Average Error: 25.8 → 14.4
Time: 8.1s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.7351387155195298 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im \cdot y.re}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -5.1209567346932005 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 2.0246643543322144 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3.237823215092107 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array}\]
\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 -1.7351387155195298 \cdot 10^{+25}:\\
\;\;\;\;\frac{x.im \cdot y.re}{{y.im}^{2}} - \frac{x.re}{y.im}\\

\mathbf{elif}\;y.im \leq -5.1209567346932005 \cdot 10^{-132}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 2.0246643543322144 \cdot 10^{-121}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\

\mathbf{elif}\;y.im \leq 3.237823215092107 \cdot 10^{+141}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{else}:\\
\;\;\;\;\frac{-x.re}{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 (<= y.im -1.7351387155195298e+25)
   (- (/ (* x.im y.re) (pow y.im 2.0)) (/ x.re y.im))
   (if (<= y.im -5.1209567346932005e-132)
     (/
      (/
       (- (* x.im y.re) (* y.im x.re))
       (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
      (sqrt (+ (* y.re y.re) (* y.im y.im))))
     (if (<= y.im 2.0246643543322144e-121)
       (- (/ x.im y.re) (/ (* y.im x.re) (pow y.re 2.0)))
       (if (<= y.im 3.237823215092107e+141)
         (/
          (/
           (- (* x.im y.re) (* y.im x.re))
           (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
          (sqrt (+ (* y.re y.re) (* y.im y.im))))
         (/ (- x.re) y.im))))))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (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 <= -1.7351387155195298e+25) {
		tmp = ((x_46_im * y_46_re) / pow(y_46_im, 2.0)) - (x_46_re / y_46_im);
	} else if (y_46_im <= -5.1209567346932005e-132) {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.0246643543322144e-121) {
		tmp = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / pow(y_46_re, 2.0));
	} else if (y_46_im <= 3.237823215092107e+141) {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = -x_46_re / 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 4 regimes

  2. if y.im < -1.7351387155195298e25

    1. Initial program 32.8

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

    2. Taylor expanded around 0 17.4

      \[\leadsto \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}}\]

    3. Simplified17.4

      \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} - \frac{x.re}{y.im}}\]

    if -1.7351387155195298e25 < y.im < -5.12095673469320054e-132 or 2.02466435433221439e-121 < y.im < 3.2378232150921068e141

    1. Initial program 15.8

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

    3. Applied add-sqr-sqrt_binary64_112315.8

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

    4. Applied associate-/r*_binary64_104515.7

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

    5. Simplified15.7

      \[\leadsto \frac{\color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\]

    if -5.12095673469320054e-132 < y.im < 2.02466435433221439e-121

    1. Initial program 24.1

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

    2. Taylor expanded around inf 10.5

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}}\]

    3. Simplified10.5

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}}\]

    if 3.2378232150921068e141 < y.im

    1. Initial program 42.1

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

    2. Taylor expanded around 0 13.4

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

    3. Simplified13.4

      \[\leadsto \color{blue}{\frac{-x.re}{y.im}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification14.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7351387155195298 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im \cdot y.re}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -5.1209567346932005 \cdot 10^{-132}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 2.0246643543322144 \cdot 10^{-121}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 3.237823215092107 \cdot 10^{+141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2020357 
(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))))