time-to-botec

Benchmark sampling in different programming languages
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temme1.js (4147B)

      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 and copyright notice are from the [Boost library]{@link http://www.boost.org/doc/libs/1_64_0/boost/math/special_functions/detail/ibeta_inverse.hpp}. The implementation has been modified for JavaScript.
     22 *
     23 * ```text
     24 * Copyright John Maddock 2006.
     25 * Copyright Paul A. Bristow 2007.
     26 *
     27 * Use, modification and distribution are subject to the
     28 * Boost Software License, Version 1.0. (See accompanying file
     29 * LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
     30 * ```
     31 */
     32 
     33 'use strict';
     34 
     35 // MODULES //
     36 
     37 var evalpoly = require( './../../../../base/tools/evalpoly' );
     38 var erfcinv = require( './../../../../base/special/erfcinv' );
     39 var sqrt = require( './../../../../base/special/sqrt' );
     40 var exp = require( './../../../../base/special/exp' );
     41 var SQRT2 = require( '@stdlib/constants/float64/sqrt-two' );
     42 
     43 
     44 // VARIABLES //
     45 
     46 // Workspaces for the polynomial coefficients:
     47 var workspace = [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ];
     48 var terms = [ 0.0, 0.0, 0.0, 0.0 ];
     49 
     50 
     51 // MAIN //
     52 
     53 /**
     54 * Carries out the first method by Temme (described in section 2).
     55 *
     56 * ## References
     57 *
     58 * -   Temme, N. M. 1992. "Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function." _Journal of Computational and Applied Mathematics_ 41 (1–2): 1638–63. doi:[10.1016/0377-0427(92)90244-R](https://doi.org/10.1016/0377-0427(92)90244-R).
     59 *
     60 * @private
     61 * @param {PositiveNumber} a - function parameter
     62 * @param {PositiveNumber} b - function parameter
     63 * @param {Probability} z - function parameter
     64 * @returns {number} function value
     65 */
     66 function temme1( a, b, z ) {
     67 	var eta0;
     68 	var eta2;
     69 	var eta;
     70 	var B2;
     71 	var B3;
     72 	var B;
     73 	var c;
     74 
     75 	// Get the first approximation for eta from the inverse error function (Eq: 2.9 and 2.10):
     76 	eta0 = erfcinv( 2.0 * z );
     77 	eta0 /= -sqrt( a / 2.0 );
     78 
     79 	terms[ 0 ] = eta0;
     80 
     81 	// Calculate powers:
     82 	B = b - a;
     83 	B2 = B * B;
     84 	B3 = B2 * B;
     85 
     86 	// Calculate correction terms:
     87 
     88 	// See eq following 2.15:
     89 	workspace[ 0 ] = -B * SQRT2 / 2;
     90 	workspace[ 1 ] = ( 1 - (2.0*B) ) / 8.0;
     91 	workspace[ 2 ] = -(B * SQRT2 / 48.0);
     92 	workspace[ 3 ] = -1.0 / 192.0;
     93 	workspace[ 4 ] = -B * SQRT2 / 3840.0;
     94 	workspace[ 5 ] = 0.0;
     95 	workspace[ 6 ] = 0.0;
     96 	terms[ 1 ] = evalpoly( workspace, eta0 );
     97 
     98 	// Eq Following 2.17:
     99 	workspace[ 0 ] = B * SQRT2 * ( (3.0*B) - 2.0) / 12.0;
    100 	workspace[ 1 ] = ( (20.0*B2) - (12.0*B) + 1.0 ) / 128.0;
    101 	workspace[ 2 ] = B * SQRT2 * ( (20.0*B) - 1.0) / 960.0;
    102 	workspace[ 3 ] = ( (16.0*B2) + (30.0*B) - 15.0) / 4608.0;
    103 	workspace[ 4 ] = B * SQRT2 * ( (21.0*B) + 32) / 53760.0;
    104 	workspace[ 5 ] = (-(32.0*B2) + 63.0) / 368640.0;
    105 	workspace[ 6 ] = -B * SQRT2 * ( (120.0*B) + 17.0) / 25804480.0;
    106 	terms[ 2 ] = evalpoly( workspace, eta0 );
    107 
    108 	// Eq Following 2.17:
    109 	workspace[ 0 ] = B * SQRT2 * ( (-75*B2) + (80.0*B) - 16.0) / 480.0;
    110 	workspace[ 1 ] = ( (-1080.0*B3) + (868.0*B2) - (90.0*B) - 45.0) / 9216.0;
    111 	workspace[ 2 ] = B * SQRT2 * ( (-1190.0*B2) + (84.0*B) + 373.0) / 53760.0;
    112 	workspace[ 3 ] = ( (-2240.0*B3)-(2508.0*B2)+(2100.0*B)-165.0 ) / 368640.0;
    113 	workspace[ 4 ] = 0.0;
    114 	workspace[ 5 ] = 0.0;
    115 	workspace[ 6 ] = 0.0;
    116 	terms[ 3 ] = evalpoly( workspace, eta0 );
    117 
    118 	// Bring them together to get a final estimate for eta:
    119 	eta = evalpoly( terms, 1.0/a );
    120 
    121 	// Now we need to convert eta to the return value `x`, by solving the appropriate quadratic equation:
    122 	eta2 = eta * eta;
    123 	c = -exp( -eta2 / 2.0 );
    124 	if ( eta2 === 0.0 ) {
    125 		return 0.5;
    126 	}
    127 	return ( 1.0 + ( eta * sqrt( ( 1.0+c ) / eta2 ) ) ) / 2.0;
    128 }
    129 
    130 
    131 // EXPORTS //
    132 
    133 module.exports = temme1;