Is super-accuracy matters?

[EdS2]

(10-06-2023 04:36 PM)Albert Chan Wrote:  

(10-06-2023 08:40 AM)J-F Garnier Wrote:  ... this has no real-life interest ...

I've been thinking about this for a while.
...
Now, I think we should shoot for better accuracy, whether or not it matters in real life.
No need to convince others!

What do you think?

It's a good question. I think the problem comes from the difference between engineer-mindset and mathematician-mindset. Each of us may adopt one or the other at different times.

When we adopt engineer-mindset we tend to think of numbers as measurements and apply to something in the physical world. Such measurements have some particular accuracy. Calculations with them will have some limit to their accuracy and computing to greater precision seems odd. In this mindset commentary from the other mindset can seem bafflingly misguided.

When we adopt mathematican-mindset we tend to think of numbers as Platonic ideals, or as abstract entities. 1E22 is exactly that number, just as 3 is exactly that number. Pi is an irrational and almost every calculator will need to round it off at some point. Questions about values of sin and cos are questions about functions which apply to any real argument, not just to angles. In this mindset commentary from the other mindset can seem bafflingly misguided.

When we count ULPs, we are in some way adopting both mindsets: we know that the inputs are limited precision and we know that the outputs are limited precision, and we'd like to get the best result, regardless of which mindset the calculation might have been performed in. Implementing a calculator is an engineering problem, at least to some extent. But implementing a calculator with more than 10 digits of precision is doing something which is more than just satisfying the users who are adopting the engineer-mindset.

Where things go wrong, I think, is when two people can only see one of those two perspectives, different ones, and are arguing at cross-purposes.