@stdlib/math-base-special-gamma1pm1/lib/lgamma_small_imp.js

Compute gamma(x+1) - 1.

/**
* @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( '@stdlib/math-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;