Hi;

How can we use a spigot algorithm to calculate the digits of a constant? I will use the Wikipedia article and flesh out the explanation a bit so that we can see it used on a real problem.

Let's see if we can get the 50th binary digit of log(2). We will use the relation:

First we multiply both sides of 1) by 2^49.

The RHS is split into two parts.

This term on the right is the important one.

We can replace 4) with

This form can be rapidly computed using mods and only needs floating point arithmetic of the single precision variety. So 3) becomes:

Okay, let's do the number crunching and let's try to get it right.

We are only interested in the fractional part so this becomes,

( You may have noticed the smearing error in the above calculation and how to repair it.)

Now we need that number in binary.

So according to our calculations the 50th - 56th binary digit of log(2) is 1111001

Checking this we find that we have agreement between our prediction in 7) and the answer from the 50th to the 56th digit.