Average Error: 41.0 → 0.6
Time: 13.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9967130846117587816834770819696132093668:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + \left(x + x\right)} - 1 \cdot \left(1 \cdot 1\right)}{\left(1 + e^{x}\right) \cdot e^{x} + 1 \cdot 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x \cdot \frac{1}{12}} \cdot e^{\log \left(\sqrt[3]{x \cdot \frac{1}{12}} \cdot \sqrt[3]{x \cdot \frac{1}{12}}\right)} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9967130846117587816834770819696132093668:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{x + \left(x + x\right)} - 1 \cdot \left(1 \cdot 1\right)}{\left(1 + e^{x}\right) \cdot e^{x} + 1 \cdot 1}}\\

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

\end{array}
double f(double x) {
        double r2928075 = x;
        double r2928076 = exp(r2928075);
        double r2928077 = 1.0;
        double r2928078 = r2928076 - r2928077;
        double r2928079 = r2928076 / r2928078;
        return r2928079;
}
↓
double f(double x) {
        double r2928080 = x;
        double r2928081 = exp(r2928080);
        double r2928082 = 0.9967130846117588;
        bool r2928083 = r2928081 <= r2928082;
        double r2928084 = r2928080 + r2928080;
        double r2928085 = r2928080 + r2928084;
        double r2928086 = exp(r2928085);
        double r2928087 = 1.0;
        double r2928088 = r2928087 * r2928087;
        double r2928089 = r2928087 * r2928088;
        double r2928090 = r2928086 - r2928089;
        double r2928091 = r2928087 + r2928081;
        double r2928092 = r2928091 * r2928081;
        double r2928093 = r2928092 + r2928088;
        double r2928094 = r2928090 / r2928093;
        double r2928095 = r2928081 / r2928094;
        double r2928096 = 0.08333333333333333;
        double r2928097 = r2928080 * r2928096;
        double r2928098 = cbrt(r2928097);
        double r2928099 = r2928098 * r2928098;
        double r2928100 = log(r2928099);
        double r2928101 = exp(r2928100);
        double r2928102 = r2928098 * r2928101;
        double r2928103 = 0.5;
        double r2928104 = 1.0;
        double r2928105 = r2928104 / r2928080;
        double r2928106 = r2928103 + r2928105;
        double r2928107 = r2928102 + r2928106;
        double r2928108 = r2928083 ? r2928095 : r2928107;
        return r2928108;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.0
Target40.5
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9967130846117588

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    5. Simplified0.0

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

    if 0.9967130846117588 < (exp x)

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0.9

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

    4. Applied add-cube-cbrt0.9

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

    6. Applied add-exp-log0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9967130846117587816834770819696132093668:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x + \left(x + x\right)} - 1 \cdot \left(1 \cdot 1\right)}{\left(1 + e^{x}\right) \cdot e^{x} + 1 \cdot 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{x \cdot \frac{1}{12}} \cdot e^{\log \left(\sqrt[3]{x \cdot \frac{1}{12}} \cdot \sqrt[3]{x \cdot \frac{1}{12}}\right)} + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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