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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 acos = require( '@stdlib/math/base/special/acos' );
var atan = require( '@stdlib/math/base/special/atan' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
// MAIN //
/**
* Computes the inverse (arc) secant of a number.
*
* ## Method
*
* ```tex
* \operatorname{asec}(x) = \operatorname{acos}(\frac{1}{x})
* ```
*
* @param {number} x - input value
* @returns {number} inverse (arc) secant
*
* @example
* var v = asec( 1.0 );
* // returns 0.0
*
* @example
* var v = asec( 2.0 );
* // returns ~1.0472
*
* @example
* var v = asec( NaN );
* // returns NaN
*/
function asec( x ) {
// Near `x = 1`, `acos( 1/x )` loses the low bits of `1-1/x` to the rounding of `1/x`, and those bits are most of the (small) result. Use `asec(x) = atan( sqrt(x^2-1) ) = atan( sqrt((x-1)*(x+1)) )` instead, where `x-1` is exact for `1 <= x <= 2`:
if ( x >= 1.0 && x <= 2.0 ) {
return atan( sqrt( ( x - 1.0 ) * ( x + 1.0 ) ) );
}
return acos( 1.0 / x );
}
// EXPORTS //
module.exports = asec;
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