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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, long comment, copyright, license, and constants are from [Cephes]{@link http://www.netlib.org/cephes}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
*
* Some software in this archive may be from the book _Methods and Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster International, 1989) or from the Cephes Mathematical Library, a commercial product. In either event, it is copyrighted by the author. What you see here may be used freely but it comes with no support or guarantee.
*
* Stephen L. Moshier
* moshier@na-net.ornl.gov
* ```
*/
'use strict';
// MODULES //
var isnan = require( '@stdlib/math/base/assert/is-nan' );
var isInteger = require( '@stdlib/math/base/assert/is-integer' );
var isNegativeZero = require( '@stdlib/math/base/assert/is-negative-zero' );
var abs = require( '@stdlib/math/base/special/abs' );
var floor = require( '@stdlib/math/base/special/floor' );
var sin = require( '@stdlib/math/base/special/sin' );
var PINF = require( '@stdlib/constants/float64/pinf' );
var NINF = require( '@stdlib/constants/float64/ninf' );
var PI = require( '@stdlib/constants/float64/pi' );
var stirlingApprox = require( './stirling_approximation.js' );
var smallApprox = require( './small_approximation.js' );
var rateval = require( './rational_pq.js' );
// MAIN //
/**
* Evaluates the gamma function.
*
* ## Method
*
* 1. Arguments \\(|x| \leq 34\\) are reduced by recurrence and the function approximated by a rational function of degree \\(6/7\\) in the interval \\((2,3)\\).
* 2. Large negative arguments are made positive using a reflection formula.
* 3. Large arguments are handled by Stirling's formula.
*
* ## Notes
*
* - Relative error:
*
* | arithmetic | domain | # trials | peak | rms |
* |:----------:|:---------:|:--------:|:-------:|:-------:|
* | DEC | -34,34 | 10000 | 1.3e-16 | 2.5e-17 |
* | IEEE | -170,-33 | 20000 | 2.3e-15 | 3.3e-16 |
* | IEEE | -33, 33 | 20000 | 9.4e-16 | 2.2e-16 |
* | IEEE | 33, 171.6 | 20000 | 2.3e-15 | 3.2e-16 |
*
* - Error for arguments outside the test range will be larger owing to error amplification by the exponential function.
*
* @param {number} x - input value
* @returns {number} function value
*
* @example
* var v = gamma( 4.0 );
* // returns 6.0
*
* @example
* var v = gamma( -1.5 );
* // returns ~2.363
*
* @example
* var v = gamma( -0.5 );
* // returns ~-3.545
*
* @example
* var v = gamma( 0.5 );
* // returns ~1.772
*
* @example
* var v = gamma( 0.0 );
* // returns Infinity
*
* @example
* var v = gamma( -0.0 );
* // returns -Infinity
*
* @example
* var v = gamma( NaN );
* // returns NaN
*/
function gamma( x ) {
var sign;
var q;
var p;
var z;
if (
(isInteger( x ) && x < 0) ||
x === NINF ||
isnan( x )
) {
return NaN;
}
if ( x === 0.0 ) {
if ( isNegativeZero( x ) ) {
return NINF;
}
return PINF;
}
if ( x > 171.61447887182298 ) {
return PINF;
}
if ( x < -170.5674972726612 ) {
return 0.0;
}
q = abs( x );
if ( q > 33.0 ) {
if ( x >= 0.0 ) {
return stirlingApprox( x );
}
p = floor( q );
// Check whether `x` is even...
if ( (p&1) === 0 ) {
sign = -1.0;
} else {
sign = 1.0;
}
z = q - p;
if ( z > 0.5 ) {
p += 1.0;
z = q - p;
}
z = q * sin( PI * z );
return sign * PI / ( abs(z)*stirlingApprox(q) );
}
// Reduce `x`...
z = 1.0;
while ( x >= 3.0 ) {
x -= 1.0;
z *= x;
}
while ( x < 0.0 ) {
if ( x > -1.0e-9 ) {
return smallApprox( x, z );
}
z /= x;
x += 1.0;
}
while ( x < 2.0 ) {
if ( x < 1.0e-9 ) {
return smallApprox( x, z );
}
z /= x;
x += 1.0;
}
if ( x === 2.0 ) {
return z;
}
x -= 2.0;
return z * rateval( x );
}
// EXPORTS //
module.exports = gamma;
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