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Date: Fri, 27 Jul 2012 14:40:08 GMT
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To: freebsd-bugs@FreeBSD.org
From: Stephen Montgomery-Smith <stephen@missouri.edu>
Cc: 
Subject: Re: bin/170206: complex arcsinh, log, etc.
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The following reply was made to PR bin/170206; it has been noted by GNATS.

From: Stephen Montgomery-Smith <stephen@missouri.edu>
To: Bruce Evans <brde@optusnet.com.au>
Cc: Stephen Montgomery-Smith <stephen@freebsd.org>,
        FreeBSD-gnats-submit@freebsd.org, freebsd-bugs@freebsd.org
Subject: Re: bin/170206: complex arcsinh, log, etc.
Date: Fri, 27 Jul 2012 09:39:55 -0500

 On 07/27/2012 09:26 AM, Bruce Evans wrote:
 > On Wed, 25 Jul 2012, Stephen Montgomery-Smith wrote:
 >
 >> This function seems to be able to compute clog with a worst case
 >> relative error of 4 or 5 ULP.
 >> ...
 >
 > I lost your previous reply about this after reading just the first part.
 > Please resend if interested.
 >
 > First part recovered by vidcontrol:
 >
 > VC> > I'm still working on testing and fixing clog.  Haven't got near
 > the more
 > VC> > complex functions.
 > VC> >
 > 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.
 >
 > Hmm.  But you need 53 more value bits after the 156.  Quadruple precision
 > gives 3 to spare.  I didn't notice that this number was exactly a power
 > of 2, but just added 15-17 for the value bits in decimal to 47 to get over
 > 60.
 
 I think one should be able to prove mathematically that if the number is 
 as small as 1e-47, only the first one or two bits of the mantissa will 
 be non-zero.  I think that if more than triple double precision is 
 needed, it is only one or two more bits more than triple double precision.