.. _erfc:

erfc
====


Computes the element-wise complementary error function of vector elements.


.. contents::
    :local:
    :depth: 1

Description
***********


The ``erfc`` function computes the complementary error function values
for elements of the input vector ``a`` and writes them to the output
vector ``y``.


The complementary error function is defined as follows:

.. math::

   \mathrm{erfc} (x) = \frac{2}{\sqrt{\pi}} \int_x^{\infty} e^{-t^2}dt


Useful relations:


.. math::

   \mathrm{erfc} (x) = 1 - \mathrm{erf} (x)


.. math::

   \Phi (x) = \frac{1}{2} \mathrm{erf} (\frac{x}{\sqrt{2}})



where


.. math::

   \Phi (x) = - \frac{1}{\sqrt{2\pi}}
   \int_0^x \exp (- frac{t^2}{2}) dt


is the cumulative normal distribution function.


.. math::

   \Phi^{-1} (x) = \sqrt{2} \mathrm{erf}^{-1} (2x - 1)


where :math:`\phi -1(x)` and ``erf-1(x)`` are the inverses to :math:`\phi (x)` and ``erf(x)``, respectively.


The following figure illustrates the relationships among ``erf`` family
functions (``erf``, ``erfc``, ``cdfnorm``).


.. figure:: /_figures/GUID-0B9A59CC-C914-429D-AF87-93B16DABD291-low.jpg
   :align: center
   :alt: erfc Family Functions Relationship

   ``erfc`` Family Functions Relationship


Useful relations for these functions:


.. math::

   \mathrm{erf} (x) + \mathrm{erfc} = 1


.. math::

   \mathrm{cdfnorm} (x) =
   \frac{1}{2}
   (1 + \mathrm{erf} (\frac{x}{\sqrt{2}})) =
   1 - frac{1}{2} \mathrm{erfc} (\frac{x}{\sqrt{2}})


.. list-table::
   :header-rows: 1

   * -  Argument
     -  Result
     -  Error Code
   * -  a > underflow
     -  +0
     -  ``status::underflow``
   * -  +\ :math:`\infty`
     -  +0
     -   
   * -  -\ :math:`\infty`
     -  +2
     -   
   * -  QNAN
     -  QNAN
     -   
   * -  SNAN
     -  QNAN
     -   


API
***


Syntax
------

**Buffer API**


.. code-block:: cpp

   namespace oneapi::mkl::vm {

   sycl::event erfc(sycl::queue & exec_queue,
         std::int64_t n,
         sycl::buffer<T> & a,
         sycl::buffer<T> & y,
         oneapi::mkl::vm::mode mode = oneapi::mkl::vm::mode::not_defined,
         oneapi::mkl::vm::error_handler<T> errhandler = {});

   }

.. code-block:: cpp

   namespace oneapi::mkl::vm {

   sycl::event erfc(sycl::queue & exec_queue,
         sycl::buffer<T> & a,
         oneapi::mkl::slice sa,
         sycl::buffer<T> & y,
         oneapi::mkl::slice sy,
         oneapi::mkl::vm::mode mode = oneapi::mkl::vm::mode::not_defined,
         oneapi::mkl::vm::error_handler<T> errhandler = {});

   }

**USM API**

.. code-block:: cpp

   namespace oneapi::mkl::vm {

   sycl::event erfc(sycl::queue & exec_queue,
         std::int64_t n,
         T const * a,
         T * y,
         std::vector<cl::sycl::event> const & depends = {},
         oneapi::mkl::vm::mode mode = oneapi::mkl::vm::mode::not_defined,
         oneapi::mkl::vm::error_handler<T> errhandler = {});

   }

.. code-block:: cpp

   namespace oneapi::mkl::vm {

   sycl::event erfc(sycl::queue & exec_queue,
         T const * a,
         oneapi::mkl::slice sa,
         T * y,
         oneapi::mkl::slice sy,
         std::vector<cl::sycl::event> const & depends = {},
         oneapi::mkl::vm::mode mode = oneapi::mkl::vm::mode::not_defined,
         oneapi::mkl::vm::error_handler<T> errhandler = {});

   }




``erfc`` supports the following precisions and devices:


.. list-table::
   :header-rows: 1

   * -  T
     -  Devices supported
   * -  ``float``
     -  Host, CPU, and GPU
   * -  ``double``
     -  Host, CPU, and GPU


Input Parameters
----------------

**Buffer API**


exec_queue
   The queue where the routine should be executed.


n
   Specifies the number of elements to be calculated.


a
   The buffer containing the input vector.


sa
   Slice selector for ``a``. See :ref:`data-types` for a description of the |O-MKL| slice type.


sy
   Slice selector for ``y``. See :ref:`data-types` for a description of the |O-MKL| slice type.


mode
   Overrides the global VM mode setting for this function call. See the :ref:`set_mode` function for possible values and their description. This is an optional parameter. The default value is ``mode::not_defined``.


errhandler
   Sets local error handling mode for this function call. See the :ref:`create_error_handler` function for arguments and their descriptions. This is an optional parameter. The local error handler is disabled by default.


**USM API**


exec_queue
   The queue where the routine should be executed.


n
   Specifies the number of elements to be calculated.


a
   Pointer to the input vector.


sa
   Slice selector for ``a``. See :ref:`data-types` for a description of the |O-MKL| slice type.


sy
   Slice selector for ``y``. See :ref:`data-types` for a description of the |O-MKL| slice type.


depends
   Vector of dependent events (to wait for input data to be ready). This is an optional parameter. The default is an empty vector.


mode
   Overrides the global VM mode setting for this function call. See the :ref:`set_mode` function for possible values and their description. This is an optional parameter. The default value is ``mode::not_defined``.


errhandler
   Sets local error handling mode for this function call. See the ref:`create_error_handler` function for arguments and their descriptions. This is an optional parameter. The local error handler is disabled by default.


Output Parameters
-----------------

**Buffer API**


y
   The buffer containing the output vector.


return value (event)
   Computation end event.


**USM API**


y
   Pointer to the output vector.


return value (event)
   Computation end event.


Examples
********

An example of how to use ``erfc`` can be found in the |O-MKL|
installation directory, under:


.. code-block::


   examples/dpcpp/vml/source/verfc.cpp