pi.txt (2358B)
1 This file describes how pi is computed by the program in 'pi.c' (see 2 the utils subdirectory). 3 4 Basically, we use Machin's formula, which is what everyone in the 5 world uses as a simple method for computing approximations to pi. 6 This works for up to a few thousand digits without too much effort. 7 Beyond that, though, it gets too slow. 8 9 Machin's formula states: 10 11 pi := 16 * arctan(1/5) - 4 * arctan(1/239) 12 13 We compute this in integer arithmetic by first multiplying everything 14 through by 10^d, where 'd' is the number of digits of pi we wanted to 15 compute. It turns out, the last few digits will be wrong, but the 16 number that are wrong is usually very small (ordinarly only 2-3). 17 Having done this, we compute the arctan() function using the formula: 18 19 1 1 1 1 1 20 arctan(1/x) := --- - ----- + ----- - ----- + ----- - ... 21 x 3 x^3 5 x^5 7 x^7 9 x^9 22 23 This is done iteratively by computing the first term manually, and 24 then iteratively dividing x^2 and k, where k = 3, 5, 7, ... out of the 25 current figure. This is then added to (or subtracted from) a running 26 sum, as appropriate. The iteration continues until we overflow our 27 available precision and the current figure goes to zero under integer 28 division. At that point, we're finished. 29 30 Actually, we get a couple extra bits of precision out of the fact that 31 we know we're computing y * arctan(1/x), by setting up the multiplier 32 as: 33 34 y * 10^d 35 36 ... instead of just 10^d. There is also a bit of cleverness in how 37 the loop is constructed, to avoid special-casing the first term. 38 Check out the code for arctan() in 'pi.c', if you are interested in 39 seeing how it is set up. 40 41 Thanks to Jason P. for this algorithm, which I assembled from notes 42 and programs found on his cool "Pile of Pi Programs" page, at: 43 44 http://www.isr.umd.edu/~jasonp/pipage.html 45 46 Thanks also to Henrik Johansson <Henrik.Johansson@Nexus.Comm.SE>, from 47 whose pi program I borrowed the clever idea of pre-multiplying by x in 48 order to avoid a special case on the loop iteration. 49 50 ------------------------------------------------------------------ 51 This Source Code Form is subject to the terms of the Mozilla Public 52 # License, v. 2.0. If a copy of the MPL was not distributed with this 53 # file, You can obtain one at http://mozilla.org/MPL/2.0/.