Average Error: 41.3 → 0.7
Time: 2.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9576068429522089919814220593252684921026:\\ \;\;\;\;\sqrt[3]{\frac{1}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9576068429522089919814220593252684921026:\\
\;\;\;\;\sqrt[3]{\frac{1}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}\\

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

\end{array}
double f(double x) {
        double r111652 = x;
        double r111653 = exp(r111652);
        double r111654 = 1.0;
        double r111655 = r111653 - r111654;
        double r111656 = r111653 / r111655;
        return r111656;
}
↓
double f(double x) {
        double r111657 = x;
        double r111658 = exp(r111657);
        double r111659 = 0.957606842952209;
        bool r111660 = r111658 <= r111659;
        double r111661 = 1.0;
        double r111662 = 1.0;
        double r111663 = r111662 / r111658;
        double r111664 = r111661 - r111663;
        double r111665 = 3.0;
        double r111666 = pow(r111664, r111665);
        double r111667 = r111661 / r111666;
        double r111668 = cbrt(r111667);
        double r111669 = 0.5;
        double r111670 = 0.08333333333333333;
        double r111671 = r111670 * r111657;
        double r111672 = r111661 / r111657;
        double r111673 = r111671 + r111672;
        double r111674 = r111669 + r111673;
        double r111675 = r111660 ? r111668 : r111674;
        return r111675;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.3
Target40.8
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.957606842952209

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    4. Simplified0.0

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.1

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

    7. Simplified0.1

      \[\leadsto \frac{1}{\sqrt[3]{\color{blue}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube0.1

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

    10. Applied cbrt-undiv0.1

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

    11. Simplified0.1

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

    if 0.957606842952209 < (exp x)

    1. Initial program 62.0

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

    2. Taylor expanded around 0 1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9576068429522089919814220593252684921026:\\ \;\;\;\;\sqrt[3]{\frac{1}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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