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* @license Apache-2.0
*
* Copyright (c) 2018 The Stdlib Authors.
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
'use strict';
// MODULES //
var expm1 = require( '@stdlib/math/base/special/expm1' );
var abs = require( '@stdlib/math/base/special/abs' );
var EPS = require( '@stdlib/constants/float64/eps' );
var PINF = require( '@stdlib/constants/float64/pinf' );
// VARIABLES //
var HUGE_VALUE = 7.09782712893383973096e+02; // 0x40862E42 0xFEFA39EF
var OVERFLOW_THRESHOLD = 7.163568913878179e+02;
// MAIN //
/**
* Computes the relative error exponential.
*
* ## Method
*
* To avoid overflow when \\( x \\) exceeds the natural logarithm of the maximum double-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 = expm1rel( 0.0 );
* // returns 1.0
*
* @example
* var v = expm1rel( 1.0 );
* // returns ~1.718
*
* @example
* var v = expm1rel( -1.0 );
* // returns ~0.632
*
* @example
* var v = expm1rel( NaN );
* // returns NaN
*/
function expm1rel( x ) {
var tmp;
if ( abs( x ) <= EPS ) {
return 1.0; // L'Hopital's Rule
}
if ( x < HUGE_VALUE ) {
return expm1( x ) / x;
}
if ( x >= OVERFLOW_THRESHOLD ) {
return PINF; // L'Hopital's Rule
}
// HUGE_VALUE <= x < OVERFLOW_THRESHOLD
tmp = expm1( x/2.0 );
return ( tmp/x ) * ( tmp+2.0 ); // 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 = expm1rel;
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