Average Error: 40.0 → 0.3
Time: 2.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.9132623240596211 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.9132623240596211 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}{x}\\

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

\end{array}
double f(double x) {
        double r92830 = x;
        double r92831 = exp(r92830);
        double r92832 = 1.0;
        double r92833 = r92831 - r92832;
        double r92834 = r92833 / r92830;
        return r92834;
}
↓
double f(double x) {
        double r92835 = x;
        double r92836 = -0.00019132623240596211;
        bool r92837 = r92835 <= r92836;
        double r92838 = exp(r92835);
        double r92839 = r92838 * r92838;
        double r92840 = 1.0;
        double r92841 = r92840 * r92840;
        double r92842 = r92839 - r92841;
        double r92843 = r92838 + r92840;
        double r92844 = r92842 / r92843;
        double r92845 = r92844 / r92835;
        double r92846 = 2.0;
        double r92847 = pow(r92835, r92846);
        double r92848 = 0.16666666666666666;
        double r92849 = r92835 * r92848;
        double r92850 = 0.5;
        double r92851 = r92849 + r92850;
        double r92852 = r92847 * r92851;
        double r92853 = r92852 + r92835;
        double r92854 = r92853 / r92835;
        double r92855 = r92837 ? r92845 : r92854;
        return r92855;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target40.5
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 < -0.00019132623240596211

    1. Initial program 0.0

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

    3. Applied flip--0.1

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

    if -0.00019132623240596211 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}{x}\]

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.9132623240596211 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020064 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

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

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