/*
	*    function evalpoly( x ) {
	*        if ( x === 0.0 ) {
	*            return c[ 0 ];
	*        }
	*        return c[0]+x*(c[1]+x*(c[2]+x*(c[3]+...+x*(c[n-2]+x*c[n-1]))));
	*    }
	*/

/*
* Notes:
*	=> sin(-x) = -sin(x)
*	=> sin(+n) = +0, where `n` is a positive integer
*	=> sin(-n) = -sin(+n) = -0, where `n` is a positive integer
*	=> cos(-x) = cos(x)
*/

/* This is a generated file. Do not edit directly. */

/* eslint-disable default-case */

/* eslint-disable max-statements, max-lines */

/* eslint-disable no-lonely-if */

/* eslint-disable no-unneeded-ternary */

/* falls through */

/**
	* Calculate the next term of the series.
	*
	* @private
	* @returns {Array} series expansion terms
	*/

/**
	* Calculate the next term of the series.
	*
	* @private
	* @returns {number} series expansion term
	*/

/**
	* Calculate the numerator and denominator of the next term of the series.
	*
	* @private
	* @returns {Array} series expansion terms
	*/

/**
	* Evaluates a polynomial.
	*
	* @private
	* @param {number} x - value at which to evaluate a polynomial
	* @returns {number} evaluated polynomial
	*/

/**
	* Evaluates the cumulative distribution function (CDF) for a Normal distribution.
	*
	* @private
	* @param {number} x - input value
	* @returns {Probability} evaluated CDF
	*
	* @example
	* var y = cdf( 2.0 );
	* // returns <number>
	*/

/**
	* Evaluates the cumulative distribution function (CDF) for a Student's t distribution.
	*
	* @private
	* @param {number} x - input value
	* @returns {Probability} evaluated CDF
	*
	* @example
	* var y = cdf( 2.0 );
	* // returns <number>
	*/

/**
	* Evaluates the cumulative distribution function (CDF) of a degenerate distribution.
	*
	* @private
	* @param {number} x - input value
	* @returns {Probability} evaluated cumulative distribution function
	*
	* @example
	* var y = cdf( 10.0 );
	* // returns <number>
	*/

/**
	* Evaluates the probability density function (PDF) for a Student's t distribution.
	*
	* @private
	* @param {number} x - input value
	* @returns {number} evaluated PDF
	*
	* @example
	* var y = pdf( 2.3 );
	* // returns <number>
	*/

/**
	* Evaluates the probability density function (PDF) for a normal distribution.
	*
	* @private
	* @param {number} x - input value
	* @returns {number} evaluated probability density function
	*
	* @example
	* var y = pdf( -3.14 );
	* // returns <number>
	*/

/**
	* Evaluates the probability density function (PDF) of a degenerate distribution.
	*
	* @private
	* @param {number} x - input value
	* @returns {number} evaluated probability density function
	*
	* @example
	* var y = pdf( 10.0 );
	* // returns <number>
	*/

/**
* @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 following copyright, license, and long comment were part of the original implementation available as part of [FDLIBM]{@link http://www.netlib.org/fdlibm/s_expm1.c}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ```
*/

/**
* @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 following copyright, license, and long comment were part of the original implementation available as part of [FDLIBM]{@link http://www.netlib.org/fdlibm/s_log1p.c}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ```
*/

/**
* @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 following copyright, license, and long comment were part of the original implementation available as part of [FreeBSD]{@link https://svnweb.freebsd.org/base/release/9.3.0/lib/msun/src/e_lgamma_r.c}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
*
* Developed at SunPro, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ```
*/

/**
* @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, 1995, 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
* ```
*/

/**
* @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_37_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)
* ```
*/

/**
* @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_37_0/boost/math/special_functions/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
* (C) Copyright Paul A. Bristow 2007.
*
* 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)
* ```
*/

/**
* @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_60_0/boost/math/special_functions/powm1.hpp}. The implementation follows the original, but 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)
* ```
*/

/**
* @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_61_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)
* ```
*/

/**
* @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_61_0/boost/math/special_functions/beta.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* (C) Copyright John Maddock 2006.
* (C) Copyright Paul A. Bristow 2007.
*
* 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)
* ```
*/

/**
* @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_62_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)
* ```
*/

/**
* @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_62_0/boost/math/special_functions/gamma.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) 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)
* ```
*/

/**
* @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/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)
* ```
*/

/**
* @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)
* ```
*/

/**
* @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/gamma.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)
* ```
*/

/**
* @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/gamma.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* Copyright John Maddock 2006-7, 2013-14.
* Copyright Paul A. Bristow 2007, 2013-14.
* Copyright Nikhar Agrawal 2013-14.
* 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)
* ```
*/

/**
* @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/lanczos.hpp}. The implementation has been modified for JavaScript.
*
* ```text
* 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)
* ```
*/

