_divideComplex, imaginary part

Average Error: 26.2 → 13.6

Time: 4.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -1.0689057415946615 \cdot 10^{65}:\\ \;\;\;\;\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.5910699936696656 \cdot 10^{174}:\\ \;\;\;\;1 \cdot \frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r133914 = x_im;
        double r133915 = y_re;
        double r133916 = r133914 * r133915;
        double r133917 = x_re;
        double r133918 = y_im;
        double r133919 = r133917 * r133918;
        double r133920 = r133916 - r133919;
        double r133921 = r133915 * r133915;
        double r133922 = r133918 * r133918;
        double r133923 = r133921 + r133922;
        double r133924 = r133920 / r133923;
        return r133924;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r133925 = y_re;
        double r133926 = -1.0689057415946615e+65;
        bool r133927 = r133925 <= r133926;
        double r133928 = -1.0;
        double r133929 = x_im;
        double r133930 = r133928 * r133929;
        double r133931 = y_im;
        double r133932 = hypot(r133925, r133931);
        double r133933 = r133930 / r133932;
        double r133934 = 1.5910699936696656e+174;
        bool r133935 = r133925 <= r133934;
        double r133936 = 1.0;
        double r133937 = r133929 * r133925;
        double r133938 = x_re;
        double r133939 = r133938 * r133931;
        double r133940 = r133937 - r133939;
        double r133941 = r133940 / r133932;
        double r133942 = r133941 / r133932;
        double r133943 = r133936 * r133942;
        double r133944 = r133929 / r133932;
        double r133945 = r133935 ? r133943 : r133944;
        double r133946 = r133927 ? r133933 : r133945;
        return r133946;
}

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 3 regimes

2. if y.re < -1.0689057415946615e+65

   1. Initial program 36.6

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

   3. Applied add-sqr-sqrt 36.6

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

   4. Applied *-un-lft-identity 36.6

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

   5. Applied times-frac 36.6

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

   6. Simplified 36.6

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

   7. Simplified 24.6

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

   9. Applied associate-*r/ 24.5

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

   10. Simplified 24.5

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

   11. Taylor expanded around -inf 18.3

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

   if -1.0689057415946615e+65 < y.re < 1.5910699936696656e+174

   1. Initial program 19.4

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

   3. Applied add-sqr-sqrt 19.4

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

   4. Applied *-un-lft-identity 19.4

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

   5. Applied times-frac 19.4

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

   6. Simplified 19.4

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

   7. Simplified 12.3

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

   9. Applied associate-*r/ 12.3

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

   10. Simplified 12.2

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

   12. Applied clear-num 12.3

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

   14. Applied *-un-lft-identity 12.3

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

   15. Applied *-un-lft-identity 12.3

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

   16. Applied *-un-lft-identity 12.3

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

   17. Applied times-frac 12.3

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

   18. Applied add-sqr-sqrt 12.3

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

   19. Applied times-frac 12.3

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

   20. Applied times-frac 12.3

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

   21. Simplified 12.3

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

   22. Simplified 12.2

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

   if 1.5910699936696656e+174 < y.re

   1. Initial program 46.2

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

   3. Applied add-sqr-sqrt 46.2

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

   4. Applied *-un-lft-identity 46.2

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

   5. Applied times-frac 46.2

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

   6. Simplified 46.2

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

   7. Simplified 33.7

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

   9. Applied associate-*r/ 33.7

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

   10. Simplified 33.7

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

   11. Taylor expanded around inf 12.8

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

4. Final simplification 13.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -1.0689057415946615 \cdot 10^{65}:\\ \;\;\;\;\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 1.5910699936696656 \cdot 10^{174}:\\ \;\;\;\;1 \cdot \frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +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))))