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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.
*/
'use strict';
// MODULES //
var gammaln = require( '@stdlib/math/base/special/gammaln' );
var ln = require( '@stdlib/math/base/special/ln' );
var LN_SQRT_TWO_PI = require( '@stdlib/constants/float64/ln-sqrt-two-pi' );
var SMALLEST_FLOAT32 = require( '@stdlib/constants/float32/smallest-normal' );
var MAX_FLOAT32 = require( '@stdlib/constants/float32/max' );
var chepolsum = require( './chepolsum.js' );
var polyvalC = require( './polyval_c.js' );
var polyvalD = require( './polyval_d.js' );
// VARIABLES //
var C6 = 0.30865217988013567769;
// MAIN //
/**
* Computes the Stirling series corresponding to asymptotic series for the logarithm of the gamma function.
*
* ```tex
* \frac{1}{12x}-\frac{1}{360x^3}\ldots; x \ge 3
* ```
*
* @private
* @param {number} x - input value
* @returns {number} function value
*/
function stirling( x ) {
var z;
if ( x < SMALLEST_FLOAT32 ) {
return MAX_FLOAT32;
}
if ( x < 1.0 ) {
return gammaln( x+1.0 ) - ( (x+0.5) * ln(x) ) + x - LN_SQRT_TWO_PI;
}
if ( x < 2.0 ) {
return gammaln( x ) - ( (x-0.5) * ln(x) ) + x - LN_SQRT_TWO_PI;
}
if ( x < 3.0 ) {
return gammaln( x-1.0 ) - ( (x-0.5) * ln(x) ) + x - LN_SQRT_TWO_PI + ln( x-1.0 ); // eslint-disable-line max-len
}
if ( x < 12.0 ) {
z = ( 18.0/( x*x ) ) - 1.0;
return chepolsum( 17, z ) / ( 12.0*x );
}
z = 1.0 / ( x * x );
if ( x < 1000.0 ) {
return polyvalC( z ) / ( C6+z ) / x;
}
return polyvalD( z ) / x;
}
// EXPORTS //
module.exports = stirling;
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