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Diffstat (limited to 'mozilla/security/nss/lib/freebl/mpi/doc/pi.txt')
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diff --git a/mozilla/security/nss/lib/freebl/mpi/doc/pi.txt b/mozilla/security/nss/lib/freebl/mpi/doc/pi.txt new file mode 100644 index 0000000..d93229c --- /dev/null +++ b/mozilla/security/nss/lib/freebl/mpi/doc/pi.txt @@ -0,0 +1,90 @@ +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 $ + + |