Average Error: 39.6 → 0.4
Time: 11.7s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0002240283144802314:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x}\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0002240283144802314:\\
\;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\

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

\end{array}
double f(double x) {
        double r1742314 = x;
        double r1742315 = exp(r1742314);
        double r1742316 = 1.0;
        double r1742317 = r1742315 - r1742316;
        double r1742318 = r1742317 / r1742314;
        return r1742318;
}
↓
double f(double x) {
        double r1742319 = x;
        double r1742320 = -0.0002240283144802314;
        bool r1742321 = r1742319 <= r1742320;
        double r1742322 = exp(r1742319);
        double r1742323 = r1742322 / r1742319;
        double r1742324 = 1.0;
        double r1742325 = r1742324 / r1742319;
        double r1742326 = r1742323 - r1742325;
        double r1742327 = 0.16666666666666666;
        double r1742328 = r1742327 * r1742319;
        double r1742329 = 0.5;
        double r1742330 = r1742328 + r1742329;
        double r1742331 = r1742330 * r1742319;
        double r1742332 = exp(r1742331);
        double r1742333 = log(r1742332);
        double r1742334 = r1742324 + r1742333;
        double r1742335 = r1742321 ? r1742326 : r1742334;
        return r1742335;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
Target38.7
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0002240283144802314

    1. Initial program 0.0

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

    3. Applied div-sub0.1

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

    if -0.0002240283144802314 < x

    1. Initial program 59.9

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0002240283144802314:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 
(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))