 Small-speed length contraction estimate

At speed $v\ll c$, an endpoint separation $L$ measured simultaneously in its own rest frame becomes $L\sqrt{1-v^2/c^2}$ in the moving frame, shorter by approximately $L{v}^2/(2c^2)$. The stable evaluation $L{\beta }^2/[1+\sqrt{1-{\beta }^2}]$ avoids subtracting nearly equal floating-point numbers.