Isn't that just Half-Lambert that Valve came up with for Half-Life?
"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. "
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[ 2.6 ms ] story [ 19.3 ms ] thread[1] https://lisyarus.github.io/blog/media/a-silly-diffuse-shadin...
[2] https://archive.org/details/DragonMagazine260_201801/DragonM...
"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