_divideComplex, imaginary part

Average Error: 26.4 → 12.6

Time: 5.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.3927648088213951 \cdot 10^{132}:\\ \;\;\;\;{\left(\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le -1.2987370292207596 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;y.re \le 6.1765680479198497 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 1.45382927677037052 \cdot 10^{189}:\\ \;\;\;\;{\left(\frac{\frac{y.re}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.im}} - \frac{y.im}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.re}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r80762 = x_im;
        double r80763 = y_re;
        double r80764 = r80762 * r80763;
        double r80765 = x_re;
        double r80766 = y_im;
        double r80767 = r80765 * r80766;
        double r80768 = r80764 - r80767;
        double r80769 = r80763 * r80763;
        double r80770 = r80766 * r80766;
        double r80771 = r80769 + r80770;
        double r80772 = r80768 / r80771;
        return r80772;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r80773 = y_re;
        double r80774 = -5.392764808821395e+132;
        bool r80775 = r80773 <= r80774;
        double r80776 = -1.0;
        double r80777 = x_im;
        double r80778 = r80776 * r80777;
        double r80779 = y_im;
        double r80780 = hypot(r80773, r80779);
        double r80781 = r80778 / r80780;
        double r80782 = 1.0;
        double r80783 = pow(r80781, r80782);
        double r80784 = -1.2987370292207596e-142;
        bool r80785 = r80773 <= r80784;
        double r80786 = r80779 * r80779;
        double r80787 = fma(r80773, r80773, r80786);
        double r80788 = r80787 / r80773;
        double r80789 = r80777 / r80788;
        double r80790 = x_re;
        double r80791 = r80787 / r80779;
        double r80792 = r80790 / r80791;
        double r80793 = r80789 - r80792;
        double r80794 = 6.17656804791985e-128;
        bool r80795 = r80773 <= r80794;
        double r80796 = r80777 * r80773;
        double r80797 = r80790 * r80779;
        double r80798 = r80796 - r80797;
        double r80799 = r80798 / r80780;
        double r80800 = r80799 / r80780;
        double r80801 = pow(r80800, r80782);
        double r80802 = 1.4538292767703705e+189;
        bool r80803 = r80773 <= r80802;
        double r80804 = sqrt(r80780);
        double r80805 = 3.0;
        double r80806 = pow(r80804, r80805);
        double r80807 = r80806 / r80777;
        double r80808 = r80773 / r80807;
        double r80809 = r80806 / r80790;
        double r80810 = r80779 / r80809;
        double r80811 = r80808 - r80810;
        double r80812 = r80811 / r80804;
        double r80813 = pow(r80812, r80782);
        double r80814 = r80777 / r80780;
        double r80815 = pow(r80814, r80782);
        double r80816 = r80803 ? r80813 : r80815;
        double r80817 = r80795 ? r80801 : r80816;
        double r80818 = r80785 ? r80793 : r80817;
        double r80819 = r80775 ? r80783 : r80818;
        return r80819;
}

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 5 regimes

2. if y.re < -5.392764808821395e+132

   1. Initial program 44.3

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

   3. Applied add-sqr-sqrt 44.3

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

   4. Applied *-un-lft-identity 44.3

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

   5. Applied times-frac 44.3

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

   6. Simplified 44.3

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

   7. Simplified 29.0

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

   9. Applied pow1 29.0

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

   10. Applied pow1 29.0

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

   11. Applied pow-prod-down 29.0

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

   12. Simplified 28.9

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

   13. Taylor expanded around -inf 14.7

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

   if -5.392764808821395e+132 < y.re < -1.2987370292207596e-142

   1. Initial program 16.4

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

   3. Applied div-sub 16.4

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

   4. Simplified 13.8

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

   5. Simplified 12.1

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

   if -1.2987370292207596e-142 < y.re < 6.17656804791985e-128

   1. Initial program 22.6

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

   3. Applied add-sqr-sqrt 22.6

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

   4. Applied *-un-lft-identity 22.6

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

   5. Applied times-frac 22.6

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

   6. Simplified 22.6

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

   7. Simplified 12.4

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

   9. Applied pow1 12.4

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

   10. Applied pow1 12.4

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

   11. Applied pow-prod-down 12.4

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

   12. Simplified 12.2

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

   if 6.17656804791985e-128 < y.re < 1.4538292767703705e+189

   1. Initial program 21.0

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

   3. Applied add-sqr-sqrt 21.0

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

   4. Applied *-un-lft-identity 21.0

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

   5. Applied times-frac 21.0

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

   6. Simplified 21.0

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

   7. Simplified 13.2

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

   9. Applied pow1 13.2

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

   10. Applied pow1 13.2

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

   11. Applied pow-prod-down 13.2

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

   12. Simplified 13.1

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

   14. Applied add-sqr-sqrt 13.3

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

   15. Applied associate-/r* 13.4

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

   17. Applied div-sub 13.4

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

   18. Applied div-sub 13.4

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

   19. Simplified 11.9

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

   20. Simplified 12.2

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

   if 1.4538292767703705e+189 < y.re

   1. Initial program 44.5

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

   3. Applied add-sqr-sqrt 44.5

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

   4. Applied *-un-lft-identity 44.5

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

   5. Applied times-frac 44.5

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

   6. Simplified 44.5

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

   7. Simplified 32.0

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

   9. Applied pow1 32.0

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

   10. Applied pow1 32.0

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

   11. Applied pow-prod-down 32.0

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

   12. Simplified 32.0

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

   13. Taylor expanded around inf 12.6

       \[\leadsto {\left(\frac{\color{blue}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\]
3. Recombined 5 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.3927648088213951 \cdot 10^{132}:\\ \;\;\;\;{\left(\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le -1.2987370292207596 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;y.re \le 6.1765680479198497 \cdot 10^{-128}:\\ \;\;\;\;{\left(\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \mathbf{elif}\;y.re \le 1.45382927677037052 \cdot 10^{189}:\\ \;\;\;\;{\left(\frac{\frac{y.re}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.im}} - \frac{y.im}{\frac{{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{3}}{x.re}}}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}\right)}^{1}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{1}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))