Decimal digits vs integer digits

[pier4r]

(02-23-2018 02:52 AM)Valentin Albillo Wrote:  I hope this will answer your original post to your satisfaction.
Regards.

V.
.

It is surely interesting content but I would have expected also a comparison with growing digits, to see the trend.

What I mean is the following. You showed that using the multiprecision arithmetic built in the basic interpreter of the hp 71 ROM (nice, by the way), you can compute my example faster than an equivalent number of digits of Pi.

That's ok, it is a fact. But it is an answer like "look, we have g(x) and f(x). When x is 1250, f(x) is smaller than g(x)". Nice, but can I use it to say "and surely, since x=1250 is pretty large, we can conclude that f(x) will be always smaller than g(x) for any x larger than 1250" ? As far as I know, no.

In other words, what is the upper bound (in terms of time and/or memory) of functions used to compute the digits of pi (or square roots) compared to functions used to compute large numbers?

This considering the limitation of the computing hardware. If we would have hardware capable of infinite precision operations (so not 16, 32, 64 or whatever bits) then my question may be not that interesting.

Instead considering the limit of the operations, so say with 16 bit registers at most, I have the feeling (since I did not dig in the formulas yet) that computing square roots or digits of pi requires more or less constant time for the same number of digits extracted. This using optimized algorithms.

Instead for large numbers the work (I think) grows and grows , as one need to consider the previous results to compute the new one, and the previous result is getting larger.