_divideComplex, imaginary part

Average Error: 25.6 → 12.8

Time: 3.9m

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -6.693395251828147 \cdot 10^{+97}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{elif}\;y.re \le 1.321565434537417 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r15540660 = x_im;
        double r15540661 = y_re;
        double r15540662 = r15540660 * r15540661;
        double r15540663 = x_re;
        double r15540664 = y_im;
        double r15540665 = r15540663 * r15540664;
        double r15540666 = r15540662 - r15540665;
        double r15540667 = r15540661 * r15540661;
        double r15540668 = r15540664 * r15540664;
        double r15540669 = r15540667 + r15540668;
        double r15540670 = r15540666 / r15540669;
        return r15540670;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r15540671 = y_re;
        double r15540672 = -6.693395251828147e+97;
        bool r15540673 = r15540671 <= r15540672;
        double r15540674 = x_im;
        double r15540675 = -r15540674;
        double r15540676 = y_im;
        double r15540677 = hypot(r15540671, r15540676);
        double r15540678 = r15540675 / r15540677;
        double r15540679 = 1.321565434537417e+152;
        bool r15540680 = r15540671 <= r15540679;
        double r15540681 = r15540674 * r15540671;
        double r15540682 = x_re;
        double r15540683 = r15540682 * r15540676;
        double r15540684 = r15540681 - r15540683;
        double r15540685 = r15540684 / r15540677;
        double r15540686 = r15540685 / r15540677;
        double r15540687 = r15540674 / r15540677;
        double r15540688 = r15540680 ? r15540686 : r15540687;
        double r15540689 = r15540673 ? r15540678 : r15540688;
        return r15540689;
}

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 < -6.693395251828147e+97

   1. Initial program 37.9

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

   2. Simplified 37.9

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

   4. Applied add-sqr-sqrt 37.9

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

   5. Applied associate-/r* 37.8

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

   7. Applied *-un-lft-identity 37.8

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

   8. Applied sqrt-prod 37.8

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

   9. Applied *-un-lft-identity 37.8

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

   10. Applied sqrt-prod 37.8

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

   11. Applied *-un-lft-identity 37.8

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

   12. Applied times-frac 37.8

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

   13. Applied times-frac 37.8

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

   14. Simplified 37.8

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

   15. Simplified 25.5

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

   16. Taylor expanded around -inf 16.8

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

   17. Simplified 16.8

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

   if -6.693395251828147e+97 < y.re < 1.321565434537417e+152

   1. Initial program 18.8

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

   2. Simplified 18.8

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

   4. Applied add-sqr-sqrt 18.8

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

   5. Applied associate-/r* 18.7

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

   7. Applied *-un-lft-identity 18.7

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

   8. Applied sqrt-prod 18.7

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

   9. Applied *-un-lft-identity 18.7

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

   10. Applied sqrt-prod 18.7

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

   11. Applied *-un-lft-identity 18.7

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

   12. Applied times-frac 18.7

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

   13. Applied times-frac 18.7

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

   14. Simplified 18.7

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

   15. Simplified 11.6

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

   if 1.321565434537417e+152 < y.re

   1. Initial program 44.0

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

   2. Simplified 44.0

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

   4. Applied add-sqr-sqrt 44.0

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

   5. Applied associate-/r* 44.0

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

   7. Applied *-un-lft-identity 44.0

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

   8. Applied sqrt-prod 44.0

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

   9. Applied *-un-lft-identity 44.0

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

   10. Applied sqrt-prod 44.0

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

   11. Applied *-un-lft-identity 44.0

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

   12. Applied times-frac 44.0

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

   13. Applied times-frac 44.0

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

   14. Simplified 44.0

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

   15. Simplified 28.8

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

   16. Taylor expanded around inf 13.1

       \[\leadsto 1 \cdot \frac{\color{blue}{x.im}}{\sqrt{y.re^2 + y.im^2}^*}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -6.693395251828147 \cdot 10^{+97}:\\ \;\;\;\;\frac{-x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{elif}\;y.re \le 1.321565434537417 \cdot 10^{+152}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re^2 + y.im^2}^*}}{\sqrt{y.re^2 + y.im^2}^*}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re^2 + y.im^2}^*}\\ \end{array}\]

Reproduce

herbie shell --seed 2019107 +o rules:numerics
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))