Date: Sun, 12 Aug 2012 23:13:57 -0000 From: Bruce Evans <brde@optusnet.com.au> To: Stephen Montgomery-Smith <stephen@missouri.edu> Cc: Diane Bruce <db@db.net>, Steve Kargl <sgk@troutmask.apl.washington.edu>, John Baldwin <jhb@freebsd.org>, David Chisnall <theraven@freebsd.org>, Bruce Evans <bde@freebsd.org>, Bruce Evans <brde@optusnet.com.au>, 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: <20120720120802.F1061@besplex.bde.org> Resent-Message-ID: <20120812231349.GG20453@server.rulingia.com> In-Reply-To: <50083E83.9090404@missouri.edu> References: <20120529045612.GB4445@server.rulingia.com> <20120711223247.GA9964@troutmask.apl.washington.edu> <20120713114100.GB83006@server.rulingia.com> <201207130818.38535.jhb@freebsd.org> <9EB2DA4F-19D7-4BA5-8811-D9451CB1D907@theravensnest.org> <C527B388-3537-406F-BA6D-2FA45B9EAA3B@FreeBSD.org> <20120713155805.GC81965@zim.MIT.EDU> <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> <20120719144432.N1596@besplex.bde.org> <50083E83.9090404@missouri.edu>
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On Thu, 19 Jul 2012, Stephen Montgomery-Smith wrote: > On 07/19/2012 01:37 AM, Bruce Evans wrote: >> ... >> problem. Complex functions should have only poles and zeros, with >> projective infinity and "projective zero" (= inverse of projective >> infinity). Real functions can and do have affine infinities and zeros >> (+-Inf and +-0), with more detailed special cases. It's just impossible >> to have useful, detailed special cases for all the ways of approaching >> complex (projective) infinity and 0. >> ... >> sign functions). Hopefully, the specification of imag(clog()) is >> that it has the same sign behaviour as atan2(), so you can just use >> atan2(). The sign conventions for both are arbitrary, but they >> shouldn't be gratuitously different. You still have to check that >> they aren't non-gratuitously different, because different conventions >> became established. > > I checked. Actually the sign conventions are not that arbitrary. But as a > mathematician I would say they are a bit useless, e.g. > atan(infinity,infinity) = pi/4 = 45 degrees > How do you know that the two infinities are the same? One could be double > the other. It should be NaN with projective infinity. > If it had been up to me, there would have been finite numbers, and nan. And > none of this -0. I think Kahan is a mathematician, and is primarily responsible for +-0. +-0 give poor man's branch cuts for real functions. >> I don't know what happens with zeros of complex inverse trig functions. >> I think they don't have many (like log()), but their real and imaginary >> parts do, and they are too general for accurate behaviour of the real >> and imaginary parts relative to themselves to fall out. > > casinh(z) is zero only when z=0, and near that point I could use Taylor's > series (but a lot of terms would be needed because the Taylot series > converges quite slowly). To get accuracy near zeros, you only need to use a series method in a very small radius. Even a linear approximation may be enough, and the main difficulty is the linear term: f() ~= f'(z0) * (z-z0) where z0 typically needs to be known to hundreds or thousands of bits of precision and the subtraction must be done in this precision. f'(z0) and the multiplkication only need a couple of extra bits. This is only easy when z0 is a dyadic rational. > I can now see that the separate cases of the real part and imaginary parts of > casinh being zero is going to be hard. I won't ask for that and will measure errors relative to the absolute value of the result. Bruce
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