+ // Recover z = 1 - x - y.
+ double z_R = 1.0 - x_R - y_R;
+ double z_G = 1.0 - x_G - y_G;
+ double z_B = 1.0 - x_B - y_B;
+
+ // Find the XYZ coordinates of D65 (white point for both Rec. 601 and 709),
+ // normalized so that Y=1.
+ double d65_X = d65_x / d65_y;
+ double d65_Y = 1.0;
+ double d65_Z = (1.0 - d65_x - d65_y) / d65_y;
+
+ // We have, for each primary (example is with red):
+ //
+ // X_R / (X_R + Y_R + Z_R) = x_R
+ // Y_R / (X_R + Y_R + Z_R) = y_R
+ // Z_R / (X_R + Y_R + Z_R) = z_R
+ //
+ // Some algebraic fiddling yields (unsurprisingly):
+ //
+ // X_R = (x_R / y_R) Y_R
+ // Z_R = (z_R / y_R) Y_R
+ //
+ // We also know that since RGB=(1,1,1) should give us the
+ // D65 illuminant, we must have
+ //
+ // X_R + X_G + X_B = D65_X
+ // Y_R + Y_G + Y_B = D65_Y
+ // Z_R + Z_G + Z_B = D65_Z
+ //
+ // But since we already know how to express Y and Z by
+ // some constant multiple of X, this reduces to
+ //
+ // k1 Y_R + k2 Y_G + k3 Y_B = D65_X
+ // Y_R + Y_G + Y_B = D65_Y
+ // k4 Y_R + k5 Y_G + k6 Y_B = D65_Z
+ //
+ // Which we can solve for (Y_R, Y_G, Y_B) by inverting a 3x3 matrix.
+
+ Matrix3x3 temp, inverted;
+ temp[0] = x_R / y_R;
+ temp[3] = x_G / y_G;
+ temp[6] = x_B / y_B;
+
+ temp[1] = 1.0;
+ temp[4] = 1.0;
+ temp[7] = 1.0;
+
+ temp[2] = z_R / y_R;
+ temp[5] = z_G / y_G;
+ temp[8] = z_B / y_B;
+
+ invert_3x3_matrix(temp, inverted);
+ float Y_R, Y_G, Y_B;
+ multiply_3x3_matrix_float3(inverted, d65_X, d65_Y, d65_Z, &Y_R, &Y_G, &Y_B);
+
+ // Now convert xyY -> XYZ.
+ double X_R = temp[0] * Y_R;
+ double Z_R = temp[2] * Y_R;
+ double X_G = temp[3] * Y_G;
+ double Z_G = temp[5] * Y_G;
+ double X_B = temp[6] * Y_B;
+ double Z_B = temp[8] * Y_B;