Average Error: 40.3 → 0.8
Time: 21.7s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 148.71336856569798:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 148.71336856569798:\\
\;\;\;\;\frac{e^{x}}{e^{x} - 1}\\

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

\end{array}
double f(double x) {
        double r9151793 = x;
        double r9151794 = exp(r9151793);
        double r9151795 = 1.0;
        double r9151796 = r9151794 - r9151795;
        double r9151797 = r9151794 / r9151796;
        return r9151797;
}
↓
double f(double x) {
        double r9151798 = x;
        double r9151799 = exp(r9151798);
        double r9151800 = 1.0;
        double r9151801 = r9151799 - r9151800;
        double r9151802 = r9151799 / r9151801;
        double r9151803 = 148.71336856569798;
        bool r9151804 = r9151802 <= r9151803;
        double r9151805 = 0.08333333333333333;
        double r9151806 = r9151805 * r9151798;
        double r9151807 = 0.5;
        double r9151808 = r9151800 / r9151798;
        double r9151809 = r9151807 + r9151808;
        double r9151810 = r9151806 + r9151809;
        double r9151811 = r9151804 ? r9151802 : r9151810;
        return r9151811;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.8
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp x) (- (exp x) 1)) < 148.71336856569798

    1. Initial program 1.3

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

    if 148.71336856569798 < (/ (exp x) (- (exp x) 1))

    1. Initial program 61.5

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

    2. Taylor expanded around 0 0.6

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

    3. Taylor expanded around 0 0.6

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 148.71336856569798:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019112 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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