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Derivation

1. Split input into 2 regimes

2. if y.im < 3.7092653838509206e+139

   1. Initial program 23.1

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

   2. Initial simplification23.1

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

   4. Applied add-sqr-sqrt23.2

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

   5. Applied *-un-lft-identity23.2

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

   6. Applied times-frac23.2

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

   7. Simplified23.1

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

   8. Simplified14.8

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

   10. Applied associate-*l/14.7

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

   11. Simplified14.7

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

   13. Applied add-sqr-sqrt14.9

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

   14. Applied associate-/r*14.9

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

   16. Applied div-sub14.9

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

   17. Applied div-sub14.9

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

   18. Applied div-sub14.9

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

   19. Simplified7.0

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

   if 3.7092653838509206e+139 < y.im

   1. Initial program 42.1

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

   2. Initial simplification42.1

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

   4. Applied add-sqr-sqrt42.1

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

   5. Applied *-un-lft-identity42.1

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

   6. Applied times-frac42.1

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

   7. Simplified42.1

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

   8. Simplified27.3

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

   10. Applied associate-*l/27.2

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

   11. Simplified27.2

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

   12. Taylor expanded around 0 14.3

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

   13. Simplified14.3

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

4. Final simplification8.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le 3.7092653838509206 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{x.im}{\sqrt{y.im^2 + y.re^2}^*}}{\frac{\sqrt{y.im^2 + y.re^2}^*}{y.re}} - \frac{\frac{\frac{x.re \cdot y.im}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 2.4m)Debug logProfile

herbie shell --seed 2018295 +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))))