On fixpoints of Higham’s function

Abstract

We study the strange behavior in floating-point arithmetic of a function proposed by Nicholas Higham, consisting of repeated square roots extraction followed by the same number of times squaring and find its fixpoints. For IEEE standard double precision floating point numbers the fixpoints have the form \[ x \in \left\{\left( 1+k\mathrm{eps}\right) ^{\frac{1}{\mathrm{eps}}},\quad k=\left[ -745:\frac{1}{2}:-\frac{1}{2},0:709\right]\right\} \cup \{0\} , \] where \mathrm{eps} is the machine epsilon."