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Derivation

1. Split input into 2 regimes

2. if y.im < -8.701618858899359e+117 or 2.8233858841363024e-133 < y.im

   1. Initial program 30.8

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt30.8

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity30.8

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   5. Applied times-frac30.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified30.8

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   7. Simplified19.8

      \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}\]
   8. Using strategy rm

   9. Applied associate-*l/19.8

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}}\]

   10. Simplified19.8

       \[\leadsto \frac{\color{blue}{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]
   11. Using strategy rm

   12. Applied div-sub19.8

       \[\leadsto \frac{\color{blue}{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*} - \frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]

   13. Applied div-sub19.8

       \[\leadsto \color{blue}{\frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*} - \frac{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}}\]
   14. Using strategy rm

   15. Applied *-un-lft-identity19.8

       \[\leadsto \frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*} - \frac{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\color{blue}{1 \cdot \sqrt{y.re^2 + y.im^2}^*}}\]

   16. Applied add-cube-cbrt20.1

       \[\leadsto \frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*} - \frac{\color{blue}{\left(\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}}{1 \cdot \sqrt{y.re^2 + y.im^2}^*}\]

   17. Applied times-frac20.1

       \[\leadsto \frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*} - \color{blue}{\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}}\]

   18. Applied add-sqr-sqrt20.2

       \[\leadsto \frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*}}{\color{blue}{\sqrt{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt{\sqrt{y.re^2 + y.im^2}^*}}} - \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]

   19. Applied div-inv20.2

       \[\leadsto \frac{\color{blue}{\left(y.re \cdot x.im\right) \cdot \frac{1}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{\sqrt{y.re^2 + y.im^2}^*} \cdot \sqrt{\sqrt{y.re^2 + y.im^2}^*}} - \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]

   20. Applied times-frac22.1

       \[\leadsto \color{blue}{\frac{y.re \cdot x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}} - \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]

   21. Applied prod-diff22.2

       \[\leadsto \color{blue}{(\left(\frac{y.re \cdot x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) \cdot \left(\frac{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}}\right) + \left(-\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1}\right))_* + (\left(-\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \left(\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1}\right) + \left(\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1}\right))_*}\]

   22. Simplified14.9

       \[\leadsto \color{blue}{\left(\frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\frac{y.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} - \frac{\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\frac{\sqrt{y.re^2 + y.im^2}^*}{y.im}}\right)} + (\left(-\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\right) \cdot \left(\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1}\right) + \left(\frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*} \cdot \frac{\sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \sqrt[3]{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{1}\right))_*\]

   23. Simplified1.4

       \[\leadsto \left(\frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{\frac{y.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} - \frac{\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\frac{\sqrt{y.re^2 + y.im^2}^*}{y.im}}\right) + \color{blue}{0}\]

   if -8.701618858899359e+117 < y.im < 2.8233858841363024e-133

   1. Initial program 19.7

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt19.7

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   4. Applied *-un-lft-identity19.7

      \[\leadsto \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   5. Applied times-frac19.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]

   6. Simplified19.7

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   7. Simplified12.0

      \[\leadsto \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}\]
   8. Using strategy rm

   9. Applied associate-*l/11.9

      \[\leadsto \color{blue}{\frac{1 \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}}\]

   10. Simplified11.9

       \[\leadsto \frac{\color{blue}{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]
   11. Using strategy rm

   12. Applied div-sub11.9

       \[\leadsto \frac{\color{blue}{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*} - \frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}}{\sqrt{y.re^2 + y.im^2}^*}\]

   13. Applied div-sub11.9

       \[\leadsto \color{blue}{\frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*} - \frac{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}}\]
   14. Using strategy rm

   15. Applied associate-/l*3.1

       \[\leadsto \frac{\color{blue}{\frac{y.re}{\frac{\sqrt{y.re^2 + y.im^2}^*}{x.im}}}}{\sqrt{y.re^2 + y.im^2}^*} - \frac{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\]
3. Recombined 2 regimes into one program.

4. Final simplification2.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -8.701618858899359 \cdot 10^{+117} \lor \neg \left(y.im \le 2.8233858841363024 \cdot 10^{-133}\right):\\ \;\;\;\;\frac{\frac{y.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} \cdot \frac{x.im}{\sqrt{\sqrt{y.re^2 + y.im^2}^*}} - \frac{\frac{x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\frac{\sqrt{y.re^2 + y.im^2}^*}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{\sqrt{y.re^2 + y.im^2}^*}{x.im}}}{\sqrt{y.re^2 + y.im^2}^*} - \frac{\frac{x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 9.0m)Debug logProfile

herbie shell --seed 2018230 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))