@stdlib/math-base-special-asin/lib/main.js

Compute the arcsine of a double-precision floating-point number.

/**
* @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( '@stdlib/math-base-assert-is-nan' );
var sqrt = require( '@stdlib/math-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 double-precision floating-point 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;