Average Error: 39.7 → 0.3
Time: 4.2s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.125225089524106296704972218769569280994 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.125225089524106296704972218769569280994 \cdot 10^{-4}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}{x}\\

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

\end{array}
double f(double x) {
        double r89369 = x;
        double r89370 = exp(r89369);
        double r89371 = 1.0;
        double r89372 = r89370 - r89371;
        double r89373 = r89372 / r89369;
        return r89373;
}
↓
double f(double x) {
        double r89374 = x;
        double r89375 = -0.00011252250895241063;
        bool r89376 = r89374 <= r89375;
        double r89377 = exp(r89374);
        double r89378 = 1.0;
        double r89379 = r89377 - r89378;
        double r89380 = 3.0;
        double r89381 = pow(r89379, r89380);
        double r89382 = cbrt(r89381);
        double r89383 = r89382 / r89374;
        double r89384 = 2.0;
        double r89385 = pow(r89374, r89384);
        double r89386 = 0.16666666666666666;
        double r89387 = r89374 * r89386;
        double r89388 = 0.5;
        double r89389 = r89387 + r89388;
        double r89390 = r89385 * r89389;
        double r89391 = r89390 + r89374;
        double r89392 = r89391 / r89374;
        double r89393 = r89376 ? r89383 : r89392;
        return r89393;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.7
Target40.1
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 < -0.00011252250895241063

    1. Initial program 0.1

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

    3. Applied add-cbrt-cube0.1

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

    4. Simplified0.1

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

    if -0.00011252250895241063 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}{x}\]

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.125225089524106296704972218769569280994 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(e^{x} - 1\right)}^{3}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 
(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))