Average Error: 41.5 → 0.3
Time: 2.7s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.0000164897469046:\\ \;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 1.0000164897469046:\\
\;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

\end{array}
double f(double x) {
        double r80751 = x;
        double r80752 = exp(r80751);
        double r80753 = 1.0;
        double r80754 = r80752 - r80753;
        double r80755 = r80752 / r80754;
        return r80755;
}
↓
double f(double x) {
        double r80756 = x;
        double r80757 = exp(r80756);
        double r80758 = 1.0000164897469046;
        bool r80759 = r80757 <= r80758;
        double r80760 = 2.0;
        double r80761 = pow(r80756, r80760);
        double r80762 = 0.16666666666666666;
        double r80763 = r80756 * r80762;
        double r80764 = 0.5;
        double r80765 = r80763 + r80764;
        double r80766 = r80761 * r80765;
        double r80767 = r80766 + r80756;
        double r80768 = r80757 / r80767;
        double r80769 = 1.0;
        double r80770 = 1.0;
        double r80771 = r80770 / r80757;
        double r80772 = r80769 - r80771;
        double r80773 = r80769 / r80772;
        double r80774 = r80759 ? r80768 : r80773;
        return r80774;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.5
Target40.9
Herbie0.3
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 1.0000164897469046

    1. Initial program 41.6

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

    2. Taylor expanded around 0 10.6

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

    3. Simplified0.3

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

    if 1.0000164897469046 < (exp x)

    1. Initial program 37.2

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied clear-num37.2

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]

    4. Simplified2.1

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.0000164897469046:\\ \;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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