Inequality plot with PGFPlots

Question

I am trying to draw the inequality plot of this inequality but did not mange to find a way. The plot I want is similar to the one that Mathematica produces. Anybody can help?

I tried this command

\addplot3 [surf,shader=interp,samples=45, domain=-2:6, y domain=-2:6]
{(3*sqrt(x^2+y)+sqrt(4*x^3-3*x^2+y)-4)/(sqrt(y+1)+sqrt(3)*sqrt(4*x^3+3*y-  1)-4)};

and then view={0}{90} but it takes a lot of time to compile and I guess there should be a way to produce a 2D plot rather than changing the angle of a 3D plot.

Answer 1

The quick and easy way is to use the CAS Sage, along with the sagetex package with code something like this:

\documentclass{standalone}
\usepackage{sagetex}
\usepackage[usenames,dvipsnames]{xcolor}
\usepackage{pgfplots}
\begin{document}
\begin{sagesilent}
H =region_plot((3*sqrt(x^2+y)+sqrt(4*x^3-3*x^2+y)-4)/(sqrt(y+1)+sqrt(3)*sqrt(4*x^3+3*y-1)-4)<=1,(x,-3,5),(y,-3,5),incol='lightblue', bordercol='gray',plot_points=500)
\end{sagesilent}
\sageplot[width=6cm]{H}
\end{document}

Run the code through a free SagemathCloud account to get this output. Documentation on region_plot can be found here on the Sage website. I think it's possible to force this type of plot through pgfplots but it would take a lot more time. If that's what you want you can find some related posts on this site (search plot and sagetex) such as the sagetex solutions for Riemann Zeta and Cantor Function.

Answer 2

You can do it with pgfplots and use compat=newest which should enable the Lua backend for faster function evaluation. However, the compile time is still several minutes.

\documentclass{article}
\pagestyle{empty}
\usepackage{pgfplots}
\pgfplotsset{compat=newest}
\begin{document}
\begin{tikzpicture}
  \begin{axis}
    [
    view={0}{90},
    samples=101,
    domain=-2:6,
    y domain=-2:6,
    restrict z to domain=-inf:1,
    xmin=-2, xmax=6,
    ymin=-2, ymax=6,
    ]

    \addplot3 [surf,shader=interp]
    {
      (3*sqrt(x^2+y)+sqrt(4*x^3-3*x^2+y)-4)
      /
      (sqrt(y+1)+sqrt(3)*sqrt(4*x^3+3*y-1)-4)
    };
  \end{axis}
\end{tikzpicture}
\end{document}