Average Error: 40.8 → 0.6
Time: 16.7s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0014285877116162612:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{\left(x + x\right) + x} + -1}{\left(e^{x} + 1\right) \cdot e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0014285877116162612:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{\left(x + x\right) + x} + -1}{\left(e^{x} + 1\right) \cdot e^{x} + 1}}\\

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

\end{array}
double f(double x) {
        double r3862835 = x;
        double r3862836 = exp(r3862835);
        double r3862837 = 1.0;
        double r3862838 = r3862836 - r3862837;
        double r3862839 = r3862836 / r3862838;
        return r3862839;
}
↓
double f(double x) {
        double r3862840 = x;
        double r3862841 = -0.0014285877116162612;
        bool r3862842 = r3862840 <= r3862841;
        double r3862843 = exp(r3862840);
        double r3862844 = r3862840 + r3862840;
        double r3862845 = r3862844 + r3862840;
        double r3862846 = exp(r3862845);
        double r3862847 = -1.0;
        double r3862848 = r3862846 + r3862847;
        double r3862849 = 1.0;
        double r3862850 = r3862843 + r3862849;
        double r3862851 = r3862850 * r3862843;
        double r3862852 = r3862851 + r3862849;
        double r3862853 = r3862848 / r3862852;
        double r3862854 = r3862843 / r3862853;
        double r3862855 = 0.08333333333333333;
        double r3862856 = r3862855 * r3862840;
        double r3862857 = r3862849 / r3862840;
        double r3862858 = 0.5;
        double r3862859 = r3862857 + r3862858;
        double r3862860 = r3862856 + r3862859;
        double r3862861 = r3862842 ? r3862854 : r3862860;
        return r3862861;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.8
Target40.4
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0014285877116162612

    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. Simplified0.0

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

    5. Simplified0.0

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

    if -0.0014285877116162612 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0014285877116162612:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{\left(x + x\right) + x} + -1}{\left(e^{x} + 1\right) \cdot e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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