Average Error: 39.8 → 1.0
Time: 15.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 1.0:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{1}{12}\right) + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 1.0:\\
\;\;\;\;\frac{e^{x}}{e^{x} - 1}\\

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

\end{array}
double f(double x) {
        double r1741887 = x;
        double r1741888 = exp(r1741887);
        double r1741889 = 1.0;
        double r1741890 = r1741888 - r1741889;
        double r1741891 = r1741888 / r1741890;
        return r1741891;
}
↓
double f(double x) {
        double r1741892 = x;
        double r1741893 = exp(r1741892);
        double r1741894 = 1.0;
        double r1741895 = r1741893 - r1741894;
        double r1741896 = r1741893 / r1741895;
        double r1741897 = 1.0;
        bool r1741898 = r1741896 <= r1741897;
        double r1741899 = cbrt(r1741892);
        double r1741900 = r1741899 * r1741899;
        double r1741901 = 0.08333333333333333;
        double r1741902 = r1741900 * r1741901;
        double r1741903 = r1741899 * r1741902;
        double r1741904 = r1741894 / r1741892;
        double r1741905 = 0.5;
        double r1741906 = r1741904 + r1741905;
        double r1741907 = r1741903 + r1741906;
        double r1741908 = r1741898 ? r1741896 : r1741907;
        return r1741908;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.3
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (exp x) (- (exp x) 1)) < 1.0

    1. Initial program 1.3

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

    2. Taylor expanded around -inf 1.3

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

    if 1.0 < (/ (exp x) (- (exp x) 1))

    1. Initial program 61.0

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

    2. Taylor expanded around 0 0.8

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

    4. Applied add-cube-cbrt0.8

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

    5. Applied associate-*r*0.8

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{e^{x}}{e^{x} - 1} \le 1.0:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x} \cdot \left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \frac{1}{12}\right) + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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