bin/170206: complex arcsinh, log, etc.

[Stephen Montgomery-Smith]

The following reply was made to PR bin/170206; it has been noted by GNATS.

From: Stephen Montgomery-Smith <stephen at missouri.edu>
To: Bruce Evans <brde at optusnet.com.au>
Cc: Stephen Montgomery-Smith <stephen at freebsd.org>,
        FreeBSD-gnats-submit at freebsd.org, freebsd-bugs at freebsd.org
Subject: Re: bin/170206: complex arcsinh, log, etc.
Date: Fri, 27 Jul 2012 15:50:14 -0500

 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.