Average Error: 39.8 → 0.3
Time: 6.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-4594045809466361}{36893488147419103232}:\\ \;\;\;\;\frac{{\left({\left(e^{\sqrt[3]{x}}\right)}^{\left(2 \cdot \sqrt[3]{x}\right)}\right)}^{\left(\sqrt[3]{x}\right)} - 1 \cdot 1}{x \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-4594045809466361}{36893488147419103232}:\\
\;\;\;\;\frac{{\left({\left(e^{\sqrt[3]{x}}\right)}^{\left(2 \cdot \sqrt[3]{x}\right)}\right)}^{\left(\sqrt[3]{x}\right)} - 1 \cdot 1}{x \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\

\end{array}
double f(double x) {
        double r80894 = x;
        double r80895 = exp(r80894);
        double r80896 = 1.0;
        double r80897 = r80895 - r80896;
        double r80898 = r80897 / r80894;
        return r80898;
}
↓
double f(double x) {
        double r80899 = x;
        double r80900 = -4594045809466361.0;
        double r80901 = 3.6893488147419103e+19;
        double r80902 = r80900 / r80901;
        bool r80903 = r80899 <= r80902;
        double r80904 = cbrt(r80899);
        double r80905 = exp(r80904);
        double r80906 = 2.0;
        double r80907 = r80906 * r80904;
        double r80908 = pow(r80905, r80907);
        double r80909 = pow(r80908, r80904);
        double r80910 = 1.0;
        double r80911 = r80910 * r80910;
        double r80912 = r80909 - r80911;
        double r80913 = exp(r80899);
        double r80914 = r80913 + r80910;
        double r80915 = r80899 * r80914;
        double r80916 = r80912 / r80915;
        double r80917 = 0.16666666666666666;
        double r80918 = pow(r80899, r80906);
        double r80919 = r80917 * r80918;
        double r80920 = 0.5;
        double r80921 = r80920 * r80899;
        double r80922 = 1.0;
        double r80923 = r80921 + r80922;
        double r80924 = r80919 + r80923;
        double r80925 = r80903 ? r80916 : r80924;
        return r80925;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target40.2
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0001245218611780339

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied associate-/l/0.1

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

    6. Applied add-cube-cbrt0.0

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

    7. Applied exp-prod0.0

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

    8. Applied add-cube-cbrt0.1

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

    9. Applied exp-prod0.1

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

    10. Applied pow-prod-down0.0

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

    11. Simplified0.0

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

    if -0.0001245218611780339 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-4594045809466361}{36893488147419103232}:\\ \;\;\;\;\frac{{\left({\left(e^{\sqrt[3]{x}}\right)}^{\left(2 \cdot \sqrt[3]{x}\right)}\right)}^{\left(\sqrt[3]{x}\right)} - 1 \cdot 1}{x \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

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