_divideComplex, imaginary part

Average Error: 26.4 → 12.9

Time: 4.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \le -5.1227572413769072 \cdot 10^{95}:\\ \;\;\;\;\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 6.69895789485502902 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 2.54946443013026016 \cdot 10^{79}:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;y.re \le 4.6230231189603044 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((double) (((double) (((double) (x_46_im * y_46_re)) - ((double) (x_46_re * y_46_im)))) / ((double) (((double) (y_46_re * y_46_re)) + ((double) (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 VAR;
	if ((y_46_re <= -5.122757241376907e+95)) {
		VAR = ((double) (((double) (-1.0 * x_46_im)) / ((double) hypot(y_46_re, y_46_im))));
	} else {
		double VAR_1;
		if ((y_46_re <= 6.698957894855029e-106)) {
			VAR_1 = ((double) (((double) (1.0 / ((double) (((double) hypot(y_46_re, y_46_im)) / ((double) (((double) (x_46_im * y_46_re)) - ((double) (x_46_re * y_46_im)))))))) / ((double) hypot(y_46_re, y_46_im))));
		} else {
			double VAR_2;
			if ((y_46_re <= 2.54946443013026e+79)) {
				VAR_2 = ((double) (((double) (x_46_im / ((double) (((double) fma(y_46_re, y_46_re, ((double) (y_46_im * y_46_im)))) / y_46_re)))) - ((double) (x_46_re / ((double) (((double) fma(y_46_re, y_46_re, ((double) (y_46_im * y_46_im)))) / y_46_im))))));
			} else {
				double VAR_3;
				if ((y_46_re <= 4.6230231189603044e+141)) {
					VAR_3 = ((double) (((double) (1.0 / ((double) (((double) hypot(y_46_re, y_46_im)) / ((double) (((double) (x_46_im * y_46_re)) - ((double) (x_46_re * y_46_im)))))))) / ((double) hypot(y_46_re, y_46_im))));
				} else {
					VAR_3 = ((double) (x_46_im / ((double) hypot(y_46_re, y_46_im))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

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 4 regimes

2. if y.re < -5.122757241376907e+95

   1. Initial program 39.9

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

   3. Applied add-sqr-sqrt 39.9

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

   4. Applied *-un-lft-identity 39.9

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

   5. Applied times-frac 39.9

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

   6. Simplified 39.9

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

   7. Simplified 27.7

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

   9. Applied associate-*r/ 27.7

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

   10. Simplified 27.6

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

   11. Taylor expanded around -inf 15.3

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

   if -5.122757241376907e+95 < y.re < 6.698957894855029e-106 or 2.54946443013026e+79 < y.re < 4.6230231189603044e+141

   1. Initial program 20.0

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

   3. Applied add-sqr-sqrt 20.0

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

   4. Applied *-un-lft-identity 20.0

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

   5. Applied times-frac 20.1

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

   6. Simplified 20.1

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

   7. Simplified 12.4

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

   9. Applied associate-*r/ 12.3

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

   10. Simplified 12.3

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

   12. Applied clear-num 12.3

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

   if 6.698957894855029e-106 < y.re < 2.54946443013026e+79

   1. Initial program 15.7

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

   3. Applied div-sub 15.7

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

   4. Simplified 14.4

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

   5. Simplified 11.8

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

   if 4.6230231189603044e+141 < y.re

   1. Initial program 44.2

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

   3. Applied add-sqr-sqrt 44.2

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

   4. Applied *-un-lft-identity 44.2

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

   5. Applied times-frac 44.2

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

   6. Simplified 44.2

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

   7. Simplified 28.3

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

   9. Applied associate-*r/ 28.2

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

   10. Simplified 28.2

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

   11. Taylor expanded around inf 13.3

       \[\leadsto \frac{\color{blue}{x.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\]
3. Recombined 4 regimes into one program.

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \le -5.1227572413769072 \cdot 10^{95}:\\ \;\;\;\;\frac{-1 \cdot x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 6.69895789485502902 \cdot 10^{-106}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \le 2.54946443013026016 \cdot 10^{79}:\\ \;\;\;\;\frac{x.im}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.re}} - \frac{x.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{y.im}}\\ \mathbf{elif}\;y.re \le 4.6230231189603044 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im \cdot y.re - x.re \cdot y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +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))))