I believe comments are vital for explaining what you’re doing and why if it’s not INSTANTLY clear. That is, even if your routine is understandable after a minute of thinking the code through, it needs a comment because it’s not immediately obvious.
In your example, I agree that you should make that its own routine for clarity, but I still would have left the reference to Newton-Raphson approximation in there, no matter how you organised it. You cannot possibly believe that everyone will immediately understand that’s what it’s doing.
I like to think of the comments as integral to the story of the code - Combining the code and the comments, you can see what a routine is doing very, very quickly, and that’s important.
Comments are the what and why of the code, and the code itself is the how. When properly used, they slot together perfectly and have the ability to speed up development and debugging.
Comments at the start of methods and classes are possibly the most useful comments out there - they reduce the need for heavy commenting within the code itself, explain who did what and why (useful for corporate coding and maintenance), and when you combine them with something like doxygen it can save you a lot of time hunting down someone else’s code in a large code-bed.
In my experience there are developers who comment and like comments, and developers who have never given comments a proper chance and therefore hate it. Often developers take on a level of arrogance that says that their code is so perfect they don’t need comments, but you cannot forget that there are developers out there at all skill levels and if you really are a hot coder, most of those guys really won’t understand why you’re doing something in a certain way.
BTW, some of us are forced to go through code reviews with idiots who are not coders, such as IT auditors and management. With the comments, they can get by without you having to sit there painfully explaining every line to them!
Give comments a chance mate! If they’re not working for you, then your comments are BAD comments!