Say that we want to approximate the function $$t\to f(t) = \sqrt{t}$$ on the interval $t\in [0,1]$. We know that polynomial approximation close to $0$ is very bad so we want to avoid that.
Instead let us say we find some polynomial which optimally in some sense approximates $f(t)$ for $t\in [1,4]$.
Then we can use the identity $\sqrt{2^{2k}t} = 2^k\sqrt{t}$
In order to calculate $f(t_0)$ the idea is to find which integer power of $4$ to multiply $t_0$ so that $4^kt_0 \in [1,4]$.
We then calculate that value and adjust by a simple subtraction / addition in the exponent part of our floating point number, since a multiplication /division by an integer power of 2 can be done that way.
The question is, how efficient will this be in comparison to methods which already exist?