title: Why don't C++ compilers do better constant folding?
body: <p>I'm investigating ways to speed up a large section of C++ code, which has automatic derivatives for computing jacobians. This involves doing some amount of work in the actual residuals, but the majority of the work (based on profiled execution time) is in calculating the jacobians. </p>
  
  <p>This surprised me, since most of the jacobians are propagated forward from 0s and 1s, so the amount of work should be 2-4x the function, not 10-12x. In order to model what a large amount of the jacobian work is like, I made a super minimal example with just a dot product (instead of sin, cos, sqrt and more that would be in a real situation) that the compiler should be able to optimize to a single return value:</p>
  
  <pre><code>#include &lt;Eigen/Core&gt;
  #include &lt;Eigen/Geometry&gt;
  
  using Array12d = Eigen::Matrix&lt;double,12,1&gt;;
  
  double testReturnFirstDot(const Array12d&amp; b)
  {
      Array12d a;
      a.array() = 0.;
      a(0) = 1.;
      return a.dot(b);
  }
  </code></pre>
  
  <p>Which should be the same as </p>
  
  <pre><code>double testReturnFirst(const Array12d&amp; b)
  {
      return b(0);
  }
  </code></pre>
  
  <p>I was disappointed to find that, without fast-math enabled, neither GCC 8.2, Clang 6 or MSVC 19 were able to make any optimizations at all over the naive dot-product with a matrix full of 0s. Even with fast-math (<a href="https://godbolt.org/z/GvPXFy" rel="noreferrer">https://godbolt.org/z/GvPXFy</a>) the optimizations are very poor in GCC and Clang (still involve multiplications and additions), and MSVC doesn't do any optimizations at all.</p>
  
  <p>I don't have a background in compilers, but is there a reason for this? I'm fairly sure that in a large proportion of scientific computations being able to do better constant propagation/folding would make more optimizations apparent, even if the constant-fold itself didn't result in a speedup.</p>
  
  <p>While I'm interested in explanations for why this isn't done on the compiler side, I'm also interested for what I can do on a practical side to make my own code faster when facing these kinds of patterns.</p>