Considering two non-floating-point numbers $x$ and $y$, we write their floating point representation as $\operatorname{fl}(x)$ and $\operatorname{fl}(y)$ respectively. By $\circ$, we denote an arithmetic operation.
Now, in my text a distinction is made between an error caused in the data and a second one caused by converting the result of an exact calculation to floating point form. This is illustrated by using the triangle inequality, where these two errors are represented by the first and second term on the right hand side, respectively:
$$ \big|\ x\circ y - \operatorname{fl} \big( \operatorname{fl}(x) \circ \operatorname{fl}(y) \big) \big| \leq \big|\ x \circ y - \operatorname{fl}(x) \circ \operatorname{fl}(y) \big| + \big| \ \operatorname{fl}(x) \circ \operatorname{fl}(y) -\operatorname{fl}\big( \operatorname{fl} \big( \operatorname{fl}(x) \circ \operatorname{fl}(y) \big) \big| $$
I want to understand what is meant by the statement $\big| \ x \circ y - \operatorname{fl}(x) \circ \operatorname{fl}(y) \big| $ is caused by an error in the data.
Supplement.
The application of the triangle inequality can be seen by denoting $a := x \circ y$, $\ b := \operatorname{fl}(x) \circ \operatorname{fl}(y)$ and $\ c := \operatorname{fl}(b)$:
$$ |a-c| = |(a-b)+(b-c)| \leq |a-c| + |b-c| \ . $$
Text: Numerical Methods for Ordinary Differential Equations, second edition (Vuik,2016), page 4; 1.4 rounding errors.
A more elaborate phrasing of the text's
error in the datacan be given by: an error caused by the (finite) representation of the data $\operatorname{fl}(x)$ and $\operatorname{fl}(y)$.