[klibertp]

Well, for me both examples are totally unreadable. I think math's is the most unreadable notation ever and even a cross between PERL and Brainfuck would be better. Mathematicians are masochists, and I refuse to follow their lead. I prefer meaningful variable names, good use of horizontal and vertical whitespace, context-free grammars, and the like. So maybe this is the difference and the reason for me perceiving the notations as more or less equivalent: I'm not biased, as most people, in favor of infix, but rather biased the other way.

Anyway, let's try doing something with your postfix example (I hope it displays alright; also, I think you made a mistake in your translation to postfix, but I left it as it is):

    c_z   x *
    s_z   y *
        s_y *
    c_y   z *
    +
        c_x *

    c_z   y *
    s_z   x *
    -
        s_x *
    -

        d_z =

Now, this is much more readable than unformatted infix version you give and that's before factoring this into smaller parts. I didn't read that much of Forth, but I'm 97% sure that it would be factored into 3 or 4 words, I think. And it also has an advantage that you read the operation in order they're going to be executed, while with infix you need to read the entire expression to know which computation occurs first.

Of course, it's only more readable for me, with my particular background knowledge and habits; I'm not saying this is or should be the same for anyone else. But, if there are people who see one form as more readable and people who see the other form as more readable, then I think that's a solid argument in favor of both forms not being drastically, qualitatively different.

The problems you point to are definitely real; it's true "the operator doesn't show which operands it has" by default (for example) and you need to go out of your way to show it. But infix notation has it's problems too, and you need to work around them as well. Like operator precedence, which is frankly a terrible idea.

> Postfix also suffers from not making it clear how many operands a given operator takes. With infix, this is always clear.

No, not always. J has words which take one argument on the left and, for example, two on the right. Or three. Or a variable number of arguments on the left (granted, they are `tied` with ` word, but still) and nothing on the right... And Ken Iverson says it's very readable! (To him, at least).

To summarize: I'm still not convinced that there is a major and unfixable difference in readability between the notations, and I still think you can make all 3 notations as readable as any other.