Average Error: 40.2 → 0.6
Time: 18.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9811907080117241:\\ \;\;\;\;\frac{\frac{e^{x}}{e^{x} \cdot e^{x} - 1}}{1 + \sqrt{e^{x}}} \cdot \left(\left(e^{x} + 1\right) \cdot \frac{e^{x} - 1}{\sqrt{e^{x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + x \cdot \frac{1}{12}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9811907080117241:\\
\;\;\;\;\frac{\frac{e^{x}}{e^{x} \cdot e^{x} - 1}}{1 + \sqrt{e^{x}}} \cdot \left(\left(e^{x} + 1\right) \cdot \frac{e^{x} - 1}{\sqrt{e^{x}} - 1}\right)\\

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

\end{array}
double f(double x) {
        double r4295417 = x;
        double r4295418 = exp(r4295417);
        double r4295419 = 1.0;
        double r4295420 = r4295418 - r4295419;
        double r4295421 = r4295418 / r4295420;
        return r4295421;
}
↓
double f(double x) {
        double r4295422 = x;
        double r4295423 = exp(r4295422);
        double r4295424 = 0.9811907080117241;
        bool r4295425 = r4295423 <= r4295424;
        double r4295426 = r4295423 * r4295423;
        double r4295427 = 1.0;
        double r4295428 = r4295426 - r4295427;
        double r4295429 = r4295423 / r4295428;
        double r4295430 = sqrt(r4295423);
        double r4295431 = r4295427 + r4295430;
        double r4295432 = r4295429 / r4295431;
        double r4295433 = r4295423 + r4295427;
        double r4295434 = r4295423 - r4295427;
        double r4295435 = r4295430 - r4295427;
        double r4295436 = r4295434 / r4295435;
        double r4295437 = r4295433 * r4295436;
        double r4295438 = r4295432 * r4295437;
        double r4295439 = r4295427 / r4295422;
        double r4295440 = 0.5;
        double r4295441 = r4295439 + r4295440;
        double r4295442 = 0.08333333333333333;
        double r4295443 = r4295422 * r4295442;
        double r4295444 = r4295441 + r4295443;
        double r4295445 = r4295425 ? r4295438 : r4295444;
        return r4295445;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.2
Target39.8
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9811907080117241

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{\frac{e^{x}}{e^{x} + 1}}{e^{x} - 1}} \cdot \left(e^{x} + 1\right)\]
    6. Using strategy rm

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

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

    8. Applied add-sqr-sqrt0.0

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

    9. Applied difference-of-squares0.0

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

    10. Applied flip-+0.0

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

    11. Applied associate-/r/0.0

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

    12. Applied times-frac0.0

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

    13. Applied associate-*l*0.0

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

    if 0.9811907080117241 < (exp x)

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9811907080117241:\\ \;\;\;\;\frac{\frac{e^{x}}{e^{x} \cdot e^{x} - 1}}{1 + \sqrt{e^{x}}} \cdot \left(\left(e^{x} + 1\right) \cdot \frac{e^{x} - 1}{\sqrt{e^{x}} - 1}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + x \cdot \frac{1}{12}\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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