Average Error: 39.8 → 0.7
Time: 17.6s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9981680912834135:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right) \cdot \sqrt[3]{\frac{1}{12} \cdot x}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9981680912834135:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right) \cdot \sqrt[3]{\frac{1}{12} \cdot x}\\

\end{array}
double f(double x) {
        double r2864098 = x;
        double r2864099 = exp(r2864098);
        double r2864100 = 1.0;
        double r2864101 = r2864099 - r2864100;
        double r2864102 = r2864099 / r2864101;
        return r2864102;
}
↓
double f(double x) {
        double r2864103 = x;
        double r2864104 = exp(r2864103);
        double r2864105 = 0.9981680912834135;
        bool r2864106 = r2864104 <= r2864105;
        double r2864107 = r2864104 * r2864104;
        double r2864108 = 1.0;
        double r2864109 = r2864107 - r2864108;
        double r2864110 = r2864104 + r2864108;
        double r2864111 = r2864109 / r2864110;
        double r2864112 = r2864104 / r2864111;
        double r2864113 = 0.5;
        double r2864114 = r2864108 / r2864103;
        double r2864115 = r2864113 + r2864114;
        double r2864116 = 0.08333333333333333;
        double r2864117 = r2864116 * r2864103;
        double r2864118 = cbrt(r2864117);
        double r2864119 = r2864118 * r2864118;
        double r2864120 = r2864119 * r2864118;
        double r2864121 = r2864115 + r2864120;
        double r2864122 = r2864106 ? r2864112 : r2864121;
        return r2864122;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.4
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9981680912834135

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    if 0.9981680912834135 < (exp x)

    1. Initial program 60.0

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

    2. Taylor expanded around 0 1.0

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

    4. Applied add-cube-cbrt1.0

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9981680912834135:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot e^{x} - 1}{e^{x} + 1}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \left(\sqrt[3]{\frac{1}{12} \cdot x} \cdot \sqrt[3]{\frac{1}{12} \cdot x}\right) \cdot \sqrt[3]{\frac{1}{12} \cdot x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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