Error

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1191905605526316e+139

   1. Initial program 43.4

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

   3. Applied add-sqr-sqrt43.4

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

   4. Applied associate-/r*43.4

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

   6. Applied add-sqr-sqrt62.9

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

   7. Applied add-sqr-sqrt62.9

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

   8. Applied swap-sqr62.9

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

   9. Applied hypot-def62.9

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

   10. Simplified43.4

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

   12. Applied *-commutative43.4

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

   13. Applied add-sqr-sqrt62.9

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

   14. Applied add-sqr-sqrt62.9

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

   15. Applied swap-sqr62.9

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

   16. Applied hypot-def62.9

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

   17. Simplified28.1

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

   18. Taylor expanded around -inf 14.0

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

   19. Simplified14.0

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

   if -1.1191905605526316e+139 < y.re < 2.0932116228373425e+217

   1. Initial program 20.9

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

   3. Applied add-sqr-sqrt20.9

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

   4. Applied associate-/r*20.8

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

   6. Applied add-sqr-sqrt41.0

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

   7. Applied add-sqr-sqrt41.0

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

   8. Applied swap-sqr41.0

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

   9. Applied hypot-def41.0

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

   10. Simplified20.8

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

   12. Applied *-commutative20.8

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

   13. Applied add-sqr-sqrt41.0

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

   14. Applied add-sqr-sqrt41.0

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

   15. Applied swap-sqr41.0

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

   16. Applied hypot-def36.3

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

   17. Simplified13.1

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

   if 2.0932116228373425e+217 < y.re

   1. Initial program 41.9

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

   3. Applied add-sqr-sqrt41.9

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

   4. Applied associate-/r*41.9

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

   6. Applied add-sqr-sqrt41.9

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

   7. Applied add-sqr-sqrt41.9

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

   8. Applied swap-sqr41.9

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

   9. Applied hypot-def41.9

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

   10. Simplified41.9

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

   12. Applied *-commutative41.9

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

   13. Applied add-sqr-sqrt41.9

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

   14. Applied add-sqr-sqrt41.9

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

   15. Applied swap-sqr41.9

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

   16. Applied hypot-def31.2

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

   17. Simplified31.1

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

   18. Taylor expanded around inf 10.0

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

4. Final simplification13.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.1191905605526316 \cdot 10^{+139}:\\ \;\;\;\;-\frac{x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{elif}\;y.re \le 2.0932116228373425 \cdot 10^{+217}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019072 +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))))