Average Error: 39.5 → 0.3
Time: 10.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00017827683498693693:\\ \;\;\;\;\frac{-1 + e^{x} \cdot e^{x}}{x + e^{x} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00017827683498693693:\\
\;\;\;\;\frac{-1 + e^{x} \cdot e^{x}}{x + e^{x} \cdot x}\\

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

\end{array}
double f(double x) {
        double r2479855 = x;
        double r2479856 = exp(r2479855);
        double r2479857 = 1.0;
        double r2479858 = r2479856 - r2479857;
        double r2479859 = r2479858 / r2479855;
        return r2479859;
}
↓
double f(double x) {
        double r2479860 = x;
        double r2479861 = -0.00017827683498693693;
        bool r2479862 = r2479860 <= r2479861;
        double r2479863 = -1.0;
        double r2479864 = exp(r2479860);
        double r2479865 = r2479864 * r2479864;
        double r2479866 = r2479863 + r2479865;
        double r2479867 = r2479864 * r2479860;
        double r2479868 = r2479860 + r2479867;
        double r2479869 = r2479866 / r2479868;
        double r2479870 = 0.16666666666666666;
        double r2479871 = r2479870 * r2479860;
        double r2479872 = 0.5;
        double r2479873 = r2479871 + r2479872;
        double r2479874 = r2479860 * r2479873;
        double r2479875 = 1.0;
        double r2479876 = r2479874 + r2479875;
        double r2479877 = r2479862 ? r2479869 : r2479876;
        return r2479877;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target38.6
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 < -0.00017827683498693693

    1. Initial program 0.0

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

    3. Applied div-inv0.0

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

    5. Applied flip--0.0

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

    6. Applied frac-times0.0

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

    7. Simplified0.0

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

    8. Simplified0.0

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

    if -0.00017827683498693693 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00017827683498693693:\\ \;\;\;\;\frac{-1 + e^{x} \cdot e^{x}}{x + e^{x} \cdot x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019135 
(FPCore (x)
  :name "Kahan's exp quotient"

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

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