Any general algorithm to find $x$ given $y$ for the elliptic curve equation $y^2 = x^3 + ax + b$, in real numbers?

Question

Is there a way to solve for $x$ given $y$, in the equation $y^2 = x^3 + ax + b$, in the real number space (not finite field).

Or is the solution too difficult to achieve? I suspect the solution will require factoring a 3rd degree polynomial.

I'm trying to animate a dot going from top to bottom of the screen while following the curve. This will require inputing the $y$ component into some function, and getting the $x$ component in return, so that I can place the dot in the correct spot on the graph.

This question is not to be confused with the question asking about finding $x$ for the elliptic curve equation, in a finite field. I'm trying to solve in the real number space, not in the finite field.

Answer 1

Consider the cubic equation $$ x^3 + \alpha x + (\beta-y^2) =0$$ The discriminant $$\Delta=-4 \alpha ^3-27 \left(y^2-\beta \right)^2$$ is always negative if $\alpha >0$. So, if this is the case, there is only one real root.

Using the hyperbolic method, this root is $$\color{blue}{x=-2 \sqrt{\frac{\alpha }{3}}\sinh \left(\frac{1}{3} \sinh ^{-1}\left(\frac 32 \sqrt{\frac3 {\alpha }}\,\,\frac{(\beta-y^2)}\alpha\right)\right)}$$

If $\alpha <0$ (still with $\Delta<0$), the solution is $$\color{blue}{x=-2 \sqrt{-\frac{\alpha }{3}}\,\,\frac{\left| \beta -y^2\right| }{\beta -y^2}\,\cosh \left(\frac{1}{3} \cosh ^{-1}\left(-\frac 32 \sqrt{-\frac3 {\alpha }} \,\,\frac{\left| \beta -y^2\right| }{\alpha }\right)\right)}$$

This is much better than using Cardano formula.

Answer 2

It is perhaps easier to solve the equation $y^2=x^3+ax+b$ of an elliptic curve for given real $x$ (and not for given $y$), by taking $y=\pm\sqrt{x^3+ax+b}$ for those real $x$, which satisfy $x^3+ax+b\ge 0$.