erfc

Computes the complementary error function value of vector elements.

Syntax

Buffer API:

namespace oneapi::mkl::vm {

sycl::event erfc(
        sycl::queue& exec_queue,
        std::int64_t n,
        sycl::buffer<T,1>& a,
        sycl::buffer<T,1>& y,
        oneapi::mkl::vm::mode mode = oneapi::mkl::vm::mode::not_defined,
        oneapi::mkl::vm::error_handler<T> errhandler = {});

} // namespace oneapi::mkl::vm

USM API:

namespace oneapi::mkl::vm {

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

} // namespace oneapi::mkl::vm

erfc supports the following precisions.

T

float

double

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:


\[\operatorname{erfc}(x) = \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} \operatorname{d \!} t\]

Useful relations:

\[\operatorname{erfc}(x) = 1 - \operatorname{erf}(x)\]
\[\Phi(x) = \frac{1}{2} \left( 1 + \operatorname{erf}\left(x / \sqrt{2}\right) \right)\]

where

\[\Phi(x) = \frac{1}{\sqrt{2\pi}} \int_0^x \exp(-t^2/2) \operatorname{d \!} t\]

is the cumulative normal distribution function.

\[\Phi^{-1}(x) = \sqrt{2} \operatorname{erf}^{-1}(2x - 1)\]

where \(\Phi^{-1}(x)\) and \(\operatorname{erf}^{-1}(x)\) are the inverses to \(\Phi(x)\) and \(\operatorname{erf}(x)\), respectively.

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

erfc Family Functions Relationship |

image0

Useful relations for these functions:

\[\operatorname{erf}(x) + \operatorname{erfc}(x) = 1\]

\[\begin{split}\operatorname{cdfnorm}(x) &= \frac{1}{2} \left(1 + \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right) \right) \\ &= 1 - \frac{1}{2} \operatorname{erfc}\left(\frac{x}{\sqrt{2}}\right)\end{split}\]

Argument

Result

Status code

a > underflow

+0

oneapi::mkl::vm::status::underflow

+∞

+0

-∞

+2

QNAN

QNAN

SNAN

QNAN

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 a containing input vector of size n.

mode

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

errhandler

Sets local error handling mode for this function call. See the 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 a to the input vector of size n.

depends

Vector of dependent events (to wait for input data to be ready).

mode

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

errhandler

Sets local error handling mode for this function call. See the 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 y containing the output vector of size n.

USM API:

y

Pointer y to the output vector of size n.

return value (event)

Event, signifying availability of computed output and status code(s).

Exceptions

For list of generated exceptions please refer to  Exceptions

Parent topic: VM Mathematical Functions

erf erfcinv