Another Splitting Technique: Splitting Around a Power of 2

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