_divideComplex, imaginary part

Average Error: 25.9 → 12.5

Time: 13.0s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -8.854189588374422409856349074066788051196 \cdot 10^{171}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 2.946859042858365719350909171488092188425 \cdot 10^{116}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r41442 = x_im;
        double r41443 = y_re;
        double r41444 = r41442 * r41443;
        double r41445 = x_re;
        double r41446 = y_im;
        double r41447 = r41445 * r41446;
        double r41448 = r41444 - r41447;
        double r41449 = r41443 * r41443;
        double r41450 = r41446 * r41446;
        double r41451 = r41449 + r41450;
        double r41452 = r41448 / r41451;
        return r41452;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r41453 = y_im;
        double r41454 = -8.854189588374422e+171;
        bool r41455 = r41453 <= r41454;
        double r41456 = x_re;
        double r41457 = y_re;
        double r41458 = hypot(r41457, r41453);
        double r41459 = r41456 / r41458;
        double r41460 = 2.9468590428583657e+116;
        bool r41461 = r41453 <= r41460;
        double r41462 = -r41453;
        double r41463 = x_im;
        double r41464 = r41457 * r41463;
        double r41465 = fma(r41456, r41462, r41464);
        double r41466 = r41465 / r41458;
        double r41467 = r41466 / r41458;
        double r41468 = -r41456;
        double r41469 = r41468 / r41458;
        double r41470 = r41461 ? r41467 : r41469;
        double r41471 = r41455 ? r41459 : r41470;
        return r41471;
}

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.im < -8.854189588374422e+171

   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. Simplified 44.5

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

   4. Applied add-sqr-sqrt 44.5

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

   5. Applied *-un-lft-identity 44.5

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

   6. Applied times-frac 44.5

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

   7. Simplified 44.5

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

   8. Simplified 30.6

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

   10. Applied associate-*r/ 30.6

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

   11. Simplified 30.5

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

   12. Taylor expanded around -inf 11.3

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

   if -8.854189588374422e+171 < y.im < 2.9468590428583657e+116

   1. Initial program 19.6

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

   2. Simplified 19.6

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

   4. Applied add-sqr-sqrt 19.6

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

   5. Applied *-un-lft-identity 19.6

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

   6. Applied times-frac 19.6

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

   7. Simplified 19.6

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

   8. Simplified 12.5

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

   10. Applied associate-*r/ 12.5

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

   11. Simplified 12.4

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

   13. Applied clear-num 12.5

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

   15. Applied *-un-lft-identity 12.5

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

   16. Applied *-un-lft-identity 12.5

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

   17. Applied times-frac 12.5

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

   18. Applied add-cube-cbrt 12.5

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

   19. Applied times-frac 12.5

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

   20. Simplified 12.5

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

   21. Simplified 12.4

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

   if 2.9468590428583657e+116 < y.im

   1. Initial program 39.9

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

   2. Simplified 39.9

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

   4. Applied add-sqr-sqrt 39.9

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

   5. Applied *-un-lft-identity 39.9

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

   6. Applied times-frac 39.9

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

   7. Simplified 39.9

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

   8. Simplified 26.5

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

   10. Applied associate-*r/ 26.5

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

   11. Simplified 26.4

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

   12. Taylor expanded around inf 14.1

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

   13. Simplified 14.1

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -8.854189588374422409856349074066788051196 \cdot 10^{171}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 2.946859042858365719350909171488092188425 \cdot 10^{116}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, -y.im, y.re \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 +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))))