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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_64_0/boost/math/special_functions/detail/lgamma_small.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006-7, 2013-14.
* (C) Copyright Paul A. Bristow 2007, 2013-14.
* (C) Copyright Nikhar Agrawal 2013-14.
* (C) Copyright Christopher Kormanyos 2013-14.
*
* 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 ln = require( './../../../../base/special/ln' );
var EPS = require( '@stdlib/constants/float64/eps' );
var rateval1 = require( './rational_p1q1.js' );
var rateval2 = require( './rational_p2q2.js' );
var rateval3 = require( './rational_p3q3.js' );
// VARIABLES //
var Y1 = 0.158963680267333984375;
var Y2 = 0.52815341949462890625;
var Y3 = 0.452017307281494140625;
// MAIN //
/**
* Evaluates the natural logarithm of the gamma function for small arguments.
*
* ## Method
*
* 1. For \\( z > 2 \\), begin by performing argument reduction until \\( z \\) is in \\(\[2,3)\\). Use the following form:
*
* ```tex
* \operatorname{gammaln}(z) = (z-2)(z+1)(Y + R(z-2))
* ```
*
* where \\( R(z-2) \\) is a rational approximation optimized for low absolute error. As long as the absolute error is small compared to the constant \\( Y \\), then any rounding error in the computation will get wiped out.
*
* 2. If \\( z < 1 \\), use recurrence to shift to \\( z \\) in the interval \\(\[1,2\]\\). Then, use one of two approximations: one for \\( z \\) in \\(\[1,1.5\]\\) and one for \\( z \\) in \\(\[1.5,2\]\\):
*
* - For \(( z \\) in \\(\[1,1.5\]\\), use
*
* ```tex
* \operatorname{gammaln}(z) = (z-1)(z-2)(Y + R(z-1))
* ```
*
* where \\( R(z-1) \\) is a rational approximation optimized for low absolute error. As long as the absolute error is small compared to the constant \\( Y \\), then any rounding error in the computation will get wiped out.
*
* - For \\( z \\) in \\(\[1.5,2\]\\), use
*
* ```tex
* \operatorname{gammaln}(z) = (2-z)(1-z)(Y + R(2-z))
* ```
*
* where \\( R(2-z) \\) is a rational approximation optimized for low absolute error. As long as the absolute error is small compared to the constant \\( Y \\), then any rounding error in the computation will get wiped out.
*
*
* ## Notes
*
* - Relative error:
*
* | function | peak | maximum deviation |
* |:--------:|:------------:|:-----------------:|
* | R(Z-2) | 4.231e-18 | 5.900e-24 |
* | R(Z-1) | 1.230011e-17 | 3.139e-021 |
* | R(2-Z) | 1.797565e-17 | 2.151e-021 |
*
*
* @private
* @param {number} z - input value
* @param {number} zm1 - `z` minus one
* @param {number} zm2 - `z` minus two
* @returns {number} function value
*/
function lgammaSmallImp( z, zm1, zm2 ) {
var prefix;
var result;
var r;
var R;
if ( z < EPS ) {
return -ln( z );
}
if ( zm1 === 0.0 || zm2 === 0.0 ) {
return 0.0;
}
result = 0.0;
if ( z > 2.0 ) {
if ( z >= 3.0 ) {
do {
z -= 1.0;
zm2 -= 1.0;
result += ln(z);
} while ( z >= 3.0 );
zm2 = z - 2.0;
}
r = zm2 * ( z+1.0 );
R = rateval1( zm2 );
result += ( r*Y1 ) + ( r*R );
return result;
}
if ( z < 1.0 ) {
result += -ln(z);
zm2 = zm1;
zm1 = z;
z += 1.0;
}
if ( z <= 1.5 ) {
r = rateval2( zm1 );
prefix = zm1 * zm2;
result += ( prefix*Y2 ) + ( prefix*r );
return result;
}
// Case: 1.5 < z <= 2
r = zm2 * zm1;
R = rateval3( -zm2 );
result += ( r*Y3 ) + ( r*R );
return result;
}
// EXPORTS //
module.exports = lgammaSmallImp;
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