if x < -3.6838407832643567e-13

1. Started with
   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   0.6
2. Using strategy rm
   0.6
3. Applied add-sqr-sqrt to get
   \[\sqrt{\frac{\color{red}{e^{2 \cdot x}} - 1}{e^{x} - 1}} \leadsto \sqrt{\frac{\color{blue}{{\left(\sqrt{e^{2 \cdot x}}\right)}^2} - 1}{e^{x} - 1}}\]
   0.5
4. Applied difference-of-sqr-1 to get
   \[\sqrt{\frac{\color{red}{{\left(\sqrt{e^{2 \cdot x}}\right)}^2 - 1}}{e^{x} - 1}} \leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + 1\right) \cdot \left(\sqrt{e^{2 \cdot x}} - 1\right)}}{e^{x} - 1}}\]
   0.1

if -3.6838407832643567e-13 < x

1. Started with
   \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
   60.6
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