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* @license Apache-2.0
*
* Copyright (c) 2022 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var asin = require( '@stdlib/math/base/special/asin' );
var atan = require( '@stdlib/math/base/special/atan' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
// MAIN //
/**
* Computes the arccosecant of a number.
*
* ## Method
*
* ```text
* \operatorname{acsc}(x) = \operatorname{asin}(\frac{1}{x})
* ```
*
* @param {number} x - input value
* @returns {number} arccosecant (in radians)
*
* @example
* var v = acsc( 1.0 );
* // returns ~1.57
*
* @example
* var v = acsc( 3.141592653589793 );
* // returns ~0.32
*
* @example
* var v = acsc( -3.141592653589793 );
* // returns ~-0.32
*
* @example
* var v = acsc( NaN );
* // returns NaN
*/
function acsc( x ) {
// Near `|x| = 1`, `asin( 1/x )` amplifies the rounding of `1/x` (the slope of `asin` is unbounded at one). Use `acsc(x) = atan( 1/sqrt(x^2-1) ) = atan( 1/sqrt((x-1)*(x+1)) )` instead, where `x-1` is exact for `1 <= x <= 2`:
if ( x >= 1.0 && x <= 2.0 ) {
return atan( 1.0 / sqrt( ( x - 1.0 ) * ( x + 1.0 ) ) );
}
if ( x <= -1.0 && x >= -2.0 ) {
return -atan( 1.0 / sqrt( ( -x - 1.0 ) * ( 1.0 - x ) ) );
}
return asin( 1.0 / x );
}
// EXPORTS //
module.exports = acsc;
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