if x < -6.025113626725801e-15

1. Started with
   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   0.7
2. Using strategy rm
   0.7
3. Applied flip-- to get
   \[\sqrt{\frac{e^{2 \cdot x} - 1}{\color{red}{e^{x} - 1}}} \leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{{\left(e^{x}\right)}^2 - {1}^2}{e^{x} + 1}}}}\]
   0.4

if -6.025113626725801e-15 < x

1. Started with
   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   60.8
2. Applied taylor to get
   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}} \leadsto \left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}\]
   0.1
3. Taylor expanded around 0 to get
   \[\color{red}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}} \leadsto \color{blue}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}}\]
   0.1
4. Applied simplify to get
   \[\color{red}{\left(\frac{1}{2} \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{1}{4} \cdot \frac{{x}^2}{\sqrt{2}}\right)\right) - \frac{1}{8} \cdot \frac{{x}^2}{{\left(\sqrt{2}\right)}^{3}}} \leadsto \color{blue}{\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)}\]
   0.2
5. Applied simplify to get
   \[\left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{red}{\frac{x \cdot x}{\sqrt{2}} \cdot \left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right)} \leadsto \left(\frac{x \cdot \frac{1}{2}}{\sqrt{2}} + \sqrt{2}\right) + \color{blue}{\left(\frac{1}{4} - \frac{\frac{1}{8}}{2}\right) \cdot \frac{{x}^2}{\sqrt{2}}}\]
   0.2