Average Error: 41.4 → 0.7
Time: 11.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9351426122859008982501904938544612377882:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9351426122859008982501904938544612377882:\\
\;\;\;\;\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}}\\

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

\end{array}
double f(double x) {
        double r59601 = x;
        double r59602 = exp(r59601);
        double r59603 = 1.0;
        double r59604 = r59602 - r59603;
        double r59605 = r59602 / r59604;
        return r59605;
}
↓
double f(double x) {
        double r59606 = x;
        double r59607 = exp(r59606);
        double r59608 = 0.9351426122859009;
        bool r59609 = r59607 <= r59608;
        double r59610 = cbrt(r59607);
        double r59611 = r59610 * r59610;
        double r59612 = sqrt(r59607);
        double r59613 = 1.0;
        double r59614 = sqrt(r59613);
        double r59615 = r59612 + r59614;
        double r59616 = r59611 / r59615;
        double r59617 = r59612 - r59614;
        double r59618 = r59610 / r59617;
        double r59619 = r59616 * r59618;
        double r59620 = 0.5;
        double r59621 = 0.08333333333333333;
        double r59622 = r59621 * r59606;
        double r59623 = 1.0;
        double r59624 = r59623 / r59606;
        double r59625 = r59622 + r59624;
        double r59626 = r59620 + r59625;
        double r59627 = r59609 ? r59619 : r59626;
        return r59627;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target41.0
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9351426122859009

    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

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

    if 0.9351426122859009 < (exp x)

    1. Initial program 61.5

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

    2. Taylor expanded around 0 1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9351426122859008982501904938544612377882:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019294 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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