Date: Sun, 12 Aug 2012 23:09:38 -0000 From: Stephen Montgomery-Smith <stephen@missouri.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: Diane Bruce <db@db.net>, John Baldwin <jhb@freebsd.org>, David Chisnall <theraven@freebsd.org>, Bruce Evans <bde@freebsd.org>, Steve Kargl <sgk@troutmask.apl.washington.edu>, David Schultz <das@freebsd.org>, Peter Jeremy <peter@rulingia.com>, Warner Losh <imp@bsdimp.com> Subject: Re: Use of C99 extra long double math functions after r236148 Message-ID: <500850AE.9030608@missouri.edu> Resent-Message-ID: <20120812230931.GO20453@server.rulingia.com> In-Reply-To: <20120720035001.W4053@besplex.bde.org> References: <20120714120432.GA70706@server.rulingia.com> <20120717084457.U3890@besplex.bde.org> <5004A5C7.1040405@missouri.edu> <5004DEA9.1050001@missouri.edu> <20120717200931.U6624@besplex.bde.org> <5006D13D.2080702@missouri.edu> <20120718205625.GA409@troutmask.apl.washington.edu> <500725F2.7060603@missouri.edu> <20120719025345.GA1376@troutmask.apl.washington.edu> <50077987.1080307@missouri.edu> <20120719032706.GA1558@troutmask.apl.washington.edu> <5007826D.7060806@missouri.edu> <5007AD41.9070000@missouri.edu> <20120719205347.T2601@besplex.bde.org> <50084322.7020401@missouri.edu> <20120720035001.W4053@besplex.bde.org>
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On 07/19/2012 01:12 PM, Bruce Evans wrote: > On Thu, 19 Jul 2012, Stephen Montgomery-Smith wrote: > >> I have the ULP down to about 1.2 now. I don't see how I can do >> better, because I have to invoke log functions twice, and probably >> each one has a ULP of about 0.6. >> >> Also I decided to use 1/2 log(x*x+y*y) when x and y are not too large. > > That's close to Apple complex.c clog(). Once you don't use hypot(), > it is clearly best to use log1p(): > > log(sqrt(x*x + y*y)) = log(|x|) + 1/2 log(1 + (y*y)/(x*x)) > = log(|x|) + 1/2 log1p((y*y)/(x*x)) > > where |x| >= |y| so that log1p()'s arg is as small as possible. > >> I am really rather proud of how I got around the large ULP when >> hypot(x,y) is close to 1. I would be glad if any of you could look at >> the code when you get a chance. > > WIll look more closely later. I see that you already use log1p() and much > more. Apple clog() uses not so much more, mainly by depending on extra > precision in hardware. The above also avoids overflow and use of hypot() > for all finite x and and y, but is probably too simple. The Apple solution has a problem. The two invocations of log might produce results that are nearly identical, but with opposite signs. Think about x = y = 1/sqrt(2).
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