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I showed it to a friend who is quite good at math, and he told me the book may be trying too hard especially in the examples variety, and how it might not be needed for comprehension's sake.
Would you still recommend this book, and if yes, in what circumstances?
They do make a good doorstop though.
Your milage will vary. I got zero value from the books. They do make a good doorstop though.
I read through volume 1 several times. I then skimmed through the arithmetics part of volume 2 and read only the section about random numbers, which is very fascinating but you will be able to live without unless you are doing simulations or cryptography/cryptoanalysis.
Whether you will need volumes 4a, b, c depends on whether your work has much to do with combinatorics. I'm a professor and I did not need most of it, but had a good laugh about the hidden gems and jokes like the exercise about Knuths personal pipe organ.
As a software engineer, you may occasionally consult volumes 3 and 1 (searching, sorting, data structures).
[Side node: If I was Knuth, I don't think I would have created three subvolumes instead of the planned single volume 4. While it is true that the material exploded, that will always be the case, and is also true for the other chapters. In his shoes, I would have worked on succession planning and trying to map out the remaining volumes so that the book can be completed one day. Under Knuths approach (esp. going back to update the existing volumes repeatedly on the way) TAOCP will unlikely ever be completed, which would be a shame.]