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

↓

\[\begin{array}{l} \mathbf{if}\;x \le 89.66429698981352:\\ \;\;\;\;\frac{\left(\frac{1}{8} \cdot {x}^{8} + \left(\frac{1}{6} \cdot {x}^{9} + \left(\frac{2}{3} \cdot {x}^{3} + 2\right)\right)\right) - \left({x}^{2} + \frac{1}{4} \cdot {x}^{6}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2}\\ \end{array}\]

Derivation

Split input into 2 regimes
if x < 89.66429698981352
1. Initial program 39.3
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Taylor expanded around 0 1.0
  \[\leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^{2}}}{2}\]
3. Using strategy rm
4. Applied add-log-exp1.0
  \[\leadsto \frac{\left(2 + \color{blue}{\log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)}\right) - {x}^{2}}{2}\]
5. Using strategy rm
6. Applied flip3--1.0
  \[\leadsto \frac{\color{blue}{\frac{{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) \cdot \left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) + \left({x}^{2} \cdot {x}^{2} + \left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right) \cdot {x}^{2}\right)}}}{2}\]
7. Applied simplify1.0
  \[\leadsto \frac{\frac{{\left(2 + \log \left(e^{\frac{2}{3} \cdot {x}^{3}}\right)\right)}^{3} - {\left({x}^{2}\right)}^{3}}{\color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot x\right) + \left(\left(x \cdot x + 2\right) + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right) \cdot \left(2 + \left(x \cdot x\right) \cdot \left(x \cdot \frac{2}{3}\right)\right)}}}{2}\]
8. Taylor expanded around 0 1.0
  \[\leadsto \frac{\color{blue}{\left(\frac{1}{8} \cdot {x}^{8} + \left(\frac{1}{6} \cdot {x}^{9} + \left(\frac{2}{3} \cdot {x}^{3} + 2\right)\right)\right) - \left({x}^{2} + \frac{1}{4} \cdot {x}^{6}\right)}}{2}\]
if 89.66429698981352 < 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
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\left(\left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)}}{2}\]
4. Applied associate-*r*0.2
  \[\leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot \left(\sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}} \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}\right)\right) \cdot \sqrt[3]{e^{-\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
Recombined 2 regimes into one program.

Error

Derivation

`if x < 89.66429698981352`

`if 89.66429698981352 < x`

Runtime