Average Error: 40.1 → 0.3
Time: 10.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.668911236678853246811343735700461365923 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \end{array}\]
\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(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\

\end{array}
double f(double x) {
        double r3911279 = x;
        double r3911280 = exp(r3911279);
        double r3911281 = 1.0;
        double r3911282 = r3911280 - r3911281;
        double r3911283 = r3911282 / r3911279;
        return r3911283;
}
↓
double f(double x) {
        double r3911284 = x;
        double r3911285 = -0.00016689112366788532;
        bool r3911286 = r3911284 <= r3911285;
        double r3911287 = exp(r3911284);
        double r3911288 = 1.0;
        double r3911289 = r3911287 - r3911288;
        double r3911290 = r3911289 / r3911284;
        double r3911291 = 1.0;
        double r3911292 = 0.16666666666666666;
        double r3911293 = r3911292 * r3911284;
        double r3911294 = 0.5;
        double r3911295 = r3911293 + r3911294;
        double r3911296 = r3911284 * r3911295;
        double r3911297 = r3911291 + r3911296;
        double r3911298 = r3911286 ? r3911290 : r3911297;
        return r3911298;
}

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

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

  4. Final simplification0.3

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

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))