if x < 540.2976f0

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   22.0
2. Using strategy rm
   22.0
3. Applied flip3-- to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   22.5
4. Applied associate-*l/ to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)} \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}}{2}\]
   22.5
5. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2}\]
   22.5
6. Applied un-div-inv to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}{2}\]
   22.5
7. Applied frac-sub to get
   \[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{{\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}}{2}\]
   22.6
8. Applied simplify to get
   \[\frac{\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^{3} - {1}^{3}\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}{2} \leadsto \frac{\frac{\color{blue}{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) + {\left(\frac{1}{\varepsilon}\right)}^2\right) - e^{x \cdot \left(1 - \varepsilon\right) - \varepsilon \cdot x} \cdot \frac{\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1}{e^{x}}}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}{2}\]
   22.6
9. Applied taylor to get
   \[\frac{\frac{\left(\frac{1}{\varepsilon} + 1\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) + {\left(\frac{1}{\varepsilon}\right)}^2\right) - e^{x \cdot \left(1 - \varepsilon\right) - \varepsilon \cdot x} \cdot \frac{\frac{\frac{1}{\varepsilon}}{\varepsilon \cdot \varepsilon} - 1}{e^{x}}}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left({\left(\frac{1}{\varepsilon}\right)}^2 + \left({1}^2 + \frac{1}{\varepsilon} \cdot 1\right)\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
   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 540.2976f0 < 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 neg-mul-1 to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{red}{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]
   0.1
4. Applied exp-prod to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-1 \cdot \left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2}\]
   0.1
5. Using strategy rm
   0.1
6. Applied add-sqr-sqrt to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(\sqrt{{\left(e^{-1}\right)}^{\left(\left(1 + \varepsilon\right) \cdot x\right)}}\right)}^2}}{2}\]
   0.1