/**
* @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)
* ```
*/

/**
* @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.
*/

/**
* @license Apache-2.0
*
* Copyright (c) 2020 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.
*/

/**
* Alias for `eval` global.
*
* @module @stdlib/utils/eval
*
* @example
* var evil = require( '@stdlib/utils/eval' );
*
* var v = evil( '5*4*3*2*1' );
* // returns 120
*/

/**
* Arbitrary constant `g` to be used in Lanczos approximation functions.
*
* @constant
* @type {number}
* @default 10.900511
* @see [Lanczos Approximation]{@link https://en.wikipedia.org/wiki/Lanczos_approximation}
*/

/**
* Arbitrary constant `g` to be used in Lanczos approximation functions.
*
* @module @stdlib/constants/math/float64-gamma-lanczos-g
* @type {number}
*
* @example
* var FLOAT64_GAMMA_LANCZOS_G = require( '@stdlib/constants/math/float64-gamma-lanczos-g' );
* // returns 10.900511
*/

/**
* Asymptotic expansions of the incomplete gamma functions when `a` is large and `x ~ a` (IEEE double precision or 10^-17).
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @returns {number} value of asymptotic expansion
*/

/**
* Calculate a continued fraction approximation.
*
* @module @stdlib/math/base/tools/continued-fraction
*
* @example
* var continuedFraction = require( '@stdlib/math/base/tools/continued-fraction' );
*
* // Continued fraction for (e-1)^(-1):
* var gen = generator()
* var out = continuedFraction( gen );
* // returns ~0.582
*
* function generator() {
*    var i = 0;
*    return function() {
*        i++;
*        return [ i, i ];
*    };
* }
*/

/**
* Calculate the Lanczos sum for the approximation of the gamma function (scaled by `exp(-g)`, where `g = 10.900511`.
*
* @module @stdlib/math/base/special/gamma-lanczos-sum-expg-scaled
*
* @example
* var gammaLanczosSumExpGScaled = require( '@stdlib/math/base/special/gamma-lanczos-sum-expg-scaled' );
*
* var v = gammaLanczosSumExpGScaled( 4.0 );
* // returns ~0.018
*
* v = gammaLanczosSumExpGScaled( -1.5 );
* // returns ~25.337
*
* v = gammaLanczosSumExpGScaled( -0.5 );
* // returns ~-12.911
*
* v = gammaLanczosSumExpGScaled( 0.5 );
* // returns ~1.772
*
* v = gammaLanczosSumExpGScaled( 0.0 );
* // returns Infinity
*
* v = gammaLanczosSumExpGScaled( NaN );
* // returns NaN
*/

/**
* Calculate the Lanczos sum for the approximation of the gamma function.
*
* @module @stdlib/math/base/special/gamma-lanczos-sum
*
* @example
* var gammaLanczosSum = require( '@stdlib/math/base/special/gamma-lanczos-sum' );
*
* var v = gammaLanczosSum( 4.0 );
* // returns ~950.366
*
* v = gammaLanczosSum( -1.5 );
* // returns ~1373366.245
*
* v = gammaLanczosSum( -0.5 );
* // returns ~-699841.735
*
* v = gammaLanczosSum( 0.5 );
* // returns ~96074.186
*
* v = gammaLanczosSum( 0.0 );
* // returns Infinity
*
* v = gammaLanczosSum( NaN );
* // returns NaN
*/

/**
* Calculate the ratio of two gamma functions.
*
* @module @stdlib/math/base/special/gamma-delta-ratio
*
* @example
* var gammaDeltaRatio = require( '@stdlib/math/base/special/gamma-delta-ratio' );
*
* var y = gammaDeltaRatio( 2.0, 3.0 );
* // returns ~0.042
*
* y = gammaDeltaRatio( 4.0, 0.5 );
* // returns 2.0
*
* y = gammaDeltaRatio( 100.0, 0.0 );
* // returns 1.0
*/

/**
* Calculates normalized Q when a is a half-integer.
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @returns {number} upper gamma fraction
*/

/**
* Calculates normalized Q when a is an integer.
*
* @private
* @param {integer} a - function parameter
* @param {number} x - function parameter
* @returns {number} upper gamma fraction
*/

/**
* Calculates the Lanczos sum approximation.
*
* @name gammaLanczosSum
* @type {Function}
* @param {number} x - input value
* @returns {number} Lanczos sum approximation
*
* @example
* var v = gammaLanczosSum( 4.0 );
* // returns ~950.366
*
* @example
* var v = gammaLanczosSum( -1.5 );
* // returns ~1373366.245
*
* @example
* var v = gammaLanczosSum( -0.5 );
* // returns ~-699841.735
*
* @example
* var v = gammaLanczosSum( 0.5 );
* // returns ~96074.186
*
* @example
* var v = gammaLanczosSum( 0.0 );
* // returns Infinity
*
* @example
* var v = gammaLanczosSum( NaN );
* // returns NaN
*/

/**
* Calculates the Lanczos sum for the approximation of the gamma function (scaled by `exp(-g)`, where `g = 10.900511`.
*
* @name gammaLanczosSumExpGScaled
* @type {Function}
* @param {number} x - input value
* @returns {number} Lanczos sum approximation
*
* @example
* var v = gammaLanczosSumExpGScaled( 4.0 );
* // returns ~0.018
*
* @example
* var v = gammaLanczosSumExpGScaled( -1.5 );
* // returns ~25.337
*
* @example
* var v = gammaLanczosSumExpGScaled( -0.5 );
* // returns ~-12.911
*
* @example
* var v = gammaLanczosSumExpGScaled( 0.5 );
* // returns ~1.772
*
* @example
* var v = gammaLanczosSumExpGScaled( 0.0 );
* // returns Infinity
*
* @example
* var v = gammaLanczosSumExpGScaled( NaN );
* // returns NaN
*/

/**
* Calculates the power term prefix `(z^a)(e^-z)` used in the non-normalized incomplete gammas.
*
* @private
* @param {number} a - function parameter
* @param {number} z - function parameter
* @returns {number} power term prefix
*/

/**
* Calculates the ratio of two gamma functions via Lanczos approximation.
*
* ## Notes
*
* -   When \\( z < \epsilon \\), we get spurious numeric overflow unless we're very careful. This can occur either inside `lanczosSum(z)` or in the final combination of terms. To avoid this, split the product up into 2 (or 3) parts:
*
*     ```tex
*     \begin{align*}
*     G(z) / G(L) &= 1 / (z \cdot G(L)) ; z < \eps, L = z + \delta = \delta \\
*     z * G(L) &= z * G(lim) \cdot (G(L)/G(lim)) ; lim = \text{largest factorial}
*     \end{align*}
*     ```
*
* @private
* @param {number} z - first gamma parameter
* @param {number} delta - difference
* @returns {number} gamma ratio
*/

/**
* Compute `exp(x) - 1`.
*
* @module @stdlib/math/base/special/expm1
*
* @example
* var expm1 = require( '@stdlib/math/base/special/expm1' );
*
* var v = expm1( 0.2 );
* // returns ~0.221
*
* v = expm1( -9.0 );
* // returns ~-0.999
*
* v = expm1( 0.0 );
* // returns 0.0
*
* v = expm1( NaN );
* // returns NaN
*/

/**
* Compute `gamma(x+1) - 1` without cancellation errors.
*
* @module @stdlib/math/base/special/gamma1pm1
*
* @example
* var gamma1pm1 = require( '@stdlib/math/base/special/gamma1pm1' );
*
* var v = gamma1pm1( 0.2 );
* // returns ~-0.082
*
* v = gamma1pm1( -5.3 );
* // returns ~-1.102
*
* v = gamma1pm1( 0.0 );
* // returns 0.0
*
* v = gamma1pm1( NaN );
* // returns NaN
*/

/**
* Compute the arcsine of a number.
*
* @module @stdlib/math/base/special/asin
*
* @example
* var asin = require( '@stdlib/math/base/special/asin' );
*
* var v = asin( 0.0 );
* // returns 0.0
*
* v = asin( 3.141592653589793/4.0 );
* // returns ~0.903
*
* v = asin( -3.141592653589793/6.0 );
* // returns ~-0.551
*
* v = asin( NaN );
* // returns NaN
*/

/**
* Compute the full upper fraction (Q) when `a` is very small.
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @param {boolean} invert - boolean indicating if the upper tail of the incomplete gamma function should be evaluated
* @returns {Array} full upper fraction (Q) and pgam
*/

/**
* Compute the value of `sin(πx)`.
*
* @module @stdlib/math/base/special/sinpi
*
* @example
* var sinpi = require( '@stdlib/math/base/special/sinpi' );
*
* var y = sinpi( 0.0 );
* // returns 0.0
*
* y = sinpi( 0.5 );
* // returns 1.0
*
* y = sinpi( 0.9 );
* // returns ~0.309
*
* y = sinpi( NaN );
* // returns NaN
*/

/**
* Computes `(z^a)*(e^-z) / gamma(a)`.
*
* @private
* @param {number} a - input value
* @param {number} z - input value
* @returns {number} function value
*/

/**
* Computes `exp(x) - 1`.
*
* ## Method
*
* 1.  Given \\(x\\), we use argument reduction to find \\(r\\) and an integer \\(k\\) such that
*
*     ```tex
*     x = k \cdot \ln(2) + r
*     ```
*
*     where
*
*     ```tex
*     |r| \leq \frac{\ln(2)}{2} \approx 0.34658
*     ```
*
*     <!-- <note> -->
*
*     A correction term \\(c\\) will need to be computed to compensate for the error in \\(r\\) when rounded to a floating-point number.
*
*     <!-- </note> -->
*
* 2.  To approximate \\(\operatorname{expm1}(r)\\), we use a special rational function on the interval \\(\[0,0.34658]\\). Since
*
*     ```tex
*     r \frac{e^r + 1}{e^r - 1} = 2 + \frac{r^2}{6} - \frac{r^4}{360} + \ldots
*     ```
*
*     we define \\(\operatorname{R1}(r^2)\\) by
*
*     ```tex
*     r \frac{e^r + 1}{e^r - 1} = 2 + \frac{r^2}{6} \operatorname{R1}(r^2)
*     ```
*
*     That is,
*
*     ```tex
*     \begin{align*}
*     \operatorname{R1}(r^2) &= \frac{6}{r} \biggl(\frac{e^r+1}{e^r-1} - \frac{2}{r}\biggr) \\
*     &= \frac{6}{r} \biggl( 1 + 2 \biggl(\frac{1}{e^r-1} - \frac{1}{r}\biggr)\biggr) \\
*     &= 1 - \frac{r^2}{60} + \frac{r^4}{2520} - \frac{r^6}{100800} + \ldots
*     \end{align*}
*     ```
*
*     We use a special Remes algorithm on \\(\[0,0.347]\\) to generate a polynomial of degree \\(5\\) in \\(r^2\\) to approximate \\(\mathrm{R1}\\). The maximum error of this polynomial approximation is bounded by \\(2^{-61}\\). In other words,
*
*     ```tex
*     \operatorname{R1}(z) \approx 1 + \mathrm{Q1} \cdot z + \mathrm{Q2} \cdot z^2 + \mathrm{Q3} \cdot z^3 + \mathrm{Q4} \cdot z^4 + \mathrm{Q5} \cdot z^5
*     ```
*
*     where
*
*     ```tex
*     \begin{align*}
*     \mathrm{Q1} &= -1.6666666666666567384\mbox{e-}2 \\
*     \mathrm{Q2} &= 3.9682539681370365873\mbox{e-}4 \\
*     \mathrm{Q3} &= -9.9206344733435987357\mbox{e-}6 \\
*     \mathrm{Q4} &= 2.5051361420808517002\mbox{e-}7 \\
*     \mathrm{Q5} &= -6.2843505682382617102\mbox{e-}9
*     \end{align*}
*     ```
*
*     where \\(z = r^2\\) and the values of \\(\mathrm{Q1}\\) to \\(\mathrm{Q5}\\) are listed in the source. The error is bounded by
*
*     ```tex
*     \biggl| 1 + \mathrm{Q1} \cdot z + \ldots + \mathrm{Q5} \cdot z - \operatorname{R1}(z) \biggr| \leq 2^{-61}
*     ```
*
*     \\(\operatorname{expm1}(r) = e^r - 1\\) is then computed by the following specific way which minimizes the accumulated rounding error
*
*     ```tex
*     \operatorname{expm1}(r) = r + \frac{r^2}{2} + \frac{r^3}{2} \biggl( \frac{3 - (\mathrm{R1} + \mathrm{R1} \cdot \frac{r}{2})}{6 - r ( 3 - \mathrm{R1} \cdot \frac{r}{2})} \biggr)
*     ```
*
*     To compensate for the error in the argument reduction, we use
*
*     ```tex
*     \begin{align*}
*     \operatorname{expm1}(r+c) &= \operatorname{expm1}(r) + c + \operatorname{expm1}(r) \cdot c \\
