Difference among approximatelyEqual and essentiallyEqual in The art of computer programming

Question

I get this code snippet from some where else. According to the webmaster, the code is picked from The art of computer programming by Knuth

Since I do not have a copy of

Answer 1

The difference is that essential equality implies approximate equality, but not vice versa. So essential equality is stronger than approximate equality.

Also essential equality is not transitive, but if a is essentially equal to b, and b is essentially equal to c, then a is approximately equal to c (for another value of epsilon).