It looks like you have typos?
(x^2+y^2)+(sin(atan(x/y))*cos(atan(x/y)))
reduces to
x^2+y^2+( (x/y) / (x^2/y^2 + 1) ) - not the equation given? Tho it's easier to see that this would be symmetrical if you rearrange it to:
x^2+y^2+( (xy) / (x^2+y^2) )
Also, if f(x,y) = x^2+y^2+( (x/y) / (x^2+y^2) )
then f(2,1) is 5.2 and f(1,2) is 5.1? - this is how I noticed the mistake. (the other reduction gives the same answer, 5.4, for both, by symmetry, as you suggest)
There's a simpler solution which produces integer ids (though they are large): 2^x & 2^y. Another solution is to multiply the xth and yth primes.
I only looked because I was curious how you proved it unique!
Also, if f(x,y) = x^2+y^2+( (x/y) / (x^2+y^2) ) then f(2,1) is 5.2 and f(1,2) is 5.1? - this is how I noticed the mistake. (the other reduction gives the same answer, 5.4, for both, by symmetry, as you suggest)
There's a simpler solution which produces integer ids (though they are large): 2^x & 2^y. Another solution is to multiply the xth and yth primes.
I only looked because I was curious how you proved it unique!