[VA] SRC #010 - Pi Day 2022 Special

[J-F Garnier]

(03-16-2022 10:30 AM)Ángel Martin Wrote:  I'm however curious: how does it respond for higher number of terms, say 10,000 or even 100,000? That's where the rubber meets the road, methinks.

With the HP41, the best results are obtained with N around 400. The accuracy starts to decline after 500, as you noted too.
I guess the reason is the accuracy of the ln(1-1/n²) quantity even with the LN1+X function.
ln(1+x) is about x-x²/2+... and with 10 digits the term (1/n²)² starts to get inaccurate (relative to 1/n²) when n is in range of 1000 or so.
When switching to the Free42 platform, I got similar (and not identical) results to Valentin's program, however without improving the accuracy of the corrected value.

For instance:
N=1E5, w/o correction:
VA : 3.14160 83615 13791 56287 28512 11516 805
JFG: 3.14160 83615 13791 56287 28668 95754 789

N=1E5, w/ correction:
VA : 3.14159 26535 89793 52207..
JFG: 3.14159 26535 89793 52207..

I guess that at N=1E5, we are still far from the limits of Free42 accuracy.

J-F