From wikipedia it looks like both the article's approximation and this method have octagonal level sets. I was wondering what the actual difference was. Lookin

From wikipedia it looks like both the article's approximation and this method have octagonal level sets. I was wondering what the actual difference was. Looking into it a bit, I came across the error graph [2] from the article[1]. The main difference seems to be that the octagon used by the article is 'too small'. The maximum error would be smaller with a larger octagon. Currently there is an undershoot of 5.2 % and an overshoot of 2.5%. To reduce the maximum error, you need undershoot and overshoot to be equal. Another notable difference is that the article's octagons have a different rotation than the alpha plus max beta algorithm. The article's octagons cross the axes perpendicularly, whilst the alternative algorithm is rotated by 360/16=22.5 degrees so it has vertices at the axes. That certainly changes at which angles the approximations are better, which might matter in practice. What I wasn't able to find quickly is the average error for the article's approximation. Would be interesting to compare that to the 2.41% for the better algorithm. [1] https://klafyvel.me/blog/articles/approximate-euclidian-norm/ https://klafyvel.me/blog/articles/approximate-euclidian-norm... [2] https://klafyvel.me/assets/blog/articles/approximate-euclidian-norm/code/output/illustration10.svg https://klafyvel.me/assets/blog/articles/approximate-euclidi...