if x < 3.63006f0

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   21.9
2. Using strategy rm
   21.9
3. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   21.9
4. Applied un-div-inv to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   21.9
5. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   21.8
6. Applied un-div-inv to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   21.9
7. Applied frac-sub to get
   \[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\frac{1}{\varepsilon} - 1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   21.9
8. Applied simplify to get
   \[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{red}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{\color{blue}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}}{2}\]
   21.9
9. Applied taylor to get
   \[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x} - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\frac{1}{\varepsilon} - 1\right)}{{\left(e^{x}\right)}^{\left(1 - \left(-1\right)\right)}}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
   0.1
10. Taylor expanded around 0 to get
    \[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
    0.1
11. Applied simplify to get
    \[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
    0.1

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12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{{x}^3 \cdot \frac{2}{3} + \left(2 - x \cdot x\right)}{2}}\]
    0.1

if 3.63006f0 < x

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   1.1
2. Using strategy rm
   1.1
3. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   1.1
4. Applied un-div-inv to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   1.1
5. Using strategy rm
   1.1
6. Applied pow1 to get
   \[\frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \color{blue}{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{1}}}{2}\]
   1.1