Average Error: 40.3 → 0.7
Time: 16.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 9.438996161322502:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(x \cdot \sqrt{\frac{1}{12}}\right) \cdot \sqrt{\frac{1}{12}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 9.438996161322502:\\
\;\;\;\;\frac{e^{x}}{e^{x} - 1}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(x \cdot \sqrt{\frac{1}{12}}\right) \cdot \sqrt{\frac{1}{12}}\\

\end{array}
double f(double x) {
        double r6236848 = x;
        double r6236849 = exp(r6236848);
        double r6236850 = 1.0;
        double r6236851 = r6236849 - r6236850;
        double r6236852 = r6236849 / r6236851;
        return r6236852;
}
↓
double f(double x) {
        double r6236853 = x;
        double r6236854 = exp(r6236853);
        double r6236855 = 1.0;
        double r6236856 = r6236854 - r6236855;
        double r6236857 = r6236854 / r6236856;
        double r6236858 = 9.438996161322502;
        bool r6236859 = r6236857 <= r6236858;
        double r6236860 = 0.5;
        double r6236861 = r6236855 / r6236853;
        double r6236862 = r6236860 + r6236861;
        double r6236863 = 0.08333333333333333;
        double r6236864 = sqrt(r6236863);
        double r6236865 = r6236853 * r6236864;
        double r6236866 = r6236865 * r6236864;
        double r6236867 = r6236862 + r6236866;
        double r6236868 = r6236859 ? r6236857 : r6236867;
        return r6236868;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.9
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp x) (- (exp x) 1)) < 9.438996161322502

    1. Initial program 1.0

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

    2. Taylor expanded around -inf 1.0

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

    if 9.438996161322502 < (/ (exp x) (- (exp x) 1))

    1. Initial program 61.2

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

    2. Taylor expanded around 0 0.6

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
    3. Using strategy rm

    4. Applied add-sqr-sqrt0.6

      \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{12}} \cdot \sqrt{\frac{1}{12}}\right)} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\]

    5. Applied associate-*l*0.6

      \[\leadsto \color{blue}{\sqrt{\frac{1}{12}} \cdot \left(\sqrt{\frac{1}{12}} \cdot x\right)} + \left(\frac{1}{x} + \frac{1}{2}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 9.438996161322502:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(x \cdot \sqrt{\frac{1}{12}}\right) \cdot \sqrt{\frac{1}{12}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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