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/**
* @license Apache-2.0
*
* Copyright (c) 2018 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.
*
*
* ## Notice
*
* The original C code, long comment, copyright, license, and constants are from [Cephes]{@link http://www.netlib.org/cephes}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright 1984, 1995, 2000 by Stephen L. Moshier
*
* Some software in this archive may be from the book _Methods and Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster International, 1989) or from the Cephes Mathematical Library, a commercial product. In either event, it is copyrighted by the author. What you see here may be used freely but it comes with no support or guarantee.
*
* Stephen L. Moshier
* moshier@na-net.ornl.gov
* ```
*/
'use strict';
// MODULES //
var isnan = require( './../../../../base/assert/is-nan' );
var sqrt = require( './../../../../base/special/sqrt' );
var PIO4 = require( '@stdlib/constants/float64/fourth-pi' );
var ratevalPQ = require( './rational_pq.js' );
var ratevalRS = require( './rational_rs.js' );
// VARIABLES //
var MOREBITS = 6.123233995736765886130e-17; // pi/2 = PIO2 + MOREBITS
// MAIN //
/**
* Computes the arcsine of a number.
*
* ## Method
*
* - A rational function of the form
*
* ```tex
* x + x^3 \frac{P(x^2)}{Q(x^2)}
* ```
*
* is used for \\(\|x\|\\) in the interval \\(\[0, 0.5\]\\). If \\(\|x\| > 0.5\\), it is transformed by the identity
*
* ```tex
* \operatorname{asin}(x) = \frac{\pi}{2} - 2 \operatorname{asin}( \sqrt{ (1-x)/2 } )
* ```
*
* ## Notes
*
* - Relative error:
*
* | arithmetic | domain | # trials | peak | rms |
* |:-----------|:-------|:---------|:--------|:--------|
* | DEC | -1, 1 | 40000 | 2.6e-17 | 7.1e-18 |
* | IEEE | -1, 1 | 10^6 | 1.9e-16 | 5.4e-17 |
*
* @param {number} x - input value
* @returns {number} arcsine (in radians)
*
* @example
* var v = asin( 0.0 );
* // returns ~0.0
*
* @example
* var v = asin( 3.141592653589793/4.0 );
* // returns ~0.903
*
* @example
* var v = asin( -3.141592653589793/6.0 );
* // returns ~-0.551
*
* @example
* var v = asin( NaN );
* // returns NaN
*/
function asin( x ) {
var sgn;
var zz;
var a;
var p;
var z;
if ( isnan( x ) ) {
return NaN;
}
if ( x > 0.0 ) {
a = x;
} else {
sgn = true;
a = -x;
}
if ( a > 1.0 ) {
return NaN;
}
if ( a > 0.625 ) {
// arcsin(1-x) = pi/2 - sqrt(2x)(1+R(x))
zz = 1.0 - a;
p = zz * ratevalRS( zz );
zz = sqrt( zz + zz );
z = PIO4 - zz;
zz = ( zz*p ) - MOREBITS;
z -= zz;
z += PIO4;
} else {
if ( a < 1.0e-8 ) {
return x;
}
zz = a * a;
z = zz * ratevalPQ( zz );
z = ( a*z ) + a;
}
return ( sgn ) ? -z : z;
}
// EXPORTS //
module.exports = asin;
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