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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 following copyright, license, and long comment were part of the original implementation available as part of [FreeBSD]{@link https://svnweb.freebsd.org/base/release/12.2.0/lib/msun/src/k_cos.c}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ```
*/
'use strict';
// MODULES //
var polyval13 = require( './polyval_c13.js' );
var polyval46 = require( './polyval_c46.js' );
// MAIN //
/**
* Computes the cosine on \\( \[-\pi/4, \pi/4] \\), where \\( \pi/4 \approx 0.785398164 \\).
*
* ## Method
*
* - Since \\( \cos(-x) = \cos(x) \\), we need only to consider positive \\(x\\).
*
* - If \\( x < 2^{-27} \\), return \\(1\\) which is inexact if \\( x \ne 0 \\).
*
* - \\( cos(x) \\) is approximated by a polynomial of degree \\(14\\) on \\( \[0,\pi/4] \\).
*
* ```tex
* \cos(x) \approx 1 - \frac{x \cdot x}{2} + C_1 \cdot x^4 + \ldots + C_6 \cdot x^{14}
* ```
*
* where the Remez error is
*
* ```tex
* \left| \cos(x) - \left( 1 - \frac{x^2}{2} + C_1x^4 + C_2x^6 + C_3x^8 + C_4x^{10} + C_5x^{12} + C_6x^{15} \right) \right| \le 2^{-58}
* ```
*
* - Let \\( C_1x^4 + C_2x^6 + C_3x^8 + C_4x^{10} + C_5x^{12} + C_6x^{14} \\), then
*
* ```tex
* \cos(x) \approx 1 - \frac{x \cdot x}{2} + r
* ```
*
* Since
*
* ```tex
* \cos(x+y) \approx \cos(x) - \sin(x) \cdot y \approx \cos(x) - x \cdot y
* ```
*
* a correction term is necessary in \\( \cos(x) \\). Hence,
*
* ```tex
* \cos(x+y) = 1 - \left( \frac{x \cdot x}{2} - (r - x \cdot y) \right)
* ```
*
* For better accuracy, rearrange to
*
* ```tex
* \cos(x+y) \approx w + \left( t + ( r - x \cdot y ) \right)
* ```
*
* where \\( w = 1 - \frac{x \cdot x}{2} \\) and \\( t \\) is a tiny correction term (\\( 1 - \frac{x \cdot x}{2} = w + t \\) exactly in infinite precision). The exactness of \\(w + t\\) in infinite precision depends on \\(w\\) and \\(t\\) having the same precision as \\(x\\).
*
* @param {number} x - input value (in radians, assumed to be bounded by ~pi/4 in magnitude)
* @param {number} y - tail of `x`
* @returns {number} cosine
*
* @example
* var v = kernelCos( 0.0, 0.0 );
* // returns ~1.0
*
* @example
* var v = kernelCos( 3.141592653589793/6.0, 0.0 );
* // returns ~0.866
*
* @example
* var v = kernelCos( 0.785, -1.144e-17 );
* // returns ~0.707
*
* @example
* var v = kernelCos( NaN, 0.0 );
* // returns NaN
*/
function kernelCos( x, y ) {
var hz;
var r;
var w;
var z;
z = x * x;
w = z * z;
r = z * polyval13( z );
r += w * w * polyval46( z );
hz = 0.5 * z;
w = 1.0 - hz;
return w + ( ((1.0-w) - hz) + ((z*r) - (x*y)) );
}
// EXPORTS //
module.exports = kernelCos;
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