If we were not to worry about loss of precision, we could in principle make \( h \) as small as possible. However, due to the computed expression in the above program example $$ f_0''=\frac{ f_h -2f_0 +f_{-h}}{h^2}=\frac{ (f_h -f_0) +(f_{-h}-f_0)}{h^2}, $$ we reach fairly quickly a limit for where loss of precision due to the subtraction of two nearly equal numbers becomes crucial.

If \( (f_{\pm h} -f_0) \) are very close, we have \( (f_{\pm h} -f_0)\approx \epsilon_M \), where \( |\epsilon_M|\le 10^{-7} \) for single and \( |\epsilon_M|\le 10^{-15} \) for double precision, respectively.

We have then $$ \left|f_0''\right|= \left|\frac{ (f_h -f_0) +(f_{-h}-f_0)}{h^2}\right|\le \frac{ 2 \epsilon_M}{h^2}. $$