Wallis' product exploration

[EdS2]

Interesting thread!

(02-07-2020 10:55 PM)pinkman Wrote:  Despite the fact that Wallis' product is really slow (and it's very deceptive), I have one question:
What do you think about calculations precisions?

I was a bit surprised that we can multiply millions of numbers and still get recognisable results. But then I had a think: a good multiplication or division algorithm will be accurate to within one unit in the last place (an ULP) or indeed, I think, to within half an ULP, because of rounding.

The average error then will be a quarter ULP per multiplication. I'm going to assume the error can be treated as random.

So after a very long series of multiplications, we'll get the exact result, plus an error term which is a random walk of millions of quarter ULPs - which will only add up to an average error on the order of thousands of ULPs, because there's a square root law in such cases.

And thousands of ULPs is only three decimal digits of error. Which is why, I think, we still recognise pi coming out as the result.