*     &\approx \operatorname{expm1}(r) + c + rc
*     \end{align*}
*     ```
*
*     Thus, \\(c + rc\\) will be added in as the correction terms for \\(\operatorname{expm1}(r+c)\\). Now, we can rearrange the term to avoid optimization screw up.
*
*     ```tex
*     \begin{align*}
*     \operatorname{expm1}(r+c) &\approx r - \biggl( \biggl( r + \biggl( \frac{r^2}{2} \biggl( \frac{\mathrm{R1} - (3 - \mathrm{R1} \cdot \frac{r}{2})}{6 - r (3 - \mathrm{R1} \cdot \frac{r}{2})} \biggr) - c \biggr) - c \biggr) - \frac{r^2}{2} \biggr) \\
*     &= r - \mathrm{E}
*     \end{align*}
*     ```
*
* 3.  To scale back to obtain \\(\operatorname{expm1}(x)\\), we have (from step 1)
*
*     ```tex
*     \operatorname{expm1}(x) = \begin{cases}
*     2^k  (\operatorname{expm1}(r) + 1) - 1 \\
*     2^k (\operatorname{expm1}(r) + (1-2^{-k}))
*     \end{cases}
*     ```
*
* ## Special Cases
*
* ```tex
* \begin{align*}
* \operatorname{expm1}(\infty) &= \infty \\
* \operatorname{expm1}(-\infty) &= -1 \\
* \operatorname{expm1}(\mathrm{NaN}) &= \mathrm{NaN}
* \end{align*}
* ```
*
*
* ## Notes
*
* -   For finite arguments, only \\(\operatorname{expm1}(0) = 0\\) is exact.
*
* -   To save one multiplication, we scale the coefficient \\(\mathrm{Qi}\\) to \\(\mathrm{Qi} \cdot {2^i}\\) and replace \\(z\\) by \\(\frac{x^2}{2}\\).
*
* -   To achieve maximum accuracy, we compute \\(\operatorname{expm1}(x)\\) by
*
*     -   if \\(x < -56 \cdot \ln(2)\\), return \\(-1.0\\) (raise inexact if \\(x\\) does not equal \\(\infty\\))
*
*     -   if \\(k = 0\\), return \\(r-\mathrm{E}\\)
*
*     -   if \\(k = -1\\), return \\(\frac{(r-\mathrm{E})-1}{2}\\)
*
*     -   if \\(k = 1\\),
*
*         -   if \\(r < -0.25\\), return \\(2((r+0.5)- \mathrm{E})\\)
*         -   else return \\(1+2(r-\mathrm{E})\\)
*
*     -   if \\(k < -2\\) or \\(k > 56\\), return \\(2^k(1-(\mathrm{E}-r)) - 1\\) (or \\(e^x-1\\))
*
*     -   if \\(k \leq 20\\), return \\(2^k((1-2^{-k})-(\mathrm{E}-r))\\)
*
*     -   else return \\(2^k(1-((\mathrm{E}+2^{-k})-r))\\)
*
* -   For IEEE 754 double, if \\(x > 7.09782712893383973096\mbox{e+}02\\), then \\(\operatorname{expm1}(x)\\) will overflow.
*
* -   The hexadecimal values listed in the source are the intended ones for the implementation constants. Decimal values may be used, provided that the compiler will convert from decimal to binary accurately enough to produce the intended hexadecimal values.
*
* -   According to an error analysis, the error is always less than \\(1\\) ulp (unit in the last place).
*
*
* @param {number} x - input value
* @returns {number} function value
*
* @example
* var v = expm1( 0.2 );
* // returns ~0.221
*
* @example
* var v = expm1( -9.0 );
* // returns ~-0.9999
*
* @example
* var v = expm1( 0.0 );
* // returns 0.0
*
* @example
* var v = expm1( NaN );
* // returns NaN
*/

/**
* Computes `gamma(x+1) - 1`.
*
* @param {number} x - input value
* @returns {number} function value
*
* @example
* var v = gammap1m1( 0.2 );
* // returns ~-0.082
*
* @example
* var v = gammap1m1( -9.2 );
* // returns ~-1.0
*
* @example
* var v = gammap1m1( 0.0 );
* // returns 0.0
*
* @example
* var v = gammap1m1( -3.0 );
* // returns NaN
*
* @example
* var v = gammap1m1( NaN );
* // returns NaN
*/

/**
* Computes the arcsine of a number.
*
* ## Method
*
* -   A rational function of the form
*
*     ```tex
*     x + x^3 \frac{P(x^2)}{Q(x^2)}
*     ```
*
*     is used for \\(\|x\|\\) in the interval \\(\[0, 0.5\]\\). If \\(\|x\| > 0.5\\), it is transformed by the identity
*
*     ```tex
*     \operatorname{asin}(x) = \frac{\pi}{2} - 2 \operatorname{asin}( \sqrt{ (1-x)/2 } )
*     ```
*
* ## Notes
*
* -   Relative error:
*
*     | arithmetic | domain | # trials | peak    | rms     |
*     |:-----------|:-------|:---------|:--------|:--------|
*     | DEC        | -1, 1  | 40000    | 2.6e-17 | 7.1e-18 |
*     | IEEE       | -1, 1  | 10^6     | 1.9e-16 | 5.4e-17 |
*
* @param {number} x - input value
* @returns {number} arcsine (in radians)
*
* @example
* var v = asin( 0.0 );
* // returns ~0.0
*
* @example
* var v = asin( 3.141592653589793/4.0 );
* // returns ~0.903
*
* @example
* var v = asin( -3.141592653589793/6.0 );
* // returns ~-0.551
*
* @example
* var v = asin( NaN );
* // returns NaN
*/

/**
* Computes the delta in `beta(a,b,x) = prefix + delta * beta(a+k,b,x)`.
*
* ## Notes
*
* Specifically, the function calculates
*
* ```tex
* \frac{ (a)(a+1)(a+2)...(a+k-1) }{ (b)(b+1)(b+2)...(b+k-1) }
* ```
*
* The function should only called with small `k`; for large `k`, it is grossly inefficient.
*
* @private
* @param {number} a - input value
* @param {number} b - input value
* @param {NonNegativeInteger} k - input value
* @returns {number} ratio value
*/

/**
* Computes the difference between `ibeta(a,b,x)` and `ibeta(a+k,b,x)`.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {Probability} y - probability equal to `1-x`
* @param {NonNegativeInteger} k - function input
* @param {boolean} normalized - boolean indicating whether to evaluate the power terms of the regularized or non-regularized incomplete beta function
* @param {(Array|TypedArray|Object)} out - output array holding the derivative as the second element
* @returns {number} difference between ibeta(a,b,x) and ibeta(a+k,b,x)
*/

/**
* Computes the leading power terms in the incomplete beta function.
*
* When normalized,
*
* ```tex
* \frac{ x^a y^b }{ \operatorname{Beta}(a,b) }
* ```
*
* and otherwise
*
* ```tex
* x^a y^b
* ```
*
* ## Notes
*
* -   Almost all of the error in the incomplete beta comes from this function, particularly when \\( a \\) and \\( b \\) are large. Computing large powers are _hard_ though, and using logarithms just leads to horrendous cancellation errors.
*
* -   For \\( l1 * l2 > 0 \\) or \\( \operatorname{min}( a, b ) < 1 \\), the two power terms both go in the same direction (toward zero or toward infinity). In this case if either term overflows or underflows, then the product of the two must do so also. Alternatively, if one exponent is less than one, then we can't productively use it to eliminate overflow or underflow from the other term.  Problems with spurious overflow/underflow can't be ruled out. In this case, but it is _very_ unlikely since one of the power terms will evaluate to a number close to 1.
*
* -   If \\( \max( \abs(l1), \abs(l2) ) < 0.5 \\), both exponents are near one and both the exponents are greater than one, and, further, these two power terms tend in opposite directions (one toward zero, the other toward infinity), so we have to combine the terms to avoid any risk of overflow or underflow. We do this by moving one power term inside the other, we have:
*
*     ```tex
*     (1 + l_1)^a \cdot (1 + l_2)^b \\
*     = ((1 + l_1) \cdot (1 + l_2)^(b/a))^a \\
*     = (1 + l_1 + l_3 + l_1*l_3)^a
*     ```
*
*     and
*
*     ```tex
*     l_3 = (1 + l_2)^(b/a) - 1 \\
*     = \exp((b/a) * \ln(1 + l_2)) - 1
*     ```
*
*     The tricky bit is deciding which term to move inside. By preference, we move the larger term inside, so that the size of the largest exponent is reduced.  However, that can only be done as long as l3 (see above) is also small.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {Probability} y - probability equal to `1-x`
* @param {boolean} normalized - boolean indicating whether to evaluate the power terms of the regularized or non-regularized incomplete beta function
* @returns {number} power terms
*/

/**
* Computes the ratio of two gamma functions.
*
* ## Notes
*
* -   Specifically, the function evaluates
*
*     ```tex
*     \frac{ \Gamma( z ) }{ \Gamma( z + \delta ) }
*     ```
*
* @param {number} z - first gamma parameter
* @param {number} delta - difference
* @returns {number} gamma ratio
*
* @example
* var y = gammaDeltaRatio( 2.0, 3.0 );
* // returns ~0.042
*
* @example
* var y = gammaDeltaRatio( 4.0, 0.5 );
* // returns ~0.516
*
* @example
* var y = gammaDeltaRatio( 100.0, 0.0 );
* // returns 1.0
*/

/**
* Computes the regularized incomplete gamma function. The upper tail is calculated via the modified Lentz's method for computing continued fractions, the lower tail using a power expansion.
*
*
* ## Notes
*
* -   When `a >= MAX_FACTORIAL` and computing the non-normalized incomplete gamma, result is rather hard to compute unless we use logs. There are really two options a) if `x` is a long way from `a` in value then we can reliably use methods 2 and 4 below in logarithmic form and go straight to the result. Otherwise we let the regularized gamma take the strain (the result is unlikely to underflow in the central region anyway) and combine with `lgamma` in the hopes that we get a finite result.
*
* @param {NonNegativeNumber} x - function parameter
* @param {PositiveNumber} a - function parameter
* @param {boolean} [regularized=true] - boolean indicating if the function should evaluate the regularized or non-regularized incomplete gamma functions
* @param {boolean} [upper=false] - boolean indicating if the function should return the upper tail of the incomplete gamma function
* @returns {number} function value
*/

/**
* Computes the value of `sin(πx)`.
*
* @param {number} x - input value
* @returns {number} function value
*
* @example
* var y = sinpi( 0.0 );
* // returns 0.0
*
* @example
* var y = sinpi( 0.5 );
* // returns 1.0
*
* @example
* var y = sinpi( 0.9 );
* // returns ~0.309
*
* @example
* var y = sinpi( NaN );
* // returns NaN
*/

/**
* Continued fraction for the incomplete beta.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {Probability} y - probability equal to `1-x`
* @returns {Function} series function
*/

/**
* Creates a function to evaluate a series expansion of the incomplete gamma function.
*
* @private
* @param {number} a1 - function parameter
* @param {number} z1 - function parameter
* @returns {Function} series function
*/

/**
* Creates a function to evaluate a series expansion of the upper incomplete gamma fraction.
*
* @private
* @param {number} a1 - function parameter
* @param {number} z1 - function parameter
* @returns {Function} series function
*/

/**
* Degenerate distribution cumulative distribution function (CDF).
*
* @module @stdlib/stats/base/dists/degenerate/cdf
*
* @example
* var cdf = require( '@stdlib/stats/base/dists/degenerate/cdf' );
*
* var y = cdf( 2.0, 5.0 );
* // returns 0.0
*
* @example
* var factory = require( '@stdlib/stats/base/dists/degenerate/cdf' ).factory;
