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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, 1987, 1989, 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 sincos = require( '@stdlib/math/base/special/sincos' ).assign;
var abs = require( '@stdlib/math/base/special/abs' );
var fmod = require( '@stdlib/math/base/special/fmod' );
var HALF_PI = require( '@stdlib/constants/float64/half-pi' );
var PI = require( '@stdlib/constants/float64/pi' );
var polyS = require( './rational_psqs.js' );
var polyF = require( './rational_pfqf.js' );
var polyG = require( './rational_pgqg.js' );
// VARIABLES //
// Array for storing sincos evaluation:
var sc = [ 0.0, 0.0 ];
// MAIN //
/**
* Computes the Fresnel integral S(x).
*
* ## Method
*
* Evaluates the Fresnel integral
*
* ```tex
* \operatorname{S}(x) = \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t
* ```
*
* The integral is evaluated by a power series for \\( x < 1 \\). For \\( x >= 1 \\) auxiliary functions \\( f(x) \\) and \\( g(x) \\) are employed such that
*
* ```tex
* \operatorname{S}(x) = \frac{1}{2} - f(x) \cos\left( \frac{\pi}{2} x^2 \right) - g(x) \sin\left( \frac{\pi}{2} x^2 \right).
* ```
*
* ## Notes
*
* - Relative error on test interval \\( \[0,10\] \\):
*
* | arithmetic | function | # trials | peak | rms |
* |:----------:|:--------:|:--------:|:--------:|:-------:|
* | IEEE | S(x) | 10000 | 2.0e-15 | 3.2e-16 |
*
* @param {number} x - input value
* @returns {number} S(x)
*
* @example
* var v = fresnels( 0.0 );
* // returns 0.0
*
* @example
* var v = fresnels( 1.0 );
* // returns ~0.438
*
* @example
* var v = fresnels( Infinity );
* // returns ~0.5
*
* @example
* var v = fresnels( -Infinity );
* // returns ~-0.5
*
* @example
* var v = fresnels( NaN );
* // returns NaN
*/
function fresnels( x ) {
var x2;
var hi;
var lo;
var xa;
var S;
var f;
var g;
var t;
var u;
xa = abs( x );
x2 = xa * xa;
if ( x2 < 2.5625 ) {
t = x2 * x2;
S = xa * x2 * polyS( t );
} else if ( xa > 1.0e16 ) {
// Above `1e16`, the oscillating terms, which are at most `1/(πx)`, are below half an ulp of `0.5`:
S = 0.5;
} else {
// Asymptotic power series auxiliary functions for large arguments...
x2 = xa * xa;
t = PI * x2;
u = 1.0 / (t * t);
t = 1.0 / t;
f = 1.0 - ( u * polyF( u ) );
g = t * polyG( u );
// `sin( π x^2 / 2 )` and `cos( π x^2 / 2 )` have period 4 in `x^2`, and forming `x*x` rounds away bits that the large argument turns into whole radians. Reduce `x^2` modulo 4 exactly instead: split `x = hi + lo` (Veltkamp, with `134217729 = 2^27+1`) so that `hi*hi`, `2*hi*lo` and `lo*lo` are exact, and reduce each term:
t = 134217729.0 * xa;
hi = t - ( t - xa );
lo = xa - hi;
t = fmod( hi*hi, 4.0 ) + fmod( 2.0*hi*lo, 4.0 ) + ( lo*lo );
sincos( HALF_PI * t, sc, 1, 0 );
t = PI * xa;
S = 0.5 - ( ( (f*sc[1]) + (g*sc[0]) ) / t );
}
if ( x < 0.0 ) {
S = -S;
}
return S;
}
// EXPORTS //
module.exports = fresnels;
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