Average Error: 40.4 → 0.4
Time: 15.5s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00012950113967971158:\\ \;\;\;\;\frac{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right)}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x} - \frac{\frac{1}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00012950113967971158:\\
\;\;\;\;\frac{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right)}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x} - \frac{\frac{1}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x}\\

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

\end{array}
double f(double x) {
        double r2878356 = x;
        double r2878357 = exp(r2878356);
        double r2878358 = 1.0;
        double r2878359 = r2878357 - r2878358;
        double r2878360 = r2878359 / r2878356;
        return r2878360;
}
↓
double f(double x) {
        double r2878361 = x;
        double r2878362 = -0.00012950113967971158;
        bool r2878363 = r2878361 <= r2878362;
        double r2878364 = exp(r2878361);
        double r2878365 = r2878364 * r2878364;
        double r2878366 = r2878364 * r2878365;
        double r2878367 = 1.0;
        double r2878368 = r2878364 + r2878367;
        double r2878369 = r2878368 + r2878365;
        double r2878370 = r2878366 / r2878369;
        double r2878371 = r2878370 / r2878361;
        double r2878372 = r2878367 / r2878369;
        double r2878373 = r2878372 / r2878361;
        double r2878374 = r2878371 - r2878373;
        double r2878375 = 0.16666666666666666;
        double r2878376 = r2878375 * r2878361;
        double r2878377 = 0.5;
        double r2878378 = r2878376 + r2878377;
        double r2878379 = r2878361 * r2878378;
        double r2878380 = r2878367 + r2878379;
        double r2878381 = r2878363 ? r2878374 : r2878380;
        return r2878381;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.4
Target39.6
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00012950113967971158

    1. Initial program 0.1

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

    3. Applied flip3--0.1

      \[\leadsto \frac{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}{x}\]

    4. Simplified0.1

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

    5. Simplified0.1

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

    7. Applied div-sub0.1

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

    8. Applied div-sub0.1

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

    if -0.00012950113967971158 < 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

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00012950113967971158:\\ \;\;\;\;\frac{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right)}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x} - \frac{\frac{1}{\left(e^{x} + 1\right) + e^{x} \cdot e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(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))