Behavior of $x^2+y^2+\frac{y}{x}=1$. In my mathematical travels, I've stumbled upon the implicit formula $y^2+x^2+\frac{y}{x}=1$ and found that every graphing program I've plugged it in to seems to believe that there is large set of points which satisfy the equation $(y^2+x^2+\frac{y}{x})^{-1}=1$ which do not satisfy the original equation and this has me quite perplexed. I suspect that this is simply a glitch in the software and this question might therefore be better suited in the CS forum but I figured I would post it here first in the event that someone may have a mathematical explanation for this bizarre behavior. Any and all insights are welcome!

rotationally symmetric about the origin; your version does not allow $x=0$ but the curve becomes a smooth variety if $(0,0)$ is included
The smooth implicit function is
$$ x^3 + x y^2 + y - x = 0  $$
with gradient
$$ \left\langle 3 x^2 + y^2 - 1, 2xy + 1 \right\rangle  $$
For large $|y|,$ solving the quadratic formula in $y$ shows $xy \approx -1.$ Indeed, large or small, for $x \neq 0$ we have
$$  y = \frac{-1 \pm \sqrt {1 + 4 x^2 - 4 x^4}}{2x} $$ and we require
$1 + 4 x^2 - 4 x^4 \geq 0.$

