if x < 61.772213f0

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   19.4
2. Using strategy rm
   19.4
3. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   19.4
4. Applied flip-- to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   19.7
5. Applied frac-times to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   20.3
6. Applied flip-+ to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   20.1
7. Applied associate-*l/ to get
   \[\frac{\color{red}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   19.4
8. Applied frac-sub to get
   \[\frac{\color{red}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
   20.2
9. Applied taylor to get
   \[\frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\frac{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]
   9.2
10. Taylor expanded around 0 to get
    \[\frac{\frac{\color{red}{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\frac{\color{blue}{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2}\]
    9.2
11. Applied simplify to get
    \[\frac{\frac{2 - \left(2 \cdot \frac{x}{{\varepsilon}^2} + 2 \cdot \frac{1}{{\varepsilon}^2}\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{2 - \frac{2}{\varepsilon} \cdot \left(\frac{1}{\varepsilon} + \frac{x}{\varepsilon}\right)}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}\]
    9.7

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12. Applied final simplification
13. Applied simplify to get
    \[\color{red}{\frac{2 - \frac{2}{\varepsilon} \cdot \left(\frac{1}{\varepsilon} + \frac{x}{\varepsilon}\right)}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}} \leadsto \color{blue}{\frac{2 - \frac{(x * \left(\frac{2}{\varepsilon}\right) + \left(\frac{2}{\varepsilon}\right))_*}{\varepsilon}}{\left(2 \cdot \left(1 - \frac{1}{\varepsilon}\right)\right) \cdot (\left(e^{(\varepsilon * x + x)_*}\right) * \left(\frac{1}{\varepsilon}\right) + \left(e^{(\varepsilon * x + x)_*}\right))_*}}\]
    1.1

if 61.772213f0 < x

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   0.1
2. Using strategy rm
   0.1
3. Applied add-cube-cbrt to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\color{red}{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\color{blue}{{\left(\sqrt[3]{\left(1 + \varepsilon\right) \cdot x}\right)}^3}}}{2}\]
   0.1
4. Applied simplify to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-{\color{red}{\left(\sqrt[3]{\left(1 + \varepsilon\right) \cdot x}\right)}}^3}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-{\color{blue}{\left(\sqrt[3]{(\varepsilon * x + x)_*}\right)}}^3}}{2}\]
   0.1