_divideComplex, imaginary part

Average Error: 26.1 → 13.1

Time: 13.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \le -9.323590362566378695338438287572931549023 \cdot 10^{102}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 1.019281839692786179500249310763930579622 \cdot 10^{117}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, -x.re, x.im \cdot y.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

C

double f(double x_re, double x_im, double y_re, double y_im) {
        double r119613 = x_im;
        double r119614 = y_re;
        double r119615 = r119613 * r119614;
        double r119616 = x_re;
        double r119617 = y_im;
        double r119618 = r119616 * r119617;
        double r119619 = r119615 - r119618;
        double r119620 = r119614 * r119614;
        double r119621 = r119617 * r119617;
        double r119622 = r119620 + r119621;
        double r119623 = r119619 / r119622;
        return r119623;
}

↓

double f(double x_re, double x_im, double y_re, double y_im) {
        double r119624 = y_im;
        double r119625 = -9.323590362566379e+102;
        bool r119626 = r119624 <= r119625;
        double r119627 = x_re;
        double r119628 = y_re;
        double r119629 = hypot(r119624, r119628);
        double r119630 = r119627 / r119629;
        double r119631 = 1.0192818396927862e+117;
        bool r119632 = r119624 <= r119631;
        double r119633 = -r119627;
        double r119634 = x_im;
        double r119635 = r119634 * r119628;
        double r119636 = fma(r119624, r119633, r119635);
        double r119637 = r119636 / r119629;
        double r119638 = r119637 / r119629;
        double r119639 = -r119630;
        double r119640 = r119632 ? r119638 : r119639;
        double r119641 = r119626 ? r119630 : r119640;
        return r119641;
}

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 < -9.323590362566379e+102

   1. Initial program 40.4

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

   2. Simplified 40.4

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

   4. Applied add-sqr-sqrt 40.4

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

   5. Applied *-un-lft-identity 40.4

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

   6. Applied times-frac 40.4

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

   7. Simplified 40.4

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

   8. Simplified 27.4

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

   10. Applied *-un-lft-identity 27.4

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

   11. Applied associate-*l* 27.4

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

   12. Simplified 27.3

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

   13. Taylor expanded around -inf 17.0

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

   if -9.323590362566379e+102 < y.im < 1.0192818396927862e+117

   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

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

   4. Applied add-sqr-sqrt 18.8

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

   5. Applied *-un-lft-identity 18.8

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

   6. Applied times-frac 18.8

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

   7. Simplified 18.8

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

   8. Simplified 11.7

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

   10. Applied *-un-lft-identity 11.7

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

   11. Applied associate-*l* 11.7

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

   12. Simplified 11.6

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

   if 1.0192818396927862e+117 < y.im

   1. Initial program 40.4

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

   2. Simplified 40.4

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

   4. Applied add-sqr-sqrt 40.4

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

   5. Applied *-un-lft-identity 40.4

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

   6. Applied times-frac 40.4

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

   7. Simplified 40.4

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

   8. Simplified 26.4

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

   10. Applied *-un-lft-identity 26.4

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

   11. Applied associate-*l* 26.4

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

   12. Simplified 26.4

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

   13. Taylor expanded around inf 15.2

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

   14. Simplified 15.2

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

4. Final simplification 13.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \le -9.323590362566378695338438287572931549023 \cdot 10^{102}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \le 1.019281839692786179500249310763930579622 \cdot 10^{117}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, -x.re, x.im \cdot y.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 +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))))