I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one.
However, I was looking for the more general case where we find the continuum between the operators themselves - add, multiply, and exponentiation or even higher.
The function could be a hyperoperation$(n,x,y)$, where $x$ and $y$ are the numbers to operate on, and $n$ is allowed to be a non-integer number for the type of operator. So $n=1$ would be 'addition', $n=2$ would be multiplication, and $n=3$ would be exponentiation. But one would also have $n=2.5$ to be the shade of grey BETWEEN multiplication and exponentiation, or $n=3.5$ or $4$ for beyond exponentiation and tetration).
I want the formula to work with real numbers, not just integers. Any thoughts?
These related posts may also be of interest:
Why are addition and multiplication commutative, but not exponentiation?
There is (at least) one option - however depending on whether a solution for the problem of "fractional iteration of logarithm" is available (aka: "Tetration". Note that in the tetration-forum there is an exchange on various approaches to this problem).
Assume for the moment that a version of that fractional iteration is available, and let us denote $ \log_b^{\circ h}(x) $ for the $h$-times iterated logarithm to some given base $b$, including the opportunity to have $h$ also fractional.
Assume also that we have this well-defined for the interval $h=-1 \cdots +1$ (so for $h=-1$ this means actually exponentiation) then $$ f_b(x,y,0) = \exp_b^{\circ 0} ( \log_b^{\circ 0} (x)+\log_b^{\circ 0} (y)) = x+y \\ f_b(x,y,1) = \exp_b^{\circ 1} ( \log_b^{\circ 1} (x)+\log_b^{\circ 1} (y)) = x \cdot y $$ and for $0<h<1$ we have then some operation in between well defined: $$ f_b(x,y,h) = \exp_b^{\circ h} ( \log_b^{\circ h} (x)+\log_b^{\circ h} (y)) = x \{\circ _h \} y $$ where the operation-symbol of the circle and the index $h$ means that interpolated operation between "+" (for $h=0$) and "*" (for $h=1$).
Now let's see, whether our assumption of the existence of such fractional iterations is justified/can be realized:
(Note: this is just one ad-hoc-approach; it does not, for instance, provide a solution comparable with one which might be expected, when we aproach the problem via the interpolation of the Ackermann-function as indicated by Daniel Geisler)
Addition, multiplication and "half multiplication" tables for $x,y=1 \ldots 10$ . I used the base $b=t^{1/t} =\sqrt2 \approx 1.414$ with $t_l=2$ resp $t_u=4$ for the logarithm and the exponentiation. For the half-exponentials/half-logarithms the formulae relate on lower fixpoint $t_l=2$ for $x \le 3$ and on upper fixpoint $t_u=4$ for $x>3$ and use the well known Schröder-mechanism for the half-iterate.
[update]: In the comment MphLee provided a link into wikipedia, where this "hyperoperation"-scheme is mentioned as due to A. Bennet (but not then generalized to fractional orders / fractional iteration heights of the $\log()$ and the $\exp()$ )
addition $h=0$ ($\text{"+"}=\circ _0$)
"half multiplication" $h=0.5$ ($\circ _{0.5}$) with base $b=\sqrt2$ using Schröder-mechanism
multiplication $h=1$ ($\text{"*"}=\circ _{1}$)
halfway between multiplication and commutative(!) exponent $h=1.5$($\text{"??"}=\circ _{1.5}$)
Commutative(!) expon $h=2$($\text{"??"}=\circ _{2}$)
(Remark: because double-log of number $1$ would be negative infinity, I replaced this by approximation and insertion of $1+1e-20$ instead of $1$)
We can even define negative-half index of operation:
and the negative $-1$- operation
And to see that different ways to define the fractional iterate of exponentiation lead to different multiplication-tables I show here the "half-multiplication" taken by base $b=2$ and the implementation via something which I called "polynomial tetration" (which is based on eigendecomposition of the truncated carlemanmatrices, and seems to approximate the Kneser-solution when the truncation allows larger matrix-sizes)
[update]: Appendix.
In the (in WP) cited article of A Bennett (1915) for the interpolation to fractional orders (there $n$ here $h$) A.Bennet misses the possibility to take another base $b$ for iterated logarithm and exponentiation, where $b \ne e$ and actually $1<b<e^{1/e}$ where we can iterate infinitely and also have a "regular" method for interpolation to fractional orders/iterates using E. Schröder's method which I've used here. See the screenshot of some paragraph in A. Bennett's short paper: