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Derivation

1. Split input into 2 regimes

2. if y.re < -1.507317256257123e+121

   1. Initial program 41.2

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

   2. Initial simplification41.2

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

   4. Applied add-sqr-sqrt41.2

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

   5. Applied associate-/r*41.2

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

   7. Applied clear-num41.2

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

   8. Taylor expanded around -inf 39.9

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

   9. Simplified39.9

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

   if -1.507317256257123e+121 < y.re

   1. Initial program 23.5

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

   2. Initial simplification23.5

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

   4. Applied add-sqr-sqrt23.5

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

   5. Applied associate-/r*23.4

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

   7. Applied clear-num23.5

      \[\leadsto \frac{\color{blue}{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{x.im \cdot y.im + x.re \cdot y.re}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]
3. Recombined 2 regimes into one program.

4. Final simplification26.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.507317256257123 \cdot 10^{+121}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}{x.im \cdot y.im + x.re \cdot y.re}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Runtime

Time bar (total: 28.5s)Debug logProfile

herbie shell --seed 2018339 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))