Math question on Newton's method and detecting actual zeros

[Han]

(Admins: If this is in the wrong forum, please feel free to move it)

This came up during a debugging process in which Newton's method (using backtracking linesearch) gave me a solution to the system

\[ \frac{x\cdot y}{x+y} = 127\times 10^{-12}, \quad \left( \frac{x+y}{x} \right)^2 = 8.377 \]

(This problem was posed on the HP Prime subforum: http://hpmuseum.org/forum/thread-7677.html)

One solution I found was: \( x=1.94043067156\times 10^{-10}, \
y=3.67576704293\times 10^{-10} \) (hopefully no typos).

On the Prime, the error for the equations are in the order of \(10^{-19} \) and \(10^{-11}\) for the first and second equations, respectively (again, assuming I made no typos copying). So my question is: should a numerical solver should treat \(1.27\times 10^{-10}\) as "significant" or 0 (especially when it comes time to check for convergence, when the tolerance for \( |f_i| \) might be set to, say, \( 10^{-10} \) -- here \( f_i \) is the i-th equation in the system, set equal to 0)?