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Diffstat (limited to 'security/nss/lib/freebl/mpi/doc/pi.txt')
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diff --git a/security/nss/lib/freebl/mpi/doc/pi.txt b/security/nss/lib/freebl/mpi/doc/pi.txt new file mode 100644 index 0000000000..a6ef91137f --- /dev/null +++ b/security/nss/lib/freebl/mpi/doc/pi.txt @@ -0,0 +1,53 @@ +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. + +------------------------------------------------------------------ + This Source Code Form is subject to the terms of the Mozilla Public + # License, v. 2.0. If a copy of the MPL was not distributed with this + # file, You can obtain one at http://mozilla.org/MPL/2.0/. |