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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