Find a formula for $f(x)$ if \[ f^{\prime}(x)=\frac{4}{1+x^{2}}, \quad f(1)=0 \] Use this formula to compute $f(0)$. Give your answer as a decimal number with at least three decimal places.

Step 1 ：We are given that the derivative of the function, \(f'(x)\), is \(\frac{4}{1+x^{2}}\).

Step 2 ：To find the original function, \(f(x)\), we need to find the antiderivative of \(f'(x)\).

Step 3 ：The antiderivative of \(\frac{4}{1+x^{2}}\) is \(4\arctan(x)\).

Step 4 ：We also know that \(f(1) = 0\), which we can use to find the constant of integration.

Step 5 ：After finding the function \(f(x)\), we substitute \(x = 0\) into the function to find \(f(0)\).

Step 6 ：Final Answer: \(f(0) = \boxed{-3.142}\)