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From: LINDSAY@tl-20b.arpa
Newsgroups: net.micro
Subject: re: n-digit Pi algorithm (again)
Message-ID: <1021@brl-tgr.ARPA>
Date: Tue, 24-Dec-85 02:33:50 EST
Article-I.D.: brl-tgr.1021
Posted: Tue Dec 24 02:33:50 1985
Date-Received: Sat, 28-Dec-85 00:46:20 EST
Sender: news@brl-tgr.ARPA
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I've been asked for the provenance of the formula I gave.  I suggest looking
up the "Gregory-Leibniz equation", which is a close relative of

arctan x = x - x**3/3 + x**5/5 ...                and dates to the 1600's.

Various people have fiddled with argument values and trig transforms:

Gauss's version     Pi = 48 arctan 1/18  + 32 arctan 1/57 - 20 arctan 1/239
Stormer's version   Pi = 24 arctan 1/8   +  8 arctan 1/57 +  4 arctan 1/239
the one I gave      Pi = 16 arctan 1/5                    -  4 arctan 1/239

In 1961, someone calculated Pi to 100,000 places, using both Gauss's version
and Stormer's version. Both gave the same answer (except for roundoff).
However, Gauss's version ran in half the time. I suspect my version is
the slowest of the three.
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