stirling.js (2096B)
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 'use strict'; 20 21 // MODULES // 22 23 var gammaln = require( './../../../../base/special/gammaln' ); 24 var ln = require( './../../../../base/special/ln' ); 25 var LN_SQRT_TWO_PI = require( '@stdlib/constants/float64/ln-sqrt-two-pi' ); 26 var SMALLEST_FLOAT32 = require( '@stdlib/constants/float32/smallest-normal' ); 27 var MAX_FLOAT32 = require( '@stdlib/constants/float32/max' ); 28 var chepolsum = require( './chepolsum.js' ); 29 var polyvalC = require( './polyval_c.js' ); 30 var polyvalD = require( './polyval_d.js' ); 31 32 33 // VARIABLES // 34 35 var C6 = 0.30865217988013567769; 36 37 38 // MAIN // 39 40 /** 41 * Computes the Stirling series corresponding to asymptotic series for the logarithm of the gamma function. 42 * 43 * ```tex 44 * \frac{1}{12x}-\frac{1}{360x^3}\ldots; x \ge 3 45 * ``` 46 * 47 * @private 48 * @param {number} x - input value 49 * @returns {number} function value 50 */ 51 function stirling( x ) { 52 var z; 53 if ( x < SMALLEST_FLOAT32 ) { 54 return MAX_FLOAT32; 55 } 56 if ( x < 1.0 ) { 57 return gammaln( x+1.0 ) - ( (x+0.5) * ln(x) ) + x - LN_SQRT_TWO_PI; 58 } 59 if ( x < 2.0 ) { 60 return gammaln( x ) - ( (x-0.5) * ln(x) ) + x - LN_SQRT_TWO_PI; 61 } 62 if ( x < 3.0 ) { 63 return gammaln( x-1.0 ) - ( (x-0.5) * ln(x) ) + x - LN_SQRT_TWO_PI + ln( x-1.0 ); // eslint-disable-line max-len 64 } 65 if ( x < 12.0 ) { 66 z = ( 18.0/( x*x ) ) - 1.0; 67 return chepolsum( 17, z ) / ( 12.0*x ); 68 } 69 z = 1.0 / ( x * x ); 70 if ( x < 1000.0 ) { 71 return polyvalC( z ) / ( C6+z ) / x; 72 } 73 return polyvalD( z ) / x; 74 } 75 76 77 // EXPORTS // 78 79 module.exports = stirling;