"To soften the diffuse contribution from local lights, the dot product from the Lambertian model is scaled by ½, add ½ and squared. The result is that this dot product, which normally lies in the range of -1 to +1, is instead in the range of 0 to 1 and has a more pleasing falloff. "

https://developer.valvesoftware.com/wiki/Half_Lambert

[1] https://lisyarus.github.io/blog/media/a-silly-diffuse-shadin...

[2] https://archive.org/details/DragonMagazine260_201801/DragonM...