Re: What to do about tgammal?
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Date: Sat, 04 Dec 2021 21:20:23 UTC
On Sat, Dec 04, 2021 at 08:40:56PM +0100, Hans Petter Selasky wrote: > On 12/4/21 19:53, Steve Kargl wrote: > > What to do about tgammal? (trim some history) > > > > 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? > I'm afraid that I don't fully understand your questions. The ULP, listed above, were computed by comparing the libm tgammal(x) against a tgammal(x) computed with MPFR. The MPFR result was configured to have 256 bits of precision. In other words, MPFR is assumed to be exact for the comparison between a 64-bit tgammal(x) and a 256-bit mpfr_gamma() function. There is no speed hack with mpfr_gamma(). % time ./tlibm_lmath -l -s 0 -x 6 -X 1755 -n 100000 tgamma Interval tested for tgammal: [6,1755] 100000 calls, 0.042575 secs, 0.42575 usecs/call count: 100000 xmu = LD80C(0xae3587b6f275c42c, 4, 2.17761377613776137760e+01L), libmu = LD80C(0xb296591784078768, 64, 2.57371418855839536160e+19L), mpfru = LD80C(0xb296591784078760, 64, 2.57371418855839536000e+19L), ULP = 8.28349 6.04 real 6.02 user 0.01 sys My test program shows 100000 libm tgammal(x) calls took about 0.04 seconds while the program takes 6 seconds to finish. Most of that time is dominated by MPFR. In general, floating point arithmetic, where a finite number is the result, is inexact. The basic binary operators, +x-/*, are specified by IEEE 754 to have an error no larger that 0.5 ULP. The mantra that I follow (and know bde followed) is to try to optimize libm functions to give the most accurate result as fast as possible. -- Steve