Average Error: 40.3 → 0.0
Time: 2.6s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0018711434362285975:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \mathbf{elif}\;x \le 9.641487078301664 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}}{1 + \frac{1}{e^{x}}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0018711434362285975:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}\\

\mathbf{elif}\;x \le 9.641487078301664 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{\frac{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}}{1 + \frac{1}{e^{x}}}}\\

\end{array}
double f(double x) {
        double r87991 = x;
        double r87992 = exp(r87991);
        double r87993 = 1.0;
        double r87994 = r87992 - r87993;
        double r87995 = r87992 / r87994;
        return r87995;
}
↓
double f(double x) {
        double r87996 = x;
        double r87997 = -0.0018711434362285975;
        bool r87998 = r87996 <= r87997;
        double r87999 = 1.0;
        double r88000 = 1.0;
        double r88001 = log(r88000);
        double r88002 = r88001 - r87996;
        double r88003 = exp(r88002);
        double r88004 = r87999 - r88003;
        double r88005 = cbrt(r88004);
        double r88006 = r87999 / r88005;
        double r88007 = r88006 / r88005;
        double r88008 = r88007 / r88005;
        double r88009 = 0.0009641487078301664;
        bool r88010 = r87996 <= r88009;
        double r88011 = 0.5;
        double r88012 = 0.08333333333333333;
        double r88013 = r88012 * r87996;
        double r88014 = r87999 / r87996;
        double r88015 = r88013 + r88014;
        double r88016 = r88011 + r88015;
        double r88017 = 2.0;
        double r88018 = exp(r88017);
        double r88019 = pow(r88018, r88002);
        double r88020 = r88019 / r87999;
        double r88021 = r87999 - r88020;
        double r88022 = exp(r87996);
        double r88023 = r88000 / r88022;
        double r88024 = r87999 + r88023;
        double r88025 = r88021 / r88024;
        double r88026 = r87999 / r88025;
        double r88027 = r88010 ? r88016 : r88026;
        double r88028 = r87998 ? r88008 : r88027;
        return r88028;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.3
Target39.8
Herbie0.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if x < -0.0018711434362285975

    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-exp-log0.0

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

    7. Applied div-exp0.0

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

    9. Applied add-cube-cbrt0.0

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

    10. Applied associate-/r*0.0

      \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}\]

    11. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}\]

    if -0.0018711434362285975 < x < 0.0009641487078301664

    1. Initial program 62.4

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

    2. Taylor expanded around 0 0.0

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

    if 0.0009641487078301664 < x

    1. Initial program 35.8

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

    3. Applied clear-num35.8

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

    4. Simplified1.1

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

    6. Applied add-exp-log1.1

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

    7. Applied div-exp0.7

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

    9. Applied flip--0.7

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

    10. Simplified0.6

      \[\leadsto \frac{1}{\frac{\color{blue}{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}}}{1 + e^{\log 1 - x}}}\]

    11. Simplified0.6

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

  4. Final simplification0.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0018711434362285975:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \mathbf{elif}\;x \le 9.641487078301664 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}}{1 + \frac{1}{e^{x}}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020025 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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