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Derivation

1. Initial program 25.5

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

3. Applied add-sqr-sqrt25.5

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

4. Applied associate-/r*25.4

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

6. Applied div-sub25.5

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

8. Applied associate-/l*23.9

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

10. Applied associate-/l*22.4

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

11. Final simplification22.4

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

Runtime

Time bar (total: 16.6s)Debug logProfile

herbie shell --seed 2018340 
(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))))