mirror of
https://github.com/rn10950/RetroZilla.git
synced 2024-11-10 01:40:17 +01:00
91 lines
3.8 KiB
Plaintext
91 lines
3.8 KiB
Plaintext
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.
|
|
|
|
------------------------------------------------------------------
|
|
***** BEGIN LICENSE BLOCK *****
|
|
Version: MPL 1.1/GPL 2.0/LGPL 2.1
|
|
|
|
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 the Initial Developer are Copyright (C) 1998, 2000
|
|
the Initial Developer. All Rights Reserved.
|
|
|
|
Contributor(s):
|
|
|
|
Alternatively, the contents of this file may be used under the terms of
|
|
either the GNU General Public License Version 2 or later (the "GPL"), or
|
|
the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
|
|
in which case the provisions of the GPL or the LGPL are applicable instead
|
|
of those above. If you wish to allow use of your version of this file only
|
|
under the terms of either the GPL or the LGPL, and not to allow others to
|
|
use your version of this file under the terms of the MPL, indicate your
|
|
decision by deleting the provisions above and replace them with the notice
|
|
and other provisions required by the GPL or the LGPL. If you do not delete
|
|
the provisions above, a recipient may use your version of this file under
|
|
the terms of any one of the MPL, the GPL or the LGPL.
|
|
|
|
***** END LICENSE BLOCK *****
|
|
|
|
$Id: pi.txt,v 1.2 2005/02/02 22:28:22 gerv%gerv.net Exp $
|
|
|
|
|