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/**
* @license Apache-2.0
*
* Copyright (c) 2026 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/12.2.0/lib/msun/src/s_erff.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.
* ```
*/
/* eslint-enable stdlib/jsdoc-no-shortcut-reference-link, stdlib/jsdoc-no-undefined-references */
'use strict';
// MODULES //
var exp = require( '@stdlib/math/base/special/exp' );
var absf = require( '@stdlib/math/base/special/absf' );
var toWordf = require( '@stdlib/number/float32/base/to-word' );
var fromWordf = require( '@stdlib/number/float32/base/from-word' );
var f32 = require( '@stdlib/number/float64/base/to-float32' );
var polyvalPP = require( './polyval_pp.js' );
var polyvalQQ = require( './polyval_qq.js' );
var polyvalPA = require( './polyval_pa.js' );
var polyvalQA = require( './polyval_qa.js' );
var polyvalRA = require( './polyval_ra.js' );
var polyvalSA = require( './polyval_sa.js' );
var polyvalRB = require( './polyval_rb.js' );
var polyvalSB = require( './polyval_sb.js' );
// VARIABLES //
var TINY = f32( 1.0e-30 );
var HALF = f32( 0.5 );
var ONE = f32( 1.0 );
var TWO = f32( 2.0 );
// 0x3f57bb00
var ERX = f32( 8.42697144e-01 );
// MAIN //
/**
* Evaluates the complementary error function for a single-precision floating-point number.
*
* ## Method
*
* 1. For \\( x \in [0, 0.84375] \\), we use the approximation
*
* ```tex
* \operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = 1 - x \cdot \frac{P(x^2)}{Q(x^2)}
* ```
*
* where \\( P \\) and \\( Q \\) are polynomials (pp and qq).
*
* 2. For \\( x \in [0.84375, 1.25] \\), we use the approximation
*
* ```tex
* \operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = 1 - \operatorname{erx} - \frac{P(x-1)}{Q(x-1)}
* ```
*
* where \\( P \\) and \\( Q \\) are polynomials (pa and qa).
*
* 3. For \\( x \in [1.25, 11] \\), we use the approximation
*
* ```tex
* \operatorname{erfc}(x) = \frac{1}{x} e^{-x^2 - 0.5625 + \frac{R(1/x^2)}{S(1/x^2)}}
* ```
*
* where \\( R \\) and \\( S \\) are rational functions.
*
* 4. For \\( x \geq 11 \\), we use the approximation
*
* ```tex
* \operatorname{erfc}(x) = 0
* ```
*
* ## Notes
*
* - Relative error:
* - \\( |x| < 0.84375 \\): \\( 2^{-31} \\)
* - \\( 0.84375 \leq |x| < 1.25 \\): \\( 2^{-31} \\)
* - \\( 1.25 \leq |x| < 2.85715 \\): \\( 2^{-30} \\)
* - \\( 2.85715 \leq |x| < 11 \\): \\( 2^{-30} \\)
*
* @param {number} x - input value
* @returns {number} function value
*
* @see [FreeBSD]{@link https://svnweb.freebsd.org/base/release/12.2.0/lib/msun/src/s_erff.c}
*
* @example
* var y = erfcf( 0.0 );
* // returns 1.0
*
* @example
* var y = erfcf( 1.0 );
* // returns ~0.1573
*
* @example
* var y = erfcf( -1.0 );
* // returns ~1.8427
*
* @example
* var y = erfcf( Infinity );
* // returns 0.0
*
* @example
* var y = erfcf( -Infinity );
* // returns 2.0
*
* @example
* var y = erfcf( NaN );
* // returns NaN
*/
function erfcf( x ) {
var xm2;
var hx;
var ix;
var ax;
var R;
var S;
var P;
var Q;
var s;
var y;
var z;
var r;
x = f32( x );
hx = toWordf( x ) | 0;
ix = (hx & 0x7fffffff) | 0; // remove sign bit
// Handle NaN and ±Infinity
if ( ix >= 0x7f800000 ) {
// ((hx >>> 31) << 1) gives 0 for positive, 2 for negative
// ONE/x gives: 1/NaN=NaN, 1/+Inf=0, 1/-Inf=-0
return f32( f32( ((hx >>> 31) << 1) ) + f32( ONE / x ) );
}
// Case: |x| < 0.84375
if ( ix < 0x3f580000 ) { // 0x3f580000 is the float32 representation of 0.84375
// Case: |x| < 2**-24
if ( ix < 0x33800000 ) {
return f32( ONE - x );
}
z = f32( x * x );
r = f32( polyvalPP( z ) );
s = f32( polyvalQQ( z ) );
y = f32( r / s );
if ( hx < 0x3e800000 ) {
// Case: x < 1/4
return f32( ONE - f32( x + f32( x * y ) ) );
}
r = f32( x * y );
r = f32( r + f32( x - HALF ) );
return f32( HALF - r );
}
// Case: 0.84375 <= |x| < 1.25
if ( ix < 0x3fa00000 ) {
ax = absf( x );
s = f32( f32( ax ) - ONE );
P = f32( polyvalPA( s ) );
Q = f32( polyvalQA( s ) );
if ( hx >= 0 ) {
z = f32( ONE - ERX );
return f32( z - f32( P / Q ) );
}
z = f32( ERX + f32( P / Q ) );
return f32( ONE + z );
}
// Case: |x| < 11
if ( ix < 0x41300000 ) {
ax = absf( x );
ax = f32( ax );
s = f32( ONE / f32( ax * ax ) );
// Case: |x| < 2.85715 ~ 1/0.35
if ( ix < 0x4036db8c ) {
R = f32( polyvalRA( s ) );
S = f32( polyvalSA( s ) );
} else {
// Case: x < -5
if ( hx < 0 && ix >= 0x40a00000 ) {
return TWO-TINY;
}
R = f32( polyvalRB( s ) );
S = f32( polyvalSB( s ) );
}
z = f32( fromWordf( (hx & 0xffffe000) >>> 0 ) );
xm2 = f32( f32( z - ax ) * f32( z + ax ) );
r = f32( exp( f32( f32( f32( -z * z ) - f32( 0.5625 ) ) +
f32( xm2 + f32( R / S ) ) ) ) );
if ( hx > 0 ) {
r = f32( r / ax );
return r;
}
return f32( TWO - f32( r / ax ) );
}
// Case: |x| >= 11
if ( hx > 0 ) {
return TINY * TINY;
}
return TWO - TINY;
}
// EXPORTS //
module.exports = erfcf;
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