Average Error: 40.1 → 0.3
Time: 19.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -4.68177674119828 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{e^{3 \cdot x} + -1}{e^{x} \cdot \frac{\left(\sqrt[3]{1 - e^{x} \cdot e^{x}} \cdot \sqrt[3]{1 - e^{x} \cdot e^{x}}\right) \cdot \sqrt[3]{1 - e^{x} \cdot e^{x}}}{1 - e^{x}} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -4.68177674119828 \cdot 10^{-05}:\\
\;\;\;\;\frac{\frac{e^{3 \cdot x} + -1}{e^{x} \cdot \frac{\left(\sqrt[3]{1 - e^{x} \cdot e^{x}} \cdot \sqrt[3]{1 - e^{x} \cdot e^{x}}\right) \cdot \sqrt[3]{1 - e^{x} \cdot e^{x}}}{1 - 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 r4484453 = x;
        double r4484454 = exp(r4484453);
        double r4484455 = 1.0;
        double r4484456 = r4484454 - r4484455;
        double r4484457 = r4484456 / r4484453;
        return r4484457;
}
↓
double f(double x) {
        double r4484458 = x;
        double r4484459 = -4.68177674119828e-05;
        bool r4484460 = r4484458 <= r4484459;
        double r4484461 = 3.0;
        double r4484462 = r4484461 * r4484458;
        double r4484463 = exp(r4484462);
        double r4484464 = -1.0;
        double r4484465 = r4484463 + r4484464;
        double r4484466 = exp(r4484458);
        double r4484467 = 1.0;
        double r4484468 = r4484466 * r4484466;
        double r4484469 = r4484467 - r4484468;
        double r4484470 = cbrt(r4484469);
        double r4484471 = r4484470 * r4484470;
        double r4484472 = r4484471 * r4484470;
        double r4484473 = r4484467 - r4484466;
        double r4484474 = r4484472 / r4484473;
        double r4484475 = r4484466 * r4484474;
        double r4484476 = r4484475 + r4484467;
        double r4484477 = r4484465 / r4484476;
        double r4484478 = r4484477 / r4484458;
        double r4484479 = 0.5;
        double r4484480 = 0.16666666666666666;
        double r4484481 = r4484480 * r4484458;
        double r4484482 = r4484479 + r4484481;
        double r4484483 = r4484458 * r4484482;
        double r4484484 = r4484483 + r4484467;
        double r4484485 = r4484460 ? r4484478 : r4484484;
        return r4484485;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target39.3
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 < -4.68177674119828e-05

    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 flip-+0.1

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

    8. Simplified0.1

      \[\leadsto \frac{\frac{e^{x \cdot 3} + -1}{e^{x} \cdot \frac{\color{blue}{1 - e^{x} \cdot e^{x}}}{1 - e^{x}} + 1}}{x}\]
    9. Using strategy rm

    10. Applied add-cube-cbrt0.1

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

    if -4.68177674119828e-05 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.68177674119828 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{e^{3 \cdot x} + -1}{e^{x} \cdot \frac{\left(\sqrt[3]{1 - e^{x} \cdot e^{x}} \cdot \sqrt[3]{1 - e^{x} \cdot e^{x}}\right) \cdot \sqrt[3]{1 - e^{x} \cdot e^{x}}}{1 - e^{x}} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 
(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))