diff --git a/internal/ceres/low_rank_inverse_hessian.cc b/internal/ceres/low_rank_inverse_hessian.cc
index 372165f..0bb71dc 100644
--- a/internal/ceres/low_rank_inverse_hessian.cc
+++ b/internal/ceres/low_rank_inverse_hessian.cc
@@ -52,7 +52,9 @@
 bool LowRankInverseHessian::Update(const Vector& delta_x,
                                    const Vector& delta_gradient) {
   const double delta_x_dot_delta_gradient = delta_x.dot(delta_gradient);
-  if (delta_x_dot_delta_gradient <= 1e-10) {
+  // Note that 1e-14 is very small, but larger values (1e-10/12) substantially
+  // weaken the performance on the NIST benchmark suite.
+  if (delta_x_dot_delta_gradient <= 1e-14) {
     VLOG(2) << "Skipping LBFGS Update, delta_x_dot_delta_gradient too small: "
             << delta_x_dot_delta_gradient;
     return false;
diff --git a/internal/ceres/polynomial.cc b/internal/ceres/polynomial.cc
index feec884..e548da0 100644
--- a/internal/ceres/polynomial.cc
+++ b/internal/ceres/polynomial.cc
@@ -137,6 +137,9 @@
   if (degree == 0) {
     LOG(WARNING) << "Trying to extract roots from a constant "
                  << "polynomial in FindPolynomialRoots";
+    // We return true with no roots, not false, as if the polynomial is constant
+    // it is correct that there are no roots. It is not the case that they were
+    // there, but that we have failed to extract them.
     return true;
   }