How to generate random points within an octahedron without discarding?

I need random points within an octahedron, uniformly distributed. I am defining an octahedron as the volume where all points satisfy abs(x) + abs(y) + abs(z) <= 1 where abs gives absolute value. IE: each of the six vertices would be on an axis, 1 away from 0,0,0. Maybe you could call it a unit octahedron.

With the definition in mind, I can naively generate a point like so:

val x: Double = nextDouble() // 0-1 range
val y = nextDouble(1.0 -x) // 1-x is upper bound, probably <1
val z = nextDouble(1.0 -(x+y))

The problem is that this leans toward small y values, and smaller z values. Clearly not an even distribution. Also clearly, all these points are in just one of eight quadrants.

I'm avoiding the discard method because this function will be called a lot, and it just seems like I should be able to do better than throwing out the majority of points.

Note that the dual of the octahedron is the cube. Because of this, I have an inkling that there might exist a simple function to translate any point within a cube to be within the octahedron, but that's just an intuition I'm still exploring.