On 12/4/21 19:53, Steve Kargl wrote:
What to do about tgammal?
A long time ago (2013-09-06), theraven@ committed a kludge that mapped
several missing long double math functions to double math functions
(e.g., tanhl(x) was mapped to tanh(x)). Over the next few years, I
(along with bde and das reviews) provided Intel 80-bit (ld80) and IEEE
128-bit (ld128) implementations for some of these functions; namely,
coshl(x), sinhl(x), tanhl(x), erfl(x), erfcl(x), and lgamma(x). The
last remaining function is tgammal(x). If one links a program that uses
tgammal(x) with libm, one sees
/usr/local/bin/ld: fcn_list.o: in function `build_fcn_list':
fcn_list.c:(.text+0x7c4): warning: tgammal has lower than advertised
precision
The warning is actually misleading. Not only does tgammal(x) have a
*MUCH* lower precision, it has a reduced domain. That is, tgammal(x)
produces +inf for x > 172 whereas tgammal(x) should produce a finite
result for values of x up to 1755 (or so). On amd64-*-freebsd,
testing 1000000 in the below intervals demonstrates pathetic accuracy.
Current implmentation via imprecise.c
Interval | Max ULP
-------------------+------------
[6,171] | 1340542.2
[1.0662,6] | 14293.3
[1.01e-17,1.0661] | 3116.1
[-1.9999,-1.0001] | 15330369.3
-------------------+------------
Well, I finally have gotten around to removing theraven@'s last kludge
for FreeBSD on systems that support ld80. This is done with a straight
forward modification of the msun/bsdsrc code. The limitation on
domain is removed and the accuracy substantially improved.
Interval | Max ULP
-------------------+----------
[6,1755] | 8.457
[1.0662,6] | 11.710
[1.01e-17,1.0661] | 11.689
[-1.9999,-1.0001] | 11.871
-------------------+----------
My modifications leverage the fact that tgamma(x) (ie., double function)
uses extend arithmetic to do the computations (approximately 85 bits of
precision). To get the Max ULP below 1 (the desired upper limit), a few
minimax polynomials need to be determined and the mystery around a few
magic numbers need to be unraveled.
Extending what I have done to an ld128 implementation requires much
more effort than I have time and energy to pursue. Someone with
interest in floating point math on ld128 system can provide an
implementation.
So, is anyone interested in seeing a massive patch?
Hi,
Do you need a implementation of tgamma() which is 100% correct, or a
so-called speed-hack version of tgamma() which is almost correct?
I've looked a bit into libm in FreeBSD and I see some functions are
implemented so that they execute quickly, instead of producing exact
results. Is this true?
--HPS
