Better HDRI sky integration formula

com.unity.render-pipelines.high-definition/HDRP/RenderPipeline/HDStringConstants.cs

         public static readonly int _ThicknessRemap = Shader.PropertyToID("_ThicknessRemap");
 
         public static readonly int _Cubemap = Shader.PropertyToID("_Cubemap");
+        public static readonly int _InvOmegaP = Shader.PropertyToID("_InvOmegaP");
         public static readonly int _SkyParam = Shader.PropertyToID("_SkyParam");
         public static readonly int _SkyIntensity = Shader.PropertyToID("_SkyIntensity");
         public static readonly int _PixelCoordToViewDirWS = Shader.PropertyToID("_PixelCoordToViewDirWS");

com.unity.render-pipelines.high-definition/HDRP/Sky/HDRISky/HDRISkyRenderer.cs

         {
             using (new ProfilingSample(builtinParams.commandBuffer, "Get hdri skybox intensity"))
             {
-                if (m_HdriSkyParams.updateHDRISkyIntensity)
+                // if (m_HdriSkyParams.updateHDRISkyIntensity)
+                    float omegaP = (Mathf.PI * 4) / (6.0f * m_HdriSkyParams.hdriSky.value.width * m_HdriSkyParams.hdriSky.value.width);
+                    m_IntegrateHDRISkyMaterial.SetFloat(HDShaderIDs._InvOmegaP, 1.0f / omegaP);
 
                     CoreUtils.SetRenderTarget(builtinParams.commandBuffer, m_IntensityTexture, ClearFlag.None);
                     CoreUtils.DrawFullScreen(builtinParams.commandBuffer, m_IntegrateHDRISkyMaterial);

com.unity.render-pipelines.high-definition/HDRP/Sky/HDRISky/IntegrateHDRISky.shader

     {
         [HideInInspector]
         _Cubemap ("", CUBE) = "white" {}
+        _InvOmegaP ("", Float) = 0
     }
 
     SubShader
             #pragma fragment Frag
             #pragma target 4.5
             #pragma only_renderers d3d11 ps4 xboxone vulkan metal switch
+#pragma enable_d3d11_debug_symbols
 
             #include "CoreRP/ShaderLibrary/Common.hlsl"
             #include "CoreRP/ShaderLibrary/Color.hlsl"
             };
 
             TextureCube<float4> _Cubemap;
+            float _InvOmegaP;
 
             Varyings Vert(Attributes input)
             {
             }
             
 
-            real IntegrateHDRISky(TEXTURECUBE_ARGS(skybox, sampler_skybox), real3 N, uint sampleCount = 8192)
+            real IntegrateHDRISky(TEXTURECUBE_ARGS(skybox, sampler_skybox), real3 N, uint sampleCount = 4096)
-                real2 randNum  = InitRandom(0.5);
+                real2 randNum  = InitRandom(0.0);
 
                 real3x3 localToWorld = GetLocalFrame(N);
 
                     real3 L;
                     
                     ImportanceSampleLambert(u, localToWorld, L, NdotL, weightOverPdf);
-                    real val = Luminance(SAMPLE_TEXTURECUBE_LOD(skybox, sampler_skybox, L, 0).rgb);
-                    acc += PI * val;
+                    
+                    real pdf = NdotL / PI;
+                    real omegaS = rcp(sampleCount) * rcp(pdf);
+                    // _InvOmegaP = rcp(FOUR_PI / (6.0 * cubemapWidth * cubemapWidth));
+                    real mipLevel = 0.5 * log2(omegaS * _InvOmegaP);
+
+                    if (NdotL > 0.0)
+                    {
+                        real val = Luminance(SAMPLE_TEXTURECUBE_LOD(skybox, sampler_skybox, L, mipLevel).rgb);
+                        acc += PI * val;
+                    }
                 }
 
                 return acc / sampleCount;
             {
-                float3 N = float3(0.0, -1.0, 0.0);
+                float3 N = float3(0.0, 1.0, 0.0);
 
                 float intensity = IntegrateHDRISky(TEXTURECUBE_PARAM(_Cubemap, s_trilinear_clamp_sampler), N);