Average Error: 26.8 → 23.1
Time: 3.8s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.024796178527634 \cdot 10^{-05} \lor \neg \left(y.im \leq 7.741980536026424 \cdot 10^{-194}\right):\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array}\]
\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}\;y.im \leq -2.024796178527634 \cdot 10^{-05} \lor \neg \left(y.im \leq 7.741980536026424 \cdot 10^{-194}\right):\\
\;\;\;\;\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}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

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

\end{array}
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
↓
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.024796178527634e-05)
         (not (<= y.im 7.741980536026424e-194)))
   (-
    (*
     (/ x.im (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (/ y.re (sqrt (+ (* y.re y.re) (* y.im y.im)))))
    (*
     (/ x.re (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (/ y.im (sqrt (+ (* y.re y.re) (* y.im y.im))))))
   (-
    (* x.im (/ y.re (+ (* y.re y.re) (* y.im y.im))))
    (/ (* y.im x.re) (+ (* y.re y.re) (* y.im y.im))))))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (((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 tmp;
	if (((y_46_im <= -2.024796178527634e-05) || !(y_46_im <= 7.741980536026424e-194))) {
		tmp = ((double) (((double) ((x_46_im / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))) * (y_46_re / ((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))))))) * (y_46_im / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))))))));
	} else {
		tmp = ((double) (((double) (x_46_im * (y_46_re / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))) - (((double) (y_46_im * x_46_re)) / ((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im)))))));
	}
	return tmp;
}

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 2 regimes

  2. if y.im < -2.02479617852763408e-5 or 7.74198053602642366e-194 < y.im

    1. Initial program 29.1

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

    3. Applied div-sub_binary6429.1

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

    5. Applied add-sqr-sqrt_binary6429.1

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

    6. Applied times-frac_binary6426.3

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

    8. Applied add-sqr-sqrt_binary6426.3

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

    9. Applied times-frac_binary6425.2

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

    if -2.02479617852763408e-5 < y.im < 7.74198053602642366e-194

    1. Initial program 21.5

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

    3. Applied div-sub_binary6421.5

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

    5. Applied *-un-lft-identity_binary6421.5

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

    6. Applied times-frac_binary6418.6

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

    7. Simplified18.6

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

  4. Final simplification23.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.024796178527634 \cdot 10^{-05} \lor \neg \left(y.im \leq 7.741980536026424 \cdot 10^{-194}\right):\\ \;\;\;\;\frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2020210 
(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))))