Average Error: 39.8 → 0.3
Time: 13.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.245218611780339091916341986987504242279 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\sqrt[3]{e^{x} + 1}} \cdot \frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\sqrt[3]{e^{x} + 1}}\right) \cdot \frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\left(\sqrt[3]{\sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.245218611780339091916341986987504242279 \cdot 10^{-4}:\\
\;\;\;\;\frac{\left(\frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\sqrt[3]{e^{x} + 1}} \cdot \frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\sqrt[3]{e^{x} + 1}}\right) \cdot \frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\left(\sqrt[3]{\sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}{x}\\

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

\end{array}
double f(double x) {
        double r66727 = x;
        double r66728 = exp(r66727);
        double r66729 = 1.0;
        double r66730 = r66728 - r66729;
        double r66731 = r66730 / r66727;
        return r66731;
}
↓
double f(double x) {
        double r66732 = x;
        double r66733 = -0.0001245218611780339;
        bool r66734 = r66732 <= r66733;
        double r66735 = r66732 + r66732;
        double r66736 = exp(r66735);
        double r66737 = 1.0;
        double r66738 = r66737 * r66737;
        double r66739 = r66736 - r66738;
        double r66740 = cbrt(r66739);
        double r66741 = exp(r66732);
        double r66742 = r66741 + r66737;
        double r66743 = cbrt(r66742);
        double r66744 = r66740 / r66743;
        double r66745 = r66744 * r66744;
        double r66746 = cbrt(r66743);
        double r66747 = r66746 * r66746;
        double r66748 = r66747 * r66746;
        double r66749 = r66740 / r66748;
        double r66750 = r66745 * r66749;
        double r66751 = r66750 / r66732;
        double r66752 = 0.5;
        double r66753 = 0.16666666666666666;
        double r66754 = r66753 * r66732;
        double r66755 = r66752 + r66754;
        double r66756 = r66732 * r66755;
        double r66757 = 1.0;
        double r66758 = r66756 + r66757;
        double r66759 = r66734 ? r66751 : r66758;
        return r66759;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target40.2
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 < -0.0001245218611780339

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

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

    6. Applied add-cube-cbrt0.1

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

    7. Applied add-cube-cbrt0.1

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

    8. Applied times-frac0.1

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

    9. Simplified0.1

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

    11. Applied add-cube-cbrt0.1

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

    if -0.0001245218611780339 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.245218611780339091916341986987504242279 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\sqrt[3]{e^{x} + 1}} \cdot \frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\sqrt[3]{e^{x} + 1}}\right) \cdot \frac{\sqrt[3]{e^{x + x} - 1 \cdot 1}}{\left(\sqrt[3]{\sqrt[3]{e^{x} + 1}} \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}\right) \cdot \sqrt[3]{\sqrt[3]{e^{x} + 1}}}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(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))