How to effectively compute eigenvalue rotation in HHL

In the HHL algorithm, how do you efficiently do the $lambda-$controlled rotation on the ancillary qubit ? It seems to me after reading around some answers that this can be done in two steps :

• First, we map $|lambdaranglemapsto |frac{1}{pi}arcsin(frac{C}{lambda})rangle$, defining $|frac{1}{pi}arcsin(frac{C}{lambda})rangle$ to be a binary representation $|frac{1}{pi}arcsin(frac{C}{lambda})rangle$ with $m$ qubits.
• Then perform a controlled rotation $U_y(|thetarangle otimes |0rangle)mapsto |thetarangle otimes big(cos(theta)|0rangle + sin{(theta})|1ranglebig)$ where $U_y$ is simply
  $$
  U_y(|thetarangle otimes |0rangle) = prod_{j=1}^m (I^{otimes^m}otimes R_y(2pitheta_j/2^j))
  $$
  i.e. a sequence of controlled rotations where we successively halve the angle of rotation conditionally of the digits of the binary representation of $theta$.

My question is the following how can one implement efficiently the first step in an environment such as Qiskit ?