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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 10.164318894064335:\\ \;\;\;\;\frac{\log \left(\frac{\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}{\sqrt[3]{e^{{x}^{2}}} \cdot \sqrt[3]{e^{{x}^{2}}}} \cdot \frac{\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}{\sqrt[3]{e^{{x}^{2}}}}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 10.164318894064335
1. Initial program 39.5
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.3
  \[\leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-log-exp1.3
  \[\leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - \color{blue}{\log \left(e^{{x}^{2}}\right)}}{2}\]
5. Applied add-log-exp1.3
  \[\leadsto \frac{\color{blue}{\log \left(e^{2 + \frac{2}{3} \cdot {x}^{3}}\right)} - \log \left(e^{{x}^{2}}\right)}{2}\]
6. Applied diff-log1.3
  \[\leadsto \frac{\color{blue}{\log \left(\frac{e^{2 + \frac{2}{3} \cdot {x}^{3}}}{e^{{x}^{2}}}\right)}}{2}\]
7. Using strategy rm
8. Applied add-cube-cbrt1.3
  \[\leadsto \frac{\log \left(\frac{e^{2 + \frac{2}{3} \cdot {x}^{3}}}{\color{blue}{\left(\sqrt[3]{e^{{x}^{2}}} \cdot \sqrt[3]{e^{{x}^{2}}}\right) \cdot \sqrt[3]{e^{{x}^{2}}}}}\right)}{2}\]
9. Applied add-cube-cbrt1.3
  \[\leadsto \frac{\log \left(\frac{\color{blue}{\left(\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}\right) \cdot \sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}}{\left(\sqrt[3]{e^{{x}^{2}}} \cdot \sqrt[3]{e^{{x}^{2}}}\right) \cdot \sqrt[3]{e^{{x}^{2}}}}\right)}{2}\]
10. Applied times-frac1.3
  \[\leadsto \frac{\log \color{blue}{\left(\frac{\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}} \cdot \sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}{\sqrt[3]{e^{{x}^{2}}} \cdot \sqrt[3]{e^{{x}^{2}}}} \cdot \frac{\sqrt[3]{e^{2 + \frac{2}{3} \cdot {x}^{3}}}}{\sqrt[3]{e^{{x}^{2}}}}\right)}}{2}\]
if 10.164318894064335 < 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
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
4. Applied associate-*l*0.4
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot \sqrt[3]{\frac{1}{\varepsilon} - 1}\right) \cdot \left(\sqrt[3]{\frac{1}{\varepsilon} - 1} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]
Recombined 2 regimes into one program.

Error

Try it out

Derivation

`if x < 10.164318894064335`

`if 10.164318894064335 < x`

Runtime

In
Out