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From: milne@icse.uci.edu (Alastair Milne)
Newsgroups: net.micro
Subject: Arctan approximation for generating pi
Message-ID: <1086@brl-tgr.ARPA>
Date: Fri, 27-Dec-85 22:16:56 EST
Article-I.D.: brl-tgr.1086
Posted: Fri Dec 27 22:16:56 1985
Date-Received: Sat, 28-Dec-85 13:25:26 EST
Sender: news@brl-tgr.ARPA
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   I must agreee with the observations about the speed of the power series
   suggested so far.  Two key points in my understanding of numeric
   approximation:

   1.  For numeric approximations which people are actually going to have to
       use in their work, avoid power series if possible.  Deadly slow, even
       if, as some will, they converge sufficiently within 5 terms or fewer.

   2.  Don't raise values to exponents unless it's unavoidable.  The
       implementations used in most languages use EXP and LN, which are
       themselves approximations, resulting in representation error,
       round-off error, and amplification of these in the
       multiplications which produce the final result.  Wirth knew what
       he was doing when he refused to put exponentiation into Pascal.


   The best way I know of to get around both of these is to use
   continued fractions.  And most fortunately, there is a very nice
   continued fraction form for arctan which requires perhaps 3 floating
   point divisions, and 2 floating-point additions.  Furthermore, it
   gives a very pleasant number of significant digits.  Incredibly few
   operations for a very good result.  Naturally, I am describing it
   like this because I can't at the moment recall its exact form
   (sorry!).  It is among some of my old numeric analysis notes.  When
   (and if) I find it, I'll send it in.  I think the people who need
   this will be quite impressed with it.  I certainly was when it was
   first shown to me, by a teacher who made sure we first endured
   coding power series evaluations.


   Alastair Milne

   PS.  Like Sheldon Meth, I am very intrigued by the apparently special
	relationship between atan(1/5) and atan(1/239).  But is it really true
	than pi, a transcendental number, is EQUAL to a linear combination of
	these two, or does it only lie within an acceptable tolerance?