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: freebsd-bugs at freebsd.org, FreeBSD-gnats-submit at freebsd.org,
        Stephen Montgomery-Smith <stephen at freebsd.org>
Subject: Re: bin/170206: complex arcsinh, log, etc.
Date: Sat, 28 Jul 2012 11:15:20 -0500

 On 07/28/2012 10:46 AM, Stephen Montgomery-Smith wrote:
 > OK.  This clog really seems to work.
 >
 > x*x + y*y - 1 is computed with a ULP less than 0.8.  The rest of the
 > errors seem to be due to the implementation of log1p.  The ULP of the
 > final answer seems to be never bigger than a little over 2.
 >
 >
 
 
 Also, I don't think the problem is due to the implementation of log1p. 
 If you do an error analysis of log(1+x) where x is about exp(-1)-1, and 
 x is correct to within 0.8 ULP, I suspect that about 2.5 ULP is the best 
 you can do for the final answer:
 
 relative_error(log(1+x)) = fabs(1/((1+x) log(1+x))) * relative_error(x)
                    = 1.58 * relative_error(x)
 
 Given that log1p has itself a ULP of about 1, and relative error in x is 
 0.8, and considering x=exp(-1)-1, this gives a ULP at around 1.58*0.8+1 
 = 2.3.  And that is what I observed.
 
 (Here "=" means approximately equal to.)