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# kernelTan

> Compute the [tangent][tangent] of a double-precision floating-point number on `[-π/4, π/4]`.

<section class="usage">

## Usage

```javascript
var kernelTan = require( '@stdlib/math/base/special/kernel-tan' );
```

#### kernelTan( x, y, k )

Computes the [tangent][tangent] of a double-precision floating-point number on `[-π/4, π/4]`.

```javascript
var out = kernelTan( 3.141592653589793/4.0, 0.0, 1 );
// returns ~1.0

out = kernelTan( 3.141592653589793/6.0, 0.0, 1 );
// returns ~0.577

out = kernelTan( 0.664, 5.288e-17, 1 );
// returns ~0.783
```

If `k = 1`, the function returns `tan(x+y)`. To return the negative inverse `-1/tan(x+y)`, set `k = -1`. 

```javascript
var out = kernelTan( 3.141592653589793/4.0, 0.0, -1 );
// returns ~-1.0
```

If either `x` or `y` is `NaN`, the function returns `NaN`.

```javascript
var out = kernelTan( NaN, 0.0, 1 );
// returns NaN

out = kernelTan( 3.0, NaN, 1 );
// returns NaN

out = kernelTan( NaN, NaN, 1 );
// returns NaN
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   For increased accuracy, the number for which the [tangent][tangent] should be evaluated can be supplied as a [double-double number][double-double-arithmetic] (i.e., a non-evaluated sum of two [double-precision floating-point numbers][ieee754] `x` and `y`).

-   As components of a [double-double number][double-double-arithmetic], the two [double-precision floating-point numbers][ieee754] `x` and `y` must satisfy 

    <!-- <equation class="equation" label="eq:double_double_inequality" align="center" raw="|y| \leq \frac{1}{2} \operatorname{ulp}(x)" alt="Inequality for the two components of a double-double number."> -->

    <div class="equation" align="center" data-raw-text="|y| \leq \frac{1}{2} \operatorname{ulp}(x)" data-equation="eq:double_double_inequality">
        <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@bb29798906e119fcb2af99e94b60407a270c9b32/lib/node_modules/@stdlib/math/base/special/kernel-tan/docs/img/equation_double_double_inequality.svg" alt="Inequality for the two components of a double-double number.">
        <br>
    </div>

    <!-- </equation> -->

    where `ulp` stands for [units in the last place][ulp].

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var linspace = require( '@stdlib/array/base/linspace' );
var binomial = require( '@stdlib/random/base/binomial' ).factory;
var PI = require( '@stdlib/constants/float64/pi' );
var kernelTan = require( '@stdlib/math/base/special/kernel-tan' );

var x = linspace( -PI/4.0, PI/4.0, 100 );
var rbinom = binomial( 1, 0.5 );

var descr;
var i;
var k;

for ( i = 0; i < x.length; i++ ) {
    k = rbinom();
    descr = ( k === 1 ) ? 'tan(%d) = %d' : '-1/tan(%d) = %d';
    console.log( descr, x[ i ], kernelTan( x[ i ], 0.0, k ) );
}
```

</section>

<!-- /.examples -->

<!-- C interface documentation. -->

* * *

<section class="c">

## C APIs

<!-- Section to include introductory text. Make sure to keep an empty line after the intro `section` element and another before the `/section` close. -->

<section class="intro">

</section>

<!-- /.intro -->

<!-- C usage documentation. -->

<section class="usage">

### Usage

```c
#include "stdlib/math/base/special/kernel_tan.h"
```

#### stdlib_base_kernel_tan( x, y, k)

Computes the [tangent][tangent] of a double-precision floating-point number on `[-π/4, π/4]`.

```c
double out = stdlib_base_kernel_tan( 3.141592653589793/4.0, 0.0, 1 );
// returns ~1.0

out = stdlib_base_kernel_tan( 3.141592653589793/6.0, 0.0, 1 );
// returns ~0.577
```

The function accepts the following arguments:

-   **x**: `[in] double` input value (in radians, assumed to be bounded by `~pi/4` in magnitude).
-   **y**: `[in] double` tail of `x`.
-   **k**: `[in] int32_t` indicates whether `tan(x+y)` (if `k = 1`) or `-1/tan(x+y)` (if `k = -1`) is returned.

```c
double stdlib_base_kernel_tan( const double x, const double y, const int32_t k );
```

</section>

<!-- /.usage -->

<!-- C API usage notes. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="notes">

### Notes

-   For increased accuracy, the number for which the [tangent][tangent] should be evaluated can be supplied as a [double-double number][double-double-arithmetic] (i.e., a non-evaluated sum of two [double-precision floating-point numbers][ieee754] `x` and `y`).

</section>

<!-- /.notes -->

<!-- C API usage examples. -->

<section class="examples">

### Examples

```c
#include "stdlib/math/base/special/kernel_tan.h"
#include <stdio.h>

int main( void ) {
    const double x[] = { -0.7853981633974483, -0.6108652381980153, -0.4363323129985824, -0.26179938779914946, -0.08726646259971649, 0.08726646259971649, 0.26179938779914935, 0.43633231299858233, 0.6108652381980153, 0.7853981633974483 };

    double out;
    int i;
    for ( i = 0; i < 10; i++ ) {
        out = stdlib_base_kernel_tan( x[ i ], 0.0, 1 );
        printf( "tan(%lf) = %lf\n", x[ i ], out );
    }
}
```

</section>

<!-- /.examples -->

</section>

<!-- /.c -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/math/base/special/kernel-cos`][@stdlib/math/base/special/kernel-cos]</span><span class="delimiter">: </span><span class="description">compute the cosine of a double-precision floating-point number on \[-π/4, π/4].</span>
-   <span class="package-name">[`@stdlib/math/base/special/kernel-sin`][@stdlib/math/base/special/kernel-sin]</span><span class="delimiter">: </span><span class="description">compute the sine of a double-precision floating-point number on \[-π/4, π/4].</span>
-   <span class="package-name">[`@stdlib/math/base/special/tan`][@stdlib/math/base/special/tan]</span><span class="delimiter">: </span><span class="description">evaluate the tangent of a number.</span>

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<section class="links">

[tangent]: https://en.wikipedia.org/wiki/Tangent

[double-double-arithmetic]: https://en.wikipedia.org/wiki/Quadruple-precision_floating-point_format#Double-double_arithmetic

[ieee754]: https://en.wikipedia.org/wiki/IEEE_floating_point

[ulp]: https://en.wikipedia.org/wiki/Unit_in_the_last_place

<!-- <related-links> -->

[@stdlib/math/base/special/kernel-cos]: https://github.com/stdlib-js/math/tree/main/base/special/kernel-cos

[@stdlib/math/base/special/kernel-sin]: https://github.com/stdlib-js/math/tree/main/base/special/kernel-sin

[@stdlib/math/base/special/tan]: https://github.com/stdlib-js/math/tree/main/base/special/tan

<!-- </related-links> -->

</section>

<!-- /.links -->
