_divideComplex, real part

Average Error: 26.2 → 11.3

Time: 7.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.374813608163854 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.re}{y.im}, \frac{y.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -4.082695455232209 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.453427998098388 \cdot 10^{-255}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.563402658894378 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -8.374813608163854e+134)
   (fma (/ x.re y.im) (/ y.re y.im) (/ x.im y.im))
   (if (<= y.im -4.082695455232209e-87)
     (/ (/ (fma y.im x.im (* x.re y.re)) (hypot y.im y.re)) (hypot y.im y.re))
     (if (<= y.im 5.453427998098388e-255)
       (+
        (/ x.re y.re)
        (* (/ y.im (* y.re y.re)) (- x.im (/ (* y.im x.re) y.re))))
       (if (<= y.im 5.563402658894378e+112)
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.im y.re))
          (hypot y.im y.re))
         (* (/ 1.0 (hypot y.im y.re)) (fma (/ x.re y.im) y.re x.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * 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 <= -8.374813608163854e+134) {
		tmp = fma((x_46_re / y_46_im), (y_46_re / y_46_im), (x_46_im / y_46_im));
	} else if (y_46_im <= -4.082695455232209e-87) {
		tmp = (fma(y_46_im, x_46_im, (x_46_re * y_46_re)) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re);
	} else if (y_46_im <= 5.453427998098388e-255) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / (y_46_re * y_46_re)) * (x_46_im - ((y_46_im * x_46_re) / y_46_re)));
	} else if (y_46_im <= 5.563402658894378e+112) {
		tmp = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_im, y_46_re)) / hypot(y_46_im, y_46_re);
	} else {
		tmp = (1.0 / hypot(y_46_im, y_46_re)) * fma((x_46_re / y_46_im), y_46_re, x_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

Derivation

1. Split input into 5 regimes

2. if y.im < -8.374813608163854e134

   1. Initial program 42.4

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

   2. Simplified 42.4

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

   3. Applied add-sqr-sqrt_binary64 42.4

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

   4. Applied *-un-lft-identity_binary64 42.4

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

   5. Applied times-frac_binary64 42.4

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

   6. Simplified 42.4

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

   7. Simplified 27.3

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

   8. Taylor expanded in y.im around inf 15.6

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

   9. Simplified 7.1

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

   if -8.374813608163854e134 < y.im < -4.0826954552322086e-87

   1. Initial program 16.5

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

   2. Simplified 16.5

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

   3. Applied add-sqr-sqrt_binary64 16.5

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

   4. Applied *-un-lft-identity_binary64 16.5

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

   5. Applied times-frac_binary64 16.6

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

   6. Simplified 16.6

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

   7. Simplified 11.9

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

   8. Applied associate-*l/_binary64 11.8

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

   9. Simplified 11.8

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

   10. Applied add-sqr-sqrt_binary64 12.0

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

   11. Applied associate-/r*_binary64 12.0

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

   12. Applied *-un-lft-identity_binary64 12.0

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

   13. Applied *-un-lft-identity_binary64 12.0

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

   14. Applied sqrt-prod_binary64 12.0

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

   15. Applied *-un-lft-identity_binary64 12.0

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

   16. Applied sqrt-prod_binary64 12.0

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

   17. Applied *-un-lft-identity_binary64 12.0

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

   18. Applied times-frac_binary64 12.0

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

   19. Applied times-frac_binary64 12.0

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

   20. Applied times-frac_binary64 12.0

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

   21. Simplified 12.0

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

   22. Simplified 11.8

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

   if -4.0826954552322086e-87 < y.im < 5.45342799809838813e-255

   1. Initial program 20.6

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

   2. Simplified 20.6

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

   3. Taylor expanded in y.re around inf 16.6

      \[\leadsto \color{blue}{\left(\frac{y.im \cdot x.im}{{y.re}^{2}} + \frac{x.re}{y.re}\right) - \frac{x.re \cdot {y.im}^{2}}{{y.re}^{3}}} \]

   4. Simplified 13.5

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

   if 5.45342799809838813e-255 < y.im < 5.56340265889437815e112

   1. Initial program 19.4

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

   2. Simplified 19.4

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

   3. Applied add-sqr-sqrt_binary64 19.4

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

   4. Applied *-un-lft-identity_binary64 19.4

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

   5. Applied times-frac_binary64 19.4

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

   6. Simplified 19.4

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

   7. Simplified 12.7

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

   8. Applied associate-*l/_binary64 12.6

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

   9. Simplified 12.6

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

   if 5.56340265889437815e112 < y.im

   1. Initial program 41.0

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

   2. Simplified 41.0

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

   3. Applied add-sqr-sqrt_binary64 41.0

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

   4. Applied *-un-lft-identity_binary64 41.0

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

   5. Applied times-frac_binary64 41.1

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

   6. Simplified 41.1

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

   7. Simplified 26.9

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

   8. Taylor expanded in y.im around inf 13.7

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

   9. Simplified 9.4

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

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.374813608163854 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.re}{y.im}, \frac{y.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -4.082695455232209 \cdot 10^{-87}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(y.im, x.im, x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq 5.453427998098388 \cdot 10^{-255}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 5.563402658894378 \cdot 10^{+112}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \mathsf{fma}\left(\frac{x.re}{y.im}, y.re, x.im\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022094 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))