Initial program 25.5
\[\frac{b \cdot c - a \cdot d}{c \cdot c + d \cdot d}\]
- Using strategy
rm Applied add-sqr-sqrt25.5
\[\leadsto \frac{b \cdot c - a \cdot d}{\color{blue}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}}\]
Applied *-un-lft-identity25.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(b \cdot c - a \cdot d\right)}}{\sqrt{c \cdot c + d \cdot d} \cdot \sqrt{c \cdot c + d \cdot d}}\]
Applied times-frac25.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{c \cdot c + d \cdot d}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}}\]
Applied simplify25.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*}} \cdot \frac{b \cdot c - a \cdot d}{\sqrt{c \cdot c + d \cdot d}}\]
Applied simplify16.4
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\frac{c \cdot b - a \cdot d}{\sqrt{c^2 + d^2}^*}}\]
- Using strategy
rm Applied div-sub16.4
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\sqrt{c^2 + d^2}^*}\right)}\]
- Using strategy
rm Applied *-un-lft-identity16.4
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \frac{a \cdot d}{\color{blue}{1 \cdot \sqrt{c^2 + d^2}^*}}\right)\]
Applied times-frac9.1
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{c \cdot b}{\sqrt{c^2 + d^2}^*} - \color{blue}{\frac{a}{1} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}}\right)\]
Applied add-sqr-sqrt9.1
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\frac{c \cdot b}{\color{blue}{\sqrt{\sqrt{c^2 + d^2}^*} \cdot \sqrt{\sqrt{c^2 + d^2}^*}}} - \frac{a}{1} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\right)\]
Applied times-frac0.8
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \left(\color{blue}{\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}} \cdot \frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}} - \frac{a}{1} \cdot \frac{d}{\sqrt{c^2 + d^2}^*}\right)\]
Applied prod-diff0.8
\[\leadsto \frac{1}{\sqrt{c^2 + d^2}^*} \cdot \color{blue}{\left((\left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(-\frac{d}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{1}\right))_* + (\left(-\frac{d}{\sqrt{c^2 + d^2}^*}\right) \cdot \left(\frac{a}{1}\right) + \left(\frac{d}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{1}\right))_*\right)}\]
Applied distribute-lft-in0.8
\[\leadsto \color{blue}{\frac{1}{\sqrt{c^2 + d^2}^*} \cdot (\left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(-\frac{d}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{1}\right))_* + \frac{1}{\sqrt{c^2 + d^2}^*} \cdot (\left(-\frac{d}{\sqrt{c^2 + d^2}^*}\right) \cdot \left(\frac{a}{1}\right) + \left(\frac{d}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{1}\right))_*}\]
Applied simplify0.7
\[\leadsto \color{blue}{\frac{(\left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(\frac{-d}{\sqrt{c^2 + d^2}^*} \cdot a\right))_*}{\sqrt{c^2 + d^2}^*}} + \frac{1}{\sqrt{c^2 + d^2}^*} \cdot (\left(-\frac{d}{\sqrt{c^2 + d^2}^*}\right) \cdot \left(\frac{a}{1}\right) + \left(\frac{d}{\sqrt{c^2 + d^2}^*} \cdot \frac{a}{1}\right))_*\]
Applied simplify0.7
\[\leadsto \frac{(\left(\frac{c}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) \cdot \left(\frac{b}{\sqrt{\sqrt{c^2 + d^2}^*}}\right) + \left(\frac{-d}{\sqrt{c^2 + d^2}^*} \cdot a\right))_*}{\sqrt{c^2 + d^2}^*} + \color{blue}{0}\]