*
* var cdf = factory( 5.0 );
*
* var y = cdf( 3.0 );
* // returns 0.0
*
* y = cdf( 6.0 );
* // returns 1.0
*/

/**
* Degenerate distribution probability density function (PDF).
*
* @module @stdlib/stats/base/dists/degenerate/pdf
*
* @example
* var pdf = require( '@stdlib/stats/base/dists/degenerate/pdf' );
*
* var y = pdf( 2.0, 0.0 );
* // returns 0.0
*
* @example
* var factory = require( '@stdlib/stats/base/dists/degenerate/pdf' ).factory;
*
* var pdf = factory( 10.0 );
*
* var y = pdf( 10.0 );
* // returns Infinity
*/

/**
* Difference between one and the smallest value greater than one that can be represented as a double-precision floating-point number.
*
* ## Notes
*
* The difference is
*
* ```tex
* \frac{1}{2^{52}}
* ```
*
* @constant
* @type {number}
* @default 2.220446049250313e-16
* @see [IEEE 754]{@link https://en.wikipedia.org/wiki/IEEE_754-1985}
* @see [Machine Epsilon]{@link https://en.wikipedia.org/wiki/Machine_epsilon}
*/

/**
* Difference between one and the smallest value greater than one that can be represented as a double-precision floating-point number.
*
* @module @stdlib/constants/math/float64-eps
* @type {number}
*
* @example
* var FLOAT64_EPSILON = require( '@stdlib/constants/math/float64-eps' );
* // returns 2.220446049250313e-16
*/

/**
* Evaluate `bˣ - 1`.
*
* @module @stdlib/math/base/special/powm1
*
* @example
* var powm1 = require( '@stdlib/math/base/special/powm1' );
*
* var y = powm1( 2.0, 3.0 );
* // returns 7.0
*
* y = powm1( 4.0, 0.5 );
* // returns 1.0
*
* y = powm1( 0.0, 100.0 );
* // returns -1.0
*
* y = powm1( 100.0, 0.0 );
* // returns 0.0
*
* y = powm1( 0.0, 0.0 );
* // returns 0.0
*
* y = powm1( 3.141592653589793, 5.0 );
* // returns ~305.0197
*
* y = powm1( NaN, 3.0 );
* // returns NaN
*
* y = powm1( 5.0, NaN );
* // returns NaN
*/

/**
* Evaluate a polynomial.
*
* @module @stdlib/math/base/tools/evalpoly
*
* @example
* var evalpoly = require( '@stdlib/math/base/tools/evalpoly' );
*
* var v = evalpoly( [3.0,2.0,1.0], 10.0 ); // 3*10^0 + 2*10^1 + 1*10^2
* // returns 123.0
*
* @example
* var evalpoly = require( '@stdlib/math/base/tools/evalpoly' );
*
* var polyval = evalpoly.factory( [3.0,2.0,1.0] );
*
* var v = polyval( 10.0 ); // => 3*10^0 + 2*10^1 + 1*10^2
* // returns 123.0
*
* v = polyval( 5.0 ); // => 3*5^0 + 2*5^1 + 1*5^2
* // returns 38.0
*/

/**
* Evaluate the beta function.
*
* @module @stdlib/math/base/special/beta
*
* @example
* var beta = require( '@stdlib/math/base/special/beta' );
*
* var v = beta( 0.0, 0.5 );
* // returns Infinity
*
* v = beta( 1.0, 1.0 );
* // returns 1.0
*
* v = beta( -1.0, 2.0 );
* // returns NaN
*
* v = beta( 5.0, 0.2 );
* // returns ~3.382
*
* v = beta( 4.0, 1.0 );
* // returns 0.25
*
* v = beta( NaN, 2.0 );
* // returns NaN
*/

/**
* 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
*/

/**
* Evaluate the incomplete beta function and its first derivative.
*
* @module @stdlib/math/base/special/kernel-betainc
*
* @example
* var kernelBetainc = require( '@stdlib/math/base/special/kernel-betainc' );
*
* var out = kernelBetainc( 0.5, 2.0, 2.0, false, false );
* // returns [ ~0.083, ~1.5 ]
*
* out = kernelBetainc( 0.2, 1.0, 2.0, true, false );
* // returns [ 0.32, 1.6 ]
*
* var arr = new Array( 2 );
* out = kernelBetainc( arr, 0.2, 1.0, 2.0, true, true );
* // returns [ 0.64, 1.6 ]
*
* var bool = ( arr === out );
* // returns true
*/

/**
* Evaluate the incomplete beta function.
*
* @module @stdlib/math/base/special/betainc
*
* @example
* var betainc = require( '@stdlib/math/base/special/betainc' );
*
* var y = betainc( 0.5, 2.0, 2.0 );
* // returns 0.5
*
* y = betainc( 0.5, 2.0, 2.0, false );
* // returns ~0.083
*
* y = betainc( 0.2, 1.0, 2.0 );
* // returns 0.36
*
* y = betainc( 0.2, 1.0, 2.0, true, true );
* // returns 0.64
*/

/**
* Evaluate the incomplete gamma function.
*
* @module @stdlib/math/base/special/gammainc
*
* @example
* var gammainc = require( '@stdlib/math/base/special/gammainc' );
*
* var v = gammainc( 6.0, 2.0 );
* // returns ~0.9826
*
* v = gammainc( 1.0, 2.0, true, true );
* // returns ~0.7358
*
* v = gammainc( 7.0, 5.0 );
* // returns ~0.8270
*
* v = gammainc( 7.0, 5.0, false );
* // returns ~19.8482
*
* v = gammainc( NaN, 2.0 );
* // returns NaN
*
* v = gammainc( 6.0, NaN );
* // returns NaN
*/

/**
* Evaluate the lower incomplete gamma integral via a series expansion and divide by `gamma(z)` to normalize.
*
* @private
* @param {number} a - function parameter
* @param {number} z - function parameter
* @returns {number} function value
*/

/**
* Evaluate the natural logarithm of \\(1+x\\).
*
* @module @stdlib/math/base/special/log1p
*
* @example
* var log1p = require( '@stdlib/math/base/special/log1p' );
*
* var v = log1p( 4.0 );
* // returns ~1.609
*
* v = log1p( -1.0 );
* // returns -Infinity
*
* v = log1p( 0.0 );
* // returns 0.0
*
* v = log1p( -0.0 );
* // returns -0.0
*
* v = log1p( -2.0 );
* // returns NaN
*
* v = log1p( NaN );
* // returns NaN
*/

/**
* Evaluate the natural logarithm of the gamma function.
*
* @module @stdlib/math/base/special/gammaln
*
* @example
* var gammaln = require( '@stdlib/math/base/special/gammaln' );
*
* var v = gammaln( 1.0 );
* // returns 0.0
*
* v = gammaln( 2.0 );
* // returns 0.0
*
* v = gammaln( 4.0 );
* // returns ~1.792
*
* v = gammaln( -0.5 );
* // returns ~1.266
*
* v = gammaln( 0.5 );
* // returns ~0.572
*
* v = gammaln( 0.0 );
* // returns Infinity
*
* v = gammaln( NaN );
* // returns NaN
*/

/**
* Evaluates `bˣ - 1`.
*
* @param {number} b - base
* @param {number} x - exponent
* @returns {number} function value
*
* @example
* var y = powm1( 2.0, 3.0 );
* // returns 7.0
*
* @example
* var y = powm1( 4.0, 0.5 );
* // returns 1.0
*
* @example
* var y = powm1( 0.0, 100.0 );
* // returns -1.0
*
* @example
* var y = powm1( 100.0, 0.0 );
* // returns 0.0
*
* @example
* var y = powm1( 0.0, 0.0 );
* // returns 0.0
*
* @example
* var y = powm1( 3.141592653589793, 5.0 );
* // returns ~305.0197
*
* @example
* var y = powm1( NaN, 3.0 );
* // returns NaN
*
* @example
* var y = powm1( 5.0, NaN );
* // returns NaN
*/

/**
* Evaluates a continued fraction expansion.
*
* ```text
*           a1
*      ---------------
*      b1 +     a2
*           ----------
*            b2 +   a3
*                -----
*                b3 + ...
* ```
*
* @private
* @param {Function} gen - function giving terms of continued fraction expansion
* @param {PositiveNumber} factor - further terms are only added as long as factor*result is smaller than the next term
* @param {PositiveInteger} maxIter - maximum number of iterations
* @returns {number} evaluated expansion
*/

/**
* Evaluates a continued fraction expansion.
*
* ```text
*      b0 +    a1
*      ---------------
*      b1 +     a2
*           ----------
*           b2 +   a3
*                -----
*                b3 + ...
* ```
*
* @private
* @param {Function} gen - function giving terms of continued fraction expansion
* @param {PositiveNumber} factor - further terms are only added as long as factor*result is smaller than the next term
* @param {PositiveInteger} maxIter - maximum number of iterations
* @returns {number} evaluated expansion
*/

/**
* Evaluates a continued fraction expansion.
*
* ```text
*      b0 +   a1
*      ---------------
*      b1 +   a2
*           ----------
*           b2 +   a3
*                -----
*                b3 + ...
* ```
*
* @private
* @param {Function} gen - function giving terms of continued fraction expansion
* @param {PositiveNumber} factor - further terms are only added as long as factor*result is smaller than the next term
* @param {PositiveInteger} maxIter - maximum number of iterations
* @returns {number} evaluated expansion
*/

/**
* Evaluates a polynomial.
*
* ## Notes
*
* -   The implementation uses [Horner's rule][horners-method] for efficient computation.
*
* [horners-method]: https://en.wikipedia.org/wiki/Horner%27s_method
*
*
* @param {NumericArray} c - polynomial coefficients sorted in ascending degree
* @param {number} x - value at which to evaluate the polynomial
* @returns {number} evaluated polynomial
*
* @example
* var v = evalpoly( [3.0,2.0,1.0], 10.0 ); // 3*10^0 + 2*10^1 + 1*10^2
* // returns 123.0
*/

/**
* Evaluates a polynomial.
*
* ## Notes
*
* -   The implementation uses [Horner's rule][horners-method] for efficient computation.
*
* [horners-method]: https://en.wikipedia.org/wiki/Horner%27s_method
*
*
* @private
* @param {number} x - value at which to evaluate the polynomial
* @returns {number} evaluated polynomial
*/

/**
* Evaluates a rational function, i.e., the ratio of two polynomials described by the coefficients stored in \\(P\\) and \\(Q\\).
*
* ## Notes
*
* -   Coefficients should be sorted in ascending degree.
* -   The implementation uses [Horner's rule][horners-method] for efficient computation.
*
* [horners-method]: https://en.wikipedia.org/wiki/Horner%27s_method
*
*
* @private
* @param {number} x - value at which to evaluate the rational function
* @returns {number} evaluated rational function
*/

/**
* Evaluates the continued fraction approximation for the supplied series generator using the modified Lentz algorithm.
*
* ## References
*
* -   Lentz, William J. 1976. "Generating bessel functions in Mie scattering calculations using continued fractions." _Applied Optics_ 15 (3): 668–71. doi:[10.1364/AO.15.000668](https://doi.org/10.1364/AO.15.000668).
*
* @param {Function} generator - function returning terms of continued fraction expansion
* @param {Object} [options] - function options
* @param {PositiveInteger} [options.maxIter=1000000] - maximum number of iterations
* @param {PositiveNumber} [options.tolerance=2.22e-16] - further terms are only added as long as the next term is greater than current term times the tolerance
* @param {boolean} [options.keep=false] - whether to keep the leading b term
* @returns {number} value of continued fraction
*
* @example
* // Continued fraction for (e-1)^(-1):
* var gen = generator()
* var out = continuedFraction( gen );
* // returns ~0.582
*
* function generator() {
*    var i = 0;
*    return function() {
*        i++;
*        return [ i, i ];
*    };
* }
*/

/**
* Evaluates the continued fraction approximation for the supplied series generator using the modified Lentz algorithm.
*
* ## References
*
* -   Lentz, William J. 1976. "Generating bessel functions in Mie scattering calculations using continued fractions." _Applied Optics_ 15 (3): 668–71. doi:[10.1364/AO.15.000668](https://doi.org/10.1364/AO.15.000668).
