Average Error: 26.0 → 16.2
Time: 10.0s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\]
↓
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.8815821022379826 \cdot 10^{+98}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.794722579529425 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -4.356524714149345 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 5.291717683969782 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.391090750179387 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array}\]
\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 -1.8815821022379826 \cdot 10^{+98}:\\
\;\;\;\;-\frac{x.re}{y.im}\\

\mathbf{elif}\;y.im \leq -6.794722579529425 \cdot 10^{-92}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\

\mathbf{elif}\;y.im \leq -4.356524714149345 \cdot 10^{-141}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\

\mathbf{elif}\;y.im \leq 5.291717683969782 \cdot 10^{-146}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\

\mathbf{elif}\;y.im \leq 7.391090750179387 \cdot 10^{+124}:\\
\;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{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 -1.8815821022379826e+98)
   (- (/ x.re y.im))
   (if (<= y.im -6.794722579529425e-92)
     (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
     (if (<= y.im -4.356524714149345e-141)
       (/
        (/
         (- (* x.im y.re) (* y.im x.re))
         (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
        (sqrt (+ (* y.re y.re) (* y.im y.im))))
       (if (<= y.im 5.291717683969782e-146)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.im 7.391090750179387e+124)
           (/
            (/
             (- (* x.im y.re) (* y.im x.re))
             (sqrt (+ (pow y.re 2.0) (pow y.im 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 <= -1.8815821022379826e+98) {
		tmp = -(x_46_re / y_46_im);
	} else if (y_46_im <= -6.794722579529425e-92) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= -4.356524714149345e-141) {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) / sqrt((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 5.291717683969782e-146) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 7.391090750179387e+124) {
		tmp = (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 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 < -1.88158210223798258e98 or 7.3910907501793867e124 < y.im

    1. Initial program 40.2

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

    2. Taylor expanded around 0 16.4

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

    3. Simplified16.4

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

    if -1.88158210223798258e98 < y.im < -6.7947225795294251e-92

    1. Initial program 15.6

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

    2. Taylor expanded around inf 33.2

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

    4. Applied unpow2_binary64_218933.2

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

    5. Applied times-frac_binary64_213031.2

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

    if -6.7947225795294251e-92 < y.im < -4.35652471414934487e-141 or 5.29171768396978226e-146 < y.im < 7.3910907501793867e124

    1. Initial program 16.5

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

    3. Applied add-sqr-sqrt_binary64_214616.5

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

    4. Applied associate-/r*_binary64_206816.4

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

    5. Simplified16.4

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

    if -4.35652471414934487e-141 < y.im < 5.29171768396978226e-146

    1. Initial program 23.5

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

    2. Taylor expanded around inf 9.2

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

    4. Applied unpow2_binary64_21899.2

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

    5. Applied associate-/r*_binary64_20685.9

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

  4. Final simplification16.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8815821022379826 \cdot 10^{+98}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -6.794722579529425 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -4.356524714149345 \cdot 10^{-141}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.im \leq 5.291717683969782 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.391090750179387 \cdot 10^{+124}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array}\]

Reproduce

herbie shell --seed 2021028 
(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))))