_divideComplex, imaginary part

Average Error: 25.6 → 12.8

Time: 15.3s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -2.349525322457489826589608031591607051529 \cdot 10^{135}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 9.70450990751971859322124590202865896249 \cdot 10^{95}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r61757 = x_im;
        double r61758 = y_re;
        double r61759 = r61757 * r61758;
        double r61760 = x_re;
        double r61761 = y_im;
        double r61762 = r61760 * r61761;
        double r61763 = r61759 - r61762;
        double r61764 = r61758 * r61758;
        double r61765 = r61761 * r61761;
        double r61766 = r61764 + r61765;
        double r61767 = r61763 / r61766;
        return r61767;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r61768 = y_im;
        double r61769 = -2.3495253224574898e+135;
        bool r61770 = r61768 <= r61769;
        double r61771 = x_re;
        double r61772 = y_re;
        double r61773 = hypot(r61772, r61768);
        double r61774 = r61771 / r61773;
        double r61775 = 9.704509907519719e+95;
        bool r61776 = r61768 <= r61775;
        double r61777 = 1.0;
        double r61778 = -r61768;
        double r61779 = x_im;
        double r61780 = r61779 * r61772;
        double r61781 = fma(r61778, r61771, r61780);
        double r61782 = r61773 / r61781;
        double r61783 = r61777 / r61782;
        double r61784 = r61783 / r61773;
        double r61785 = -r61774;
        double r61786 = r61776 ? r61784 : r61785;
        double r61787 = r61770 ? r61774 : r61786;
        return r61787;
}

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 < -2.3495253224574898e+135

   1. Initial program 41.2

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

   2. Simplified 41.2

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

   4. Applied add-sqr-sqrt 41.2

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

   5. Applied *-un-lft-identity 41.2

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

   6. Applied times-frac 41.2

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

   7. Simplified 41.2

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

   8. Simplified 27.1

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

   10. Applied associate-*r/ 27.0

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

   11. Simplified 27.0

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

   12. Taylor expanded around -inf 13.8

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

   if -2.3495253224574898e+135 < y.im < 9.704509907519719e+95

   1. Initial program 18.7

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

   2. Simplified 18.7

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

   4. Applied add-sqr-sqrt 18.7

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

   5. Applied *-un-lft-identity 18.7

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

   6. Applied times-frac 18.7

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

   7. Simplified 18.7

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

   8. Simplified 11.6

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

   10. Applied associate-*r/ 11.6

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

   11. Simplified 11.5

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

   13. Applied clear-num 11.6

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

   14. Simplified 11.6

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

   if 9.704509907519719e+95 < y.im

   1. Initial program 39.2

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

   2. Simplified 39.2

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

   4. Applied add-sqr-sqrt 39.2

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

   5. Applied *-un-lft-identity 39.2

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

   6. Applied times-frac 39.2

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

   7. Simplified 39.2

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

   8. Simplified 26.6

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

   10. Applied associate-*r/ 26.6

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

   11. Simplified 26.6

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

   12. Taylor expanded around inf 16.3

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

   13. Simplified 16.3

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

4. Final simplification 12.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -2.349525322457489826589608031591607051529 \cdot 10^{135}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \le 9.70450990751971859322124590202865896249 \cdot 10^{95}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(-y.im, x.re, x.im \cdot y.re\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +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))))