The motivation was to find a path to evaluate the miminum of the factorial for $x>0$.
To start I introduce the equation $ax=x!$ a good value is $a=2$ .The second idea is to introduce the inversed factorial function $ay=1/y!$.
In fact putting the graph we have two value for $a=2$ which are truncated $x=0.4429,y=0.5619$
Summing we get $x+y=1.0048$ . Then the next step is to add the remainders of $x+y$ with respect to the value one to the coefficient $2$ so $a=2+0.0048$ and we get an algorithm very simple .I cannot justify it but if we pursue this kind of algorithm adding or removes it if the value of $x+y$ is more or less than one does it tends to one in summing ? Or in other words does the sum of the roots in the two equation (with at each steps a new coefficient ) is equal to one if go to infinity ?
Ps:We can call this algorithm an algorithm of trial/errors .