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This file describes how pi is computed by the program in 'pi.c' (see
the utils subdirectory).
Basically, we use Machin's formula, which is what everyone in the
world uses as a simple method for computing approximations to pi.
This works for up to a few thousand digits without too much effort.
Beyond that, though, it gets too slow.
Machin's formula states:
pi := 16 * arctan(1/5) - 4 * arctan(1/239)
We compute this in integer arithmetic by first multiplying everything
through by 10^d, where 'd' is the number of digits of pi we wanted to
compute. It turns out, the last few digits will be wrong, but the
number that are wrong is usually very small (ordinarly only 2-3).
Having done this, we compute the arctan() function using the formula:
1 1 1 1 1
arctan(1/x) := --- - ----- + ----- - ----- + ----- - ...
x 3 x^3 5 x^5 7 x^7 9 x^9
This is done iteratively by computing the first term manually, and
then iteratively dividing x^2 and k, where k = 3, 5, 7, ... out of the
current figure. This is then added to (or subtracted from) a running
sum, as appropriate. The iteration continues until we overflow our
available precision and the current figure goes to zero under integer
division. At that point, we're finished.
Actually, we get a couple extra bits of precision out of the fact that
we know we're computing y * arctan(1/x), by setting up the multiplier
as:
y * 10^d
... instead of just 10^d. There is also a bit of cleverness in how
the loop is constructed, to avoid special-casing the first term.
Check out the code for arctan() in 'pi.c', if you are interested in
seeing how it is set up.
Thanks to Jason P. for this algorithm, which I assembled from notes
and programs found on his cool "Pile of Pi Programs" page, at:
http://www.isr.umd.edu/~jasonp/pipage.html
Thanks also to Henrik Johansson <Henrik.Johansson@Nexus.Comm.SE>, from
whose pi program I borrowed the clever idea of pre-multiplying by x in
order to avoid a special case on the loop iteration.
------------------------------------------------------------------
The contents of this file are subject to the Mozilla Public
License Version 1.1 (the "License"); you may not use this file
except in compliance with the License. You may obtain a copy of
the License at http://www.mozilla.org/MPL/
Software distributed under the License is distributed on an "AS
IS" basis, WITHOUT WARRANTY OF ANY KIND, either express or
implied. See the License for the specific language governing
rights and limitations under the License.
The Original Code is the MPI Arbitrary Precision Integer Arithmetic
library.
The Initial Developer of the Original Code is
Michael J. Fromberger <sting@linguist.dartmouth.edu>
Portions created by Michael J. Fromberger are
Copyright (C) 1998, 2000 Michael J. Fromberger. All Rights Reserved.
Contributor(s):
Alternatively, the contents of this file may be used under the
terms of the GNU General Public License Version 2 or later (the
"GPL"), in which case the provisions of the GPL are applicable
instead of those above. If you wish to allow use of your
version of this file only under the terms of the GPL and not to
allow others to use your version of this file under the MPL,
indicate your decision by deleting the provisions above and
replace them with the notice and other provisions required by
the GPL. If you do not delete the provisions above, a recipient
may use your version of this file under either the MPL or the GPL.
$Id$
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