Initial program 25.5
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
Initial simplification25.5
\[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}\]
- Using strategy
rm Applied add-sqr-sqrt25.5
\[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]
Applied *-un-lft-identity25.5
\[\leadsto \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
Applied times-frac25.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]
Simplified25.5
\[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
Simplified16.3
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}\]
- Using strategy
rm Applied div-sub16.3
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\left(\frac{x.im \cdot y.re}{\sqrt{y.im^2 + y.re^2}^*} - \frac{x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}\right)}\]
- Using strategy
rm Applied *-un-lft-identity16.3
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \left(\frac{x.im \cdot y.re}{\sqrt{y.im^2 + y.re^2}^*} - \frac{x.re \cdot y.im}{\color{blue}{1 \cdot \sqrt{y.im^2 + y.re^2}^*}}\right)\]
Applied times-frac9.0
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \left(\frac{x.im \cdot y.re}{\sqrt{y.im^2 + y.re^2}^*} - \color{blue}{\frac{x.re}{1} \cdot \frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}}\right)\]
Applied add-sqr-sqrt9.1
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \left(\frac{x.im \cdot y.re}{\color{blue}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}}} - \frac{x.re}{1} \cdot \frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}\right)\]
Applied times-frac0.8
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \left(\color{blue}{\frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}} - \frac{x.re}{1} \cdot \frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}\right)\]
Applied prod-diff0.8
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\left((\left(\frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) \cdot \left(\frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) + \left(-\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.re}{1}\right))_* + (\left(-\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}\right) \cdot \left(\frac{x.re}{1}\right) + \left(\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.re}{1}\right))_*\right)}\]
Applied distribute-rgt-in0.8
\[\leadsto \color{blue}{(\left(\frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) \cdot \left(\frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) + \left(-\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.re}{1}\right))_* \cdot \frac{1}{\sqrt{y.im^2 + y.re^2}^*} + (\left(-\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}\right) \cdot \left(\frac{x.re}{1}\right) + \left(\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.re}{1}\right))_* \cdot \frac{1}{\sqrt{y.im^2 + y.re^2}^*}}\]
Simplified0.8
\[\leadsto (\left(\frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) \cdot \left(\frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) + \left(-\frac{y.im}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{x.re}{1}\right))_* \cdot \frac{1}{\sqrt{y.im^2 + y.re^2}^*} + \color{blue}{0}\]
Final simplification0.8
\[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot (\left(\frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) \cdot \left(\frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\right) + \left(\left(-x.re\right) \cdot \frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}\right))_*\]
