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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 acosh = require( '@stdlib/math/base/special/acosh' );
var log1p = require( '@stdlib/math/base/special/log1p' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
// MAIN //
/**
* Computes the hyperbolic arcsecant of a number.
*
* ## Method
*
* ```tex
* \operatorname{asech}(x) = \operatorname{acosh}(\frac{1}{x})
* ```
*
* @param {number} x - input value
* @returns {number} hyperbolic arcsecant
*
* @example
* var v = asech( 1.0 );
* // returns 0.0
*
* @example
* var v = asech( 0.5 );
* // returns ~1.317
*
* @example
* var v = asech( NaN );
* // returns NaN
*/
function asech( x ) {
var d;
if ( x >= 0.5 && x <= 1.0 ) {
// `acosh( 1/x )` loses the low bits of `1/x - 1` to the rounding of `1/x`, which near one is most of the result, so there use `asech(x) = ln( (1+sqrt(1-x^2))/x ) = log1p( ((1-x) + sqrt((1-x)*(1+x)))/x )`, where `1-x` is exact:
d = 1.0 - x;
return log1p( ( d + sqrt( d * ( 1.0 + x ) ) ) / x );
}
return acosh( 1.0 / x );
}
// EXPORTS //
module.exports = asech;
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