Average Error: 41.0 → 0.6
Time: 12.6s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9967130846117587816834770819696132093668:\\ \;\;\;\;\frac{e^{x}}{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} - 1 \cdot \left(1 \cdot 1\right)} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9967130846117587816834770819696132093668:\\
\;\;\;\;\frac{e^{x}}{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} - 1 \cdot \left(1 \cdot 1\right)} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\

\mathbf{else}:\\
\;\;\;\;e^{\log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\\

\end{array}
double f(double x) {
        double r5543840 = x;
        double r5543841 = exp(r5543840);
        double r5543842 = 1.0;
        double r5543843 = r5543841 - r5543842;
        double r5543844 = r5543841 / r5543843;
        return r5543844;
}
↓
double f(double x) {
        double r5543845 = x;
        double r5543846 = exp(r5543845);
        double r5543847 = 0.9967130846117588;
        bool r5543848 = r5543846 <= r5543847;
        double r5543849 = r5543846 * r5543846;
        double r5543850 = r5543849 * r5543846;
        double r5543851 = 1.0;
        double r5543852 = r5543851 * r5543851;
        double r5543853 = r5543851 * r5543852;
        double r5543854 = r5543850 - r5543853;
        double r5543855 = r5543846 / r5543854;
        double r5543856 = r5543846 * r5543851;
        double r5543857 = r5543852 + r5543856;
        double r5543858 = r5543849 + r5543857;
        double r5543859 = r5543855 * r5543858;
        double r5543860 = 0.08333333333333333;
        double r5543861 = r5543860 * r5543845;
        double r5543862 = cbrt(r5543861);
        double r5543863 = r5543862 * r5543862;
        double r5543864 = log(r5543863);
        double r5543865 = exp(r5543864);
        double r5543866 = r5543865 * r5543862;
        double r5543867 = 1.0;
        double r5543868 = r5543867 / r5543845;
        double r5543869 = 0.5;
        double r5543870 = r5543868 + r5543869;
        double r5543871 = r5543866 + r5543870;
        double r5543872 = r5543848 ? r5543859 : r5543871;
        return r5543872;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.0
Target40.5
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9967130846117588

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

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

    if 0.9967130846117588 < (exp x)

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0.9

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

    4. Applied add-cube-cbrt0.9

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

    6. Applied add-exp-log0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9967130846117587816834770819696132093668:\\ \;\;\;\;\frac{e^{x}}{\left(e^{x} \cdot e^{x}\right) \cdot e^{x} - 1 \cdot \left(1 \cdot 1\right)} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right)} \cdot \sqrt[3]{\frac{1}{12} \cdot x} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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