Date: Fri, 27 Jul 2012 15:50:14 -0500 From: Stephen Montgomery-Smith <stephen@missouri.edu> To: Bruce Evans <brde@optusnet.com.au> Cc: freebsd-bugs@freebsd.org, FreeBSD-gnats-submit@freebsd.org, Stephen Montgomery-Smith <stephen@freebsd.org> Subject: Re: bin/170206: complex arcsinh, log, etc. Message-ID: <5012FF06.4030501@missouri.edu> In-Reply-To: <20120727233939.A7820@besplex.bde.org> References: <201207270247.q6R2lkeR021134@wilberforce.math.missouri.edu> <20120727233939.A7820@besplex.bde.org>
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On 07/27/2012 09:26 AM, Bruce Evans wrote: > VC> > For clog, the worst case that I've found so far has x^2+y^2-1 ~= > 1e-47: > VC> > > VC> > x = > 0.999999999999999555910790149937383830547332763671875000000000 > VC> > y = > VC> > 0.0000000298023223876953091912775497878893005143652317201485857367516 > VC> > (need high precision decimal or these rounded to 53 bits > binary) > VC> > x^2+y^2-1 = 1.0947644252537633366591637369e-47 > VC> VC> That is exactly 2^(-156). So maybe triple quad precision really > is enough. Furthermore, if you use the computation (x-1)*(x+1)*y*y (assuming as you do x>y>0), only double precision is necessary. This is proved in the paper "Implementing Complex Elementary Functions Using Exception Handling" by Hull, Fairgrieve, Tang, ACM Transactions on Mathematical Software, Vol 20, No 2, 1994. They give a bound on the error, which I think can be interpreted as being around 3.9 ULP. And I think you will see that your example does not contradict their theorem. Because in your example, x-1 will be rather small. So to get reasonable ULP (reasonable meaning 4 rather than 1), double precision is all you need.
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