## I Calculate the Digits of Pi – Very Slowly

Years ago I wrote an assembly language routine that drew circles with  a pen plotter.  This entailed approximating the arc of a circle with a series of small horizontal and vertical steps.  I was proud of the method because its critical loop used only additions and multiplications by 2 (bit shift left in assembly language) and was very fast.

So I thought this summer that if I can draw a circle fast, I ought to be able to calculate $\pi$ fast.  I wrote a routine in the Python programming language, optimized it as best I could and here are my results.   (My method also allowed me to calculate $\sqrt {2}$ – also very slowly)

Calculating Pi Data

Here are two python program fragments that show the algorithm.  The first is the original version which was written for clarity to stay close to the mathematics and the second was optimized for speed – not that it helped much.

Pi Calculation - First Try

Pi Calculation – “Optimized”

It is fairly easy to see where the error comes from.  By staying just outside of the circle – at most one vertical or horizontal step, the area is overestimated by at most $2\pi r\sqrt{2}$.  The calculated area is at most $\pi r^2+2\pi r\sqrt{2}$.  Dividing by $r^2$ gives $\pi + \frac{2\pi\sqrt{2}}{r}$.  Increasing $r$ ten times moves the decimal point over one.

My discovery of “drawing” conic sections by looking only one step ahead was satisfying and at least years ago useful.   I was sobered by the sad slow algorithm that I devised to calculate $\pi$.  A method with linear efficiency is really slow.  I now have more than a theoretical understanding of this fact.