- result[6] = inv_det * (m[3] * m[7] - m[6] * m[4]);
- result[7] = inv_det * (m[6] * m[1] - m[0] * m[7]);
- result[8] = inv_det * (m[0] * m[4] - m[3] * m[1]);
+ // Round to the minimum number of bits we have measured earlier.
+ // The card will do this for us anyway, but if we know what the real z
+ // is, we can pick a better total_weight below.
+ z = lrintf(z * num_subtexels) * inv_num_subtexels;
+
+ // Choose total weight w so that we minimize total squared error
+ // for the effective weights:
+ //
+ // e = (w(1-z) - a)² + (wz - b)²
+ //
+ // Differentiating by w and setting equal to zero:
+ //
+ // 2(w(1-z) - a)(1-z) + 2(wz - b)z = 0
+ // w(1-z)² - a(1-z) + wz² - bz = 0
+ // w((1-z)² + z²) = a(1-z) + bz
+ // w = (a(1-z) + bz) / ((1-z)² + z²)
+ //
+ // If z had infinite precision, this would simply reduce to w = w1 + w2.
+ *total_weight = from_fp64<DestFloat>((w1 + z * (w2 - w1)) / (z * z + (1 - z) * (1 - z)));
+
+ if (sum_sq_error != NULL) {
+ float err1 = to_fp64(*total_weight) * (1 - z) - w1;
+ float err2 = to_fp64(*total_weight) * z - w2;
+ *sum_sq_error = err1 * err1 + err2 * err2;
+ }