Error in Ulps of the Multiplication or Division by a Correctly-Rounded Function or Constant in Binary Floating-Point Arithmetic

Abstract

Assume we use a binary floating-point arithmetic and that $\operatorname{RN}$ is the round-to-nearest function. Also assume that $c$ is a constant or a real function of one or more variables, and that we have at our disposal a correctly rounded implementation of $c$ , say $\hat{c}= \operatorname{RN}(c)$ . For evaluating $x \cdot c$ (resp. $ x / c$ or $c / x$ ), the natural way is to replace it by $\operatorname{RN}(x \cdot \hat{c})$ (resp. $ \operatorname{RN}(x / \hat{c})$ or $\operatorname{RN}(\hat{c}/ x)$ ), that is, to call function $\hat{c}$ and to perform a floating-point multiplication or division. This can be generalized to the approximation of $n/d$ by $\operatorname{RN}(\hat{n}/\hat{d})$ and the approximation of $n \cdot d$ by $\operatorname{RN}(\hat{n} \cdot \hat{d})$ , where $\hat{n} = \operatorname{RN}(n)$ and $\hat{d} = \operatorname{RN}(d)$ , and $n$ and $d$ are functions for which we have at our disposal a correctly rounded implementation. We discuss tight error bounds in ulps of such approximations. From our results, one immediately obtains tight error bounds for calculations such as $\mathtt {x * pi}$ , $\mathtt {ln(2)/x}$ , $\mathtt {x/(y+z)}$ , $\mathtt {(x+y)*z}$ , $\mathtt {x/sqrt(y)}$ , $\mathtt {sqrt(x)/{y}}$ , $\mathtt {(x+y)(z+t)}$ , $\mathtt {(x+y)/(z+t)}$ , $\mathtt {(x+y)/(zt)}$ , etc. in floating-point arithmetic.