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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 380.16770422358724:\\ \;\;\;\;\frac{\left(2 + \sqrt[3]{{\left(x \cdot \left(\frac{2}{3} \cdot x\right)\right)}^{3}} \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}\right) \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 380.16770422358724
1. Initial program 39.6
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.4
  \[\leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-cube-cbrt1.4
  \[\leadsto \frac{\left(2 + \frac{2}{3} \cdot \color{blue}{\left(\left(\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}\right) \cdot \sqrt[3]{{x}^{3}}\right)}\right) - {x}^{2}}{2}\]
5. Applied associate-*r*1.4
  \[\leadsto \frac{\left(2 + \color{blue}{\left(\frac{2}{3} \cdot \left(\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{{x}^{3}}\right)\right) \cdot \sqrt[3]{{x}^{3}}}\right) - {x}^{2}}{2}\]
6. Applied simplify1.4
  \[\leadsto \frac{\left(2 + \color{blue}{\left(\left(x \cdot \frac{2}{3}\right) \cdot x\right)} \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\]
7. Using strategy rm
8. Applied add-cbrt-cube1.4
  \[\leadsto \frac{\left(2 + \color{blue}{\sqrt[3]{\left(\left(\left(x \cdot \frac{2}{3}\right) \cdot x\right) \cdot \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right)\right) \cdot \left(\left(x \cdot \frac{2}{3}\right) \cdot x\right)}} \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\]
9. Applied simplify1.4
  \[\leadsto \frac{\left(2 + \sqrt[3]{\color{blue}{{\left(x \cdot \left(\frac{2}{3} \cdot x\right)\right)}^{3}}} \cdot \sqrt[3]{{x}^{3}}\right) - {x}^{2}}{2}\]
if 380.16770422358724 < 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. Using strategy rm
3. Applied add-cube-cbrt0.2
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
4. Applied simplify0.2
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}\right)} \cdot \sqrt[3]{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\]
5. Applied simplify0.2
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}\right) \cdot \color{blue}{\sqrt[3]{\frac{\frac{1}{\varepsilon} - 1}{{\left(e^{x}\right)}^{\left(\varepsilon + 1\right)}}}}}{2}\]
Recombined 2 regimes into one program.

Error

Derivation

`if x < 380.16770422358724`

`if 380.16770422358724 < x`

Runtime