Average Error: 39.6 → 0.6
Time: 13.7s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -731.1697091805538093467475846409797668457:\\ \;\;\;\;\sqrt{\frac{e^{x}}{x}} \cdot \sqrt{\frac{e^{x}}{x}} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -731.1697091805538093467475846409797668457:\\
\;\;\;\;\sqrt{\frac{e^{x}}{x}} \cdot \sqrt{\frac{e^{x}}{x}} - \frac{1}{x}\\

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

\end{array}
double f(double x) {
        double r6782788 = x;
        double r6782789 = exp(r6782788);
        double r6782790 = 1.0;
        double r6782791 = r6782789 - r6782790;
        double r6782792 = r6782791 / r6782788;
        return r6782792;
}
↓
double f(double x) {
        double r6782793 = x;
        double r6782794 = -731.1697091805538;
        bool r6782795 = r6782793 <= r6782794;
        double r6782796 = exp(r6782793);
        double r6782797 = r6782796 / r6782793;
        double r6782798 = sqrt(r6782797);
        double r6782799 = r6782798 * r6782798;
        double r6782800 = 1.0;
        double r6782801 = r6782800 / r6782793;
        double r6782802 = r6782799 - r6782801;
        double r6782803 = 0.5;
        double r6782804 = 0.16666666666666666;
        double r6782805 = r6782793 * r6782804;
        double r6782806 = r6782803 + r6782805;
        double r6782807 = r6782806 * r6782793;
        double r6782808 = 1.0;
        double r6782809 = r6782807 + r6782808;
        double r6782810 = r6782795 ? r6782802 : r6782809;
        return r6782810;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target40.0
Herbie0.6
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -731.1697091805538

    1. Initial program 0

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

    3. Applied div-sub0

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

    5. Applied add-sqr-sqrt0

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

    if -731.1697091805538 < x

    1. Initial program 59.4

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

    2. Taylor expanded around 0 0.9

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

    3. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -731.1697091805538093467475846409797668457:\\ \;\;\;\;\sqrt{\frac{e^{x}}{x}} \cdot \sqrt{\frac{e^{x}}{x}} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))