Why don't C++ compilers do better constant folding?

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.

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:

#include <Eigen/Core>
#include <Eigen/Geometry>

using Array12d = Eigen::Matrix<double,12,1>;

double testReturnFirstDot(const Array12d& b)
{
    Array12d a;
    a.array() = 0.;
    a(0) = 1.;
    return a.dot(b);
}

Which should be the same as

double testReturnFirst(const Array12d& b)
{
    return b(0);
}

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 (https://godbolt.org/z/GvPXFy) the optimizations are very poor in GCC and Clang (still involve multiplications and additions), and MSVC doesn't do any optimizations at all.

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.

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.

This is because Eigen explicitly vectorize your code as 3 vmulpd, 2 vaddpd and 1 horizontal reduction within the remaining 4 component registers (this assumes AVX, with SSE only you'll get 6 mulpd and 5 addpd). With -ffast-math GCC and clang are allowed to remove the last 2 vmulpd and vaddpd (and this is what they do) but they cannot really replace the remaining vmulpd and horizontal reduction that have been explicitly generated by Eigen.

So what if you disable Eigen's explicit vectorization by defining EIGEN_DONT_VECTORIZE? Then you get what you expected (https://godbolt.org/z/UQsoeH) but other pieces of code might become much slower.

If you want to locally disable explicit vectorization and are not afraid of messing with Eigen's internal, you can introduce a DontVectorize option to Matrix and disable vectorization by specializing traits<> for this Matrix type:

static const int DontVectorize = 0x80000000;

namespace Eigen {
namespace internal {

template<typename _Scalar, int _Rows, int _Cols, int _MaxRows, int _MaxCols>
struct traits<Matrix<_Scalar, _Rows, _Cols, DontVectorize, _MaxRows, _MaxCols> >
: traits<Matrix<_Scalar, _Rows, _Cols> >
{
  typedef traits<Matrix<_Scalar, _Rows, _Cols> > Base;
  enum {
    EvaluatorFlags = Base::EvaluatorFlags & ~PacketAccessBit
  };
};

}
}

using ArrayS12d = Eigen::Matrix<double,12,1,DontVectorize>;

Full example there: https://godbolt.org/z/bOEyzv

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.

They have no other choice unfortunately. Since IEEE floats have signed zeros, adding 0.0 is not an identity operation:

-0.0 + 0.0 = 0.0 // Not -0.0!

Similarly, multiplying by zero does not always yield zero:

0.0 * Infinity = NaN // Not 0.0!

So the compilers simply cannot perform these constant folds in the dot product while retaining IEEE float compliance - for all they know, your input might contain signed zeros and/or infinities.

You will have to use -ffast-math to get these folds, but that may have undesired consequences. You can get more fine-grained control with specific flags (from http://gcc.gnu.org/wiki/FloatingPointMath). According to the above explanation, adding the following two flags should allow the constant folding:
-ffinite-math-only, -fno-signed-zeros

Indeed, you get the same assembly as with -ffast-math this way: https://godbolt.org/z/vGULLA. You only give up the signed zeros (probably irrelevant), NaNs and the infinities. Presumably, if you were to still produce them in your code, you would get undefined behavior, so weigh your options.

As for why your example is not optimized better even with -ffast-math: That is on Eigen. Presumably they have vectorization on their matrix operations, which are much harder for compilers to see through. A simple loop is properly optimized with these options: https://godbolt.org/z/OppEhY