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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 and copyright notice are from the [Boost library]{@link http://www.boost.org/doc/libs/1_37_0/boost/math/special_functions/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
* (C) Copyright Paul A. Bristow 2007.
*
* Use, modification and distribution are subject to the
* Boost Software License, Version 1.0. (See accompanying file
* LICENSE or copy at http://www.boost.org/LICENSE_1_0.txt)
* ```
*/
'use strict';
// MODULES //
var evalpoly = require( '@stdlib/math/base/tools/evalpoly' );
var log1pmx = require( '@stdlib/math/base/special/log1pmx' );
var erfc = require( '@stdlib/math/base/special/erfc' );
var sqrt = require( '@stdlib/math/base/special/sqrt' );
var exp = require( '@stdlib/math/base/special/exp' );
var TWO_PI = require( '@stdlib/constants/float64/two-pi' );
var polyvalC0 = require( './polyval_c0.js' );
var polyvalC1 = require( './polyval_c1.js' );
var polyvalC2 = require( './polyval_c2.js' );
var polyvalC3 = require( './polyval_c3.js' );
var polyvalC4 = require( './polyval_c4.js' );
var polyvalC5 = require( './polyval_c5.js' );
var polyvalC6 = require( './polyval_c6.js' );
var polyvalC7 = require( './polyval_c7.js' );
var polyvalC8 = require( './polyval_c8.js' );
// VARIABLES //
// Pre-allocate workspace array:
var workspace = [ 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 ]; // WARNING: not thread safe
// MAIN //
/**
* Asymptotic expansions of the incomplete gamma functions when `a` is large and `x ~ a` (IEEE double precision or 10^-17).
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @returns {number} value of asymptotic expansion
*/
function igammaTemmeLarge( a, x ) {
var result;
var sigma;
var phi;
var y;
var z;
sigma = ( x-a ) / a;
phi = -log1pmx( sigma );
y = a * phi;
z = sqrt( 2.0 * phi );
if ( x < a ) {
z = -z;
}
workspace[ 0 ] = polyvalC0( z );
workspace[ 1 ] = polyvalC1( z );
workspace[ 2 ] = polyvalC2( z );
workspace[ 3 ] = polyvalC3( z );
workspace[ 4 ] = polyvalC4( z );
workspace[ 5 ] = polyvalC5( z );
workspace[ 6 ] = polyvalC6( z );
workspace[ 7 ] = polyvalC7( z );
workspace[ 8 ] = polyvalC8( z );
workspace[ 9 ] = -0.00059676129019274625;
result = evalpoly( workspace, 1.0/a );
result *= exp( -y ) / sqrt( TWO_PI * a );
if ( x < a ) {
result = -result;
}
result += erfc( sqrt(y) ) / 2.0;
return result;
}
// EXPORTS //
module.exports = igammaTemmeLarge;
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