\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 178.4372176880553126920858630910515785217:\\ \;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}

↓

\begin{array}{l}
\mathbf{if}\;x \le 178.4372176880553126920858630910515785217:\\
\;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}{2}\\

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

\end{array}

double f(double x, double eps) {
        double r35487 = 1.0;
        double r35488 = eps;
        double r35489 = r35487 / r35488;
        double r35490 = r35487 + r35489;
        double r35491 = r35487 - r35488;
        double r35492 = x;
        double r35493 = r35491 * r35492;
        double r35494 = -r35493;
        double r35495 = exp(r35494);
        double r35496 = r35490 * r35495;
        double r35497 = r35489 - r35487;
        double r35498 = r35487 + r35488;
        double r35499 = r35498 * r35492;
        double r35500 = -r35499;
        double r35501 = exp(r35500);
        double r35502 = r35497 * r35501;
        double r35503 = r35496 - r35502;
        double r35504 = 2.0;
        double r35505 = r35503 / r35504;
        return r35505;
}

↓

double f(double x, double eps) {
        double r35506 = x;
        double r35507 = 178.4372176880553;
        bool r35508 = r35506 <= r35507;
        double r35509 = 8.0;
        double r35510 = r35506 * r35506;
        double r35511 = r35506 * r35509;
        double r35512 = 12.0;
        double r35513 = r35511 - r35512;
        double r35514 = r35510 * r35513;
        double r35515 = r35509 + r35514;
        double r35516 = cbrt(r35515);
        double r35517 = 2.0;
        double r35518 = r35516 / r35517;
        double r35519 = 1.0;
        double r35520 = eps;
        double r35521 = r35520 - r35519;
        double r35522 = r35506 * r35521;
        double r35523 = exp(r35522);
        double r35524 = r35523 / r35520;
        double r35525 = r35524 + r35523;
        double r35526 = r35519 * r35525;
        double r35527 = r35519 / r35520;
        double r35528 = r35527 - r35519;
        double r35529 = r35519 + r35520;
        double r35530 = r35529 * r35506;
        double r35531 = -r35530;
        double r35532 = exp(r35531);
        double r35533 = r35528 * r35532;
        double r35534 = r35526 - r35533;
        double r35535 = r35534 / r35517;
        double r35536 = r35508 ? r35518 : r35535;
        return r35536;
}

Derivation

Split input into 2 regimes
if x < 178.4372176880553
1. Initial program 39.0
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.3
  \[\leadsto \frac{\color{blue}{\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-cbrt-cube1.3
  \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right) \cdot \left(\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)\right) \cdot \left(\left(0.6666666666666667406815349750104360282421 \cdot {x}^{3} + 2\right) - 1 \cdot {x}^{2}\right)}}}{2}\]
5. Simplified1.3
  \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(2 + \left(x \cdot x\right) \cdot \left(0.6666666666666667406815349750104360282421 \cdot x - 1\right)\right)}^{3}}}}{2}\]
6. Taylor expanded around 0 1.3
  \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(8 \cdot {x}^{3} + 8\right) - 12 \cdot {x}^{2}}}}{2}\]
7. Simplified1.3
  \[\leadsto \frac{\sqrt[3]{\color{blue}{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}}{2}\]
if 178.4372176880553 < x
1. Initial program 0.2
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around inf 0.2
  \[\leadsto \frac{\color{blue}{\left(1 \cdot \frac{e^{x \cdot \varepsilon - 1 \cdot x}}{\varepsilon} + 1 \cdot e^{x \cdot \varepsilon - 1 \cdot x}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
3. Simplified0.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right)} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
Recombined 2 regimes into one program.
Final simplification1.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 178.4372176880553126920858630910515785217:\\ \;\;\;\;\frac{\sqrt[3]{8 + \left(x \cdot x\right) \cdot \left(x \cdot 8 - 12\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{e^{x \cdot \left(\varepsilon - 1\right)}}{\varepsilon} + e^{x \cdot \left(\varepsilon - 1\right)}\right) - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

could not determine a ground truth for program body (more)

Error

Try it out

Derivation

`if x < 178.4372176880553`

`if 178.4372176880553 < x`

Reproduce

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