Average Error: 41.4 → 0.5
Time: 13.3s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{1} + \sqrt{e^{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\
\;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{1} + \sqrt{e^{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}{\sqrt{e^{x}} - \sqrt{1}}\\

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

\end{array}
double f(double x) {
        double r5127324 = x;
        double r5127325 = exp(r5127324);
        double r5127326 = 1.0;
        double r5127327 = r5127325 - r5127326;
        double r5127328 = r5127325 / r5127327;
        return r5127328;
}
↓
double f(double x) {
        double r5127329 = x;
        double r5127330 = exp(r5127329);
        double r5127331 = 0.966199829895273;
        bool r5127332 = r5127330 <= r5127331;
        double r5127333 = cbrt(r5127330);
        double r5127334 = r5127333 * r5127333;
        double r5127335 = 1.0;
        double r5127336 = sqrt(r5127335);
        double r5127337 = sqrt(r5127330);
        double r5127338 = r5127336 + r5127337;
        double r5127339 = r5127334 / r5127338;
        double r5127340 = cbrt(r5127334);
        double r5127341 = cbrt(r5127333);
        double r5127342 = r5127340 * r5127341;
        double r5127343 = r5127337 - r5127336;
        double r5127344 = r5127342 / r5127343;
        double r5127345 = r5127339 * r5127344;
        double r5127346 = 0.08333333333333333;
        double r5127347 = r5127329 * r5127346;
        double r5127348 = 1.0;
        double r5127349 = r5127348 / r5127329;
        double r5127350 = 0.5;
        double r5127351 = r5127349 + r5127350;
        double r5127352 = r5127347 + r5127351;
        double r5127353 = r5127332 ? r5127345 : r5127352;
        return r5127353;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target41.1
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.966199829895273

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}}\]
    8. Using strategy rm

    9. Applied add-cube-cbrt0.0

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

    10. Applied cbrt-prod0.0

      \[\leadsto \frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\color{blue}{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}}{\sqrt{e^{x}} - \sqrt{1}}\]

    if 0.966199829895273 < (exp x)

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{1} + \sqrt{e^{x}}} \cdot \frac{\sqrt[3]{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}} \cdot \sqrt[3]{\sqrt[3]{e^{x}}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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