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

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.