I take it you think there is something intrinsically wrong with that idea?

I find it very hard, without bracket counting, to see exactly what the '+' and '/' bind to. With the more traditional:

(-b + sqrt(bb-4ac)) / (2a)

I find in only a glance I can tell what everything is binding to.

It must be nice to live in a world with only 4 infix operators and expressions that have only 3 infix operators.

For example, lots of folks think that sqrt should be a prefix operator, not yet another function. I suppose you're going to assume that the top bar will serve as parentheses.

BTW "-b + sqrt(bb-4ac) / 2a" is the interesting expression. Is it "(-b + sqrt(bb-4ac)) / (2a)" or "-b + (sqrt(bb-4ac)) / 2a)" And, are you certain what "bb-4ac" means? (There's at least one major language where it doesn't mean "(bb)-(4ac)".)

And that's how the exceptions swallow the rule. And, it's also how we get infix programming languages where that's definitely not true, and so on. Where should we make the switch?

Also, only four? What about set operations?

  (/ (- (sqrt (discriminant a b c)) b)
     (* 2 a))