Average Error: 41.4 → 0.5
Time: 16.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{1}{\left(1 - \sqrt{\frac{1}{e^{x}}}\right) \cdot \left(\sqrt{\frac{1}{e^{x}}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\
\;\;\;\;\frac{1}{\left(1 - \sqrt{\frac{1}{e^{x}}}\right) \cdot \left(\sqrt{\frac{1}{e^{x}}} + 1\right)}\\

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

\end{array}
double f(double x) {
        double r4660853 = x;
        double r4660854 = exp(r4660853);
        double r4660855 = 1.0;
        double r4660856 = r4660854 - r4660855;
        double r4660857 = r4660854 / r4660856;
        return r4660857;
}
↓
double f(double x) {
        double r4660858 = x;
        double r4660859 = exp(r4660858);
        double r4660860 = 0.966199829895273;
        bool r4660861 = r4660859 <= r4660860;
        double r4660862 = 1.0;
        double r4660863 = 1.0;
        double r4660864 = r4660863 / r4660859;
        double r4660865 = sqrt(r4660864);
        double r4660866 = r4660862 - r4660865;
        double r4660867 = r4660865 + r4660862;
        double r4660868 = r4660866 * r4660867;
        double r4660869 = r4660862 / r4660868;
        double r4660870 = 0.08333333333333333;
        double r4660871 = r4660862 / r4660858;
        double r4660872 = fma(r4660858, r4660870, r4660871);
        double r4660873 = 0.5;
        double r4660874 = r4660872 + r4660873;
        double r4660875 = r4660861 ? r4660869 : r4660874;
        return r4660875;
}

Error

Bits error versus x

Target

Original41.4
Target41.1
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.966199829895273

    1. Initial program 0.0

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

    3. Applied clear-num0.0

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
    4. Using strategy rm

    5. Applied div-sub63.2

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

    6. Simplified0.0

      \[\leadsto \frac{1}{\color{blue}{1} - \frac{1}{e^{x}}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt0.0

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

    9. Applied *-un-lft-identity0.0

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

    10. Applied difference-of-squares0.0

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

    if 0.966199829895273 < (exp x)

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.8

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

    3. Simplified0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{1}{\left(1 - \sqrt{\frac{1}{e^{x}}}\right) \cdot \left(\sqrt{\frac{1}{e^{x}}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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