\(\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + e^{\log \left(\frac{{x}^2}{\sqrt{2}}\right) + \log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}\)
- Started with
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
26.6
- Applied taylor to get
\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leadsto \left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}\]
0.3
- Taylor expanded around 0 to get
\[\color{red}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}}\]
0.3
- Applied simplify to get
\[\color{red}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}} \leadsto \color{blue}{\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}\]
0.3
- Using strategy
rm 0.3
- Applied add-exp-log to get
\[\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \frac{x \cdot x}{\sqrt{2}} \cdot \color{red}{\left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)} \leadsto \left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \frac{x \cdot x}{\sqrt{2}} \cdot \color{blue}{e^{\log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}}\]
0.3
- Applied add-exp-log to get
\[\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{red}{\frac{x \cdot x}{\sqrt{2}}} \cdot e^{\log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)} \leadsto \left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{blue}{e^{\log \left(\frac{x \cdot x}{\sqrt{2}}\right)}} \cdot e^{\log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}\]
1.2
- Applied prod-exp to get
\[\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{red}{e^{\log \left(\frac{x \cdot x}{\sqrt{2}}\right)} \cdot e^{\log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}} \leadsto \left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{blue}{e^{\log \left(\frac{x \cdot x}{\sqrt{2}}\right) + \log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}}\]
1.3
- Applied simplify to get
\[\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + e^{\color{red}{\log \left(\frac{x \cdot x}{\sqrt{2}}\right) + \log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}} \leadsto \left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + e^{\color{blue}{\log \left(\frac{{x}^2}{\sqrt{2}}\right) + \log \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}}\]
1.3
