Date: Sat, 4 Dec 2021 10:53:52 -0800 From: Steve Kargl <sgk@troutmask.apl.washington.edu> To: freebsd-hackers@freebsd.org, freebsd-current@freebsd.org Subject: What to do about tgammal? Message-ID: <20211204185352.GA20452@troutmask.apl.washington.edu>
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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? -- Steve
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