Average Error: 40.2 → 0.3
Time: 2.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.8533246565845013 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.8533246565845013 \cdot 10^{-4}:\\
\;\;\;\;\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}\\

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

\end{array}
double f(double x) {
        double r75669 = x;
        double r75670 = exp(r75669);
        double r75671 = 1.0;
        double r75672 = r75670 - r75671;
        double r75673 = r75672 / r75669;
        return r75673;
}
↓
double f(double x) {
        double r75674 = x;
        double r75675 = -0.00018533246565845013;
        bool r75676 = r75674 <= r75675;
        double r75677 = exp(r75674);
        double r75678 = sqrt(r75677);
        double r75679 = 1.0;
        double r75680 = sqrt(r75679);
        double r75681 = r75678 + r75680;
        double r75682 = r75678 - r75680;
        double r75683 = r75674 / r75682;
        double r75684 = r75681 / r75683;
        double r75685 = 0.5;
        double r75686 = 0.16666666666666666;
        double r75687 = r75674 * r75686;
        double r75688 = r75685 + r75687;
        double r75689 = r75674 * r75688;
        double r75690 = 1.0;
        double r75691 = r75689 + r75690;
        double r75692 = r75676 ? r75684 : r75691;
        return r75692;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.2
Target40.7
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 < -0.00018533246565845013

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.1

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

    6. Applied associate-/l*0.1

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

    if -0.00018533246565845013 < 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. Using strategy rm

    4. Applied associate-+r+0.4

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

    5. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8533246565845013 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020021 
(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))