Average Error: 39.9 → 0.3
Time: 15.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001509743064020555:\\ \;\;\;\;\frac{-1 + e^{x} \cdot e^{x}}{x \cdot \left(1 + e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0001509743064020555:\\
\;\;\;\;\frac{-1 + e^{x} \cdot e^{x}}{x \cdot \left(1 + e^{x}\right)}\\

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

\end{array}
double f(double x) {
        double r10485787 = x;
        double r10485788 = exp(r10485787);
        double r10485789 = 1.0;
        double r10485790 = r10485788 - r10485789;
        double r10485791 = r10485790 / r10485787;
        return r10485791;
}
↓
double f(double x) {
        double r10485792 = x;
        double r10485793 = -0.0001509743064020555;
        bool r10485794 = r10485792 <= r10485793;
        double r10485795 = -1.0;
        double r10485796 = exp(r10485792);
        double r10485797 = r10485796 * r10485796;
        double r10485798 = r10485795 + r10485797;
        double r10485799 = 1.0;
        double r10485800 = r10485799 + r10485796;
        double r10485801 = r10485792 * r10485800;
        double r10485802 = r10485798 / r10485801;
        double r10485803 = 0.16666666666666666;
        double r10485804 = r10485803 * r10485792;
        double r10485805 = 0.5;
        double r10485806 = r10485804 + r10485805;
        double r10485807 = r10485792 * r10485806;
        double r10485808 = r10485807 + r10485799;
        double r10485809 = r10485794 ? r10485802 : r10485808;
        return r10485809;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.1
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 < -0.0001509743064020555

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    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. Simplified0.1

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

    if -0.0001509743064020555 < 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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001509743064020555:\\ \;\;\;\;\frac{-1 + e^{x} \cdot e^{x}}{x \cdot \left(1 + e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 
(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))