Consider computing $$ x = 10^{-8}, \qquad y = \sqrt{1+x^2} - 1 $$
x = 1.0e-8
y = sqrt(1.0 + x**2) - 1.0
print "%20.10e" % y
0.0000000000e+00
The result is zero due to round-off error. An equivalent expression is $$ y = \frac{x^2}{\sqrt{1 + x^2} + 1} $$
y = x**2/(sqrt(1.0+x**2) + 1.0)
print "%20.10e" % y
5.0000000000e-17