sRGB Approximations for HLSL

The sRGB color space is non-linear. Many transformations to and from the space require the RGB components to be mapped to a linear form.

The "official" transformation for each RGB component value in the range [0,1] is:

  if (C_srgb <= 0.04045)
      C_lin = C_srgb / 12.92;
  else
      C_lin = pow((C_srgb + 0.055) / 1.055, 2.4);

This is often approximated using the "gamma 2.2" formula:

  C_lin_1 = pow(C_srgb, 2.2);

This works fine, but is fairly inaccurate. The graph below uses the right-hand axis for the absolute difference:

For example, if the values are quantized to eight bits, for the sRGB component value 197/255, the linear output is 145/255 instead of 142/255. Can we do better?

In fact, if we simply change the "magic number" to 2.233333... we get better results:

  C_lin_2 = pow(C_srgb, 2.233333333);

However, the "pow" functionality is either prohibitively expensive or non-existent on many platforms. So, if we limit ourselves to simple arithmetic, a good approximation I found is the cubic:

  C_lin_3 = 0.012522878 * C_srgb +
            0.682171111 * C_srgb * C_srgb +
            0.305306011 * C_srgb * C_srgb * C_srgb;

This can be computed in HLSL using:

  float3 RGB = sRGB * (sRGB * (sRGB * 0.305306011 + 0.682171111) + 0.012522878);

The reverse transformation (from linear to sRGB) is more problematic:

Again, the "official" transformation is piecewise:

  if (C_lin <= 0.0031308)
    C_srgb = C_lin * 12.92;
  else
    C_srgb = 1.055 * pow(C_lin, 1.0 / 2.4) - 0.055;

This is usually poorly approximated with the inverse of the computation of "C_lin_1":

  C_srgb_1 = pow(C_lin, 0.4545454545);

In fact, the linear portion of the official graph is tiny, so an almost-perfect approximation is:

  C_srgb_2 = max(1.055 * pow(C_lin, 0.416666667) - 0.055, 0);

The clamp ("max(..., 0)") is free on many platforms, but the formula does use the "pow" functionality. If we assume we only have square-root operations at our disposal, a good approximation I found was:

  C_srgb_3 = 0.585122381 * sqrt(C_lin) +
             0.783140355 * sqrt(sqrt(C_lin)) -
             0.368262736 * sqrt(sqrt(sqrt(C_lin)));

This can be computed in HLSL using:

  float3 S1 = sqrt(RGB);
  float3 S2 = sqrt(S1);
  float3 S3 = sqrt(S2);
  float3 sRGB = 0.585122381 * S1 + 0.783140355 * S2 - 0.368262736 * S3;

An even better approximation (at the cost of an additional 'mad') is:

  float3 S1 = sqrt(RGB);
  float3 S2 = sqrt(S1);
  float3 S3 = sqrt(S2);
  float3 sRGB = 0.662002687 * S1 + 0.684122060 * S2 - 0.323583601 * S3 - 0.0225411470 * RGB;

Depending on your platform architecture, this may be faster using multiplication by a constant matrix for the final step.