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* @license Apache-2.0
*
* Copyright (c) 2022 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.
*
* ## Implementation Notes
*
* The function currently uses the double-precision exponential function `exp` due to the
* single-precision `expf` function not yet being merged into the main stdlib repository.
* This may result in slightly higher precision than pure single-precision computation.
* Once `expf` becomes available, this implementation should be updated to use it.
*
* ## Notice
*
* The following copyright, license, and long comment were part of the original implementation available as part of [SciPy]{@link https://github.com/scipy/scipy/blob/ed14bf0a66440a4d164581499fda662121963a56/scipy/special/Faddeeva.cc}. The implementation follows the original, but has been modified for JavaScript.
*
* ```text
* Copyright (c) 2012 Massachusetts Institute of Technology
*
* Permission is hereby granted, free of charge, to any person obtaining
* a copy of this software and associated documentation files (the
* "Software"), to deal in the Software without restriction, including
* without limitation the rights to use, copy, modify, merge, publish,
* distribute, sublicense, and/or sell copies of the Software, and to
* permit persons to whom the Software is furnished to do so, subject to
* the following conditions:
*
* The above copyright notice and this permission notice shall be
* included in all copies or substantial portions of the Software.
*
* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND
* NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE
* LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION
* OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
* WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
* ```
*/
'use strict';
// MODULES //
var exp = require( '@stdlib/math/base/special/exp' );
var PINF = require( '@stdlib/constants/float32/pinf' );
var isnanf = require( '@stdlib/math/base/assert/is-nanf' );
var Float32Array = require( '@stdlib/array/float32' );
var erfcxfY100 = require( './erfcxf_y100.js' );
// VARIABLES //
var INV_SQRT_PI = 0.56418958; // 1 / sqrt(pi) - float32 precision
// MAIN //
/**
* Float32 exponential function that matches scipy's erfcx behavior with float32 inputs.
*
* @private
* @param {number} x - input value
* @returns {number} float32 result
*/
function expf( x ) {
// SciPy's erfcx with float32 inputs still computes in double precision internally
// But converts the final result to float32. We should match this behavior.
var result = exp( x );
var f32 = new Float32Array( [ result ] ); // Convert result to float32
return f32[ 0 ];
}
/**
* Evaluates the scaled complementary error function (single-precision).
*
* ```tex
* \operatorname{erfcx}(x) = \exp{x^2} \cdot \operatorname{erfc}(x)
* ```
*
* ## Notes
*
* - This is a **single-precision (float32)** version of the scaled complementary error function. For double-precision computations, use `@stdlib/math/base/special/erfcx`.
* - Use this function to replace expressions containing `exp{x^2} erfc(x)` in order to avoid errors due to underflow or overflow.
* - For expressions of the form `\exp{-x^2} erfcx(x)`, use the complementary error function `erfc(x)` instead, as this substitution maintains accuracy by avoiding roundoff errors for large values of `x`.
* - Input and output values are computed and returned with single-precision floating-point accuracy.
*
* @param {number} x - input value (treated as single-precision)
* @returns {number} evaluated scaled complementary error function (single-precision)
*
* @example
* var y = erfcxf( 0.0 );
* // returns 1.0
*
* @example
* var y = erfcxf( 1.0 );
* // returns ~0.4276
*
* @example
* var y = erfcxf( -1.0 );
* // returns ~5.01
*
* @example
* var y = erfcxf( 50.0 );
* // returns ~0.011
*
* @example
* var y = erfcxf( -50.0 );
* // returns +Infinity
*
* @example
* var y = erfcxf( NaN );
* // returns NaN
*/
function erfcxf( x ) {
var x2;
if ( isnanf( x ) ) {
return x;
}
if ( x >= 0.0 ) {
if ( x > 50.0 ) { // continued-fraction expansion is faster
if ( x > 5.0e7 ) { // 1-term expansion, important to avoid overflow
return INV_SQRT_PI / x;
}
x2 = x * x;
// 5-term expansion (rely on compiler for CSE), simplified from: INV_SQRT_PI / (x+0.5/(x+1/(x+1.5/(x+2/x))))
return INV_SQRT_PI * ( ( x2 * (x2+4.5) ) + 2.0 ) / ( x * ( ( x2*(x2+5.0) ) + 3.75 ) ); // eslint-disable-line max-len
}
return erfcxfY100( 400.0/(4.0+x) );
}
// Use original thresholds - scipy's erfcx with float32 inputs still uses double precision internally
if ( x < -26.7 ) {
return PINF;
}
x2 = x * x;
if ( x < -6.1 ) {
return 2.0 * expf( x2 );
}
return ( 2.0*expf( x2 ) ) - erfcxfY100( 400.0/(4.0-x) );
}
// EXPORTS //
module.exports = erfcxf;
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