Initial program 58.5
\[e^{x} - 1\]
- Using strategy
rm Applied add-sqr-sqrt58.6
\[\leadsto \color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - 1\]
Applied difference-of-sqr-158.6
\[\leadsto \color{blue}{\left(\sqrt{e^{x}} + 1\right) \cdot \left(\sqrt{e^{x}} - 1\right)}\]
Taylor expanded around 0 0.4
\[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\frac{1}{2} \cdot x + \left(\frac{1}{48} \cdot {x}^{3} + \frac{1}{8} \cdot {x}^{2}\right)\right)}\]
Simplified0.4
\[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\left(\left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right) + \frac{1}{2}\right) \cdot x\right)}\]
- Using strategy
rm Applied flip-+0.4
\[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \left(\color{blue}{\frac{\left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) \cdot \left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}}{x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right) - \frac{1}{2}}} \cdot x\right)\]
Applied associate-*l/0.4
\[\leadsto \left(\sqrt{e^{x}} + 1\right) \cdot \color{blue}{\frac{\left(\left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) \cdot \left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x}{x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right) - \frac{1}{2}}}\]
Applied flip3-+0.4
\[\leadsto \color{blue}{\frac{{\left(\sqrt{e^{x}}\right)}^{3} + {1}^{3}}{\sqrt{e^{x}} \cdot \sqrt{e^{x}} + \left(1 \cdot 1 - \sqrt{e^{x}} \cdot 1\right)}} \cdot \frac{\left(\left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) \cdot \left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x}{x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right) - \frac{1}{2}}\]
Applied frac-times0.4
\[\leadsto \color{blue}{\frac{\left({\left(\sqrt{e^{x}}\right)}^{3} + {1}^{3}\right) \cdot \left(\left(\left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) \cdot \left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right)\right) - \frac{1}{2} \cdot \frac{1}{2}\right) \cdot x\right)}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}} + \left(1 \cdot 1 - \sqrt{e^{x}} \cdot 1\right)\right) \cdot \left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right) - \frac{1}{2}\right)}}\]
Simplified0.4
\[\leadsto \frac{\color{blue}{\left(\left(1 + \sqrt{e^{x}} \cdot e^{x}\right) \cdot x\right) \cdot \left(\left(\left(\frac{1}{8} + x \cdot \frac{1}{48}\right) \cdot x\right) \cdot \left(\left(\frac{1}{8} + x \cdot \frac{1}{48}\right) \cdot x\right) + \frac{-1}{4}\right)}}{\left(\sqrt{e^{x}} \cdot \sqrt{e^{x}} + \left(1 \cdot 1 - \sqrt{e^{x}} \cdot 1\right)\right) \cdot \left(x \cdot \left(\frac{1}{48} \cdot x + \frac{1}{8}\right) - \frac{1}{2}\right)}\]
Simplified0.4
\[\leadsto \frac{\left(\left(1 + \sqrt{e^{x}} \cdot e^{x}\right) \cdot x\right) \cdot \left(\left(\left(\frac{1}{8} + x \cdot \frac{1}{48}\right) \cdot x\right) \cdot \left(\left(\frac{1}{8} + x \cdot \frac{1}{48}\right) \cdot x\right) + \frac{-1}{4}\right)}{\color{blue}{\left(\left(\frac{1}{8} + \frac{1}{48} \cdot x\right) \cdot x + \frac{-1}{2}\right) + \left(e^{x} - \sqrt{e^{x}}\right) \cdot \left(\left(\frac{1}{8} + \frac{1}{48} \cdot x\right) \cdot x + \frac{-1}{2}\right)}}\]
Final simplification0.4
\[\leadsto \frac{\left(\left(\left(\frac{1}{8} + \frac{1}{48} \cdot x\right) \cdot x\right) \cdot \left(\left(\frac{1}{8} + \frac{1}{48} \cdot x\right) \cdot x\right) + \frac{-1}{4}\right) \cdot \left(\left(1 + e^{x} \cdot \sqrt{e^{x}}\right) \cdot x\right)}{\left(\frac{-1}{2} + \left(\frac{1}{8} + \frac{1}{48} \cdot x\right) \cdot x\right) + \left(\frac{-1}{2} + \left(\frac{1}{8} + \frac{1}{48} \cdot x\right) \cdot x\right) \cdot \left(e^{x} - \sqrt{e^{x}}\right)}\]
