Average Error: 40.1 → 0.3
Time: 10.5s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00016601036140039243:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00016601036140039243:\\
\;\;\;\;\frac{e^{x} - 1}{x}\\

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

\end{array}
double f(double x) {
        double r1312418 = x;
        double r1312419 = exp(r1312418);
        double r1312420 = 1.0;
        double r1312421 = r1312419 - r1312420;
        double r1312422 = r1312421 / r1312418;
        return r1312422;
}
↓
double f(double x) {
        double r1312423 = x;
        double r1312424 = -0.00016601036140039243;
        bool r1312425 = r1312423 <= r1312424;
        double r1312426 = exp(r1312423);
        double r1312427 = 1.0;
        double r1312428 = r1312426 - r1312427;
        double r1312429 = r1312428 / r1312423;
        double r1312430 = 0.16666666666666666;
        double r1312431 = r1312430 * r1312423;
        double r1312432 = 0.5;
        double r1312433 = r1312431 + r1312432;
        double r1312434 = r1312423 * r1312433;
        double r1312435 = exp(r1312434);
        double r1312436 = log(r1312435);
        double r1312437 = r1312427 + r1312436;
        double r1312438 = r1312425 ? r1312429 : r1312437;
        return r1312438;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target39.3
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 < -0.00016601036140039243

    1. Initial program 0.1

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

    2. Taylor expanded around inf 0.1

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

    if -0.00016601036140039243 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

    5. Applied add-log-exp0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00016601036140039243:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 
(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))