Happy Pi Day!

[Gerson W. Barbosa]

Thank you all for the Pi-related links and comments!

The first program in the directory, PId, computes d digits of π using the simple formula

\[\pi=4\cdot \arctan (1)\]

Arctan(1) is calculated using the formula previously presented here,

\[\frac{\pi }{4}= 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots +\frac{1}{2n-3}-\frac{1}{2n-1}+\frac{1}{4n+\frac{1^{2}}{n+\frac{2^{2}}{4n+\frac{3^{2}}{n+\frac{4^{2}}{4n+...​ }}}}}\]

using n equal number of terms of the regular series and of the continued fraction. The second program (PIn) computes the sum of such n terms. Its slowness for large number of digits or terms is due to the limitation of the HP-49G/50g when dealing with the resulting very long algebraic expression before it is eventually expanded. Freeing more memory might help.

The third program, PIdLF, is a LongFloat Library version of PId.

PiM and PiMlf, its equivalent LongFloat version, compute π using Machin's formula:

\[\pi = 16\cdot \arctan \frac{1}{5}-4\cdot \arctan \frac{1}{239}\]

That's the same formula used by William Shanks to compute π about 150 years ago, after spending 15 years in the task, done by hand, without using any computing devices whatsoever. The last 180 digits being wrong is not impressive, the first 527 digits being correct is. That's why we have used two different methods. As we can see, but for for a rounding in the last digits in the fourth printing, all digits match.

Gerson.