From owner-freebsd-bugs@FreeBSD.ORG  Fri Jul 27 20:50:16 2012
Return-Path: <owner-freebsd-bugs@FreeBSD.ORG>
Delivered-To: freebsd-bugs@freebsd.org
Received: from mx1.freebsd.org (mx1.freebsd.org [69.147.83.52])
	by hub.freebsd.org (Postfix) with ESMTP id 385C4106564A;
	Fri, 27 Jul 2012 20:50:16 +0000 (UTC)
	(envelope-from stephen@missouri.edu)
Received: from wilberforce.math.missouri.edu (wilberforce.math.missouri.edu
	[128.206.184.213])
	by mx1.freebsd.org (Postfix) with ESMTP id F10C08FC08;
	Fri, 27 Jul 2012 20:50:15 +0000 (UTC)
Received: from [127.0.0.1] (wilberforce.math.missouri.edu [128.206.184.213])
	by wilberforce.math.missouri.edu (8.14.5/8.14.5) with ESMTP id
	q6RKoEHQ022529; Fri, 27 Jul 2012 15:50:14 -0500 (CDT)
	(envelope-from stephen@missouri.edu)
Message-ID: <5012FF06.4030501@missouri.edu>
Date: Fri, 27 Jul 2012 15:50:14 -0500
From: Stephen Montgomery-Smith <stephen@missouri.edu>
User-Agent: Mozilla/5.0 (X11; Linux i686;
	rv:14.0) Gecko/20120714 Thunderbird/14.0
MIME-Version: 1.0
To: Bruce Evans <brde@optusnet.com.au>
References: <201207270247.q6R2lkeR021134@wilberforce.math.missouri.edu>
	<20120727233939.A7820@besplex.bde.org>
In-Reply-To: <20120727233939.A7820@besplex.bde.org>
Content-Type: text/plain; charset=ISO-8859-1; format=flowed
Content-Transfer-Encoding: 7bit
Cc: freebsd-bugs@freebsd.org, FreeBSD-gnats-submit@freebsd.org,
	Stephen Montgomery-Smith <stephen@freebsd.org>
Subject: Re: bin/170206: complex arcsinh, log, etc.
X-BeenThere: freebsd-bugs@freebsd.org
X-Mailman-Version: 2.1.5
Precedence: list
List-Id: Bug reports <freebsd-bugs.freebsd.org>
List-Unsubscribe: <http://lists.freebsd.org/mailman/listinfo/freebsd-bugs>,
	<mailto:freebsd-bugs-request@freebsd.org?subject=unsubscribe>
List-Archive: <http://lists.freebsd.org/pipermail/freebsd-bugs>
List-Post: <mailto:freebsd-bugs@freebsd.org>
List-Help: <mailto:freebsd-bugs-request@freebsd.org?subject=help>
List-Subscribe: <http://lists.freebsd.org/mailman/listinfo/freebsd-bugs>,
	<mailto:freebsd-bugs-request@freebsd.org?subject=subscribe>
X-List-Received-Date: Fri, 27 Jul 2012 20:50:16 -0000

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.