Average Error: 26.2 → 24.4
Time: 3.1s
Precision: 64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;x.im \le -9.63064641952166374 \cdot 10^{-28}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;x.im \le 1.1328506141572114 \cdot 10^{123}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array}\]
\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}\;x.im \le -9.63064641952166374 \cdot 10^{-28}:\\
\;\;\;\;x.im \cdot \frac{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}\\

\mathbf{elif}\;x.im \le 1.1328506141572114 \cdot 10^{123}:\\
\;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{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}\\

\end{array}
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 ((x_46_im <= -9.630646419521664e-28)) {
		VAR = ((double) (((double) (x_46_im * ((double) (y_46_re / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))) - ((double) (((double) (x_46_re * y_46_im)) / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))));
	} else {
		double VAR_1;
		if ((x_46_im <= 1.1328506141572114e+123)) {
			VAR_1 = ((double) (((double) (((double) (x_46_im * y_46_re)) / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))) - ((double) (x_46_re * ((double) (y_46_im / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))))));
		} else {
			VAR_1 = ((double) (((double) (((double) (x_46_im / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))) * ((double) (y_46_re / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))))) - ((double) (((double) (x_46_re * y_46_im)) / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))));
		}
		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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if x.im < -9.630646419521664e-28

    1. Initial program 31.9

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

    3. Applied div-sub31.9

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

    5. Applied *-un-lft-identity31.9

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

    6. Applied times-frac30.2

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

    7. Simplified30.2

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

    if -9.630646419521664e-28 < x.im < 1.1328506141572114e+123

    1. Initial program 21.2

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

    3. Applied div-sub21.2

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

    5. Applied *-un-lft-identity21.2

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

    6. Applied times-frac19.7

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

    7. Simplified19.7

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

    if 1.1328506141572114e+123 < x.im

    1. Initial program 36.6

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

    3. Applied div-sub36.6

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

    5. Applied add-sqr-sqrt36.6

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

    6. Applied times-frac33.0

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

  4. Final simplification24.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \le -9.63064641952166374 \cdot 10^{-28}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;x.im \le 1.1328506141572114 \cdot 10^{123}:\\ \;\;\;\;\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2020130 
(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))))