Average Error: 41.4 → 0.5
Time: 11.1s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{1}{\left(1 - \sqrt{\frac{1}{e^{x}}}\right) \cdot \left(\sqrt{\frac{1}{e^{x}}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\
\;\;\;\;\frac{1}{\left(1 - \sqrt{\frac{1}{e^{x}}}\right) \cdot \left(\sqrt{\frac{1}{e^{x}}} + 1\right)}\\

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

\end{array}
double f(double x) {
        double r4676246 = x;
        double r4676247 = exp(r4676246);
        double r4676248 = 1.0;
        double r4676249 = r4676247 - r4676248;
        double r4676250 = r4676247 / r4676249;
        return r4676250;
}
↓
double f(double x) {
        double r4676251 = x;
        double r4676252 = exp(r4676251);
        double r4676253 = 0.966199829895273;
        bool r4676254 = r4676252 <= r4676253;
        double r4676255 = 1.0;
        double r4676256 = 1.0;
        double r4676257 = r4676256 / r4676252;
        double r4676258 = sqrt(r4676257);
        double r4676259 = r4676255 - r4676258;
        double r4676260 = r4676258 + r4676255;
        double r4676261 = r4676259 * r4676260;
        double r4676262 = r4676255 / r4676261;
        double r4676263 = 0.08333333333333333;
        double r4676264 = r4676263 * r4676251;
        double r4676265 = r4676255 / r4676251;
        double r4676266 = 0.5;
        double r4676267 = r4676265 + r4676266;
        double r4676268 = r4676264 + r4676267;
        double r4676269 = r4676254 ? r4676262 : r4676268;
        return r4676269;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Recombined 2 regimes into one program.

  4. Final simplification0.5

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

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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