_divideComplex, imaginary part

Average Error: 26.4 → 23.8

Time: 3.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} = -inf.0 \lor \neg \left(\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le -2.4410351 \cdot 10^{-318}\right):\\ \;\;\;\;y.re \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} - \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

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 (((((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)))))) <= -inf.0) || !(((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)))))) <= -2.4410350770643e-318))) {
		VAR = ((double) (((double) (y_46_re * ((double) (x_46_im / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))) - ((double) (((double) (y_46_im / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))) * ((double) (x_46_re / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))))))));
	} else {
		VAR = ((double) (((double) (((double) (((double) (x_46_im * y_46_re)) - ((double) (x_46_re * y_46_im)))) / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))) / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))));
	}
	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 2 regimes

2. if (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))) < -inf.0 or -2.4410351e-318 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))

   1. Initial program 33.8

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

   3. Applied div-sub 33.8

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

   4. Simplified 32.6

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

   5. Simplified 31.2

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

   7. Applied add-sqr-sqrt 31.2

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

   8. Applied *-un-lft-identity 31.2

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

   9. Applied times-frac 31.2

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

   10. Applied associate-*r* 30.4

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

   11. Simplified 30.4

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

   if -inf.0 < (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))) < -2.4410351e-318

   1. Initial program 0.7

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

   3. Applied add-sqr-sqrt 0.7

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

   4. Applied associate-/r* 0.5

      \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 23.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} = -inf.0 \lor \neg \left(\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \le -2.4410351 \cdot 10^{-318}\right):\\ \;\;\;\;y.re \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} - \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020184 
(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))))