In the machine a number is represented as $$ \begin{equation} fl(x)= x(1+\epsilon) \tag{7} \end{equation} $$

where \( |\epsilon| \leq \epsilon_M \) and \( \epsilon \) is given by the specified precision, \( 10^{-7} \) for single and \( 10^{-16} \) for double precision, respectively. \( \epsilon_M \) is the given precision. In case of a subtraction \( a=b-c \), we have $$ \begin{equation} fl(a)=fl(b)-fl(c) = a(1+\epsilon_a), \tag{8} \end{equation} $$ or $$ \begin{equation} fl(a)=b(1+\epsilon_b)-c(1+\epsilon_c), \tag{9} \end{equation} $$