Average Error: 41.2 → 0.9
Time: 2.1s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

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

\end{array}
double f(double x) {
        double r67832 = x;
        double r67833 = exp(r67832);
        double r67834 = 1.0;
        double r67835 = r67833 - r67834;
        double r67836 = r67833 / r67835;
        return r67836;
}
↓
double f(double x) {
        double r67837 = x;
        double r67838 = exp(r67837);
        double r67839 = 0.0;
        bool r67840 = r67838 <= r67839;
        double r67841 = 1.0;
        double r67842 = 1.0;
        double r67843 = r67842 / r67838;
        double r67844 = r67841 - r67843;
        double r67845 = r67841 / r67844;
        double r67846 = 0.5;
        double r67847 = 0.08333333333333333;
        double r67848 = r67847 * r67837;
        double r67849 = r67841 / r67837;
        double r67850 = r67848 + r67849;
        double r67851 = r67846 + r67850;
        double r67852 = r67840 ? r67845 : r67851;
        return r67852;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.2
Target40.8
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.0

    1. Initial program 0

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

    3. Applied clear-num0

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

    4. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.3

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

    2. Taylor expanded around 0 1.4

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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