Average Error: 40.0 → 0.3
Time: 17.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00011016449422371341:\\ \;\;\;\;\left(e^{x} - 1\right) \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot x + \left(\left(x \cdot \frac{1}{6}\right) \cdot x + 1\right)\\ \end{array}\]
double f(double x) {
        double r4702280 = x;
        double r4702281 = exp(r4702280);
        double r4702282 = 1.0;
        double r4702283 = r4702281 - r4702282;
        double r4702284 = r4702283 / r4702280;
        return r4702284;
}
↓
double f(double x) {
        double r4702285 = x;
        double r4702286 = -0.00011016449422371341;
        bool r4702287 = r4702285 <= r4702286;
        double r4702288 = exp(r4702285);
        double r4702289 = 1.0;
        double r4702290 = r4702288 - r4702289;
        double r4702291 = r4702289 / r4702285;
        double r4702292 = r4702290 * r4702291;
        double r4702293 = 0.5;
        double r4702294 = r4702293 * r4702285;
        double r4702295 = 0.16666666666666666;
        double r4702296 = r4702285 * r4702295;
        double r4702297 = r4702296 * r4702285;
        double r4702298 = r4702297 + r4702289;
        double r4702299 = r4702294 + r4702298;
        double r4702300 = r4702287 ? r4702292 : r4702299;
        return r4702300;
}


\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00011016449422371341:\\
\;\;\;\;\left(e^{x} - 1\right) \cdot \frac{1}{x}\\

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

\end{array}

Error

Bits error versus x

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 < -0.00011016449422371341

    1. Initial program 0.1

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

    3. Applied div-inv0.1

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

    if -0.00011016449422371341 < x

    1. Initial program 60.2

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

    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

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}\]
    4. Using strategy rm

    5. Applied distribute-lft-in0.5

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

    6. Applied associate-+r+0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00011016449422371341:\\ \;\;\;\;\left(e^{x} - 1\right) \cdot \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot x + \left(\left(x \cdot \frac{1}{6}\right) \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019102 
(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))