Error

Derivation

1. Split input into 3 regimes

2. if y.im < -2.36398613215644e+156

   1. Initial program 43.7

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

   2. Applied simplify43.7

      \[\leadsto \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt43.7

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

   5. Applied associate-/r*43.7

      \[\leadsto \color{blue}{\frac{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]
   6. Using strategy rm

   7. Applied fma-udef43.7

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

   8. Applied hypot-def43.7

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

   10. Applied fma-udef43.7

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

   11. Applied hypot-def28.1

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

   12. Taylor expanded around -inf 11.3

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

   13. Applied simplify11.3

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

   if -2.36398613215644e+156 < y.im < 2.4347391390427925e+164

   1. Initial program 19.0

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

   2. Applied simplify19.0

      \[\leadsto \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt19.0

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

   5. Applied associate-/r*18.9

      \[\leadsto \color{blue}{\frac{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]
   6. Using strategy rm

   7. Applied fma-udef18.9

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

   8. Applied hypot-def18.9

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

   10. Applied fma-udef18.9

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

   11. Applied hypot-def12.6

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

   if 2.4347391390427925e+164 < y.im

   1. Initial program 43.7

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

   2. Applied simplify43.7

      \[\leadsto \color{blue}{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]
   3. Using strategy rm

   4. Applied add-sqr-sqrt43.7

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

   5. Applied associate-/r*43.7

      \[\leadsto \color{blue}{\frac{\frac{(x.im \cdot y.im + \left(y.re \cdot x.re\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]
   6. Using strategy rm

   7. Applied fma-udef43.7

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

   8. Applied hypot-def43.7

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

   10. Applied fma-udef43.7

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

   11. Applied hypot-def29.0

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

   12. Taylor expanded around inf 12.8

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

Runtime

Time bar (total: 20.9s)Debug logProfile

herbie shell --seed '#(1070131407 1246090267 3027482374 2150728003 2026520792 2347815650)' +o rules:numerics
(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))))