Average Error: 41.2 → 1.0
Time: 5.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\]
\frac{e^{x}}{e^{x} - 1}
↓
\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}
double f(double x) {
        double r66137 = x;
        double r66138 = exp(r66137);
        double r66139 = 1.0;
        double r66140 = r66138 - r66139;
        double r66141 = r66138 / r66140;
        return r66141;
}
↓
double f(double x) {
        double r66142 = x;
        double r66143 = exp(r66142);
        double r66144 = 2.0;
        double r66145 = pow(r66142, r66144);
        double r66146 = 0.16666666666666666;
        double r66147 = r66142 * r66146;
        double r66148 = 0.5;
        double r66149 = r66147 + r66148;
        double r66150 = r66145 * r66149;
        double r66151 = r66150 + r66142;
        double r66152 = r66143 / r66151;
        return r66152;
}

Error

Bits error versus x

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Your Program's Arguments

Results

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Target

Original41.2
Target40.8
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Initial program 41.2

    \[\frac{e^{x}}{e^{x} - 1}\]

  2. Taylor expanded around 0 11.7

    \[\leadsto \frac{e^{x}}{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}\]

  3. Simplified1.0

    \[\leadsto \frac{e^{x}}{\color{blue}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}}\]

  4. Final simplification1.0

    \[\leadsto \frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))