@Article{Hogan:2014:FRM,
  author =       "Robin J. Hogan",
  title =        "Fast Reverse-Mode Automatic Differentiation Using
                  Expression Templates in C++",
  journal =      "{ACM} Transactions on Mathematical Software",
  volume =       40,
  number =       4,
  year    =      2014,
  month   =      jun,
  pages =        "26:1--26:24",
  url =          "http://doi.acm.org/10.1145/2560359",
  accepted =     "9 December 2013",
  abstract =     "
                  Gradient-based optimization problems are encountered in
                  many fields, but the associated task of differentiating
                  large computer algorithms can be formidable. The
                  operator-overloading approach to performing reverse-mode
                  automatic differentiation is the most convenient for the
                  user but current implementations are typically 10-35 times
                  slower than the original algorithm. In this paper a fast
                  new operator-overloading method is presented that uses
                  the \textit{expression template} programming technique
                  in C++ to
                  provide a compile-time representation of each mathematical
                  expression as a computational graph that can be efficiently
                  traversed in either direction. Benchmarking with four
                  different numerical algorithms shows this approach to be
                  2.6--9 times faster than current operator-overloading
                  libraries, and 1.3--7.7 times more efficient in memory
                  usage. It is typically less than 4 times the computational
                  cost of the original algorithm, although poorer performance
                  is found for all libraries in the case of simple loops
                  containing no mathematical functions. An implementation
                  is freely available in the \textit{Adept} C++ software
                  library.",
}