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Derivation

1. Split input into 2 regimes

2. if x.im < -3.7549061336177695e+50 or 3.160825419620763e+142 < x.im

   1. Initial program 33.6

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

   3. Applied add-sqr-sqrt33.6

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

   4. Applied *-un-lft-identity33.6

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

   5. Applied times-frac33.6

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

   6. Simplified33.6

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

   7. Simplified28.1

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

   9. Applied associate-*l/28.0

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

   10. Simplified28.0

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

   12. Applied div-sub28.0

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

   14. Applied *-un-lft-identity28.0

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

   15. Applied times-frac10.2

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

   16. Simplified10.2

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

   if -3.7549061336177695e+50 < x.im < 3.160825419620763e+142

   1. Initial program 21.1

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

   3. Applied add-sqr-sqrt21.1

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

   4. Applied *-un-lft-identity21.1

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

   5. Applied times-frac21.1

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

   6. Simplified21.1

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

   7. Simplified10.6

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

   9. Applied associate-*l/10.5

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

   10. Simplified10.5

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

   12. Applied div-sub10.5

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

   14. Applied *-un-lft-identity10.5

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

   15. Applied times-frac2.3

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

   16. Simplified2.3

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

4. Final simplification4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \le -3.7549061336177695 \cdot 10^{+50} \lor \neg \left(x.im \le 3.160825419620763 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\sqrt{y.re^2 + y.im^2}^*} - \frac{y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{\sqrt{y.re^2 + y.im^2}^*} - x.re \cdot \frac{y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2018362 +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))))

Details

Time bar (total: 25.2s)Debug log