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From: ags@seaman.cc.purdue.edu (Dave Seaman)
Newsgroups: comp.sys.handhelds
Subject: Re: Interestring Property of HP-* Calculators
Keywords: 48SX,COS
Message-ID: <15610@mentor.cc.purdue.edu>
Date: 24 Oct 90 17:29:36 GMT
References: <2821@uc.msc.umn.edu>
Sender: news@mentor.cc.purdue.edu
Reply-To: ags@seaman.cc.purdue.edu (Dave Seaman)
Organization: Purdue University
Lines: 37

In article <2821@uc.msc.umn.edu> fin@norge.unet.umn.edu (Craig A. Finseth) writes:
>This article describes an interesting *artifact*[1] of the mechanism
>use for computing COS and ACOS.  It has been tested on an HP-48SX and
>TI-PROCALC (don't ask).  It is *NOT* a bug report!

It is not a property of HP calculators or of the "mechanism" used for computing
COS and ACOS.  It is merely a property of floating point arithmetic in general.

The fixed point of COS (computed in degrees) and of ACOS (computed in degrees)
is the same.  The common value, to 20 places, is

	x0 = 0.99984774153108811295

which means that the HP48 computed the best 12-digit approximation,

	x1 = 0.999847741531

where the error x1-x0 is approximately -8.8 * 10^-14.  Letting

	g(x) = 180/pi * arccos(x)

we get the linear approximation

	g(x1) ~= g(x0) + (x1-x0)*g'(x0)

where the g'(x0) term is approximately -3283.47, which explains why g(x1)-g(x0)
is unexpectedly large.  The linear approximation gives 

	g(x1) ~= 0.999847741820

to twelve places, which agrees with the HP48 result to the very last digit.
There is no way that any twelve-digit decimal-based machine could possibly give
better results for this example than the HP48 does.

--
Dave Seaman	  					
ags@seaman.cc.purdue.edu