_divideComplex, imaginary part

Average Error: 26.5 → 4.8

Time: 3.9s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -2.2616143508082277 \cdot 10^{79} \lor \neg \left(y.re \le 1.03827084215452011 \cdot 10^{-35}\right):\\ \;\;\;\;1 \cdot \left(\frac{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{\frac{x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{\frac{x.re}{1}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}\right)\\ \end{array}\]

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im)));
}

↓

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double temp;
	if (((y_46_re <= -2.2616143508082277e+79) || !(y_46_re <= 1.0382708421545201e-35))) {
		temp = (1.0 * (((x_46_im / (hypot(y_46_re, y_46_im) / y_46_re)) / hypot(y_46_re, y_46_im)) - (((x_46_re * y_46_im) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))));
	} else {
		temp = (1.0 * ((((x_46_im * y_46_re) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im)) - ((x_46_re / 1.0) / (hypot(y_46_re, y_46_im) / (y_46_im / hypot(y_46_re, y_46_im))))));
	}
	return temp;
}

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 2 regimes

2. if y.re < -2.2616143508082277e+79 or 1.0382708421545201e-35 < y.re

   1. Initial program 33.6

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 33.6

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

   4. Applied *-un-lft-identity 33.6

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

   5. Applied times-frac 33.6

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

   6. Simplified 33.6

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

   7. Simplified 23.3

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

   9. Applied *-un-lft-identity 23.3

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

   10. Applied associate-*l* 23.3

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

   11. Simplified 23.2

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

   13. Applied div-sub 23.2

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

   14. Applied div-sub 23.2

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

   16. Applied associate-/l* 7.8

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

   if -2.2616143508082277e+79 < y.re < 1.0382708421545201e-35

   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. Using strategy rm

   3. Applied add-sqr-sqrt 19.6

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

   4. Applied *-un-lft-identity 19.6

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

   5. Applied times-frac 19.6

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

   6. Simplified 19.6

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

   7. Simplified 11.8

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

   9. Applied *-un-lft-identity 11.8

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

   10. Applied associate-*l* 11.8

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

   11. Simplified 11.7

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

   13. Applied div-sub 11.7

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

   14. Applied div-sub 11.7

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

   16. Applied *-un-lft-identity 11.7

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

   17. Applied times-frac 1.7

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

   18. Applied associate-/l* 1.9

       \[\leadsto 1 \cdot \left(\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \color{blue}{\frac{\frac{x.re}{1}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 4.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -2.2616143508082277 \cdot 10^{79} \lor \neg \left(y.re \le 1.03827084215452011 \cdot 10^{-35}\right):\\ \;\;\;\;1 \cdot \left(\frac{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{\frac{x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;1 \cdot \left(\frac{\frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{\frac{x.re}{1}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))