_divideComplex, imaginary part

Average Error: 26.1 → 12.4

Time: 6.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.24214216730030625 \cdot 10^{138}:\\ \;\;\;\;{\left(\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 3.2259502147858646 \cdot 10^{139}:\\ \;\;\;\;{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 1.9590575859470631 \cdot 10^{185}:\\ \;\;\;\;{\left(\frac{\frac{y.re}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.im}} - \frac{y.im}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.re}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r110806 = x_im;
        double r110807 = y_re;
        double r110808 = r110806 * r110807;
        double r110809 = x_re;
        double r110810 = y_im;
        double r110811 = r110809 * r110810;
        double r110812 = r110808 - r110811;
        double r110813 = r110807 * r110807;
        double r110814 = r110810 * r110810;
        double r110815 = r110813 + r110814;
        double r110816 = r110812 / r110815;
        return r110816;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r110817 = y_re;
        double r110818 = -5.242142167300306e+138;
        bool r110819 = r110817 <= r110818;
        double r110820 = -1.0;
        double r110821 = x_im;
        double r110822 = r110820 * r110821;
        double r110823 = y_im;
        double r110824 = hypot(r110817, r110823);
        double r110825 = r110822 / r110824;
        double r110826 = 1.0;
        double r110827 = pow(r110825, r110826);
        double r110828 = 3.2259502147858646e+139;
        bool r110829 = r110817 <= r110828;
        double r110830 = r110821 * r110817;
        double r110831 = x_re;
        double r110832 = r110831 * r110823;
        double r110833 = r110830 - r110832;
        double r110834 = r110833 / r110824;
        double r110835 = r110834 / r110824;
        double r110836 = pow(r110835, r110826);
        double r110837 = 1.959057585947063e+185;
        bool r110838 = r110817 <= r110837;
        double r110839 = sqrt(r110824);
        double r110840 = 3.0;
        double r110841 = pow(r110839, r110840);
        double r110842 = r110841 / r110821;
        double r110843 = r110817 / r110842;
        double r110844 = r110841 / r110831;
        double r110845 = r110823 / r110844;
        double r110846 = r110843 - r110845;
        double r110847 = r110846 / r110839;
        double r110848 = pow(r110847, r110826);
        double r110849 = r110821 / r110824;
        double r110850 = pow(r110849, r110826);
        double r110851 = r110838 ? r110848 : r110850;
        double r110852 = r110829 ? r110836 : r110851;
        double r110853 = r110819 ? r110827 : r110852;
        return r110853;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Derivation

1. Split input into 4 regimes

2. if y.re < -5.242142167300306e+138

   1. Initial program 43.7

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

   3. Applied add-sqr-sqrt 43.7

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

   4. Applied *-un-lft-identity 43.7

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

   5. Applied times-frac 43.7

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

   6. Simplified 43.7

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   7. Simplified 28.3

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
   8. Using strategy rm

   9. Applied pow1 28.3

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{{\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   10. Applied pow1 28.3

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]

   11. Applied pow-prod-down 28.3

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   12. Simplified 28.3

       \[\leadsto {\color{blue}{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}^{1}\]

   13. Taylor expanded around -inf 13.9

       \[\leadsto {\left(\frac{\color{blue}{-1 \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]

   if -5.242142167300306e+138 < y.re < 3.2259502147858646e+139

   1. Initial program 19.1

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

   3. Applied add-sqr-sqrt 19.1

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

   4. Applied *-un-lft-identity 19.1

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

   5. Applied times-frac 19.1

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

   6. Simplified 19.1

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   7. Simplified 12.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
   8. Using strategy rm

   9. Applied pow1 12.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{{\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   10. Applied pow1 12.2

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]

   11. Applied pow-prod-down 12.2

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   12. Simplified 12.1

       \[\leadsto {\color{blue}{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}^{1}\]

   if 3.2259502147858646e+139 < y.re < 1.959057585947063e+185

   1. Initial program 41.8

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

   3. Applied add-sqr-sqrt 41.8

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

   4. Applied *-un-lft-identity 41.8

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

   5. Applied times-frac 41.9

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

   6. Simplified 41.9

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   7. Simplified 21.7

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
   8. Using strategy rm

   9. Applied pow1 21.7

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{{\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   10. Applied pow1 21.7

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]

   11. Applied pow-prod-down 21.7

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   12. Simplified 21.6

       \[\leadsto {\color{blue}{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}^{1}\]
   13. Using strategy rm

   14. Applied add-sqr-sqrt 21.8

       \[\leadsto {\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\color{blue}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}\right)}^{1}\]

   15. Applied associate-/r* 21.8

       \[\leadsto {\color{blue}{\left(\frac{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}}^{1}\]
   16. Using strategy rm

   17. Applied div-sub 21.8

       \[\leadsto {\left(\frac{\frac{\color{blue}{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\]

   18. Applied div-sub 21.8

       \[\leadsto {\left(\frac{\color{blue}{\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}} - \frac{\frac{x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\]

   19. Simplified 15.6

       \[\leadsto {\left(\frac{\color{blue}{\frac{y.re}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.im}}} - \frac{\frac{x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\]

   20. Simplified 13.0

       \[\leadsto {\left(\frac{\frac{y.re}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.im}} - \color{blue}{\frac{y.im}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.re}}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\]

   if 1.959057585947063e+185 < y.re

   1. Initial program 43.0

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

   3. Applied add-sqr-sqrt 43.0

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

   4. Applied *-un-lft-identity 43.0

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

   5. Applied times-frac 43.0

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

   6. Simplified 43.0

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

   7. Simplified 31.0

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}\]
   8. Using strategy rm

   9. Applied pow1 31.0

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \color{blue}{{\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   10. Applied pow1 31.0

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1}\right)}^{1}} \cdot {\left(\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]

   11. Applied pow-prod-down 31.0

       \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot 1} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}}\]

   12. Simplified 31.0

       \[\leadsto {\color{blue}{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}^{1}\]

   13. Taylor expanded around inf 12.2

       \[\leadsto {\left(\frac{\color{blue}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.24214216730030625 \cdot 10^{138}:\\ \;\;\;\;{\left(\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 3.2259502147858646 \cdot 10^{139}:\\ \;\;\;\;{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 1.9590575859470631 \cdot 10^{185}:\\ \;\;\;\;{\left(\frac{\frac{y.re}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.im}} - \frac{y.im}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.re}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020100 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))