Average Error: 40.5 → 0.7
Time: 11.1s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0025466853162572313:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{1 + \sqrt{e^{x}}} \cdot \frac{\sqrt{e^{x}}}{\sqrt{e^{x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0025466853162572313:\\
\;\;\;\;\frac{\sqrt{e^{x}}}{1 + \sqrt{e^{x}}} \cdot \frac{\sqrt{e^{x}}}{\sqrt{e^{x}} - 1}\\

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

\end{array}
double f(double x) {
        double r2538510 = x;
        double r2538511 = exp(r2538510);
        double r2538512 = 1.0;
        double r2538513 = r2538511 - r2538512;
        double r2538514 = r2538511 / r2538513;
        return r2538514;
}
↓
double f(double x) {
        double r2538515 = x;
        double r2538516 = -0.0025466853162572313;
        bool r2538517 = r2538515 <= r2538516;
        double r2538518 = exp(r2538515);
        double r2538519 = sqrt(r2538518);
        double r2538520 = 1.0;
        double r2538521 = r2538520 + r2538519;
        double r2538522 = r2538519 / r2538521;
        double r2538523 = r2538519 - r2538520;
        double r2538524 = r2538519 / r2538523;
        double r2538525 = r2538522 * r2538524;
        double r2538526 = r2538520 / r2538515;
        double r2538527 = 0.5;
        double r2538528 = r2538526 + r2538527;
        double r2538529 = 0.08333333333333333;
        double r2538530 = r2538529 * r2538515;
        double r2538531 = r2538528 + r2538530;
        double r2538532 = r2538517 ? r2538525 : r2538531;
        return r2538532;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.5
Target40.1
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0025466853162572313

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

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

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied difference-of-squares0.0

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

    8. Applied exp-prod0.0

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

    9. Applied log-pow0.0

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

    10. Applied add-sqr-sqrt0.0

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

    11. Applied times-frac0.0

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

    12. Simplified0.0

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

    if -0.0025466853162572313 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 1.0

      \[\leadsto \color{blue}{\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)}\]
    3. Using strategy rm

    4. Applied *-commutative1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0025466853162572313:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{1 + \sqrt{e^{x}}} \cdot \frac{\sqrt{e^{x}}}{\sqrt{e^{x}} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019137 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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