Initial program 25.7
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
- Using strategy
rm Applied add-sqr-sqrt25.7
\[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
Applied *-un-lft-identity25.7
\[\leadsto \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
Applied times-frac25.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
Applied simplify25.7
\[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
Applied simplify16.7
\[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}\]
- Using strategy
rm Applied div-sub16.7
\[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*} - \frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}\right)}\]
- Using strategy
rm Applied div-inv16.7
\[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*} - \color{blue}{\left(x.re \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}}\right)\]
Applied add-sqr-sqrt16.8
\[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(\frac{y.re \cdot x.im}{\color{blue}{\sqrt{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt{\sqrt{y.re^2 + y.im^2}^*}}} - \left(x.re \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}\right)\]
Applied times-frac9.5
\[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(\color{blue}{\frac{y.re}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}} - \left(x.re \cdot y.im\right) \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}\right)\]
Applied prod-diff9.6
\[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left((\left(\frac{y.re}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \left(\frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) + \left(-\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(x.re \cdot y.im\right)\right))_* + (\left(-\frac{1}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \left(x.re \cdot y.im\right) + \left(\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(x.re \cdot y.im\right)\right))_*\right)}\]
Applied distribute-lft-in9.6
\[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot (\left(\frac{y.re}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \left(\frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) + \left(-\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(x.re \cdot y.im\right)\right))_* + \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot (\left(-\frac{1}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \left(x.re \cdot y.im\right) + \left(\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(x.re \cdot y.im\right)\right))_*}\]
Applied simplify9.5
\[\leadsto \color{blue}{\frac{(\left(\frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \left(\frac{y.re}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) + \left(\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(-y.im\right)\right))_*}{\sqrt{y.re^2 + y.im^2}^*}} + \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot (\left(-\frac{1}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \left(x.re \cdot y.im\right) + \left(\frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(x.re \cdot y.im\right)\right))_*\]
Applied simplify1.0
\[\leadsto \frac{(\left(\frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \left(\frac{y.re}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) + \left(\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(-y.im\right)\right))_*}{\sqrt{y.re^2 + y.im^2}^*} + \color{blue}{\frac{0}{\sqrt{y.re^2 + y.im^2}^*}}\]
