Really let down by these short sentences.

“Here’s where it gets interesting.”

“That’s not a ____. It’s a _____.”

I’ll gladly read this if the author does an editing pass and makes it read more naturally.

That's a nice article, thanks! I learned something new from it (somehow I didn't know that typical finite difference error plot is linearly V-shaped).

tang library looks cool, and I wonder whether it will become a modern full-fledged math library independent of ancient BLAS and solver libs... I wish it would, but that would require huge effort.

A couple of nitpicks for the article:

> The pathological function barely hits 10⁻⁵.

Huh? The pathological function's error plot goes below 10⁻⁸.

> What does that buy you? Watch two gradient descent particles race down Himmelblau's function

The default value in the visualisation for h is 0.5. Of course it performs poorly! Switch to h=0.1 and the difference is gone. I wonder if there exists a more significant demonstration

let e^2 = 0.

the dual numbers are D = {x + ye | x,y in R}.

let f(z) be some polynomial.

Then by binomial theorem f(z + e) - f(z) = z^n + n*z^(n-1)*e + O(e^2) - z^n = f'(z)*e.

And so df = df/dz * e

Neato introduction, but the text reeks of LLMs so I stopped reading.