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