Taylor series approximation for $(x^{2}+(1-x)^{2})^{y} - (1 - x)^{2y}$

Question

It is said $1-(1-x)^y \approx yx$

Assuming $yx<<1$ and using Taylor series approximation.

$x$ can be $10^{-15}$ and $y$ can be $10^{9}$ so $yx<<1$

My question is can we simplify

$(x^{2}+(1-x)^{2})^{y} - (1 - x)^{2y}$ to something similar to $yx^{2}$

Any help will be appreciated.

Thanks