Average Error: 40.6 → 0.3
Time: 20.3s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0001013307789888238:\\ \;\;\;\;\frac{\frac{e^{3 \cdot x} + -1}{\sqrt[3]{\left(e^{x} + 1\right) \cdot \left(\left(e^{x} + 1\right) \cdot \left(e^{x} + 1\right)\right)} \cdot e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0001013307789888238:\\
\;\;\;\;\frac{\frac{e^{3 \cdot x} + -1}{\sqrt[3]{\left(e^{x} + 1\right) \cdot \left(\left(e^{x} + 1\right) \cdot \left(e^{x} + 1\right)\right)} \cdot e^{x} + 1}}{x}\\

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

\end{array}
double f(double x) {
        double r3384472 = x;
        double r3384473 = exp(r3384472);
        double r3384474 = 1.0;
        double r3384475 = r3384473 - r3384474;
        double r3384476 = r3384475 / r3384472;
        return r3384476;
}
↓
double f(double x) {
        double r3384477 = x;
        double r3384478 = -0.0001013307789888238;
        bool r3384479 = r3384477 <= r3384478;
        double r3384480 = 3.0;
        double r3384481 = r3384480 * r3384477;
        double r3384482 = exp(r3384481);
        double r3384483 = -1.0;
        double r3384484 = r3384482 + r3384483;
        double r3384485 = exp(r3384477);
        double r3384486 = 1.0;
        double r3384487 = r3384485 + r3384486;
        double r3384488 = r3384487 * r3384487;
        double r3384489 = r3384487 * r3384488;
        double r3384490 = cbrt(r3384489);
        double r3384491 = r3384490 * r3384485;
        double r3384492 = r3384491 + r3384486;
        double r3384493 = r3384484 / r3384492;
        double r3384494 = r3384493 / r3384477;
        double r3384495 = 0.5;
        double r3384496 = 0.16666666666666666;
        double r3384497 = r3384477 * r3384496;
        double r3384498 = r3384495 + r3384497;
        double r3384499 = r3384498 * r3384477;
        double r3384500 = r3384486 + r3384499;
        double r3384501 = r3384479 ? r3384494 : r3384500;
        return r3384501;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.6
Target39.8
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 < -0.0001013307789888238

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    7. Applied add-cbrt-cube0.1

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

    if -0.0001013307789888238 < x

    1. Initial program 60.3

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0001013307789888238:\\ \;\;\;\;\frac{\frac{e^{3 \cdot x} + -1}{\sqrt[3]{\left(e^{x} + 1\right) \cdot \left(\left(e^{x} + 1\right) \cdot \left(e^{x} + 1\right)\right)} \cdot e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019142 
(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))