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Derivation

1. Split input into 3 regimes

2. if y.re < -3.297785523352951e+180

   1. Initial program 42.7

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

   3. Applied add-sqr-sqrt42.7

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

   4. Applied *-un-lft-identity42.7

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

   5. Applied times-frac42.7

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

   6. Simplified42.7

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

   7. Simplified29.9

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

   9. Applied associate-*r/29.9

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

   10. Taylor expanded around -inf 12.6

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

   11. Simplified12.6

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

   if -3.297785523352951e+180 < y.re < 4.7704829791947515e+204

   1. Initial program 22.0

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

   3. Applied add-sqr-sqrt22.0

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

   4. Applied *-un-lft-identity22.0

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

   5. Applied times-frac22.0

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

   6. Simplified22.0

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

   7. Simplified13.2

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

   9. Applied associate-*r/13.2

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

   11. Applied sub-neg13.2

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

   12. Applied distribute-rgt-in13.2

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

   13. Simplified5.1

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

   if 4.7704829791947515e+204 < y.re

   1. Initial program 41.3

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

   3. Applied add-sqr-sqrt41.3

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

   4. Applied *-un-lft-identity41.3

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

   5. Applied times-frac41.3

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

   6. Simplified41.3

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

   7. Simplified30.0

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

   9. Applied associate-*r/30.0

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

   10. Taylor expanded around inf 10.5

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

4. Final simplification6.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -3.297785523352951 \cdot 10^{+180}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{elif}\;y.re \le 4.7704829791947515 \cdot 10^{+204}:\\ \;\;\;\;\frac{x.re \cdot \frac{-y.im}{\sqrt{y.re^2 + y.im^2}^*} + \frac{1}{\sqrt{y.re^2 + y.im^2}^*} \cdot \left(x.im \cdot y.re\right)}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 41.5s)Debug logProfile

herbie shell --seed 2018257 +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))))