Average Error: 26.2 → 14.7
Time: 8.1s
Precision: binary64
\[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.236807978306482 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{0.5}}\\ \mathbf{if}\;y.re \leq -4.270742155339291 \cdot 10^{-94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.818593708289327 \cdot 10^{-120}:\\ \;\;\;\;\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4153068053837126 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array}\\ \end{array} \]
\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 -8.236807978306482 \cdot 10^{+73}:\\
\;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_0 := \frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{0.5}}\\
\mathbf{if}\;y.re \leq -4.270742155339291 \cdot 10^{-94}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;y.re \leq 3.818593708289327 \cdot 10^{-120}:\\
\;\;\;\;\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}\\

\mathbf{elif}\;y.re \leq 6.4153068053837126 \cdot 10^{+150}:\\
\;\;\;\;t_0\\

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


\end{array}\\


\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.re -8.236807978306482e+73)
   (- (/ x.im y.re) (/ (* x.re y.im) (pow y.re 2.0)))
   (let* ((t_0
           (/
            (/
             (- (* y.re x.im) (* x.re y.im))
             (sqrt (+ (pow y.re 2.0) (pow y.im 2.0))))
            (pow (+ (* y.re y.re) (* y.im y.im)) 0.5))))
     (if (<= y.re -4.270742155339291e-94)
       t_0
       (if (<= y.re 3.818593708289327e-120)
         (- (/ (* y.re x.im) (pow y.im 2.0)) (/ x.re y.im))
         (if (<= y.re 6.4153068053837126e+150) t_0 (/ x.im y.re)))))))
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_re <= -8.236807978306482e+73) {
		tmp = (x_46_im / y_46_re) - ((x_46_re * y_46_im) / pow(y_46_re, 2.0));
	} else {
		double t_0 = (((y_46_re * x_46_im) - (x_46_re * y_46_im)) / sqrt(pow(y_46_re, 2.0) + pow(y_46_im, 2.0))) / pow(((y_46_re * y_46_re) + (y_46_im * y_46_im)), 0.5);
		double tmp_1;
		if (y_46_re <= -4.270742155339291e-94) {
			tmp_1 = t_0;
		} else if (y_46_re <= 3.818593708289327e-120) {
			tmp_1 = ((y_46_re * x_46_im) / pow(y_46_im, 2.0)) - (x_46_re / y_46_im);
		} else if (y_46_re <= 6.4153068053837126e+150) {
			tmp_1 = t_0;
		} else {
			tmp_1 = x_46_im / y_46_re;
		}
		tmp = tmp_1;
	}
	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.re < -8.236807978306482e73

    1. Initial program 37.2

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

    2. Taylor expanded around inf 16.6

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

    if -8.236807978306482e73 < y.re < -4.27074215533929121e-94 or 3.81859370828932731e-120 < y.re < 6.41530680538371258e150

    1. Initial program 16.8

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

    3. Applied add-sqr-sqrt_binary6416.8

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

    4. Applied associate-/r*_binary6416.7

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

    5. Simplified16.7

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

    7. Applied pow1_binary6416.7

      \[\leadsto \frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{\color{blue}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{1}}}} \]

    8. Applied sqrt-pow1_binary6416.7

      \[\leadsto \frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\color{blue}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{\left(\frac{1}{2}\right)}}} \]

    9. Simplified16.7

      \[\leadsto \frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{\color{blue}{0.5}}} \]
    10. Using strategy rm

    11. Applied sub-neg_binary6416.7

      \[\leadsto \frac{\frac{\color{blue}{y.re \cdot x.im + \left(-x.re \cdot y.im\right)}}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{0.5}} \]

    12. Simplified16.7

      \[\leadsto \frac{\frac{y.re \cdot x.im + \color{blue}{y.im \cdot \left(-x.re\right)}}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{0.5}} \]

    if -4.27074215533929121e-94 < y.re < 3.81859370828932731e-120

    1. Initial program 21.8

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

    2. Taylor expanded around 0 11.1

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

    if 6.41530680538371258e150 < y.re

    1. Initial program 45.3

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

    2. Taylor expanded around inf 14.7

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

  4. Final simplification14.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.236807978306482 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -4.270742155339291 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{0.5}}\\ \mathbf{elif}\;y.re \leq 3.818593708289327 \cdot 10^{-120}:\\ \;\;\;\;\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.4153068053837126 \cdot 10^{+150}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - x.re \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{{\left(y.re \cdot y.re + y.im \cdot y.im\right)}^{0.5}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Reproduce

herbie shell --seed 2021207 
(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))))