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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]{http://www.boost.org/doc/libs/1_64_0/boost/math/special_functions/beta.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
*
* 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 isnan = require( './../../../../base/assert/is-nan' );
var log1p = require( './../../../../base/special/log1p' );
var sqrt = require( './../../../../base/special/sqrt' );
var abs = require( './../../../../base/special/abs' );
var exp = require( './../../../../base/special/exp' );
var pow = require( './../../../../base/special/pow' );
var E = require( '@stdlib/constants/float64/e' );
var EPSILON = require( '@stdlib/constants/float64/eps' );
var lanczosSumExpGScaled = require( './lanczos_sum_expg_scaled.js' ); // Lanczos approximation scaled by exp(G)
// VARIABLES //
var G = 10.90051099999999983936049829935654997826;
// MAIN //
/**
* Evaluate the beta function.
*
* @param {NonNegativeNumber} a - input value
* @param {NonNegativeNumber} b - input value
* @returns {number} evaluated beta function
*
* @example
* var v = beta( 0.0, 0.5 );
* // returns Infinity
*
* @example
* var v = beta( 1.0, 1.0 );
* // returns 1.0
*
* @example
* var v = beta( -1.0, 2.0 );
* // returns NaN
*
* @example
* var v = beta( 5.0, 0.2 );
* // returns ~3.382
*
* @example
* var v = beta( 4.0, 1.0 );
* // returns 0.25
*
* @example
* var v = beta( NaN, 2.0 );
* // returns NaN
*/
function beta( a, b ) {
var ambh;
var agh;
var bgh;
var cgh;
var res;
var tmp;
var c;
if ( isnan( a ) || isnan( b ) ) {
return NaN;
}
if ( a < 0.0 || b < 0.0 ) {
return NaN;
}
if ( b === 1.0 ) {
return 1.0 / a;
}
if ( a === 1.0 ) {
return 1.0 / b;
}
c = a + b;
if ( c < EPSILON ) {
res = c / a;
res /= b;
return res;
}
// Special cases:
if ( c === a && b < EPSILON ) {
return 1.0 / b;
}
if ( c === b && a < EPSILON ) {
return 1.0 / a;
}
if ( a < b ) {
// Swap `a` and `b`:
tmp = b;
b = a;
a = tmp;
}
// Lanczos calculation:
agh = a + G - 0.5;
bgh = b + G - 0.5;
cgh = c + G - 0.5;
res = lanczosSumExpGScaled( a ) * ( lanczosSumExpGScaled( b )/lanczosSumExpGScaled( c ) ); // eslint-disable-line max-len
ambh = a - 0.5 - b;
if ( ( abs( b*ambh ) < ( cgh*100.0 ) ) && a > 100.0 ) {
// Special case where the base of the power term is close to 1; compute `(1+x)^y` instead:
res *= exp( ambh * log1p( -b/cgh ) );
} else {
res *= pow( agh/cgh, ambh );
}
if ( cgh > 1.0e10 ) {
// This avoids possible overflow, but appears to be marginally less accurate:
res *= pow( (agh/cgh)*(bgh/cgh), b );
} else {
res *= pow( (agh*bgh)/(cgh*cgh), b );
}
res *= sqrt( E/bgh);
return res;
}
// EXPORTS //
module.exports = beta;
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