Available kernels
----------------------------------

When constructing a :cpp:class:`dcgp::kernel_set` we can use the following names to add the corresponding
kernels to the set.

---------------------------------------------------------------------------

.. cssclass:: table-bordered table-striped

   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |Kernel name     |  Function              |  Definition                                                                           |
   +================+========================+=======================================================================================+
   | **Basic operations**                                                                                                            |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"sum"           |addition                | :math:`\sum_i x_i`                                                                    |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"diff"          |subtraction             | :math:`x_1 - \sum_{i=2} x_i`                                                          |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"mul"           |multiplication          | :math:`\prod_i x_i`                                                                   |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"div"           |division                | :math:`x_1 / \prod_{i=2} x_i`                                                         |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"pdiv"          |protected division      | :math:`x_1 / \prod_{i=2} x_i` or 1 if NaN                                             |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   | **Unary non linearities (ignoring all inputs after the first one)**                                                             |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"sin"           |sine                    | :math:`\sin x_1`                                                                      |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"cos"           |cosine                  | :math:`\cos x_1`                                                                      |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"log"           |natural logarithm       | :math:`\log x_1`                                                                      |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"exp"           |exponential             | :math:`e^{x_1}`                                                                       |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"gaussian"      |gaussian                | :math:`e^{-x_1^2}`                                                                    |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"sqrt"          |square root             | :math:`\sqrt{x_1}`                                                                    |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"psqrt"         |protected square root   | :math:`\sqrt{|x_1|}`                                                                  |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   | **Non linearities suitable also for dCGPANN**                                                                                   |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"sig"           |sigmoid                 | :math:`\frac{1}{1 + e^{-\sum_i x_i}}`                                                 |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"tanh"          |hyperbolic tangent      | :math:`\tanh \left(\sum_i x_i\right)`                                                 |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"ReLu"          |rectified linear unit   | :math:`\sum_i x_i` if positive, 0 otherwise                                           |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"ELU"           |exp linear unit         | :math:`\sum_i x_i` if positive, :math:`e^{\sum_i x_i} - 1` otherwise                  |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"ISRU"          |Inverse square root unit| :math:`\frac{\sum_i x_i}{\sqrt{1 + \left(\sum_i x_i\right)^2}}`                       |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"sum"           |addition                | :math:`\sum_i x_i`                                                                    |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"sin_nu"        |sine (non unary)        | :math:`\sin(\sum_i x_i)`                                                              |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"cos_nu"        |cosine (non unary)      | :math:`\cos(\sum_i x_i)`                                                              |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"gaussian_nu"   |gaussian (non unary)    | :math:`e^{\sum_i x_i}`                                                                |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"abs"           |absolute value          | :math:`\vert\sum_i x_i\vert`                                                          |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"inv_sum"       |inverse sum             | :math:`- \sum_i x_i`                                                                  |
   +----------------+------------------------+---------------------------------------------------------------------------------------+
   |"step"          |step function           | 1 if positive, 0 otherwise                                                            |
   +----------------+------------------------+---------------------------------------------------------------------------------------+