Transforming data so that the range around the median is more accurate

Suppose I have some floats that are normal distributed around 0. I need to serialize this into a uint8, but I would like to "give more" of the uint8 to the center of the distribution, and lose resolution around the edges.

For example: 127 would correspond to 0.0 and 255 to 1.0. But 191 would not be 0.5 — instead, it would be something like 0.3 because we're stretching it so that most of the numbers correspond to values near 0.

In practice, I'm actually going to have a random uint32 being generated and converted to an float. But when testing a linear mapping, the extremes (near -1.0 and 1.0) came up too frequently, and I'd like to center this around 0.0.

I'm aware that I can use the Box–Muller transform, but that's actually not suitable here because:

1. We can cap out at -1.0 and 1.0, no need to have an unbounded output.

2. We only have one number to sample from, not two.

Thanks