Performance anomalies when plotting regions

Question

There are large differences in how long it takes to plot regions which, in my eyes, should be computationally very similar.

The following two work fine:

RegionPlot[ParametricRegion[{{x, y}, x + y^(4/3) >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}]]
RegionPlot[ParametricRegion[{{x, 1+y}, x + y^(5/3) >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}]]

However, the following takes so long that that I eventually had to abort:

RegionPlot[ParametricRegion[{{x, 1+y}, x + y^(4/3) >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}]]

Why should the first one be so much easier to compute than the last one? The only difference is a simple translation. How can I work around this?

Answer 1

• Region

Region[ParametricRegion[{{x, 1 + y}, 
   x + y^(4/3) >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}]]

• Or replace y^(4/3) with CubeRoot[y^4].

RegionPlot[
 ParametricRegion[{{x, 1 + y}, 
   x + CubeRoot[y^4] >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}]]

Edit

For arbitrary rational powers, e.g. p/q=12/5, we can use Surd[y^p,q].

p = 12; q = 5;
reg = ParametricRegion[{{x, 1 + y}, 
   x + Surd[y^p, q] >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}]
reg // RegionPlot

Answer 2

Clear["Global`*"]

rgn = ParametricRegion[{{x, 1 + y}, 
    x + y^(4/3) >= 1 && 0 <= x <= 1 && 0 <= y <= 1}, {x, y}];

Region[rgn, Axes -> True] // AbsoluteTiming

RegionPlot[{x, y} ∈ rgn, {x, 0, 1}, {y, 1, 
   2}] // AbsoluteTiming