Average Error: 39.6 → 0.6
Time: 28.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9698251240266461:\\ \;\;\;\;\frac{\frac{e^{x}}{\sqrt[3]{\log \left(e^{e^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}}{\sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{12} \cdot x + \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9698251240266461:\\
\;\;\;\;\frac{\frac{e^{x}}{\sqrt[3]{\log \left(e^{e^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}}{\sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}\\

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

\end{array}
double f(double x) {
        double r2588195 = x;
        double r2588196 = exp(r2588195);
        double r2588197 = 1.0;
        double r2588198 = r2588196 - r2588197;
        double r2588199 = r2588196 / r2588198;
        return r2588199;
}
↓
double f(double x) {
        double r2588200 = x;
        double r2588201 = exp(r2588200);
        double r2588202 = 0.9698251240266461;
        bool r2588203 = r2588201 <= r2588202;
        double r2588204 = 1.0;
        double r2588205 = r2588201 - r2588204;
        double r2588206 = exp(r2588205);
        double r2588207 = log(r2588206);
        double r2588208 = cbrt(r2588207);
        double r2588209 = r2588208 * r2588208;
        double r2588210 = r2588201 / r2588209;
        double r2588211 = r2588210 / r2588208;
        double r2588212 = 0.08333333333333333;
        double r2588213 = r2588212 * r2588200;
        double r2588214 = r2588204 / r2588200;
        double r2588215 = r2588213 + r2588214;
        double r2588216 = 0.5;
        double r2588217 = r2588215 + r2588216;
        double r2588218 = r2588203 ? r2588211 : r2588217;
        return r2588218;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target39.2
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9698251240266461

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

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

    8. Applied add-cube-cbrt0.0

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

    9. Applied associate-/r*0.0

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

    if 0.9698251240266461 < (exp x)

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.9

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

    4. Applied associate-+r+0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9698251240266461:\\ \;\;\;\;\frac{\frac{e^{x}}{\sqrt[3]{\log \left(e^{e^{x} - 1}\right)} \cdot \sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}}{\sqrt[3]{\log \left(e^{e^{x} - 1}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{12} \cdot x + \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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