Initial program 15.7
\[1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}\]
- Using strategy
rm Applied flip--15.7
\[\leadsto \color{blue}{\frac{1 \cdot 1 - \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)} \cdot \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}}\]
Simplified15.2
\[\leadsto \frac{\color{blue}{1 \cdot \left(1 - 0.5\right) - 0.5 \cdot \frac{1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
- Using strategy
rm Applied associate-*r/15.2
\[\leadsto \frac{1 \cdot \left(1 - 0.5\right) - \color{blue}{\frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied flip3--15.2
\[\leadsto \frac{1 \cdot \color{blue}{\frac{{1}^{3} - {0.5}^{3}}{1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)}} - \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied associate-*r/15.2
\[\leadsto \frac{\color{blue}{\frac{1 \cdot \left({1}^{3} - {0.5}^{3}\right)}{1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)}} - \frac{0.5 \cdot 1}{\mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied frac-sub15.2
\[\leadsto \frac{\color{blue}{\frac{\left(1 \cdot \left({1}^{3} - {0.5}^{3}\right)\right) \cdot \mathsf{hypot}\left(1, x\right) - \left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \left(0.5 \cdot 1\right)}{\left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Simplified15.2
\[\leadsto \frac{\frac{\color{blue}{1 \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right)}}{\left(1 \cdot 1 + \left(0.5 \cdot 0.5 + 1 \cdot 0.5\right)\right) \cdot \mathsf{hypot}\left(1, x\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Simplified15.2
\[\leadsto \frac{\frac{1 \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)\right)}{\color{blue}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
- Using strategy
rm Applied flip3-+15.2
\[\leadsto \frac{\frac{1 \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - 0.5 \cdot \color{blue}{\frac{{\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3} + {\left(1 \cdot 1\right)}^{3}}{\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)}}\right)}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied associate-*r/15.2
\[\leadsto \frac{\frac{1 \cdot \left(\left({1}^{3} - {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right) - \color{blue}{\frac{0.5 \cdot \left({\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3} + {\left(1 \cdot 1\right)}^{3}\right)}{\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)}}\right)}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied flip--15.2
\[\leadsto \frac{\frac{1 \cdot \left(\color{blue}{\frac{{1}^{3} \cdot {1}^{3} - {0.5}^{3} \cdot {0.5}^{3}}{{1}^{3} + {0.5}^{3}}} \cdot \mathsf{hypot}\left(1, x\right) - \frac{0.5 \cdot \left({\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3} + {\left(1 \cdot 1\right)}^{3}\right)}{\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)}\right)}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied associate-*l/15.3
\[\leadsto \frac{\frac{1 \cdot \left(\color{blue}{\frac{\left({1}^{3} \cdot {1}^{3} - {0.5}^{3} \cdot {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right)}{{1}^{3} + {0.5}^{3}}} - \frac{0.5 \cdot \left({\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3} + {\left(1 \cdot 1\right)}^{3}\right)}{\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)}\right)}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Applied frac-sub15.4
\[\leadsto \frac{\frac{1 \cdot \color{blue}{\frac{\left(\left({1}^{3} \cdot {1}^{3} - {0.5}^{3} \cdot {0.5}^{3}\right) \cdot \mathsf{hypot}\left(1, x\right)\right) \cdot \left(\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)\right) - \left({1}^{3} + {0.5}^{3}\right) \cdot \left(0.5 \cdot \left({\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3} + {\left(1 \cdot 1\right)}^{3}\right)\right)}{\left({1}^{3} + {0.5}^{3}\right) \cdot \left(\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)\right)}}}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Simplified15.4
\[\leadsto \frac{\frac{1 \cdot \frac{\color{blue}{\left(-\left({1}^{3} + {0.5}^{3}\right) \cdot 0.5\right) \cdot \left({1}^{6} + {\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3}\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(0.5 + 1\right)\right) + \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right)\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \left({1}^{6} + \left(-{0.5}^{6}\right)\right)\right)}}{\left({1}^{3} + {0.5}^{3}\right) \cdot \left(\left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Simplified15.4
\[\leadsto \frac{\frac{1 \cdot \frac{\left(-\left({1}^{3} + {0.5}^{3}\right) \cdot 0.5\right) \cdot \left({1}^{6} + {\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3}\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(0.5 + 1\right)\right) + \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right)\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \left({1}^{6} + \left(-{0.5}^{6}\right)\right)\right)}{\color{blue}{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(0.5 + 1\right)\right) + \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right)\right) \cdot \left({1}^{3} + {0.5}^{3}\right)}}}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
Final simplification15.4
\[\leadsto \frac{\frac{1 \cdot \frac{\left(-\left({1}^{3} + {0.5}^{3}\right) \cdot 0.5\right) \cdot \left({1}^{6} + {\left(0.5 \cdot \left(0.5 + 1\right)\right)}^{3}\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(0.5 + 1\right)\right) + \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right)\right) \cdot \left(\mathsf{hypot}\left(1, x\right) \cdot \left({1}^{6} + \left(-{0.5}^{6}\right)\right)\right)}{\left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1 - 0.5 \cdot \left(0.5 + 1\right)\right) + \left(0.5 \cdot \left(0.5 + 1\right)\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right)\right)\right) \cdot \left({1}^{3} + {0.5}^{3}\right)}}{\mathsf{hypot}\left(1, x\right) \cdot \left(0.5 \cdot \left(0.5 + 1\right) + 1 \cdot 1\right)}}{1 + \sqrt{0.5 \cdot \left(1 + \frac{1}{\mathsf{hypot}\left(1, x\right)}\right)}}\]
