3D Texture and normals

I want to calculate normals for a heightmap generated by sampling a 3D noise texture/volume on an arbitrary plane. I need to store the height and data for calculating the normal in 1 Color(float4). Right now I've used float4.w for the height, and float4.xyz to store the "axis slopes" respectively. Using the face(plane) normal in the shader, how can I convert these axis slopes into a surface normal?

I'm sure this is possible but I'm not sure how much it will cost per fragment and I don't think I have the math skills to figure it out.