Average Error: 39.4 → 0.3
Time: 13.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00010173923815578526:\\ \;\;\;\;\frac{-1 + e^{x + x}}{x + x \cdot \left(e^{x + x} \cdot e^{x}\right)} \cdot \left(\left(1 - e^{x}\right) + \sqrt[3]{e^{x}} \cdot \left(e^{x} \cdot \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00010173923815578526:\\
\;\;\;\;\frac{-1 + e^{x + x}}{x + x \cdot \left(e^{x + x} \cdot e^{x}\right)} \cdot \left(\left(1 - e^{x}\right) + \sqrt[3]{e^{x}} \cdot \left(e^{x} \cdot \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)\right)\right)\\

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

\end{array}
double f(double x) {
        double r1622839 = x;
        double r1622840 = exp(r1622839);
        double r1622841 = 1.0;
        double r1622842 = r1622840 - r1622841;
        double r1622843 = r1622842 / r1622839;
        return r1622843;
}
↓
double f(double x) {
        double r1622844 = x;
        double r1622845 = -0.00010173923815578526;
        bool r1622846 = r1622844 <= r1622845;
        double r1622847 = -1.0;
        double r1622848 = r1622844 + r1622844;
        double r1622849 = exp(r1622848);
        double r1622850 = r1622847 + r1622849;
        double r1622851 = exp(r1622844);
        double r1622852 = r1622849 * r1622851;
        double r1622853 = r1622844 * r1622852;
        double r1622854 = r1622844 + r1622853;
        double r1622855 = r1622850 / r1622854;
        double r1622856 = 1.0;
        double r1622857 = r1622856 - r1622851;
        double r1622858 = cbrt(r1622851);
        double r1622859 = r1622858 * r1622858;
        double r1622860 = r1622851 * r1622859;
        double r1622861 = r1622858 * r1622860;
        double r1622862 = r1622857 + r1622861;
        double r1622863 = r1622855 * r1622862;
        double r1622864 = 0.5;
        double r1622865 = 0.16666666666666666;
        double r1622866 = r1622844 * r1622865;
        double r1622867 = r1622864 + r1622866;
        double r1622868 = r1622844 * r1622867;
        double r1622869 = r1622868 + r1622856;
        double r1622870 = r1622846 ? r1622863 : r1622869;
        return r1622870;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.4
Target38.6
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 < -0.00010173923815578526

    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 flip3-+0.1

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

    7. Applied associate-*r/0.1

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

    8. Applied associate-/r/0.1

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

    9. Simplified0.1

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

    11. Applied add-cube-cbrt0.1

      \[\leadsto \frac{e^{x + x} + -1}{\left(e^{x + x} \cdot e^{x}\right) \cdot x + x} \cdot \left(e^{x} \cdot \color{blue}{\left(\left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right) \cdot \sqrt[3]{e^{x}}\right)} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\]

    12. Applied associate-*r*0.1

      \[\leadsto \frac{e^{x + x} + -1}{\left(e^{x + x} \cdot e^{x}\right) \cdot x + x} \cdot \left(\color{blue}{\left(e^{x} \cdot \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)\right) \cdot \sqrt[3]{e^{x}}} + \left(1 \cdot 1 - e^{x} \cdot 1\right)\right)\]

    if -0.00010173923815578526 < x

    1. Initial program 60.3

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00010173923815578526:\\ \;\;\;\;\frac{-1 + e^{x + x}}{x + x \cdot \left(e^{x + x} \cdot e^{x}\right)} \cdot \left(\left(1 - e^{x}\right) + \sqrt[3]{e^{x}} \cdot \left(e^{x} \cdot \left(\sqrt[3]{e^{x}} \cdot \sqrt[3]{e^{x}}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019153 
(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))