Derivation

Split input into 3 regimes
if y.re < -1.609443697480765e+108
1. Initial program 40.9
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt40.9
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity40.9
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac40.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Applied simplify40.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Applied simplify27.8
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt28.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}}\]
10. Applied *-un-lft-identity28.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\color{blue}{1 \cdot (x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\]
11. Applied times-frac28.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\right)}\]
12. Taylor expanded around -inf 12.6
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(-\left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\right)}\]
13. Applied simplify8.9
  \[\leadsto \color{blue}{\frac{(\left(\frac{y.im}{y.re}\right) \cdot \left(-x.im\right) + \left(-x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
if -1.609443697480765e+108 < y.re < 714.2898411289591
1. Initial program 17.8
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt17.8
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity17.8
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac17.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Applied simplify17.9
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Applied simplify10.3
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
if 714.2898411289591 < y.re
1. Initial program 32.7
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt32.7
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity32.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac32.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Applied simplify32.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Applied simplify22.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied add-cube-cbrt22.5
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\color{blue}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}}\]
10. Applied *-un-lft-identity22.5
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\color{blue}{1 \cdot (x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}}{\left(\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\]
11. Applied times-frac22.5
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt[3]{\sqrt{y.re^2 + y.im^2}^*}}\right)}\]
12. Taylor expanded around inf 16.4
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\left(x.re + \frac{y.im \cdot x.im}{y.re}\right)}\]
13. Applied simplify14.5
  \[\leadsto \color{blue}{\frac{(\left(\frac{y.im}{y.re}\right) \cdot x.im + x.re)_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
Recombined 3 regimes into one program.

Error

Derivation

`if y.re < -1.609443697480765e+108`

`if -1.609443697480765e+108 < y.re < 714.2898411289591`

`if 714.2898411289591 < y.re`

Runtime