\((\left(\frac{b}{\sqrt{c^2 + d^2}^*}\right) * \left(\frac{d}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*}\right))_*\)
- Started with
\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
12.6
- Using strategy
rm 12.6
- Applied add-sqr-sqrt to get
\[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
12.6
- Applied simplify to get
\[\frac{a \cdot c + b \cdot d}{{\color{red}{\left(\sqrt{{c}^2 + {d}^2}\right)}}^2} \leadsto \frac{a \cdot c + b \cdot d}{{\color{blue}{\left(\sqrt{c^2 + d^2}^*\right)}}^2}\]
8.0
- Using strategy
rm 8.0
- Applied add-cube-cbrt to get
\[\color{red}{\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\right)}^3}\]
8.3
- Applied taylor to get
\[{\left(\sqrt[3]{\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\right)}^3 \leadsto {\left(\sqrt[3]{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\right)}^3\]
8.3
- Taylor expanded around 0 to get
\[{\left(\sqrt[3]{\color{red}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}}\right)}^3 \leadsto {\left(\sqrt[3]{\color{blue}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}}\right)}^3\]
8.3
- Applied simplify to get
\[{\left(\sqrt[3]{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\right)}^3 \leadsto (\left(\frac{b}{\sqrt{c^2 + d^2}^*}\right) * \left(\frac{d}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{a}{\sqrt{c^2 + d^2}^*} \cdot \frac{c}{\sqrt{c^2 + d^2}^*}\right))_*\]
0.4
- Applied final simplification
- Removed slow pow expressions
