Average Error: 40.3 → 0.7
Time: 11.3s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 9.438996161322502:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(x \cdot \sqrt{\frac{1}{12}}\right) \cdot \sqrt{\frac{1}{12}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 9.438996161322502:\\
\;\;\;\;\frac{e^{x}}{e^{x} - 1}\\

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

\end{array}
double f(double x) {
        double r1404349 = x;
        double r1404350 = exp(r1404349);
        double r1404351 = 1.0;
        double r1404352 = r1404350 - r1404351;
        double r1404353 = r1404350 / r1404352;
        return r1404353;
}
↓
double f(double x) {
        double r1404354 = x;
        double r1404355 = exp(r1404354);
        double r1404356 = 1.0;
        double r1404357 = r1404355 - r1404356;
        double r1404358 = r1404355 / r1404357;
        double r1404359 = 9.438996161322502;
        bool r1404360 = r1404358 <= r1404359;
        double r1404361 = 0.5;
        double r1404362 = r1404356 / r1404354;
        double r1404363 = r1404361 + r1404362;
        double r1404364 = 0.08333333333333333;
        double r1404365 = sqrt(r1404364);
        double r1404366 = r1404354 * r1404365;
        double r1404367 = r1404366 * r1404365;
        double r1404368 = r1404363 + r1404367;
        double r1404369 = r1404360 ? r1404358 : r1404368;
        return r1404369;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.9
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp x) (- (exp x) 1)) < 9.438996161322502

    1. Initial program 1.0

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

    2. Taylor expanded around inf 1.0

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

    if 9.438996161322502 < (/ (exp x) (- (exp x) 1))

    1. Initial program 61.2

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

    2. Taylor expanded around 0 0.6

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

    4. Applied add-sqr-sqrt0.6

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

    5. Applied associate-*l*0.6

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 9.438996161322502:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(x \cdot \sqrt{\frac{1}{12}}\right) \cdot \sqrt{\frac{1}{12}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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