Re: What to do about tgammal?

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