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From: bco@ist.UUCP (Brian Collins)
Newsgroups: net.math
Subject: Calculating Pi to Excessive Accuracy
Message-ID: <423@ist.UUCP>
Date: Tue, 6-Aug-85 18:52:12 EDT
Article-I.D.: ist.423
Posted: Tue Aug  6 18:52:12 1985
Date-Received: Sat, 3-Aug-85 06:56:44 EDT
Organization: Imperial Software Technology, London, England
Lines: 36
Xpath: icdoc ivax

Following Ken Yap's posting, which soaks up 2 hours of CPU to
calculate Pi to 1000 decimal places, may I offer the
following sh/bc script that takes just over 6 mins on my
Pyramid.
[ I can't be bothered to shar it just for 9 lines! ]

bc <<!
	/* Computes Pi to $1 places */
	t = 3 * 10^${1-100}
	for (i = 1; t != 0; i++) {
		p = p + t
		t = t * (2 * i - 1) * (2 * i - 1) / (8 * i * (2 * i + 1))
	}
	p
!

The cost is guaranteed better than quadratic in the number of
decimal places, since each term is less than a quarter of the
previous one.

The recurrence relation is easily deduced from first principles
by:
	integrating the
		binomial expansion of the
			differential of
				asin( x)
	one term at a time, and using this to compute
		asin( 0.5)
	which is (of course)
		pi / 6

Any advance on 6 minutes?

-- 
Brian Collins		...mcvax!ist!bco (bco@ist)
Imperial Software Technology, London