_divideComplex, imaginary part

Average Error: 26.5 → 12.7

Time: 9.8s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.091373264773737 \cdot 10^{+128}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.im \leq -4.23510218339596 \cdot 10^{-173}:\\ \;\;\;\;\begin{array}{l} t_1 := \sqrt{\mathsf{hypot}\left(y.im, y.re\right)}\\ \frac{\frac{1}{t_1} \cdot t_0}{t_1} \end{array}\\ \mathbf{elif}\;y.im \leq 8.403007832793524 \cdot 10^{-245}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 9.437094843246655 \cdot 10^{+101}:\\ \;\;\;\;\frac{t_0}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \left(\frac{y.re \cdot x.im}{y.im} - x.re\right)\\ \end{array}\\ \end{array} \]

FPCore

(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 -6.091373264773737e+128)
   (- (/ x.re y.im))
   (let* ((t_0 (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.im y.re))))
     (if (<= y.im -4.23510218339596e-173)
       (let* ((t_1 (sqrt (hypot y.im y.re)))) (/ (* (/ 1.0 t_1) t_0) t_1))
       (if (<= y.im 8.403007832793524e-245)
         (- (/ x.im y.re) (/ x.re (/ (* y.re y.re) y.im)))
         (if (<= y.im 9.437094843246655e+101)
           (/ t_0 (hypot y.im y.re))
           (* (/ 1.0 (hypot y.im y.re)) (- (/ (* y.re x.im) y.im) x.re))))))))

C

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 <= -6.091373264773737e+128) {
		tmp = -(x_46_re / y_46_im);
	} else {
		double t_0 = ((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_im, y_46_re);
		double tmp_1;
		if (y_46_im <= -4.23510218339596e-173) {
			double t_1_2 = sqrt(hypot(y_46_im, y_46_re));
			tmp_1 = ((1.0 / t_1_2) * t_0) / t_1_2;
		} else if (y_46_im <= 8.403007832793524e-245) {
			tmp_1 = (x_46_im / y_46_re) - (x_46_re / ((y_46_re * y_46_re) / y_46_im));
		} else if (y_46_im <= 9.437094843246655e+101) {
			tmp_1 = t_0 / hypot(y_46_im, y_46_re);
		} else {
			tmp_1 = (1.0 / hypot(y_46_im, y_46_re)) * (((y_46_re * x_46_im) / y_46_im) - x_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

Derivation

1. Split input into 5 regimes

2. if y.im < -6.09137326477373719e128

   1. Initial program 42.6

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

   2. Simplified 42.6

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

   3. Taylor expanded in y.re around 0 15.8

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

   4. Simplified 15.8

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

   if -6.09137326477373719e128 < y.im < -4.2351021833959599e-173

   1. Initial program 17.4

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

   2. Simplified 17.4

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

   3. Applied add-sqr-sqrt_binary64 17.4

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

   4. Applied *-un-lft-identity_binary64 17.4

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

   5. Applied times-frac_binary64 17.5

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

   6. Simplified 17.4

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   7. Simplified 12.4

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

   8. Applied add-sqr-sqrt_binary64 12.6

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

   9. Applied *-un-lft-identity_binary64 12.6

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

   10. Applied times-frac_binary64 12.6

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

   11. Applied associate-*l*_binary64 12.6

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

   12. Simplified 12.6

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

   13. Applied associate-*r/_binary64 12.5

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

   if -4.2351021833959599e-173 < y.im < 8.4030078327935245e-245

   1. Initial program 25.3

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

   2. Simplified 25.3

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

   3. Applied add-sqr-sqrt_binary64 25.3

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

   4. Applied *-un-lft-identity_binary64 25.3

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

   5. Applied times-frac_binary64 25.3

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

   6. Simplified 25.3

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   7. Simplified 13.6

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

   8. Applied add-sqr-sqrt_binary64 13.8

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

   9. Applied *-un-lft-identity_binary64 13.8

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

   10. Applied times-frac_binary64 13.8

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

   11. Applied associate-*l*_binary64 13.8

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

   12. Simplified 13.8

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

   13. Applied frac-times_binary64 13.7

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

   14. Taylor expanded in y.re around inf 8.9

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

   15. Simplified 10.7

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

   if 8.4030078327935245e-245 < y.im < 9.4370948432466548e101

   1. Initial program 17.8

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

   2. Simplified 17.8

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

   3. Applied add-sqr-sqrt_binary64 17.8

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

   4. Applied *-un-lft-identity_binary64 17.8

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

   5. Applied times-frac_binary64 17.8

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

   6. Simplified 17.8

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   7. Simplified 11.5

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

   8. Applied add-sqr-sqrt_binary64 11.7

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

   9. Applied *-un-lft-identity_binary64 11.7

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

   10. Applied times-frac_binary64 11.8

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

   11. Applied associate-*l*_binary64 11.8

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

   12. Simplified 11.7

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

   13. Applied frac-times_binary64 11.6

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

   14. Applied pow1/2_binary64 11.6

       \[\leadsto \frac{1 \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\sqrt{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{0.5}}} \]

   15. Applied pow1/2_binary64 11.6

       \[\leadsto \frac{1 \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\color{blue}{{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{0.5}} \cdot {\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{0.5}} \]

   16. Applied pow-sqr_binary64 11.4

       \[\leadsto \frac{1 \cdot \frac{y.re \cdot x.im - x.re \cdot y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\color{blue}{{\left(\mathsf{hypot}\left(y.im, y.re\right)\right)}^{\left(2 \cdot 0.5\right)}}} \]

   if 9.4370948432466548e101 < y.im

   1. Initial program 39.9

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

   2. Simplified 39.9

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

   3. Applied add-sqr-sqrt_binary64 39.9

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)} \cdot \sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}}} \]

   4. Applied *-un-lft-identity_binary64 39.9

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

   5. Applied times-frac_binary64 39.9

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

   6. Simplified 39.9

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   7. Simplified 26.7

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

   8. Taylor expanded in y.re around 0 14.0

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\left(\frac{y.re \cdot x.im}{y.im} - x.re\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 12.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.091373264773737 \cdot 10^{+128}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.23510218339596 \cdot 10^{-173}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\sqrt{\mathsf{hypot}\left(y.im, y.re\right)}}\\ \mathbf{elif}\;y.im \leq 8.403007832793524 \cdot 10^{-245}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 9.437094843246655 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \left(\frac{y.re \cdot x.im}{y.im} - x.re\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2021340 
(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))))