Checking Map Being Holomorphic, using Cauchy-Riemann Equation

Question

I'm reading Algebraic Curves and Riemann Surface by Rick Miranda. And I was doing the exercise at the end of the first section. I let $X = \mathbb{R}^2$. Andd I have $U$ be any subset of $X$. I define a chart $$\phi_{U}(x,y) = \frac{x}{1+ \sqrt{x^2 + y^2}} + i \frac{y}{1 + \sqrt{x^2 + y^2}}$$. I need to prove that any two charts defined like this are compatible. In this case, I need to show that $\phi_{U} \circ \phi_{U}^{-1}$ is holomorphic. Then I tried to first compute the four partial derivatives to use Cauchy-Riemann equation. I have $$\frac{\partial u}{\partial x} = \frac{1+\sqrt{x^2 + y^2} - \frac{x^2}{\sqrt{x^2 + y^2}}}{(1+ \sqrt{x^2 + y^2})^2}$$And we also have $$\frac{\partial u}{\partial y} = \frac{\frac{-y}{\sqrt{x^2 + y^2}}}{(1+ \sqrt{x^2 + y^2})^2}$$Then I have $$\frac{\partial v}{\partial y} = \frac{1+\sqrt{x^2 + y^2} - \frac{y^2}{\sqrt{x^2 + y^2}}}{(1+ \sqrt{x^2 + y^2})^2}$$ And we have $$\frac{\partial v}{\partial x} = \frac{\frac{-x}{\sqrt{x^2 + y^2}}}{(1+ \sqrt{x^2 + y^2})^2}$$ But from my computation so far, I can only conclude that it satisfies Cauchy-Riemann equation only when $x = y$. Could anyone tell me what's wrong with my proof so far? Thanks so much!
UPDATE So we let $z = x+iy$. Then we have $\phi_{U} \circ \phi_{U}^{-1} = z$, according to the definition of inverse function. Thus, it must be holomorphic. Then according to the definition, any two charts defined like this are compatible with each other.