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 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.

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