*
* @param {Function} generator - function returning terms of continued fraction expansion
* @param {Object} [options] - function options
* @param {PositiveInteger} [options.maxIter=1000] - maximum number of iterations
* @param {PositiveNumber} [options.tolerance=2.22e-16] - further terms are only added as long as the next term is greater than current term times the tolerance
* @param {boolean} [options.keep=false] - whether to keep the leading b term
* @returns {number} value of continued fraction
*
* @example
* // Continued fraction for (e-1)^(-1):
* var gen = generator();
* var out = continuedFraction( gen );
* // returns ~0.582
*
* function* generator() {
*    var i = 0;
*    while ( true ) {
*        i++;
*        yield [ i, i ];
*    }
* }
*/

/**
* Evaluates the cumulative distribution function (CDF) for a Normal distribution with mean `mu` and standard deviation `sigma` at a value `x`.
*
* @param {number} x - input value
* @param {number} mu - mean
* @param {NonNegativeNumber} sigma - standard deviation
* @returns {Probability} evaluated cumulative distribution function
*
* @example
* var y = cdf( 2.0, 0.0, 1.0 );
* // returns ~0.977
*
* @example
* var y = cdf( -1.0, -1.0, 2.0 );
* // returns 0.5
*
* @example
* var y = cdf( -1.0, 4.0, 2.0 );
* // returns ~0.006
*
* @example
* var y = cdf( NaN, 0.0, 1.0 );
* // returns NaN
*
* @example
* var y = cdf( 0.0, NaN, 1.0 );
* // returns NaN
*
* @example
* var y = cdf( 0.0, 0.0, NaN );
* // returns NaN
*
* @example
* // Negative standard deviation:
* var y = cdf( 2.0, 0.0, -1.0 );
* // returns NaN
*/

/**
* Evaluates the cumulative distribution function (CDF) for a Student's t distribution with degrees of freedom `v` at a value `x`.
*
* @param {number} x - input value
* @param {PositiveNumber} v - degrees of freedom
* @returns {Probability} evaluated CDF
*
* @example
* var y = cdf( 2.0, 0.1 );
* // returns ~0.611
*
* @example
* var y = cdf( 1.0, 2.0 );
* // returns ~0.789
*
* @example
* var y = cdf( -1.0, 4.0 );
* // returns ~0.187
*
* @example
* var y = cdf( NaN, 1.0 );
* // returns NaN
*
* @example
* var y = cdf( 0.0, NaN );
* // returns NaN
*
* @example
* var y = cdf( 2.0, -1.0 );
* // returns NaN
*/

/**
* Evaluates the cumulative distribution function (CDF) for a degenerate distribution with mean value `mu`.
*
* @param {number} x - input value
* @param {number} mu - constant value of distribution
* @returns {Probability} evaluated cumulative distribution function
*
* @example
* var y = cdf( 2.0, 3.0 );
* // returns 0.0
*
* @example
* var y = cdf( 4.0, 3.0 );
* // returns 1.0
*
* @example
* var y = cdf( 3.0, 3.0 );
* // returns 1.0
*
* @example
* var y = cdf( NaN, 0.0 );
* // returns NaN
*
* @example
* var y = cdf( 0.0, NaN );
* // returns NaN
*/

/**
* Evaluates the incomplete beta function and its first derivative.
*
* @param {(Array|TypedArray|Object)} [out] - output array
* @param {Probability} x - function input
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {boolean} regularized - boolean indicating if the function should evaluate the regularized boolean beta function
* @param {boolean} upper - boolean indicating if the function should return the upper tail of the incomplete beta function instead
* @returns {(Array|TypedArray|Object)} function value and first derivative
*
* @example
* var out = kernelBetainc( 0.5, 2.0, 2.0, false, false );
* // returns [ ~0.083, ~1.5 ]
*
* @example
* var out = kernelBetainc( 0.2, 1.0, 2.0, true, false );
* // returns [ 0.36, 1.6 ]
*
* @example
* var arr = new Array( 2 );
* var out = kernelBetainc( arr, 0.2, 1.0, 2.0, true, true );
* // returns [ 0.64, 1.6 ]
*
* var bool = ( arr === out );
* // returns true
*/

/**
* Evaluates the incomplete beta function and its first derivative. This function divides up the input range and selects the right implementation method for each domain.
*
* @private
* @param {(Array|TypedArray|Object)} out - output array
* @param {Probability} x - function input
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {boolean} regularized - boolean indicating if the function should evaluate the regularized boolean beta function
* @param {boolean} upper - boolean indicating if the function should return the upper tail of the incomplete beta function instead
* @returns {(Array|TypedArray|Object)} function value and first derivative
*
* @example
* var out = ibetaImp( new Array( 2 ), 0.5, 2.0, 2.0, false, false );
* // returns [ ~0.083, ~1.5 ]
*
* @example
* var out = ibetaImp( new Array( 2 ), 0.2, 1.0, 2.0, false, true );
* // returns [ 0.32, 1.6 ]
*
* @example
* var out = ibetaImp( new Array( 2 ), 0.2, 1.0, 2.0, true, true );
* // returns [ 0.64, 1.6 ]
*/

/**
* Evaluates the incomplete beta function.
*
* @param {Probability} x - function parameter
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {boolean} [regularized=true] - boolean indicating if the function should evaluate the regularized or non-regularized incomplete beta function
* @param {boolean} [upper=false] - boolean indicating if the function should return the upper tail of the incomplete beta function
* @returns {number} function value
*
* @example
* var y = betainc( 0.5, 2.0, 2.0 );
* // returns 0.5
*
* @example
* var y = betainc( 0.5, 2.0, 2.0, false );
* // returns ~0.083
*
* @example
* var y = betainc( 0.2, 1.0, 2.0 );
* // returns 0.36
*/

/**
* Evaluates the incomplete beta via the continued fraction representation.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {Probability} y - probability equal to `1-x`
* @param {boolean} normalized - boolean indicating whether to evaluate the power terms of the regularized or non-regularized incomplete beta function
* @param {(Array|TypedArray|Object)} out - output array holding the derivative as the second element
* @returns {number} incomplete beta value
*/

/**
* Evaluates the natural logarithm of \\(1+x\\).
*
* ## Method
*
* 1.  Argument Reduction: find \\(k\\) and \\(f\\) such that
*
*     ```tex
*     1+x = 2^k (1+f)
*     ```
*
*     where
*
*     ```tex
*     \frac{\sqrt{2}}{2} < 1+f < \sqrt{2}
*     ```
*
*     <!-- <note> -->
*
*     If \\(k=0\\), then \\(f=x\\) is exact. However, if \\(k \neq 0\\), then \\(f\\) may not be representable exactly. In that case, a correction term is needed. Let
*
*     ```tex
*     u = \operatorname{round}(1+x)
*     ```
*
*     and
*
*     ```tex
*     c = (1+x) - u
*     ```
*
*     then
*
*     ```tex
*     \ln (1+x) - \ln u \approx \frac{c}{u}
*     ```
*
*     We can thus proceed to compute \\(\ln(u)\\), and add back the correction term \\(c/u\\).
*
*     <!-- </note> -->
*
*     <!-- <note> -->
*
*     When \\(x > 2^{53}\\), one can simply return \\(\ln(x)\\).
*
*     <!-- </note> -->
*
* 2.  Approximation of \\(\operatorname{log1p}(f)\\). Let
*
*     ```tex
*     s = \frac{f}{2+f}
*     ```
*
*     based on
*
*     ```tex
*     \begin{align*}
*     \ln 1+f &= \ln (1+s) - \ln (1-s) \\
*             &= 2s + \frac{2}{3} s^3 + \frac{2}{5} s^5 + ... \\
*             &= 2s + sR \\
*     \end{align*}
*     ```
*
*     We use a special Reme algorithm on \\(\[0,0.1716\]\\) to generate a polynomial of degree \\(14\\) to approximate \\(R\\). The maximum error of this polynomial approximation is bounded by \\(2^{-58.45}\\). In other words,
*
*     ```tex
*     R(z) \approx \mathrm{Lp}_1 s^2 + \mathrm{Lp}_2 s^4 + \mathrm{Lp}_3 s^6 + \mathrm{Lp}_4 s^8 + \mathrm{Lp}_5 s^{10} + \mathrm{Lp}_6 s^{12} + \mathrm{Lp}_7 s^{14}
*     ```
*
*     and
*
*     ```tex
*     | \mathrm{Lp}_1 s^2 + \ldots + \mathrm{Lp}_7 s^14 - R(z) | \leq 2^{-58.45}
*     ```
*
*     <!-- <note> -->
*
*     The values of \\(Lp1\\) to \\(Lp7\\) may be found in the source.
*
*     <!-- </note> -->
*
*     Note that
*
*     ```tex
*     \begin{align*}
*     2s &= f - sf \\
*        &= f - \frac{f^2}{2} + s \frac{f^2}{2} \\
*     \end{align*}
*     ```
*
*     In order to guarantee error in \\(\ln\\) below \\(1\ \mathrm{ulp}\\), we compute the log by
*
*     ```tex
*     \operatorname{log1p}(f) = f - \biggl(\frac{f^2}{2} - s\biggl(\frac{f^2}{2}+R\biggr)\biggr)
*     ```
*
* 3.  Finally,
*
*     ```tex
*     \begin{align*}
*     \operatorname{log1p}(x) &= k \cdot \mathrm{ln2} + \operatorname{log1p}(f) \\
*     &= k \cdot \mathrm{ln2}_{hi}+\biggl(f-\biggl(\frac{f^2}{2}-\biggl(s\biggl(\frac{f^2}{2}+R\biggr)+k \cdot \mathrm{ln2}_{lo}\biggr)\biggr)\biggr) \\
*     \end{align*}
*     ```
*
*     Here \\(\mathrm{ln2}\\) is split into two floating point numbers:
*
*     ```tex
*     \mathrm{ln2}_{hi} + \mathrm{ln2}_{lo}
*     ```
*
*     where \\(n \cdot \mathrm{ln2}_{hi}\\) is always exact for \\(|n| < 2000\\).
*
*
* ## Special Cases
*
* -   \\(\operatorname{log1p}(x) = \mathrm{NaN}\\) with signal if \\(x < -1\\) (including \\(-\infty\\))
* -   \\(\operatorname{log1p}(+\infty) = +\infty\\)
* -   \\(\operatorname{log1p}(-1) = -\infty\\) with signal
* -   \\(\operatorname{log1p}(\mathrm{NaN})= \mathrm{NaN}\\) with no signal
*
*
* ## Notes
*
* -   According to an error analysis, the error is always less than \\(1\\) ulp (unit in the last place).
*
* -   The hexadecimal values are the intended ones for the used constants. The decimal values may be used, provided that the compiler will convert from decimal to binary accurately enough to produce the hexadecimal values shown.
*
* -   Assuming \\(\ln(x)\\) is accurate, the following algorithm can be used to evaluate \\(\operatorname{log1p}(x)\\) to within a few ULP:
*
*     ```javascript
*     var u = 1.0 + x;
*     if ( u === 1.0 ) {
*         return x;
*     } else {
*         return ln(u) * (x/(u-1.0));
*     }
*     ```
*
*     See HP-15C Advanced Functions Handbook, p.193.
*
*
* @param {number} x - input value
* @returns {number} the natural logarithm of `1+x`
*
* @example
* var v = log1p( 4.0 );
* // returns ~1.609
*
* @example
* var v = log1p( -1.0 );
* // returns -Infinity
*
* @example
* var v = log1p( 0.0 );
* // returns 0.0
*
* @example
* var v = log1p( -0.0 );
* // returns -0.0
*
* @example
* var v = log1p( -2.0 );
* // returns NaN
*
* @example
* var v = log1p( NaN );
* // returns NaN
*/

/**
* 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
*/

/**
* Evaluates the natural logarithm of the gamma function.
*
* ## Method
*
* 1.  Argument reduction for \\(0 < x \leq 8\\). Since \\(\Gamma(1+s) = s \Gamma(s)\\), for \\(x \in \[0,8]\\), we may reduce \\(x\\) to a number in \\(\[1.5,2.5]\\) by
