Introduction to Numerical Analysis, Stoer, Chapter: Error Analysis, Page 8
if $|y|<\frac{eps}{\beta}|x|$ where $eps = 0.5\times 10^{1-t}$ then $$fl(x+y)=x+^*y=x$$ where $fl(x)=$ normalized floating point number closest to $x$ and $fl(x)=x(1+\epsilon)$ with $|\epsilon|\leq eps$.
absolute error is given by $$|fl(x)-x|\leq 0.5\times \beta^{e(1-t)} $$.
To prove above equality i have assumed that $\beta=10$ so: $$|fl(x+y)-x|=|(x+y)(1+\epsilon)-x|\leq|x+y||1+\epsilon|+|x|=|x|(3+2|\epsilon|)\leq 10^{e+1}(3+10^{1-t}) $$
but it is not true! PLEASE Help to prove it :)