Average Error: 41.5 → 0.7
Time: 3.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\
\;\;\;\;\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 r100940 = x;
        double r100941 = exp(r100940);
        double r100942 = 1.0;
        double r100943 = r100941 - r100942;
        double r100944 = r100941 / r100943;
        return r100944;
}
↓
double f(double x) {
        double r100945 = x;
        double r100946 = exp(r100945);
        double r100947 = 0.8526603959516881;
        bool r100948 = r100946 <= r100947;
        double r100949 = cbrt(r100946);
        double r100950 = r100949 * r100949;
        double r100951 = sqrt(r100946);
        double r100952 = 1.0;
        double r100953 = sqrt(r100952);
        double r100954 = r100951 + r100953;
        double r100955 = r100950 / r100954;
        double r100956 = r100951 - r100953;
        double r100957 = r100949 / r100956;
        double r100958 = r100955 * r100957;
        double r100959 = 0.5;
        double r100960 = 0.08333333333333333;
        double r100961 = r100960 * r100945;
        double r100962 = 1.0;
        double r100963 = r100962 / r100945;
        double r100964 = r100961 + r100963;
        double r100965 = r100959 + r100964;
        double r100966 = r100948 ? r100958 : r100965;
        return r100966;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.5
Target41.0
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.8526603959516881

    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 0.8526603959516881 < (exp x)

    1. Initial program 62.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.8526603959516880770763691543834283947945:\\ \;\;\;\;\frac{\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt[3]{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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