*
*     ```tex
*     \operatorname{lgamma}(1+s) = \ln(s) + \operatorname{lgamma}(s)
*     ```
*
*     For example,
*
*     ```tex
*     \begin{align*}
*     \operatorname{lgamma}(7.3) &= \ln(6.3) + \operatorname{lgamma}(6.3) \\
*     &= \ln(6.3 \cdot 5.3) + \operatorname{lgamma}(5.3) \\
*     &= \ln(6.3 \cdot 5.3 \cdot 4.3 \cdot 3.3 \cdot2.3) + \operatorname{lgamma}(2.3)
*     \end{align*}
*     ```
*
* 2.  Compute a polynomial approximation of \\(\mathrm{lgamma}\\) around its minimum (\\(\mathrm{ymin} = 1.461632144968362245\\)) to maintain monotonicity. On the interval \\(\[\mathrm{ymin} - 0.23, \mathrm{ymin} + 0.27]\\) (i.e., \\(\[1.23164,1.73163]\\)), we let \\(z = x - \mathrm{ymin}\\) and use
*
*     ```tex
*     \operatorname{lgamma}(x) = -1.214862905358496078218 + z^2 \cdot \operatorname{poly}(z)
*     ```
*
*     where \\(\operatorname{poly}(z)\\) is a \\(14\\) degree polynomial.
*
* 3.  Compute a rational approximation in the primary interval \\(\[2,3]\\). Let \\( s = x - 2.0 \\). We can thus use the approximation
*
*     ```tex
*     \operatorname{lgamma}(x) = \frac{s}{2} + s\frac{\operatorname{P}(s)}{\operatorname{Q}(s)}
*     ```
*
*     with accuracy
*
*     ```tex
*     \biggl|\frac{\mathrm{P}}{\mathrm{Q}} - \biggr(\operatorname{lgamma}(x)-\frac{s}{2}\biggl)\biggl| < 2^{-61.71}
*     ```
*
*     The algorithms are based on the observation
*
*     ```tex
*     \operatorname{lgamma}(2+s) = s(1 - \gamma) + \frac{\zeta(2) - 1}{2} s^2 - \frac{\zeta(3) - 1}{3} s^3 + \ldots
*     ```
*
*     where \\(\zeta\\) is the zeta function and \\(\gamma = 0.5772156649...\\) is the Euler-Mascheroni constant, which is very close to \\(0.5\\).
*
* 4.  For \\(x \geq 8\\),
*
*     ```tex
*     \operatorname{lgamma}(x) \approx \biggl(x-\frac{1}{2}\biggr) \ln(x) - x + \frac{\ln(2\pi)}{2} + \frac{1}{12x} - \frac{1}{360x^3} + \ldots
*     ```
*
*     which can be expressed
*
*     ```tex
*     \operatorname{lgamma}(x) \approx \biggl(x-\frac{1}{2}\biggr)(\ln(x)-1)-\frac{\ln(2\pi)-1}{2} + \ldots
*     ```
*
*     Let \\(z = \frac{1}{x}\\). We can then use the approximation
*
*     ```tex
*     f(z) = \operatorname{lgamma}(x) - \biggl(x-\frac{1}{2}\biggr)(\ln(x)-1)
*     ```
*
*     by
*
*     ```tex
*     w = w_0 + w_1 z + w_2 z^3 + w_3 z^5 + \ldots + w_6 z^{11}
*     ```
*
*     where
*
*     ```tex
*     |w - f(z)| < 2^{-58.74}
*     ```
*
* 5.  For negative \\(x\\), since
*
*     ```tex
*     -x \Gamma(-x) \Gamma(x) = \frac{\pi}{\sin(\pi x)}
*     ```
*
*     where \\(\Gamma\\) is the gamma function, we have
*
*     ```tex
*     \Gamma(x) = \frac{\pi}{\sin(\pi x)(-x)\Gamma(-x)}
*     ```
*
*     Since \\(\Gamma(-x)\\) is positive,
*
*     ```tex
*     \operatorname{sign}(\Gamma(x)) = \operatorname{sign}(\sin(\pi x))
*     ```
*
*     for \\(x < 0\\). Hence, for \\(x < 0\\),
*
*     ```tex
*     \mathrm{signgam} = \operatorname{sign}(\sin(\pi x))
*     ```
*
*     and
*
*     ```tex
*     \begin{align*}
*     \operatorname{lgamma}(x) &= \ln(|\Gamma(x)|) \\
*     &= \ln\biggl(\frac{\pi}{|x \sin(\pi x)|}\biggr) - \operatorname{lgamma}(-x)
*     \end{align*}
*     ```
*
*     <!-- <note> -->
*
*     Note that one should avoid computing \\(\pi (-x)\\) directly in the computation of \\(\sin(\pi (-x))\\).
*
*     <!-- </note> -->
*
*
* ## Special Cases
*
* ```tex
* \begin{align*}
* \operatorname{lgamma}(2+s) &\approx s (1-\gamma) & \mathrm{for\ tiny\ s} \\
* \operatorname{lgamma}(x) &\approx -\ln(x) & \mathrm{for\ tiny\ x} \\
* \operatorname{lgamma}(1) &= 0 & \\
* \operatorname{lgamma}(2) &= 0 & \\
* \operatorname{lgamma}(0) &= \infty & \\
* \operatorname{lgamma}(\infty) &= \infty & \\
* \operatorname{lgamma}(-\mathrm{integer}) &= \pm \infty
* \end{align*}
* ```
*
*
* @param {number} x - input value
* @returns {number} function value
*
* @example
* var v = gammaln( 1.0 );
* // returns 0.0
*
* @example
* var v = gammaln( 2.0 );
* // returns 0.0
*
* @example
* var v = gammaln( 4.0 );
* // returns ~1.792
*
* @example
* var v = gammaln( -0.5 );
* // returns ~1.266
*
* @example
* var v = gammaln( 0.5 );
* // returns ~0.572
*
* @example
* var v = gammaln( 0.0 );
* // returns Infinity
*
* @example
* var v = gammaln( NaN );
* // returns NaN
*/

/**
* Evaluates the probability density function (PDF) for a Student's t distribution with degrees of freedom `v` at a value `x`.
*
* @param {number} x - input value
* @param {PositiveNumber} v - degrees of freedom
* @returns {number} evaluated PDF
*
* @example
* var y = pdf( 0.3, 4.0 );
* // returns ~0.355
*
* @example
* var y = pdf( 2.0, 0.7 );
* // returns ~0.058
*
* @example
* var y = pdf( -1.0, 0.5 );
* // returns ~0.118
*
* @example
* var y = pdf( 0.0, NaN );
* // returns NaN
*
* @example
* var y = pdf( NaN, 2.0 );
* // returns NaN
*
* @example
* var y = pdf( 2.0, -1.0 );
* // returns NaN
*/

/**
* Evaluates the probability density function (PDF) for a degenerate distribution centered at `mu`.
*
* @param {number} x - input value
* @param {number} mu - constant value of the distribution
* @returns {number} evaluated probability density function
*
* @example
* var y = pdf( 2.0, 3.0 );
* // returns 0.0
*
* @example
* var y = pdf( 3.0, 3.0 );
* // returns Infinity
*
* @example
* var y = pdf( NaN, 0.0 );
* // returns NaN
*
* @example
* var y = pdf( 0.0, NaN );
* // returns NaN
*/

/**
* Evaluates the probability density function (PDF) for a normal distribution with mean `mu` and standard deviation `sigma` at a value `x`.
*
* @param {number} x - input value
* @param {number} mu - mean
* @param {NonNegativeNumber} sigma - standard deviation
* @returns {number} evaluated probability density function
*
* @example
* var y = pdf( 2.0, 0.0, 1.0 );
* // returns ~0.054
*
* @example
* var y = pdf( -1.0, 4.0, 2.0 );
* // returns ~0.009
*
* @example
* var y = pdf( NaN, 0.0, 1.0 );
* // returns NaN
*
* @example
* var y = pdf( 0.0, NaN, 1.0 );
* // returns NaN
*
* @example
* var y = pdf( 0.0, 0.0, NaN );
* // returns NaN
*
* @example
* // Negative standard deviation:
* var y = pdf( 2.0, 0.0, -1.0 );
* // returns NaN
*
* @example
* var y = pdf( 2.0, 8.0, 0.0 );
* // returns 0.0
*
* @example
* var y = pdf( 8.0, 8.0, 0.0 );
* // returns Infinity
*/

/**
* For integer arguments we can relate the incomplete beta to the complement of the binomial distribution cdf and use this finite sum.
*
* @private
* @param {NonNegativeInteger} n - number of trials
* @param {NonNegativeInteger} k - function input
* @param {Probability} x - function input
* @param {Probability} y - probability equal to `1-x`
* @returns {number} sum
*/

/**
* Generate a linearly spaced numeric array.
*
* @module @stdlib/math/utils/linspace
*
* @example
* var linspace = require( '@stdlib/math/utils/linspace' );
*
* var arr = linspace( 0, 100, 6 );
* // returns [ 0, 20, 40, 60, 80, 100 ]
*/

/**
* Generates a function for evaluating a polynomial.
*
* ## Notes
*
* -   The compiled function uses [Horner's rule][horners-method] for efficient computation.
*
* [horners-method]: http://en.wikipedia.org/wiki/Horner%27s_method
*
*
* @param {NumericArray} c - polynomial coefficients sorted in ascending degree
* @returns {Function} function for evaluating a polynomial
*
* @example
* var polyval = factory( [3.0,2.0,1.0] );
*
* var v = polyval( 10.0 ); // => 3*10^0 + 2*10^1 + 1*10^2
* // returns 123.0
*
* v = polyval( 5.0 ); // => 3*5^0 + 2*5^1 + 1*5^2
* // returns 38.0
*/

/**
* Generates a linearly spaced numeric array.
*
* @param {number} x1 - first array value
* @param {number} x2 - last array value
* @param {NonNegativeInteger} [len=100] - length of output array
* @throws {TypeError} first argument must be numeric
* @throws {TypeError} second argument must be numeric
* @throws {TypeError} third argument must be a nonnegative integer
* @returns {Array} linearly spaced numeric array
*
* @example
* var arr = linspace( 0, 100, 6 );
* // returns [ 0, 20, 40, 60, 80, 100 ]
*/

/**
* Incomplete beta series.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {NonNegativeInteger} s0 - initial value
* @param {boolean} normalized - boolean indicating whether to evaluate the power terms of the regularized or non-regularized incomplete beta function
* @param {(Array|TypedArray|Object)} out - output array holding the derivative as the second element
* @param {Probability} y - probability equal to `1-x`
* @returns {number} function value
*/

/**
* Maximum double-precision floating-point number.
*
* ## Notes
*
* The maximum is given by
*
* ```tex
* 2^{1023} (2 - 2^{-52})
* ```
*
* @constant
* @type {number}
* @default 1.7976931348623157e+308
* @see [IEEE 754]{@link http://en.wikipedia.org/wiki/IEEE_754-1985}
*/

/**
* Maximum double-precision floating-point number.
*
* @module @stdlib/constants/math/float64-max
* @type {number}
*
* @example
* var FLOAT64_MAX = require( '@stdlib/constants/math/float64-max' );
* // returns 1.7976931348623157e+308
*/

/**
* Natural logarithm of the maximum double-precision floating-point number.
*
* ## Notes
*
* The natural logarithm of the maximum is given by
*
* ```tex
* \ln \left( 2^{1023} (2 - 2^{-52}) \right)
* ```
*
* @constant
* @type {number}
* @default 709.782712893384
* @see [IEEE 754]{@link http://en.wikipedia.org/wiki/IEEE_754-1985}
*/

/**
* Natural logarithm of the maximum double-precision floating-point number.
*
* @module @stdlib/constants/math/float64-max-ln
* @type {number}
*
* @example
* var FLOAT64_MAX_LN = require( '@stdlib/constants/math/float64-max-ln' );
* // returns 709.782712893384
*/

/**
* Natural logarithm of the smallest normalized double-precision floating-point number.
*
* ## Notes
*
* The number has the value
*
* ```tex
* -\ln \left( 2^{1023-1} \right)
* ```
*
* @constant
* @type {number}
* @default -708.3964185322641
* @see [IEEE 754]{@link http://en.wikipedia.org/wiki/IEEE_754-1985}
*/

/**
* Natural logarithm of the smallest normalized double-precision floating-point number.
*
* @module @stdlib/constants/math/float64-min-ln
* @type {number}
*
* @example
* var FLOAT64_MIN_LN = require( '@stdlib/constants/math/float64-min-ln' );
* // returns -708.3964185322641
*/

/**
* Normal distribution cumulative distribution function (CDF).
*
* @module @stdlib/stats/base/dists/normal/cdf
*
* @example
* var cdf = require( '@stdlib/stats/base/dists/normal/cdf' );
*
* var y = cdf( 2.0, 0.0, 1.0 );
* // returns ~0.977
*
* var myCDF = cdf.factory( 10.0, 2.0 );
* y = myCDF( 10.0 );
* // returns 0.5
*/

/**
