[VA] SRC #010 - Pi Day 2022 Special

[Fernando del Rey]

Thanks, Valentín, I have found your post rather interesting and entertaining.

Perhaps many members of this forum would prefer that you give them a challenge rather than an article, but your challenges are typically way out of my reach, so I liked this article-style SRC.

I entered all your code in Free42 Decimal on my iPhone 11 and ran all the cases in your post, and a few more cases. As expected, everything worked fine and the program executions times were extremely fast, even for N=1,000,000.

Then I decided to try to run all the code in a physical HP-42S. What I found is that you can run PX and PN up to N=100 in relatively short times (seconds, not minutes). Even when using the solver with the wrapper program PXEQ, you can start with a small N value (say, N=10) to get a first estimate of X, and then gradually progress to higher N values (N=20, 50, 100), letting the resulting X value from the previous iteration be the initial guess of the next iteration. In that way, you get a relatively fast (in time) convergence, even for N=100.

All the results are still meaningful with N=100 on a physical HP-42S and the trends can be distinguished (results approaching e or Pi), even if the convergence of the infinite product function is very slow. The result of program PN with the correction terms added is surprisingly good for N=100, with an error of only 2.9e-7!

Now, I wonder if you would have been able to derive this function and the corrections terms, and to write a similar article back in 1988, using no computer and just a physical HP-42. Or do you absolutely need the speed and increased accuracy of Free42?

My guess is that you would have managed to make the same discovery in 1988 with a physical HP-42S, intuition, and a lot of patience. What do you think?