Average Error: 39.8 → 0.3
Time: 19.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001556646497825041:\\ \;\;\;\;\frac{\log \left(e^{e^{x} \cdot e^{x} - 1}\right)}{\left(e^{x} + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0001556646497825041:\\
\;\;\;\;\frac{\log \left(e^{e^{x} \cdot e^{x} - 1}\right)}{\left(e^{x} + 1\right) \cdot x}\\

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

\end{array}
double f(double x) {
        double r2848618 = x;
        double r2848619 = exp(r2848618);
        double r2848620 = 1.0;
        double r2848621 = r2848619 - r2848620;
        double r2848622 = r2848621 / r2848618;
        return r2848622;
}
↓
double f(double x) {
        double r2848623 = x;
        double r2848624 = -0.0001556646497825041;
        bool r2848625 = r2848623 <= r2848624;
        double r2848626 = exp(r2848623);
        double r2848627 = r2848626 * r2848626;
        double r2848628 = 1.0;
        double r2848629 = r2848627 - r2848628;
        double r2848630 = exp(r2848629);
        double r2848631 = log(r2848630);
        double r2848632 = r2848626 + r2848628;
        double r2848633 = r2848632 * r2848623;
        double r2848634 = r2848631 / r2848633;
        double r2848635 = 0.16666666666666666;
        double r2848636 = r2848635 * r2848623;
        double r2848637 = 0.5;
        double r2848638 = r2848636 + r2848637;
        double r2848639 = r2848638 * r2848623;
        double r2848640 = r2848639 + r2848628;
        double r2848641 = r2848625 ? r2848634 : r2848640;
        return r2848641;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target38.9
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 < -0.0001556646497825041

    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. Using strategy rm

    6. Applied add-log-exp0.1

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

    7. Applied add-log-exp0.1

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

    8. Applied diff-log0.1

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

    9. Simplified0.0

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

    if -0.0001556646497825041 < x

    1. Initial program 60.0

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001556646497825041:\\ \;\;\;\;\frac{\log \left(e^{e^{x} \cdot e^{x} - 1}\right)}{\left(e^{x} + 1\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(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))