* Normal distribution probability density function (PDF).
*
* @module @stdlib/stats/base/dists/normal/pdf
*
* @example
* var pdf = require( '@stdlib/stats/base/dists/normal/pdf' );
*
* var y = pdf( 2.0, 0.0, 1.0 );
* // returns ~0.054
*
* var myPDF = pdf.factory( 10.0, 2.0 );
* y = myPDF( 10.0 );
* // returns ~0.199
*/

/**
* One fourth times the mathematical constant `π`.
*
* @constant
* @type {number}
* @default 7.85398163397448309616e-1
* @see [Wikipedia]{@link https://en.wikipedia.org/wiki/Pi}
*/

/**
* One fourth times the mathematical constant `π`.
*
* @module @stdlib/constants/math/float64-fourth-pi
* @type {number}
*
* @example
* var FOURTH_PI = require( '@stdlib/constants/math/float64-fourth-pi' );
* // returns 7.85398163397448309616e-1
*/

/**
* One half times the mathematical constant `π`.
*
* @constant
* @type {number}
* @default 1.5707963267948966
* @see [Wikipedia]{@link https://en.wikipedia.org/wiki/Pi}
*/

/**
* One half times the mathematical constant `π`.
*
* @module @stdlib/constants/math/float64-half-pi
* @type {number}
*
* @example
* var HALF_PI = require( '@stdlib/constants/math/float64-half-pi' );
* // returns 1.5707963267948966
*/

/**
* One half times the natural logarithm of 2.
*
* @module @stdlib/constants/math/float64-half-ln-two
* @type {number}
*
* @example
* var HALF_LN2 = require( '@stdlib/constants/math/float64-half-ln-two' );
* // returns 3.46573590279972654709e-01
*/

/**
* One half times the natural logarithm of 2.
*
* ```tex
* \frac{\ln 2}{2}
* ```
*
* @constant
* @type {number}
* @default 3.46573590279972654709e-01
*/

/**
* Return the maximum absolute value.
*
* @module @stdlib/math/base/special/maxabs
*
* @example
* var maxabs = require( '@stdlib/math/base/special/maxabs' );
*
* var v = maxabs( 3.14, -4.2 );
* // returns 4.2
*
* v = maxabs( 5.9, 3.14, 4.2 );
* // returns 5.9
*
* v = maxabs( 3.14, NaN );
* // returns NaN
*
* v = maxabs( +0.0, -0.0 );
* // returns +0.0
*/

/**
* Return the minimum absolute value.
*
* @module @stdlib/math/base/special/minabs
*
* @example
* var minabs = require( '@stdlib/math/base/special/minabs' );
*
* var v = minabs( -3.14, 4.2 );
* // returns 3.14
*
* v = min( 5.9, 3.14, 4.2 );
* // returns 3.14
*
* v = min( 3.14, NaN );
* // returns NaN
*
* v = min( +0.0, -0.0 );
* // returns +0.0
*/

/**
* Return the minimum value.
*
* @module @stdlib/math/base/special/min
*
* @example
* var min = require( '@stdlib/math/base/special/min' );
*
* var v = min( 3.14, 4.2 );
* // returns 3.14
*
* v = min( 5.9, 3.14, 4.2 );
* // returns 3.14
*
* v = min( 3.14, NaN );
* // returns NaN
*
* v = min( +0.0, -0.0 );
* // returns -0.0
*/

/**
* Returns a function for evaluating the cumulative distribution function (CDF) for a Normal distribution.
*
* @param {number} mu - mean
* @param {NonNegativeNumber} sigma - standard deviation
* @returns {Function} function to evaluate the cumulative distribution function
*
* @example
* var cdf = factory( 10.0, 2.0 );
* var y = cdf( 10.0 );
* // returns 0.5
*
* y = cdf( 12.0 );
* // returns ~0.841
*/

/**
* Returns a function for evaluating the cumulative distribution function (CDF) for a Student's t distribution with degrees of freedom `v`.
*
* @param {PositiveNumber} v - degrees of freedom
* @returns {Function} CDF
*
* @example
* var cdf = factory( 0.5 );
* var y = cdf( 3.0 );
* // returns ~0.816
*
* y = cdf( 1.0 );
* // returns ~0.699
*/

/**
* Returns a function for evaluating the cumulative distribution function (CDF) of a degenerate distribution centered at a provided mean value.
*
* @param {number} mu - constant value of distribution
* @returns {Function} function to evaluate the cumulative distribution function
*
* @example
* var cdf = factory( 5.0 );
*
* var y = cdf( 3.0 );
* // returns 0.0
*
* y = cdf( 6.0 );
* // returns 1.0
*
* y = cdf( NaN );
* // returns NaN
*/

/**
* Returns a function for evaluating the probability density function (PDF) for a Student's t distribution with `v` degrees of freedom.
*
* @param {PositiveNumber} v - degrees of freedom
* @returns {Function} PDF
*
* @example
* var pdf = factory( 1.0 );
* var y = pdf( 3.0 );
* // returns ~0.032
*
* y = pdf( 1.0 );
* // returns ~0.159
*/

/**
* Returns a function for evaluating the probability density function (PDF) for a normal distribution.
*
* @param {number} mu - mean
* @param {NonNegativeNumber} sigma - standard deviation
* @returns {Function} function to evaluate the probability density function
*
* @example
* var pdf = factory( 10.0, 2.0 );
* var y = pdf( 10.0 );
* // returns ~0.199
*
* y = pdf( 5.0 );
* // returns ~0.009
*/

/**
* Returns a function for evaluating the probability density function (PDF) of a degenerate distribution centered at a provided mean value.
*
* @param {number} mu - value at which to center the distribution
* @returns {Function} function to evaluate the probability density function
*
* @example
* var pdf = factory( 5.0 );
*
* var y = pdf( 0.0 );
* // returns 0.0
*
* y = pdf( 5.0 );
* // returns Infinity
*/

/**
* Returns the maximum absolute value.
*
* @param {number} [x] - first number
* @param {number} [y] - second number
* @param {...number} [args] - numbers
* @returns {number} maximum absolute value
*
* @example
* var v = maxabs( 3.14, -4.2 );
* // returns 4.2
*
* @example
* var v = maxabs( 5.9, 3.14, 4.2 );
* // returns 5.9
*
* @example
* var v = maxabs( 3.14, NaN );
* // returns NaN
*
* @example
* var v = maxabs( +0.0, -0.0 );
* // returns +0.0
*/

/**
* Returns the minimum absolute value.
*
* @param {number} [x] - first number
* @param {number} [y] - second number
* @param {...number} [args] - numbers
* @returns {number} minimum absolute value
*
* @example
* var v = minabs( -3.14, 4.2 );
* // returns 3.14
*
* @example
* var v = minabs( 5.9, 3.14, 4.2 );
* // returns 3.14
*
* @example
* var v = minabs( 3.14, NaN );
* // returns NaN
*
* @example
* var v = minabs( +0.0, -0.0 );
* // returns +0.0
*/

/**
* Returns the minimum value.
*
* @param {number} [x] - first number
* @param {number} [y] - second number
* @param {...number} [args] - numbers
* @returns {number} minimum value
*
* @example
* var v = min( 3.14, 4.2 );
* // returns 3.14
*
* @example
* var v = min( 5.9, 3.14, 4.2 );
* // returns 3.14
*
* @example
* var v = min( 3.14, NaN );
* // returns NaN
*
* @example
* var v = min( +0.0, -0.0 );
* // returns -0.0
*/

/**
* Series approximation to the incomplete beta.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {number} result - initial result value
* @returns {Function} series function
*/

/**
* Series representation for upper fraction when `z` is small.
*
* @private
* @param {number} a - function parameter
* @param {number} x - function parameter
* @returns {Function} series function
*/

/**
* Smallest positive single-precision floating-point normal number.
*
* @module @stdlib/constants/math/float32-smallest-normal
* @type {number}
*
* @example
* var FLOAT32_SMALLEST_NORMAL = require( '@stdlib/constants/math/float32-smallest-normal' );
* // returns 1.1754943508222875e-38
*/

/**
* Square root of double-precision floating-point epsilon.
*
* @module @stdlib/constants/math/float64-sqrt-eps
* @type {number}
*
* @example
* var FLOAT64_SQRT_EPSILON = require( '@stdlib/constants/math/float64-sqrt-eps' );
* // returns 0.14901161193847656e-7
*/

/**
* Square root of double-precision floating-point epsilon.
*
* ```tex
* \sqrt{\frac{1}{2^{52}}}
* ```
*
* @constant
* @type {number}
* @default 0.14901161193847656e-7
* @see [IEEE 754]{@link https://en.wikipedia.org/wiki/IEEE_754-1985}
* @see [Machine Epsilon]{@link https://en.wikipedia.org/wiki/Machine_epsilon}
*/

/**
* Student's t distribution cumulative distribution function (CDF).
*
* @module @stdlib/stats/base/dists/t/cdf
*
* @example
* var cdf = require( '@stdlib/stats/base/dists/t/cdf' );
*
* var y = cdf( 2.0, 0.1 );
* // returns ~0.611
*
* y = cdf( 1.0, 2.0 );
* // returns ~0.789
*
* y = cdf( -1.0, 4.0 );
* // returns ~0.187
*
* var mycdf = cdf.factory( 0.5 );
* y = mycdf( 3.0 );
* // returns ~0.816
*
* y = mycdf( 1.0 );
* // returns ~0.699
*/

/**
* Student's t distribution probability density function (PDF).
*
* @module @stdlib/stats/base/dists/t/pdf
*
* @example
* var pdf = require( '@stdlib/stats/base/dists/t/pdf' );
*
* var y = pdf( 3.0, 1.0 );
* // returns ~0.032
*
* var myPDF = pdf.factory( 3.0 );
* y = myPDF( 1.0 );
* // returns ~0.207
*/

/**
* Sum the elements of the series given by the supplied function.
*
* @module @stdlib/math/base/tools/sum-series
*
* @example
* var sumSeries = require( '@stdlib/math/base/tools/sum-series' );
*
* var gen = geometricSeriesClosure( 0.9 )
* var out = sumSeries( gen );
* // returns 10
*
* function geometricSeriesClosure( x ) {
*     var exponent = -1;
*     return function() {
*         exponent += 1;
*         return Math.pow( x, exponent );
*     };
* }
*/

/**
* Sum the elements of the series given by the supplied function.
*
* @param {Function} generator - series function
* @param {Object} [options] - function options
* @param {PositiveInteger} [options.maxTerms=1000000] - maximum number of terms to be added
* @param {PositiveNumber} [options.tolerance=2.22e-16] - further terms are only added as long as the next term is greater than current term times the tolerance
* @param {number} [options.initialValue=0] - initial value of the resulting sum
* @returns {number} sum of all series terms
*
* @example
* var gen = geometricSeriesClosure( 0.9 )
* var out = sumSeries( gen );
* // returns 10.0
*
* function geometricSeriesClosure( x ) {
*     var exponent = -1;
*     return function() {
*         exponent += 1;
*         return Math.pow( x, exponent );
*     };
* }
*/

/**
* Sum the elements of the series given by the supplied function.
*
* @param {Function} generator - series function
* @param {Object} [options] - function options
* @param {PositiveInteger} [options.maxTerms=1000000] - maximum number of terms to be added
* @param {PositiveNumber} [options.tolerance=2.22e-16] - further terms are only added as long as the next term is greater than current term times the tolerance
* @param {number} [options.initialValue=0] - initial value of the resulting sum
* @returns {number} sum of all series terms
*
* @example
* var gen = geometricSeriesGenerator( 0.9 );
* var out = sumSeries( gen );
* // returns 10.0
*
* function* geometricSeriesGenerator( x ) {
*     var exponent = 0;
*     while ( true ) {
*         yield Math.pow( x, exponent );
