Derivation

Split input into 3 regimes
if y.re < -5.305038240296897e+170
1. Initial program 43.7
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt43.7
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity43.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac43.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified43.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified28.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied add-sqr-sqrt28.2
  \[\leadsto \color{blue}{\left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}}\right)} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}\]
10. Applied associate-*l*28.2
  \[\leadsto \color{blue}{\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}\right)}\]
11. Taylor expanded around -inf 12.7
  \[\leadsto \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \color{blue}{\left(-1 \cdot x.re\right)}\right)\]
12. Simplified12.7
  \[\leadsto \sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \color{blue}{\left(-x.re\right)}\right)\]
if -5.305038240296897e+170 < y.re < 3.45232150826022e+116
1. Initial program 19.0
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt19.0
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity19.0
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac19.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified19.0
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified12.2
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/12.1
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}}\]
10. Simplified12.1
  \[\leadsto \frac{\color{blue}{\frac{(y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]
if 3.45232150826022e+116 < y.re
1. Initial program 39.7
  \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
2. Using strategy rm
3. Applied add-sqr-sqrt39.7
  \[\leadsto \frac{x.re \cdot y.re + x.im \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
4. Applied *-un-lft-identity39.7
  \[\leadsto \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
5. Applied times-frac39.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
6. Simplified39.7
  \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
7. Simplified27.1
  \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}\]
8. Using strategy rm
9. Applied associate-*l/27.0
  \[\leadsto \color{blue}{\frac{1 \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}}\]
10. Simplified27.0
  \[\leadsto \frac{\color{blue}{\frac{(y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]
11. Taylor expanded around 0 14.9
  \[\leadsto \frac{\color{blue}{x.re}}{\sqrt{y.re^2 + y.im^2}^*}\]
Recombined 3 regimes into one program.
Final simplification12.6
\[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.305038240296897 \cdot 10^{+170}:\\ \;\;\;\;\left(-\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \left(\sqrt{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot x.re\right)\\ \mathbf{elif}\;y.re \le 3.45232150826022 \cdot 10^{+116}:\\ \;\;\;\;\frac{\frac{(y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Error

Derivation

`if y.re < -5.305038240296897e+170`

`if -5.305038240296897e+170 < y.re < 3.45232150826022e+116`

`if 3.45232150826022e+116 < y.re`

Runtime