Between the ops

[Xorter]

Hmm, Xorter, I'm not sure I do actually understand what you're after. But well, in case this is in line...          

I've one time considered the relation between \( a + b \) and \( a * b \) as interpolatable. The key was that \( a * b = \log( \exp(a) + \exp(b) ) \)   and the addition can be written as \( \log^0( \exp^0(a) + \exp^0(b) ) \) compared with the multiplication \( a * b = \log^1( \exp^1(a) + \exp^1(b) ) \) . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of \( \log() \) and \( \exp() \) .                     
Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the \( \log() \) and \( \exp() \) -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz.

Gottfried

Gottfried!
I am afraid your topic is not so relevant or close to mine, to this one. But never mind (but please, open a new topic for it). Maybe I was not so clear.
Of course, I am answering for you according to my knowledge.
If I understand your point, then x[.5]y should be equal to log^o.5(exp^o.5(x)+exp^o.5(y)) (between the addition and multiplication), right?
3+3 = 6
3*3 = 9
And look at this function:
https://www.dropbox.com/s/xr0swdwug44abt...5.jpg?dl=0

The fractional (half-)iterate of exp and log functions according to whose graph I created 3[.5]3 is about 4.62197 < 6 which is a paradox if we suppose that x[z]y < x[w]y if z<w, right?
(2exp^o.5(3) = ~11.9136)
I am afraid the formula we would like to use is wrong.
To be honest I thought of the operators between operators, like between two negations...