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Derivation

1. Split input into 2 regimes

2. if y.im < 5.1781749674137205e+95

   1. Initial program 22.6

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

   2. Taylor expanded around -inf 22.6

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

   if 5.1781749674137205e+95 < y.im

   1. Initial program 39.0

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

   3. Applied add-sqr-sqrt39.0

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

   4. Applied associate-/r*39.0

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

   5. Taylor expanded around 0 38.5

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

4. Final simplification25.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le 5.1781749674137205 \cdot 10^{+95}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.im \cdot y.im + y.re \cdot y.re}}\\ \end{array}\]

Runtime

Time bar (total: 19.9s)Debug logProfile

herbie shell --seed 2018235 
(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))))