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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 1.5115703837855707:\\ \;\;\;\;\frac{\left(2 + e^{\sqrt[3]{{\left(\log \left(\sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)}\right)\right)}^{3}}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 1.5115703837855707
1. Initial program 38.9
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.1
  \[\leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-cube-cbrt1.1
  \[\leadsto \frac{\left(2 + \color{blue}{\left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}}\right) - {x}^{2}}{2}\]
5. Using strategy rm
6. Applied add-exp-log1.1
  \[\leadsto \frac{\left(2 + \color{blue}{e^{\log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right)}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\]
7. Using strategy rm
8. Applied add-cbrt-cube1.1
  \[\leadsto \frac{\left(2 + e^{\color{blue}{\sqrt[3]{\left(\log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) \cdot \log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right)\right) \cdot \log \left(\sqrt[3]{\frac{2}{3} \cdot {x}^{3}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right)}}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\]
9. Applied simplify1.1
  \[\leadsto \frac{\left(2 + e^{\sqrt[3]{\color{blue}{{\left(\log \left(\sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)} \cdot \sqrt[3]{\left(x \cdot \frac{2}{3}\right) \cdot \left(x \cdot x\right)}\right)\right)}^{3}}}} \cdot \sqrt[3]{\frac{2}{3} \cdot {x}^{3}}\right) - {x}^{2}}{2}\]
if 1.5115703837855707 < x
1. Initial program 0.4
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Using strategy rm
3. Applied add-cube-cbrt0.4
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\]
4. Applied associate-*r*0.4
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
Recombined 2 regimes into one program.

Error

Derivation

`if x < 1.5115703837855707`

`if 1.5115703837855707 < x`

Runtime