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Derivation

1. Split input into 2 regimes

2. if y.im < -9538347154527392.0 or 3.40271269432902e+126 < y.im

   1. Initial program 35.3

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

   2. Initial simplification35.3

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt35.3

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

   5. Applied *-un-lft-identity35.3

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

   6. Applied times-frac35.3

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

   7. Simplified35.3

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

   8. Simplified23.4

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

   10. Applied associate-*l/23.3

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

   11. Simplified23.3

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

   13. Applied div-sub23.3

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

   14. Applied div-sub23.3

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

   16. Applied add-sqr-sqrt23.4

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

   17. Applied add-sqr-sqrt23.5

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

   18. Applied times-frac8.3

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

   19. Applied times-frac8.3

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

   20. Simplified8.1

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

   21. Simplified7.9

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

   if -9538347154527392.0 < y.im < 3.40271269432902e+126

   1. Initial program 18.2

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]

   2. Initial simplification18.2

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt18.2

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

   5. Applied *-un-lft-identity18.2

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

   6. Applied times-frac18.2

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

   7. Simplified18.2

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

   8. Simplified11.1

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

   10. Applied associate-*l/11.0

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

   11. Simplified11.0

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

   13. Applied div-sub11.0

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

   14. Applied div-sub11.0

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

   16. Applied add-sqr-sqrt11.2

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

   17. Applied times-frac2.7

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

4. Final simplification4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -9538347154527392.0 \lor \neg \left(y.im \le 3.40271269432902 \cdot 10^{+126}\right):\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{x.re}{\sqrt{y.im^2 + y.re^2}^*} \cdot \frac{y.im}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{y.re}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{y.im^2 + y.re^2}^*} - \frac{\frac{x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 29.3s)Debug logProfile

herbie shell --seed 2018296 +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))))