(We can directly go to the "if" part to get one answer, but we need to solve "else" part too to get all possible answers which though is not asked in question)
Solving the else part:
$\frac{x}{2} + \frac{3}{2x} = \frac{x^2+3}{2x}$
So, the new value of $x$ will be $\frac{x^2+3}{2x}$ and we need it equal to $x$.
$\frac{x^2+3}{2x} = x \\ \implies x^2 + 3 = 2x^2 \\ \implies x^2 = 3 \\ \implies x = 1.732 $
Now solving the if part.
abs(x*x - 3) < 0.01
So, $x^2 - 3 < 0.01 \text { and } -\left(x^2 - 3\right) < 0.01\\ \implies x^2 < 3.01 \text{ and } x^2 > 2.99\\ \implies x < 1.735 \text { and }x > 1.729$
Corrected to $2$ decimal places answer should be $1.73$ or $1.74$.