time-to-botec

Benchmark sampling in different programming languages
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fresnel.js (4349B)

      1 /**
      2 * @license Apache-2.0
      3 *
      4 * Copyright (c) 2018 The Stdlib Authors.
      5 *
      6 * Licensed under the Apache License, Version 2.0 (the "License");
      7 * you may not use this file except in compliance with the License.
      8 * You may obtain a copy of the License at
      9 *
     10 *    http://www.apache.org/licenses/LICENSE-2.0
     11 *
     12 * Unless required by applicable law or agreed to in writing, software
     13 * distributed under the License is distributed on an "AS IS" BASIS,
     14 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
     15 * See the License for the specific language governing permissions and
     16 * limitations under the License.
     17 *
     18 *
     19 * ## Notice
     20 *
     21 * 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.
     22 *
     23 * ```text
     24 * Copyright 1984, 1987, 1989, 2000 by Stephen L. Moshier
     25 *
     26 * 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.
     27 *
     28 * Stephen L. Moshier
     29 * moshier@na-net.ornl.gov
     30 * ```
     31 */
     32 
     33 'use strict';
     34 
     35 // MODULES //
     36 
     37 var sincos = require( './../../../../base/special/sincos' );
     38 var abs = require( './../../../../base/special/abs' );
     39 var HALF_PI = require( '@stdlib/constants/float64/half-pi' );
     40 var PI = require( '@stdlib/constants/float64/pi' );
     41 var polyS = require( './rational_psqs.js' );
     42 var polyC = require( './rational_pcqc.js' );
     43 var polyF = require( './rational_pfqf.js' );
     44 var polyG = require( './rational_pgqg.js' );
     45 
     46 
     47 // VARIABLES //
     48 
     49 // Array for storing sincos evaluation:
     50 var sc = [ 0.0, 0.0 ]; // WARNING: not thread safe
     51 
     52 
     53 // MAIN //
     54 
     55 /**
     56 * Computes the Fresnel integrals S(x) and C(x).
     57 *
     58 * ## Method
     59 *
     60 * Evaluates the Fresnel integrals
     61 *
     62 * ```tex
     63 * \begin{align*}
     64 * \operatorname{S}(x) &= \int_0^x \sin\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t, \\
     65 * \operatorname{C}(x) &= \int_0^x \cos\left(\frac{\pi}{2} t^2\right)\,\mathrm{d}t.
     66 * \end{align*}
     67 * ```
     68 *
     69 * The integrals are evaluated by a power series for \\( x < 1 \\). For \\( x >= 1 \\) auxiliary functions \\( f(x) \\) and \\( g(x) \\) are employed such that
     70 *
     71 * ```tex
     72 * \begin{align*}
     73 * \operatorname{C}(x) &= \frac{1}{2} + f(x) \sin\left( \frac{\pi}{2} x^2 \right) - g(x) \cos\left( \frac{\pi}{2} x^2 \right), \\
     74 * \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).
     75 * \end{align*}
     76 * ```
     77 *
     78 * ## Notes
     79 *
     80 * -   Relative error on test interval \\( \[0,10\] \\):
     81 *
     82 *     | arithmetic | function | # trials | peak    | rms     |
     83 *     |:----------:|:--------:|:--------:|:-------:|:-------:|
     84 *     | IEEE       | S(x)     | 10000    | 2.0e-15 | 3.2e-16 |
     85 *     | IEEE       | C(x)     | 10000    | 1.8e-15 | 3.3e-16 |
     86 *
     87 * @private
     88 * @param {(Array|TypedArray|Object)} out - destination array
     89 * @param {number} x - input value
     90 * @returns {(Array|TypedArray|Object)} S(x) and C(x)
     91 *
     92 * @example
     93 * var v = fresnel( [ 0.0, 0.0 ], 0.0 );
     94 * // returns [ 0.0, 0.0 ]
     95 *
     96 * @example
     97 * var v = fresnel( [ 0.0, 0.0 ], 1.0 );
     98 * // returns [ ~0.438, ~0.780 ]
     99 *
    100 * @example
    101 * var v = fresnel( [ 0.0, 0.0 ], Infinity );
    102 * // returns [ ~0.5, ~0.5 ]
    103 *
    104 * @example
    105 * var v = fresnel( [ 0.0, 0.0 ], -Infinity );
    106 * // returns [ ~-0.5, ~-0.5 ]
    107 *
    108 * @example
    109 * var v = fresnel( [ 0.0, 0.0 ], NaN );
    110 * // returns [ NaN, NaN ]
    111 */
    112 function fresnel( out, x ) {
    113 	var x2;
    114 	var xa;
    115 	var f;
    116 	var g;
    117 	var t;
    118 	var u;
    119 
    120 	xa = abs( x );
    121 	x2 = xa * xa;
    122 	if ( x2 < 2.5625 ) {
    123 		t = x2 * x2;
    124 		out[ 0 ] = xa * x2 * polyS( t );
    125 		out[ 1 ] = xa * polyC( t );
    126 	} else if ( xa > 36974.0 ) {
    127 		out[ 1 ] = 0.5;
    128 		out[ 0 ] = 0.5;
    129 	} else {
    130 		// Asymptotic power series auxiliary functions for large arguments...
    131 		x2 = xa * xa;
    132 		t = PI * x2;
    133 		u = 1.0 / (t * t);
    134 		t = 1.0 / t;
    135 		f = 1.0 - ( u * polyF( u ) );
    136 		g = t * polyG( u );
    137 		t = HALF_PI * x2;
    138 		sincos( sc, t );
    139 		t = PI * xa;
    140 		out[ 1 ] = 0.5 + ( ( (f*sc[0]) - (g*sc[1]) ) / t );
    141 		out[ 0 ] = 0.5 - ( ( (f*sc[1]) + (g*sc[0]) ) / t );
    142 	}
    143 	if ( x < 0.0 ) {
    144 		out[ 1 ] = -out[ 1 ];
    145 		out[ 0 ] = -out[ 0 ];
    146 	}
    147 	return out;
    148 }
    149 
    150 
    151 // EXPORTS //
    152 
    153 module.exports = fresnel;