In this section, assume \(\beta=2\).

Now given a floating point \(x\), we want to form two floating point numbers \(x_{h}\) and \(x_{l}\) such that:

- \(x_{h}\) is a multiple of \(2^{h}\)
- \(|x_{l}|\le2^{h}\)

The assumption is that \(|x|\le2^{h+p-1}\)

An algorithm to do this is:

- Compute \(\sigma=2^{p+h}\)
- \(y=\RN(\sigma+x)\)
- \(x_{h}=\RN(y-\sigma)\)
- \(x_{l}=\RN(x-x_{h})\)