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