Average Error: 40.4 → 1.0
Time: 34.1s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1.0}\]
↓
\[\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}\]
\frac{e^{x}}{e^{x} - 1.0}
↓
\frac{e^{x}}{x + \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot \left(x \cdot x\right)}
double f(double x) {
        double r5752680 = x;
        double r5752681 = exp(r5752680);
        double r5752682 = 1.0;
        double r5752683 = r5752681 - r5752682;
        double r5752684 = r5752681 / r5752683;
        return r5752684;
}
↓
double f(double x) {
        double r5752685 = x;
        double r5752686 = exp(r5752685);
        double r5752687 = 0.5;
        double r5752688 = 0.16666666666666666;
        double r5752689 = r5752688 * r5752685;
        double r5752690 = r5752687 + r5752689;
        double r5752691 = r5752685 * r5752685;
        double r5752692 = r5752690 * r5752691;
        double r5752693 = r5752685 + r5752692;
        double r5752694 = r5752686 / r5752693;
        return r5752694;
}

Error

Bits error versus x

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Results

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Target

Original40.4
Target40.0
Herbie1.0
\[\frac{1.0}{1.0 - e^{-x}}\]

Derivation

  1. Initial program 40.4

    \[\frac{e^{x}}{e^{x} - 1.0}\]

  2. Taylor expanded around 0 12.0

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

  3. Simplified1.0

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

  4. Final simplification1.0

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

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))