It is the same as solving
S+delta=P*(1-x)
where 0<=delta<1 is arbitrary. The solution is then,
x=1-(S+delta)/P
It's important to keep in mind if you are solving this numerically that delta=0 will be a numerically unstable choice because the resulting x will take you close to the discontinuity of the floor function.
Let's test with delta=0.5:
S = 10;
P = 13;
delta=0.5;
x=1-(S+delta)/P
S == floor(P*(1-x)) %check
