if x < 0.47897726f0

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   21.8
2. Using strategy rm
   21.8
3. Applied exp-neg to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   21.8
4. Applied flip-- to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   22.0
5. Applied frac-times to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot \frac{1}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
   22.4
6. Applied flip-+ to get
   \[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right)} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   22.3
7. Applied associate-*l/ to get
   \[\frac{\color{red}{\frac{{1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2}{1 - \frac{1}{\varepsilon}} \cdot e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
   21.8
8. Applied frac-sub to get
   \[\frac{\color{red}{\frac{\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}}{1 - \frac{1}{\varepsilon}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
   22.4
9. Applied taylor to get
   \[\frac{\frac{\left(\left({1}^2 - {\left(\frac{1}{\varepsilon}\right)}^2\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - \left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot 1\right)}{\left(1 - \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
   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
    \[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{(\left(x \cdot \frac{2}{3}\right) * \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}}\]
    0.1

if 0.47897726f0 < x

1. Started with
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
   1.5
2. Using strategy rm
   1.5
3. Applied pow1 to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{{\left(e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{1}}}{2}\]
   1.5
4. Applied pow1 to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot {\left(e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{{\left(\frac{1}{\varepsilon} - 1\right)}^{1}} \cdot {\left(e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{1}}{2}\]
   1.5
5. Applied pow-prod-down to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{{\left(\frac{1}{\varepsilon} - 1\right)}^{1} \cdot {\left(e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{1}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}^{1}}}{2}\]
   1.5
6. Applied simplify to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{red}{\left(\left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}\right)}}^{1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{blue}{\left(\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}\right)}}^{1}}{2}\]
   1.5
7. Applied taylor to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\frac{\frac{1}{\varepsilon} - 1}{e^{(\varepsilon * x + x)_*}}\right)}^{1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}\right)}^{1}}{2}\]
   1.5
8. Taylor expanded around 0 to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{red}{\left(\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}\right)}}^{1}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\color{blue}{\left(\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}\right)}}^{1}}{2}\]
   1.5
9. Applied simplify to get
   \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - {\left(\frac{1}{e^{(\varepsilon * x + x)_*} \cdot \varepsilon} - \frac{1}{e^{(\varepsilon * x + x)_*}}\right)}^{1}}{2} \leadsto \left(\frac{e^{\left(1 - \varepsilon\right) \cdot \left(-x\right)}}{\frac{2}{1 + \frac{1}{\varepsilon}}} - \frac{e^{-(\varepsilon * x + x)_*}}{2 \cdot \varepsilon}\right) + \frac{e^{-(\varepsilon * x + x)_*}}{2}\]
   1.3

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10. Applied final simplification