Average Error: 25.8 → 23.2
Time: 3.8s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.936337481574621 \cdot 10^{-09} \lor \neg \left(y.re \leq 2.547251396127817 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - y.im \cdot \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]
\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.re \leq -1.936337481574621 \cdot 10^{-09} \lor \neg \left(y.re \leq 2.547251396127817 \cdot 10^{-98}\right):\\
\;\;\;\;\frac{y.re \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - y.im \cdot \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{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) (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 <= -1.936337481574621e-09) || !(y_46_re <= 2.547251396127817e-98))) {
		VAR = (((double) (((double) (y_46_re * (x_46_im / ((double) sqrt(((double) (((double) (y_46_re * y_46_re)) + ((double) (y_46_im * y_46_im))))))))) - ((double) (y_46_im * (x_46_re / ((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)))))));
	} else {
		VAR = ((((double) (((double) (y_46_re * x_46_im)) - ((double) (y_46_im * x_46_re)))) / ((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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if y.re < -1.93633748157462091e-9 or 2.5472513961278171e-98 < y.re

    1. Initial program Error: 29.0 bits

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

    3. Applied add-sqr-sqrtError: 29.0 bits

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

    4. Applied associate-/r*Error: 29.0 bits

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

    6. Applied div-subError: 29.0 bits

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

    7. SimplifiedError: 26.1 bits

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

    8. SimplifiedError: 24.9 bits

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

    if -1.93633748157462091e-9 < y.re < 2.5472513961278171e-98

    1. Initial program Error: 20.5 bits

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

    3. Applied add-sqr-sqrtError: 20.5 bits

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

    4. Applied associate-/r*Error: 20.4 bits

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

  4. Final simplificationError: 23.2 bits

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.936337481574621 \cdot 10^{-09} \lor \neg \left(y.re \leq 2.547251396127817 \cdot 10^{-98}\right):\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} - y.im \cdot \frac{x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020200 
(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))))