GPU Procedural planets: How to convert grayscale highmap to normal map and use it as normal map on the GPU?

Hello. I am working on procedural generation of full scale planets.

I am already implemented GPU Perlin Noise and use it to generate diffuse texture of surface and displacement map.

But also, I need somehow to use this generated gray-scale high-map as normal map.

Is it possible to transform gray scale map into normal map right inside the shader?

Of course I can extract height map and convert it into normal map on CPU by using simple algorithm, but CPU is too slow for this task. I need to generate it right inside shader and not sure how it is possible, because in the shader I can't check neighbors of pixel. Or I can?

Or maybe I can use gray-scale map as "bump map" without converting to normal map? As far as I understand, "bump mapping" it is simpler technique of normal mapping which use only gray scale texture (single channel, not a 3 channel like for normal map). In result, I need to achieve a bump\normal maps for mountains like on this picture from "Outerra" using GPU noise I have (I already have height in each point).