Average Error: 40.1 → 0.3
Time: 9.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.668911236678853246811343735700461365923 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.668911236678853246811343735700461365923 \cdot 10^{-4}:\\
\;\;\;\;\frac{e^{x} - 1}{x}\\

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

\end{array}
double f(double x) {
        double r5617487 = x;
        double r5617488 = exp(r5617487);
        double r5617489 = 1.0;
        double r5617490 = r5617488 - r5617489;
        double r5617491 = r5617490 / r5617487;
        return r5617491;
}
↓
double f(double x) {
        double r5617492 = x;
        double r5617493 = -0.00016689112366788532;
        bool r5617494 = r5617492 <= r5617493;
        double r5617495 = exp(r5617492);
        double r5617496 = 1.0;
        double r5617497 = r5617495 - r5617496;
        double r5617498 = r5617497 / r5617492;
        double r5617499 = 1.0;
        double r5617500 = 0.16666666666666666;
        double r5617501 = r5617492 * r5617500;
        double r5617502 = 0.5;
        double r5617503 = r5617501 + r5617502;
        double r5617504 = r5617492 * r5617503;
        double r5617505 = r5617499 + r5617504;
        double r5617506 = r5617494 ? r5617498 : r5617505;
        return r5617506;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target40.5
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 < -0.00016689112366788532

    1. Initial program 0.0

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

    if -0.00016689112366788532 < 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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.668911236678853246811343735700461365923 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))