# Performance anomalies when plotting regions

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?

• 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 • Thanks, both of these indeed work and fix the issue in this particular case! However, I would like to be able to do further things with the Region (e.g. intersect it with other Regions, call Plot3D using the region as the domain, etc.) and not all of this works when using the first of the two fixes. For the second fix, I do not know how to generalize it to accommodate arbitrary rational powers greater 1. 4/3 and 5/3 were specific examples, but in general the exponent is actually the result of some other computation. Mar 17 at 16:39
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
` 