Date: Thu, 7 Mar 2019 11:54:36 -0800 From: Steve Kargl <sgk@troutmask.apl.washington.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-numerics@freebsd.org Subject: Re: Update ENTERI() macro Message-ID: <20190307195436.GA20856@troutmask.apl.washington.edu> In-Reply-To: <20190307163958.V1293@besplex.bde.org> References: <20190304212159.GA12587@troutmask.apl.washington.edu> <20190305153243.Y1349@besplex.bde.org> <20190306055201.GA40298@troutmask.apl.washington.edu> <20190306225811.P2731@besplex.bde.org> <20190306184829.GA44023@troutmask.apl.washington.edu> <20190307061214.R4911@besplex.bde.org> <20190306214233.GA23159@troutmask.apl.washington.edu> <20190307144315.N932@besplex.bde.org> <20190307044447.GA16298@troutmask.apl.washington.edu> <20190307163958.V1293@besplex.bde.org>
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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:
> >> On Wed, 6 Mar 2019, Steve Kargl wrote:
> >> ...
> >> But does it preserve NaN bits as much as possible, and produce the same
> >> NaN bits as double precision after conversion to double precision? The
> >> report spells NaN and Inf using C99's fuzzy bad names. Even the C99
> >> specification spells NaN and Inf correctly in Appendix F.
> >>
> >> The complex hyperbolic functions have better comments, so are almost
> >> an easier place to start.
> >
> > I suppose it preserves the NaN bits as well as cexpf and cexp as
> > it uses the same algorthim.
>
> > ...
> > long double complex
> > cexpl (long double complex z)
> > {
> > long double c, exp_x, s, x, y;
> > uint64_t lx, ly;
> > uint16_t hx, hy, ix, iy;
> >
> > ENTERI();
> >
> > x = creall(z);
> > y = cimagl(z);
> >
> > EXTRACT_LDBL80_WORDS(hx, lx, x);
> > EXTRACT_LDBL80_WORDS(hy, ly, y);
>
> I think you recently changed to this from your new macro.
Yes, the new macro is gone.
> >
> > ix = hx&0x7fff;
> > iy = hy&0x7fff;
>
> Some style bugs. s_cexp.c doesn't use the fdlibm'ish style of sometimes
> no spaces around '&'. It puts the ix and iy calculations immediately
> after the extractions. They shouldn't be separated since they are
> also extractions. c_cexp.c extracts in a different order, and doesn't
> use iy. You reordered it a bit to use sincos(). I don't like the
> reordering much. But old s_cexp.c has its own style bug of a separating
> the extraction for y but not even following its usual style of a blank
> line before comments near the extraction of x.
I may unorder the re-ordering to the old order. If I understand
one of the comments in the GCC bugzilla report, the optimization of
converting
a = cos(x);
b = sin(x):
to
sincos(x, &b, &a).
Actually, does "a = cos(x); b = sin(x)" goes to cexp(cmplx(0,x)) in
the middle-end of the compiler. The backend then can recognize
cexp(cmplx(0,x)) and converts this to sincos(x, &b, &a). I can't
verify this for FreeBSD, because libm isn't C99 compliant. GCC
requires a C99 compliant to activate the optimization.
I fixed the extraction for hx, and hy.
> > /* cexp(0 + I y) = cos(y) + I sin(y) */
> > if ((ix | lx) == 0) {
> > sincosl(y, &s, &c);
> > RETURNI(CMPLXL(c, s));
> > }
> >
> > /* cexp(x + I 0) = exp(x) + I 0 */
> > if ((iy | ly) == 0)
> > RETURNI(CMPLXL(expl(x), y));
> >
> > if (iy == 0x7fff) {
>
> s_cexp.c actually uses a clobbered hy here. Perhaps fix this too (compilers
> mostly ignore program order for initializing variables like iy and optimize
> to an expression involving the unclobbered hy here if that is optimal. The
> extra variables are just for readability or unreadability).
>
> > if (ix < 0x7fff ||
> > (ix == 0x7fff && (lx & 0x7fffffffffffffffULL) != 0)) {
> > /* cexp(finite|NaN +- I Inf|NaN) = NaN + I NaN */
> > RETURNI(CMPLXL(y - y, y - y));
>
> Looks OK.
>
> I think ix and iy are only used here, so it is clearer to not have
> variables for them -- just use (hx & 0x7fff), etc.
>
> y - y and similar expressions usually give the right NaN (y = Inf -> dNaN
> and y = NaN -> d(y)).
>
> IIRC, the only differences for ld128 is that then normalization bit is
> hidden so masking out the top bit in the mantissa is not needed, but the
> mantissa is in 2 64-bit words.
>
> > } else if (hx & 0x8000) {
>
> This is clearest as it is with an explicit mask.
>
> > /*
> > * exp_ovfl = 11356.5234062941439497, 0x400c, 0xb17217f7d1cf79ac
> > * cexp_ovfl = 22755.3287906024445633, 0x400d, 0xb1c6a8573de9768c
> > */
> > 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.
> 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().
./testl -a 11357 0.1
libm: Re(cexpf) = inf
mpfr: Re(cexpf) = 1.90658369228334884925e+4932
libm: Im(cexpf) = 1.91296449568717353572e+4931
mpfr: Im(cexpf) = 1.91296449568717353558e+4931
troutmask:kargl[430] ./testl -a 11357 1.57
libm: Re(cexpf) = 1.52588659766018377153e+4929
mpfr: Re(cexpf) = 1.52588659766018377147e+4929
libm: Im(cexpf) = inf
mpfr: Im(cexpf) = 1.91615588587371861485e+4932
>
> > } else {
> > /*
> > * Cases covered here:
> > * - x < exp_ovfl and exp(x) won't overflow (common case)
> > * - x > cexp_ovfl, so exp(x) * s overflows for all s > 0
> > * - x = +-Inf (generated by exp())
> > * - x = NaN (spurious inexact exception from y)
> > */
> > exp_x = expl(x);
> > sincosl(y, &s, &c);
> > RETURNI(CMPLXL(exp_x * c, exp_x * s));
> > }
> > }
>
> I think the last clause can be simplfied and even optimized by using the
> kernel in all cases:
> - the kernel produces exp_x in the form kexp_x * 2**k where kexp_x is near
> 1. Just multiply kexp_x by c and s, and later multiply by 2**k. This
> cost an extra multiplication by 2**k (for the real and imaginary parts
> separately), but avoids other work so may be faster.
> - hopefully kexp_x is actually >= 1 (and < 2). Otherwise we have to
> worry about spurious underflow
> - the above method seems to avoid spurious underflow automatically, and
> doesn't even mention this (multiplying by exp_x may underflow, but not
> spuriously since c and s are <= 1).
You might be right about using the kernel. I'll leave that
to others on freebsd-numerics@; although you and I seem to be
the only subscribers.
--
Steve
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