Error

Derivation

1. Split input into 3 regimes

2. if y.re < -4.086380967133727e+141

   1. Initial program 43.2

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

   2. Initial simplification43.2

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

   4. Applied add-sqr-sqrt43.2

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

   5. Applied *-un-lft-identity43.2

      \[\leadsto \frac{\color{blue}{1 \cdot (x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   6. Applied times-frac43.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   7. Simplified43.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   8. Simplified28.7

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}\]
   9. Using strategy rm

   10. Applied associate-*l/28.6

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}}\]

   11. Simplified28.6

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

   12. Taylor expanded around -inf 14.5

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

   13. Simplified14.5

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

   if -4.086380967133727e+141 < y.re < 1.5389869704006215e+74

   1. Initial program 19.1

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

   2. Initial simplification19.1

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

   4. Applied add-sqr-sqrt19.1

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

   5. Applied *-un-lft-identity19.1

      \[\leadsto \frac{\color{blue}{1 \cdot (x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   6. Applied times-frac19.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   7. Simplified19.2

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   8. Simplified11.7

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}\]
   9. Using strategy rm

   10. Applied associate-*l/11.6

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}}\]

   11. Simplified11.6

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

   13. Applied add-sqr-sqrt11.8

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

   14. Applied add-sqr-sqrt12.0

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

   15. Applied *-un-lft-identity12.0

       \[\leadsto \frac{\frac{\color{blue}{1 \cdot (y.re \cdot x.re + \left(x.im \cdot y.im\right))_*}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}}\]

   16. Applied times-frac12.1

       \[\leadsto \frac{\color{blue}{\frac{1}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{(y.re \cdot x.re + \left(x.im \cdot y.im\right))_*}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*} \cdot \sqrt{\sqrt{y.im^2 + y.re^2}^*}}\]

   17. Applied times-frac12.1

       \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{\frac{(y.re \cdot x.re + \left(x.im \cdot y.im\right))_*}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}\]

   18. Simplified11.9

       \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{\frac{(y.re \cdot x.re + \left(x.im \cdot y.im\right))_*}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}}{\sqrt{\sqrt{y.im^2 + y.re^2}^*}}\]

   19. Simplified11.7

       \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{(y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}\]
   20. Using strategy rm

   21. Applied associate-*r/11.7

       \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot (y.im \cdot x.im + \left(y.re \cdot x.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}\]

   if 1.5389869704006215e+74 < y.re

   1. Initial program 36.4

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

   2. Initial simplification36.4

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

   4. Applied add-sqr-sqrt36.4

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

   5. Applied *-un-lft-identity36.4

      \[\leadsto \frac{\color{blue}{1 \cdot (x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*} \cdot \sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   6. Applied times-frac36.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}}\]

   7. Simplified36.4

      \[\leadsto \color{blue}{\frac{1}{\sqrt{y.im^2 + y.re^2}^*}} \cdot \frac{(x.re \cdot y.re + \left(x.im \cdot y.im\right))_*}{\sqrt{(y.im \cdot y.im + \left(y.re \cdot y.re\right))_*}}\]

   8. Simplified24.8

      \[\leadsto \frac{1}{\sqrt{y.im^2 + y.re^2}^*} \cdot \color{blue}{\frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}\]
   9. Using strategy rm

   10. Applied associate-*l/24.7

       \[\leadsto \color{blue}{\frac{1 \cdot \frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_*}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}}\]

   11. Simplified24.7

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

   12. Taylor expanded around inf 17.9

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

4. Final simplification13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -4.086380967133727 \cdot 10^{+141}:\\ \;\;\;\;\frac{-x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{elif}\;y.re \le 1.5389869704006215 \cdot 10^{+74}:\\ \;\;\;\;\frac{(y.im \cdot x.im + \left(x.re \cdot y.re\right))_* \cdot \frac{1}{\sqrt{y.im^2 + y.re^2}^*}}{\sqrt{y.im^2 + y.re^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\sqrt{y.im^2 + y.re^2}^*}\\ \end{array}\]

Runtime

Time bar (total: 16.4s)Debug logProfile

herbie shell --seed 2018285 +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))))