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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.
*/
'use strict';
// MODULES //
var expm1f = require( '@stdlib/math/base/special/expm1f' );
var absf = require( '@stdlib/math/base/special/absf' );
var f32 = require( '@stdlib/number/float64/base/to-float32' );
var EPS = require( '@stdlib/constants/float32/eps' );
var PINF = require( '@stdlib/constants/float32/pinf' );
var MAXLOGF = require( '@stdlib/constants/float32/max-ln' );
// VARIABLES //
var ONE = f32( 1.0 );
var TWO = f32( 2.0 );
var OVERFLOW_THRESHOLD = f32( 93.25821685791016 );
// MAIN //
/**
* Computes the relative error exponential of a single-precision floating-point number.
*
* ## Method
*
* To avoid overflow when \\( x \\) exceeds the natural logarithm of the maximum single-precision floating-point number, we can derive the following identity:
*
* ```tex
* \begin{align*}
* \frac{e^x - 1}{x} &= \frac{(e^{x/2} - 1)(e^{x/2} + 1)}{x} \\
* &= \frac{(e^{x/2} - 1)(e^{x/2} - 1 + 2)}{x} \\
* &= \frac{\operatorname{expm1}(x/2) \cdot (\operatorname{expm1}(x/2) + 2)}{x} \\
* &= \frac{\operatorname{expm1}(x/2)}{x} \cdot (\operatorname{expm1}(x/2) + 2)
* \end{align*}
* ```
*
* @param {number} x - input value
* @returns {number} function value
*
* @example
* var v = expm1relf( 0.0 );
* // returns 1.0
*
* @example
* var v = expm1relf( 1.0 );
* // returns ~1.718
*
* @example
* var v = expm1relf( -1.0 );
* // returns ~0.632
*
* @example
* var v = expm1relf( NaN );
* // returns NaN
*/
function expm1relf( x ) {
var tmp;
x = f32( x ); // ensure type consistency (values may arrive as `float64`s)
if ( absf( x ) <= EPS ) {
return ONE; // L'Hopital's Rule
}
if ( x < MAXLOGF ) {
return f32( expm1f( x ) / x );
}
if ( x >= OVERFLOW_THRESHOLD ) {
return PINF; // L'Hopital's Rule
}
// MAXLOGF <= x < OVERFLOW_THRESHOLD
tmp = expm1f( f32( x/TWO ) );
return f32( f32( tmp/x ) * f32( tmp+TWO ) ); // note: we keep the `+2` term in order to prevent compiler optimizations from rearranging terms, even though `tmp + 2 = tmp` due to the limits of floating-point precision
}
// EXPORTS //
module.exports = expm1relf;
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