_divideComplex, imaginary part

Average Error: 26.2 → 13.1

Time: 17.8s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -4.012444222764070938318713205906098803793 \cdot 10^{97}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 1.124934690533360950950546020938866421399 \cdot 10^{193}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r3768728 = x_im;
        double r3768729 = y_re;
        double r3768730 = r3768728 * r3768729;
        double r3768731 = x_re;
        double r3768732 = y_im;
        double r3768733 = r3768731 * r3768732;
        double r3768734 = r3768730 - r3768733;
        double r3768735 = r3768729 * r3768729;
        double r3768736 = r3768732 * r3768732;
        double r3768737 = r3768735 + r3768736;
        double r3768738 = r3768734 / r3768737;
        return r3768738;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r3768739 = y_re;
        double r3768740 = -4.012444222764071e+97;
        bool r3768741 = r3768739 <= r3768740;
        double r3768742 = x_im;
        double r3768743 = -r3768742;
        double r3768744 = y_im;
        double r3768745 = hypot(r3768744, r3768739);
        double r3768746 = r3768743 / r3768745;
        double r3768747 = 1.124934690533361e+193;
        bool r3768748 = r3768739 <= r3768747;
        double r3768749 = 1.0;
        double r3768750 = r3768742 * r3768739;
        double r3768751 = x_re;
        double r3768752 = r3768751 * r3768744;
        double r3768753 = r3768750 - r3768752;
        double r3768754 = r3768745 / r3768753;
        double r3768755 = r3768749 / r3768754;
        double r3768756 = r3768755 / r3768745;
        double r3768757 = r3768742 / r3768745;
        double r3768758 = r3768748 ? r3768756 : r3768757;
        double r3768759 = r3768741 ? r3768746 : r3768758;
        return r3768759;
}

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 < -4.012444222764071e+97

   1. Initial program 39.6

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

   2. Simplified 39.6

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

   4. Applied add-sqr-sqrt 39.6

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

   5. Applied associate-/r* 39.6

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

   7. Applied *-un-lft-identity 39.6

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

   8. Applied sqrt-prod 39.6

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

   9. Applied div-inv 39.6

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

   10. Applied times-frac 39.6

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

   11. Simplified 39.6

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

   12. Simplified 39.1

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

   14. Applied associate-*r/ 27.6

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

   15. Simplified 27.6

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

   16. Taylor expanded around -inf 16.2

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

   17. Simplified 16.2

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

   if -4.012444222764071e+97 < y.re < 1.124934690533361e+193

   1. Initial program 20.8

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

   2. Simplified 20.8

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

   4. Applied add-sqr-sqrt 20.8

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

   5. Applied associate-/r* 20.7

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

   7. Applied *-un-lft-identity 20.7

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

   8. Applied sqrt-prod 20.7

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

   9. Applied div-inv 20.8

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

   10. Applied times-frac 21.0

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

   11. Simplified 21.0

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

   12. Simplified 20.7

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

   14. Applied associate-*r/ 12.7

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

   15. Simplified 12.6

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

   17. Applied clear-num 12.7

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

   if 1.124934690533361e+193 < y.re

   1. Initial program 41.6

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

   2. Simplified 41.6

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

   4. Applied add-sqr-sqrt 41.6

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

   5. Applied associate-/r* 41.6

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

   7. Applied *-un-lft-identity 41.6

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

   8. Applied sqrt-prod 41.6

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

   9. Applied div-inv 41.6

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

   10. Applied times-frac 41.6

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

   11. Simplified 41.6

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

   12. Simplified 41.6

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

   14. Applied associate-*r/ 30.7

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

   15. Simplified 30.6

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

   16. Taylor expanded around inf 10.3

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -4.012444222764070938318713205906098803793 \cdot 10^{97}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \le 1.124934690533360950950546020938866421399 \cdot 10^{193}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 +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))))