Is there a closed form function from $\mathbb{R}$ to base-4 with only 0 and 1?

Question

Is there a closed form of the function that converts a real number to the base $4$ number with the same base-$2$ binary string?

For example, the number $45$ is $101101$ in binary, base $2$. Interpreted in base $4$ it is $1*4^5+0*4^4+1*4^3+1*4^2+0*4^1+1*4^0=1105$, so $F_{(45)}=1105$