\(\frac{(\left(x \cdot \frac{2}{3}\right) * \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}\)
- Started with
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
22.7
- Using strategy
rm 22.7
- Applied neg-sub0 to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{red}{-\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{\color{blue}{0 - \left(1 + \varepsilon\right) \cdot x}}}{2}\]
22.7
- Applied exp-diff to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{red}{e^{0 - \left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot \color{blue}{\frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
22.7
- Applied flip-- to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\left(\frac{1}{\varepsilon} - 1\right)} \cdot \frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1}} \cdot \frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
22.9
- Applied frac-times to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{red}{\frac{{\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2}{\frac{1}{\varepsilon} + 1} \cdot \frac{e^{0}}{e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \color{blue}{\frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2}\]
23.2
- Applied exp-neg to get
\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{red}{e^{-\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \color{blue}{\frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
23.2
- Applied un-div-inv to get
\[\frac{\color{red}{\left(1 + \frac{1}{\varepsilon}\right) \cdot \frac{1}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2} \leadsto \frac{\color{blue}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}{2}\]
23.2
- Applied frac-sub to get
\[\frac{\color{red}{\frac{1 + \frac{1}{\varepsilon}}{e^{\left(1 - \varepsilon\right) \cdot x}} - \frac{\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}}{\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}}}}{2} \leadsto \frac{\color{blue}{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}}{2}\]
23.2
- Applied taylor to get
\[\frac{\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right) - e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left({\left(\frac{1}{\varepsilon}\right)}^2 - {1}^2\right) \cdot e^{0}\right)}{e^{\left(1 - \varepsilon\right) \cdot x} \cdot \left(\left(\frac{1}{\varepsilon} + 1\right) \cdot e^{\left(1 + \varepsilon\right) \cdot x}\right)}}{2} \leadsto \frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2}\]
0.2
- Taylor expanded around 0 to get
\[\frac{\color{red}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2} \leadsto \frac{\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}}{2}\]
0.2
- Applied simplify to get
\[\frac{\left(2 + \frac{2}{3} \cdot {x}^{3}\right) - {x}^2}{2} \leadsto \frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}\]
0.2
- Applied final simplification
- Applied simplify to get
\[\color{red}{\frac{\left(\frac{2}{3} \cdot x\right) \cdot \left(x \cdot x\right) + \left(2 - x \cdot x\right)}{2}} \leadsto \color{blue}{\frac{(\left(x \cdot \frac{2}{3}\right) * \left(x \cdot x\right) + \left(2 - x \cdot x\right))_*}{2}}\]
0.2
