Average Error: 41.4 → 0.5
Time: 13.1s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{1} + \sqrt{e^{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\
\;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{1} + \sqrt{e^{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}{\sqrt{e^{x}} - \sqrt{1}}\\

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

\end{array}
double f(double x) {
        double r6039503 = x;
        double r6039504 = exp(r6039503);
        double r6039505 = 1.0;
        double r6039506 = r6039504 - r6039505;
        double r6039507 = r6039504 / r6039506;
        return r6039507;
}
↓
double f(double x) {
        double r6039508 = x;
        double r6039509 = exp(r6039508);
        double r6039510 = 0.966199829895273;
        bool r6039511 = r6039509 <= r6039510;
        double r6039512 = cbrt(r6039509);
        double r6039513 = r6039512 * r6039512;
        double r6039514 = 1.0;
        double r6039515 = sqrt(r6039514);
        double r6039516 = sqrt(r6039509);
        double r6039517 = r6039515 + r6039516;
        double r6039518 = r6039513 / r6039517;
        double r6039519 = cbrt(r6039513);
        double r6039520 = cbrt(r6039512);
        double r6039521 = r6039519 * r6039520;
        double r6039522 = r6039516 - r6039515;
        double r6039523 = r6039521 / r6039522;
        double r6039524 = r6039518 * r6039523;
        double r6039525 = 0.08333333333333333;
        double r6039526 = r6039508 * r6039525;
        double r6039527 = 1.0;
        double r6039528 = r6039527 / r6039508;
        double r6039529 = 0.5;
        double r6039530 = r6039528 + r6039529;
        double r6039531 = r6039526 + r6039530;
        double r6039532 = r6039511 ? r6039524 : r6039531;
        return r6039532;
}

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 add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    6. Applied add-cube-cbrt0.0

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}}}{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}\]

    7. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.0

      \[\leadsto \frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{\color{blue}{\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}}}}{\sqrt{e^{x}} - \sqrt{1}}\]

    10. Applied cbrt-prod0.0

      \[\leadsto \frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}}{\sqrt{e^{x}} - \sqrt{1}}\]

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{1} + \sqrt{e^{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

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)))