*         exponent += 1;
*     }
* }
*/

/**
* Sums elements of the series expansion of the lower incomplete gamma function.
*
* ## Method
*
* -   Multiply result by `((z^a) * (e^-z) / a)` to get the full lower incomplete integral.
* -   Divide by `tgamma(a)` to get the normalized value.
*
* @private
* @param {number} a - function parameter
* @param {number} z - function parameter
* @param {number} initialValue - initial value of the resulting sum
* @returns {number} sum of terms of lower gamma series
*/

/**
* Test for native `function*()` support.
*
* @module @stdlib/assert/has-generator-support
*
* @example
* var hasGeneratorSupport = require( '@stdlib/assert/has-generator-support' );
*
* var bool = hasGeneratorSupport();
* // returns <boolean>
*/

/**
* Tests for native `function*()` support.
*
* @returns {boolean} boolean indicating if an environment has native `function*()` support
*
* @example
* var bool = hasGeneratorSupport();
* // returns <boolean>
*/

/**
* The mathematical constant `π` times `2`.
*
* @constant
* @type {number}
* @default 6.283185307179586
* @see [Wikipedia]{@link https://en.wikipedia.org/wiki/Pi}
*/

/**
* The mathematical constant `π` times `2`.
*
* @module @stdlib/constants/math/float64-two-pi
* @type {number}
*
* @example
* var TWO_PI = require( '@stdlib/constants/math/float64-two-pi' );
* // returns 6.283185307179586
*/

/**
* The smallest positive single-precision floating-point normal number.
*
* ## Notes
*
* The number has the value
*
* ```tex
* \frac{1}{2^{127-1}}
* ```
*
* which corresponds to the bit sequence
*
* ```binarystring
* 0 00000001 00000000000000000000000
* ```
*
* @constant
* @type {number}
* @default 1.1754943508222875e-38
* @see [IEEE 754]{@link https://en.wikipedia.org/wiki/IEEE_754-1985}
*/

/**
* This is DiDonato and Morris's BGRAT routine, see Eq's 9 through 9.6.
*
* @private
* @param {NonNegativeNumber} a - function parameter
* @param {NonNegativeNumber} b - function parameter
* @param {Probability} x - function parameter
* @param {Probability} y - probability equal to `1-x`
* @param {NonNegativeInteger} s0 - initial value
* @param {number} mult - initial value
* @param {boolean} normalized - boolean indicating whether to evaluate the regularized or non-regularized incomplete beta function
* @returns {number} function value
*/

/***/

// -0.2929 < x < 0.41422

// 0.7316 <= x <=  0.9

// 0x3FD62E42 0xFEFA39EF

// 0x3FF0A2B2 0x3F3BAB73

// 0x3ff00000 - (0x200000>>k) = 1072693248 - (0x200000>>k) => 0x200000 = 0 00000000010 00000000000000000000

// 0xBC50C7CAA48A971F => TT = -(tail of TF)

// 1.7316 <= x < 2

// 2*N+1

// Add a source directive for debugging:

// Add k to y's exponent:

// Address case where `f` cannot be represented exactly...

// And from from 9.2:

// Any number raised to zero (including 0) is always 1 => b^0 - 1 = 0

// Apply a bit mask (0 00000000000 11111111111111111111) to remove the exponent:

// Approximation of log1p(f)...

// Argument reduction (reduce to [0,2))...

// Argument reduction...

// As `delta` is a small integer, we can use a finite product:

// Avoid exceeding the maximum stack size on V8 :(. Note that the choice of `500` was empirically determined...

// Begin by evaluating the next Pn from Eq 9.4:

// Begin by testing whether we're in the "bad" zone where the result will be near 0.5 and the usual series and continued fractions are slow to converge:

// Bit shift to isolate the exponent and then subtract the bias:

// Both a,b < 1:

// Both a,b >= 1:

// Build the output array...

// Calculate Q via finite sum for half integer a:

// Calculate Q via finite sum:

// Calculate the increment:

// Can we do better here?

// Case A: Iterate over generator object created by a generator function...

// Case B: Repeatedly call function...

// Case: 1.5 < z <= 2

// Changeover criterion chosen to give a changeover at Q ~ 0.33:

// Changeover here occurs when P ~ 0.75 or Q ~ 0.25:

// Check for over/underflow in the power terms:

// Check if `f=x` can be represented exactly (no need for correction terms), allowing us to bypass argument reduction...

// Check if argument reduction is needed and if we can just return a small value approximation requiring less computation but with equivalent accuracy...

// Check if u significand is less than sqrt(2) significand => 0x6a09e => 01101010000010011110

// Close all the parentheses...

// Close the function:

// Code generation. Start with the function definition...

// Combine power terms with Lanczos approximation:

// Combine with the leftover terms from the Lanczos approximation:

// Compute P:

// Compute Q:

// Compute square root of the result and then square it:

// Compute the 4th root of the result then square it twice:

// Correction term is zero:

// Correction term...

// Create the function body...

// Create the function in the global scope:

// Derivative not set...

// EXPORTS //

// Exponentiation would yield a complex result...

// Extend to a few very special cases...

// Extract the more significant bits from |x|:

// FUNCTIONS //

// Filter out huge and non-finite arguments...

// Finally the normal case:

// First base near 1 only:

// First term underflows so we need to start at the mode of the distribution and work outwards:

// General case. Direct computation is most accurate, but use various fallbacks for different parts of the problem domain:

// General case:

// If `x == 0`, return the first coefficient...

// If `x` is an integer, the mod is an integer...

// If `x` is even, recognize that `(-b)**x == (b)**x`...

// If no coefficients, the function always returns 0...

// If x < 2, use lgamma(x) = lgamma(x+1) - log(x)

// If x equals 1 or 2, return 0

// If |x| < 2**-70, return -ln(|x|)

// If |x| >= 2**52, must be -integer

// In each of the following, we further reduce to [-π/4,π/4)...

// Incomplete beta power term, combined with the Lanczos approximation:

// Lanczos calculation:

// Limit chosen so that we use Temme's expansion only if the result would be larger than about 10^-6. Below that the regular series and continued fractions converge OK, and if we use Temme's method we get increasing errors from the dominant erfc term as it's (inexact) argument increases in magnitude.

// MAIN //

// MODULES //

// No good/quick approximation for ln(b)*x, so we have to evaluate...

// Non-normalized, just compute the power:

// Normalize u by setting the exponent to 1023 (bias) => 0x3ff00000 => 0 01111111111 00000000000000000000

// Normalize u/2 by setting the exponent to 1022 (bias-1 => 2**-1 = 1/2) => 0x3fe00000 => 0 01111111110 00000000000000000000

// Now an initial value for J, see 9.6: gammainc( u, b, regularized, upper )

// Now we can start to pull things together and evaluate the sum in Eq 9:

// Now we want Jn from Jn-1 using Eq 9.6:

// OK, starting slightly above the mode didn't work, we'll have to sum the terms the old fashioned way:

// Oops, under/overflow, sidestep if we can:

// Otherwise, evaluate the polynomial...

// Overflow, return an arbitrarily large value:

// Pull it together with Eq 9:

// Relate to the binomial distribution and use a finite sum:

// Repeatedly call function...

// Second base near 1 only:

// Series summation from 0 to k-1:

// Set y = |x|:

// Sidestep on a, and then use the series representation:

// Sidestep so we can use the series representation:

// Since x is so small that P is necessarily very small too, use http://functions.wolfram.com/GammaBetaErf/GammaRegularized/06/01/05/01/01/

// Some values we'll need later, these are Eq 9.1:

// Some variables we'll need...

// Special case for large a and a ~ z:

// Special case see: http://functions.wolfram.com/GammaBetaErf/BetaRegularized/03/01/01/

// Special case where the base of the power term is close to 1; compute `(1+x)^y` instead:

// Special case: 0

// Special cases:

// Special cases: NaN, +-infinity

// Split up the calculation to avoid spurious overflow:

// Subtract hu significand from next largest hu => 0 00000000001 00000000000000000000 => 0x00100000 => 1048576

// Swap `a` and `b`:

// TODO: consider extracting as a constant

// TODO: consider moving to pkg

// This avoids possible overflow, but appears to be marginally less accurate:

// This is method 2 below, done in logs, we're really outside the range of this method, but since the result is almost certainly infinite, we should probably be OK:

// This is method 2 below, done in logs:

// This is method 4 below, done in logs:

// This isn't very sophisticated, or accurate, but it does work:

// Treat a < 1 as a special case because our Lanczos approximations are optimized against the factorials with a > 1, and for high precision types very small values of `a` can give rather erroneous results for gamma:

// Try http://functions.wolfram.com/06.06.06.0039.01

// Use a simple two-term Taylor series...

// VARIABLES //

// Value at N = 0

// WARNING: not thread safe

// We have an arcsine distribution:

// We need logs, and this *will* cancel:

// We need the quantity Pn. Unfortunately, this is computed recursively and requires a full history of all the previous values. No choice but to declare a big table and hope it's big enough...

// When the base of the exponent is very near 1 we get really gross errors unless extra care is taken:

// Zero raised to any number (except 0) is always zero => 0^x - 1 = -1

// `l1` and `l2` are the base of the exponents minus one:

// arcsin(1-x) = pi/2 - sqrt(2x)(1+R(x))

// asm type annotation

// eslint-disable-line default-case

// eslint-disable-line max-len

// eslint-disable-line no-eval

// eslint-disable-line no-new-func

// eslint-disable-line no-plusplus

// eslint-disable-line no-unused-vars

// exact

// gammaln(1+s) = ln(s) + gammaln(s)

// handle +-0 (IEEE 754-2008 spec)

// handles +-0 (IEEE 754-2008)

// if x <= -56*ln(2)

// if x > sqrt(2)/2-1 => ~-0.2929

// if |f| < 2**-20

// if |x| < 1.5*ln(2)

// if |x| < 2**-29

// if |x| < sqrt(2)-1 => ~0.41422

// if |x| > 0.5*ln(2)

// if |x| >= 56*ln(2)

// if |x| >= 709.78...

// max value => 1048575

// pi/2 = PIO2 + MOREBITS

// positive unbiased exponent

// see 9.3.

// sign preserving

// suffice to return exp(x)-1

// t*ln2_hi is exact here

// t=1-2^-k

// t=2^-k

// use a bitwise OR to cast `k` to an integer (see also asm.js type annotations: http://asmjs.org/spec/latest/#annotations)

// x is now in primary range...
