Derivation

Split input into 3 regimes
if y.re < -2.227620296543906e+84
1. Initial program 37.2
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt37.2
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity37.2
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac37.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Applied simplify37.2
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Applied simplify25.0
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt25.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}}\]
10. Applied *-un-lft-identity25.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\color{blue}{1 \cdot (x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\]
11. Applied times-frac25.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\right)}\]
12. Taylor expanded around -inf 14.5
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(-\left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\right)}\]
13. Applied simplify11.4
  \[\leadsto \color{blue}{\frac{(\left(\frac{y.im}{y.re}\right) \cdot \left(-x.im\right) + \left(-x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
if -2.227620296543906e+84 < y.re < 2.168394667707026e+41
1. Initial program 18.3
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt18.3
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity18.3
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac18.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Applied simplify18.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Applied simplify11.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
if 2.168394667707026e+41 < y.re
1. Initial program 35.4
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt35.4
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity35.4
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac35.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Applied simplify35.4
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Applied simplify23.9
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt24.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}}\]
10. Applied *-un-lft-identity24.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\color{blue}{1 \cdot (x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\]
11. Applied times-frac24.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\right)}\]
12. Taylor expanded around inf 15.2
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(x.re + \frac{y.im \cdot x.im}{y.re}\right)}\]
13. Applied simplify12.3
  \[\leadsto \color{blue}{\frac{(\left(\frac{y.im}{y.re}\right) \cdot x.im + x.re)_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if y.re < -2.227620296543906e+84`

`if -2.227620296543906e+84 < y.re < 2.168394667707026e+41`

`if 2.168394667707026e+41 < y.re`

Runtime