05:36:22PM +1100, Bruce Evans wrote:
> >> On Wed, 6 Mar 2019, Steve Kargl wrote:
> >>
> >>> On Thu, Mar 07, 2019 at 03:09:06PM +1100, Bruce Evans wrote:
> >* ...
> >>> 	/*
> >>> 	 *  exp_ovfl = 11356.5234062941439497, 0x400c, 0xb17217f7d1cf79ac
> >>> 	 * cexp_ovfl = 22755.3287906024445633, 0x400d, 0xb1c6a8573de9768c
> >>> 	 */
> 
> I checked that these are correct.  Any rounding should be down for
> exp_ovfl and up for cexp_ovfl.
> 
> >>> 	if ((hx == 0x400c && lx > 0xb17217f7d1cf79acULL) ||
> >>> 	    (hx == 0x400d && lx < 0xb1c6a8573de9768cULL)) {
> >>
> >> Don't duplicate the hex constants in the comment.  The limits can be fuzzy
> >> so don't need so many decimal digits or nonzero nonzero bits or LD80C()
> >> to round them (s_cexp.c uses only 1 decimal digit and doesn't test the
> >> lower 32 bits in the mantissa; here we have more bits than we need in lx).
> >
> > Being fuzzy was the movitation for the new macro, I was only using
> > the upper 32 bits of lx in the original cexpl() I posted.
> 
> IIRC, that was done by extracting into a 32-bit lx.  This could be written
> as (lx >> 32) > 0xb17217f7 (or a fuzzier value), and it might be better
> to do that anyway.

This is what I do in my WIP ccosh.  There are 3 thresholds
of ~22.9, ~11356, and ~22755.  I'm casting the result to
uint32_t, so something like (uint32_t)(lx >> 32) > 0xb17000000.
Do I need the cast and/or do you prefer it? 


> >> Maybe not use bit-fiddling at all for this (in other precisions too).
> >> The more important exp() functions check thresholds in FP.  Here x can
> >> be NaN so FP comparisons with it don't work right.  That seems to be
> >> the only reason to check in bits here.
> >>
> >>> 		/*
> >>> 		 * x is between exp_ovfl and cexp_ovfl, so we must scale to
> >>> 		 * avoid overflow in exp(x).
> >>> 		 */
> >>> 		RETURNI(__ldexp_cexpl(z, 1));
> >>
> >> Did your tests cover this case, and does it use the fixed version of the
> >> old __ldexp_cexpl() in k_expl.h?
> >
> > No.  The test I reported was restricted to -11350 < x < 11350.
> > Spot checked x outside that range.  I had the conditional wrong
> > for x < -exp_ovfl.  Also, the comment is actaully a little optimistic.
> > One cannot avoid overflow.  For example, x = 11357 is just above
> > exp_ovfl, and exp(x) = inf.  __ldexp_cexpl() does scaling and exploits
> > |cos(y)| <= 1 and |sin(y)| <= 1 to try to extend the range of cexpl().
> 
> I understand this better now.  __ldexp_cexpl() doesn't work for all large
> x, so callers must only use it for a range of values above cexp_ovl.
> The correctness of this is unobvious.  It depends on sin(y) and cos(y)
> never underflowing to 0.  Otherwise, this method would give Inf * 0 = NaN
> for finite x and y, and the scaling method would give finite * 0 * 2**k = 0.

cos(y) can never be 0 and sin(y) is only 0 at y = 0 (see sinpi, cospi
implementations).  y=0 is a special case and handled separately.  We
can have Inf*subnormal when y is near 0.

-- 
Steve