Average Error: 41.5 → 0.8
Time: 10.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 3.936159656535852086796232633999635795103 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{12} \cdot x + \frac{1}{2}\right) + \frac{1}{x}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 3.936159656535852086796232633999635795103 \cdot 10^{-90}:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

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

\end{array}
double f(double x) {
        double r83466 = x;
        double r83467 = exp(r83466);
        double r83468 = 1.0;
        double r83469 = r83467 - r83468;
        double r83470 = r83467 / r83469;
        return r83470;
}
↓
double f(double x) {
        double r83471 = x;
        double r83472 = exp(r83471);
        double r83473 = 3.936159656535852e-90;
        bool r83474 = r83472 <= r83473;
        double r83475 = 1.0;
        double r83476 = 1.0;
        double r83477 = r83476 / r83472;
        double r83478 = r83475 - r83477;
        double r83479 = r83475 / r83478;
        double r83480 = 0.08333333333333333;
        double r83481 = r83480 * r83471;
        double r83482 = 0.5;
        double r83483 = r83481 + r83482;
        double r83484 = r83475 / r83471;
        double r83485 = r83483 + r83484;
        double r83486 = r83474 ? r83479 : r83485;
        return r83486;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.5
Target41.0
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 3.936159656535852e-90

    1. Initial program 0

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

    3. Applied clear-num0

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

    4. Simplified0

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

    if 3.936159656535852e-90 < (exp x)

    1. Initial program 61.6

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

    2. Taylor expanded around 0 1.2

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

    3. Simplified1.2

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 3.936159656535852086796232633999635795103 \cdot 10^{-90}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{12} \cdot x + \frac{1}{2}\right) + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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