\((\left(\frac{b}{1}\right) * \left(\frac{\frac{d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{c}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{\sqrt{c^2 + d^2}^*}\right))_*\)
- Started with
\[\frac{a \cdot c + b \cdot d}{{c}^2 + {d}^2}\]
12.5
- Using strategy
rm 12.5
- Applied add-sqr-sqrt to get
\[\frac{a \cdot c + b \cdot d}{\color{red}{{c}^2 + {d}^2}} \leadsto \frac{a \cdot c + b \cdot d}{\color{blue}{{\left(\sqrt{{c}^2 + {d}^2}\right)}^2}}\]
12.5
- 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
- Applied taylor to get
\[\frac{a \cdot c + b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
7.9
- Taylor expanded around 0 to get
\[\color{red}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \color{blue}{\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
7.9
- Using strategy
rm 7.9
- Applied *-un-lft-identity to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\color{red}{\left(\sqrt{c^2 + d^2}^*\right)}}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{{\color{blue}{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}}^2}\]
7.9
- Applied square-prod to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{\color{red}{{\left(1 \cdot \sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b \cdot d}{\color{blue}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
7.9
- Applied times-frac to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{red}{\frac{b \cdot d}{{1}^2 \cdot {\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{blue}{\frac{b}{{1}^2} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
4.2
- Applied simplify to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{red}{\frac{b}{{1}^2}} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \color{blue}{\frac{b}{1}} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
4.2
- Applied taylor to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}\]
4.2
- Taylor expanded around 0 to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \color{red}{\frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}} \leadsto \frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \color{blue}{\frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2}}\]
4.2
- Applied simplify to get
\[\frac{c \cdot a}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} + \frac{b}{1} \cdot \frac{d}{{\left(\sqrt{c^2 + d^2}^*\right)}^2} \leadsto (\left(\frac{b}{1}\right) * \left(\frac{\frac{d}{\sqrt{c^2 + d^2}^*}}{\sqrt{c^2 + d^2}^*}\right) + \left(\frac{c}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{\sqrt{c^2 + d^2}^*}\right))_*\]
0.8
- Applied final simplification
- Removed slow pow expressions
