Average Error: 40.0 → 0.3
Time: 12.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00022073346315206534:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00022073346315206534:\\
\;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\

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

\end{array}
double f(double x) {
        double r3869369 = x;
        double r3869370 = exp(r3869369);
        double r3869371 = 1.0;
        double r3869372 = r3869370 - r3869371;
        double r3869373 = r3869372 / r3869369;
        return r3869373;
}
↓
double f(double x) {
        double r3869374 = x;
        double r3869375 = -0.00022073346315206534;
        bool r3869376 = r3869374 <= r3869375;
        double r3869377 = exp(r3869374);
        double r3869378 = r3869377 / r3869374;
        double r3869379 = 1.0;
        double r3869380 = r3869379 / r3869374;
        double r3869381 = r3869378 - r3869380;
        double r3869382 = 0.16666666666666666;
        double r3869383 = r3869382 * r3869374;
        double r3869384 = 0.5;
        double r3869385 = r3869383 + r3869384;
        double r3869386 = r3869385 * r3869374;
        double r3869387 = r3869379 + r3869386;
        double r3869388 = r3869376 ? r3869381 : r3869387;
        return r3869388;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target39.2
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 < -0.00022073346315206534

    1. Initial program 0.0

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

    3. Applied div-sub0.1

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

    if -0.00022073346315206534 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00022073346315206534:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(\frac{1}{6} \cdot x + \frac{1}{2}\right) \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019164 
(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))