_divideComplex, real part

Average Error: 26.2 → 15.3

Time: 11.7s

Precision: binary64

Cost: 40707

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.266447775026859 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.7218745403061054 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq -2.2148262768226444 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\ \mathbf{elif}\;y.re \leq -3084.182004279282:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq -9.83945259871375 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 1.9690354844540234 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.4163550870851477 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array}\]

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.re -1.266447775026859e+143)
   (/ x.re y.re)
   (if (<= y.re -4.7218745403061054e+135)
     (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0)))
     (if (<= y.re -2.2148262768226444e+22)
       (*
        (/ 1.0 (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
        (/
         (+ (* y.re x.re) (* x.im y.im))
         (sqrt (+ (pow y.im 2.0) (pow y.re 2.0)))))
       (if (<= y.re -3084.182004279282)
         (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0)))
         (if (<= y.re -9.83945259871375e-91)
           (/
            (/
             (+ (* y.re x.re) (* x.im y.im))
             (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
            (sqrt (+ (* y.re y.re) (* y.im y.im))))
           (if (<= y.re 1.9690354844540234e-126)
             (+ (/ x.im y.im) (/ (* y.re x.re) (pow y.im 2.0)))
             (if (<= y.re 1.4163550870851477e+142)
               (/
                (/
                 (+ (* y.re x.re) (* x.im y.im))
                 (sqrt (+ (pow y.im 2.0) (pow y.re 2.0))))
                (sqrt (+ (* y.re y.re) (* y.im y.im))))
               (/ x.re y.re)))))))))

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_re <= -1.266447775026859e+143) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -4.7218745403061054e+135) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	} else if (y_46_re <= -2.2148262768226444e+22) {
		tmp = (1.0 / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) * (((y_46_re * x_46_re) + (x_46_im * y_46_im)) / sqrt(pow(y_46_im, 2.0) + pow(y_46_re, 2.0)));
	} else if (y_46_re <= -3084.182004279282) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	} else if (y_46_re <= -9.83945259871375e-91) {
		tmp = (((y_46_re * x_46_re) + (x_46_im * y_46_im)) / 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_re <= 1.9690354844540234e-126) {
		tmp = (x_46_im / y_46_im) + ((y_46_re * x_46_re) / pow(y_46_im, 2.0));
	} else if (y_46_re <= 1.4163550870851477e+142) {
		tmp = (((y_46_re * x_46_re) + (x_46_im * y_46_im)) / 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_re;
	}
	return tmp;
}

Error

Bits error versus x.re

Bits error versus x.im

Bits error versus y.re

Bits error versus y.im

Alternatives

Alternative 1
Error	26.8
Cost	79424

\[\frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 2
Error	44.3
Cost	72512

\[\frac{\sqrt{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 3
Error	26.7
Cost	60224

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

Alternative 4
Error	26.7
Cost	59840

\[\frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 5
Error	26.7
Cost	59328

\[\frac{1}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 6
Error	46.3
Cost	47232

\[\frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{y.im + 0.5 \cdot \frac{{y.re}^{2}}{y.im}}\]

Alternative 7
Error	26.7
Cost	46528

\[\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt[3]{{y.re}^{2} + {y.im}^{2}} \cdot \sqrt[3]{{y.re}^{2} + {y.im}^{2}}}}{\sqrt[3]{y.re \cdot y.re + y.im \cdot y.im}}\]

Alternative 8
Error	39.4
Cost	40128

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

Alternative 9
Error	26.2
Cost	39744

\[\frac{1}{\sqrt{{y.re}^{2} + {y.im}^{2}}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}\]

Alternative 10
Error	53.0
Cost	34176

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

Alternative 11
Error	26.7
Cost	33856

\[\frac{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im} \cdot \sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}{\frac{{y.re}^{2} + {y.im}^{2}}{\sqrt[3]{x.re \cdot y.re + x.im \cdot y.im}}}\]

Alternative 12
Error	26.7
Cost	33856

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

Alternative 13
Error	44.2
Cost	26944

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

Alternative 14
Error	44.2
Cost	26944

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

Alternative 15
Error	26.2
Cost	26944

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

Alternative 16
Error	47.0
Cost	26816

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{\frac{{y.re}^{4} - {y.im}^{4}}{{y.re}^{2} - {y.im}^{2}}}\]

Alternative 17
Error	26.2
Cost	26816

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

Alternative 18
Error	41.8
Cost	26496

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

Alternative 19
Error	40.4
Cost	26432

\[e^{\log \left(\frac{x.re \cdot y.re + x.im \cdot y.im}{{y.re}^{2} + {y.im}^{2}}\right)}\]

Alternative 20
Error	28.5
Cost	26432

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{e^{\log \left({y.re}^{2} + {y.im}^{2}\right)}}\]

Alternative 21
Error	51.2
Cost	21504

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

Alternative 22
Error	55.0
Cost	15168

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{{y.re}^{6} + {y.im}^{6}} \cdot \left(\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)\right)\]

Alternative 23
Error	41.1
Cost	14656

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

Alternative 24
Error	51.8
Cost	14656

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

Alternative 25
Error	49.1
Cost	14144

\[\frac{x.re \cdot y.re + x.im \cdot y.im}{{y.re}^{4} - {y.im}^{4}} \cdot \left(y.re \cdot y.re - y.im \cdot y.im\right)\]

Alternative 26
Error	45.0
Cost	13824

\[\frac{\sqrt[3]{{\left(x.re \cdot y.re + x.im \cdot y.im\right)}^{3}}}{y.re \cdot y.re + y.im \cdot y.im}\]

Alternative 27
Error	26.4
Cost	13760

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

Alternative 28
Error	26.3
Cost	13760

\[\frac{1}{\frac{{y.re}^{2} + {y.im}^{2}}{x.re \cdot y.re + x.im \cdot y.im}}\]

Alternative 29
Error	45.2
Cost	13760

\[\frac{e^{\log \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im}\]

Alternative 30
Error	39.5
Cost	13376

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

Alternative 31
Error	39.2
Cost	13376

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

Alternative 32
Error	47.5
Cost	7552

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

Alternative 33
Error	46.5
Cost	7424

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

Alternative 34
Error	43.9
Cost	7040

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

Alternative 35
Error	34.7
Cost	7040

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

Alternative 36
Error	34.8
Cost	7040

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

Alternative 37
Error	47.3
Cost	6976

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

Alternative 38
Error	26.2
Cost	960

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

Alternative 39
Error	39.2
Cost	704

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

Alternative 40
Error	39.5
Cost	704

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

Alternative 41
Error	37.6
Cost	192

\[\frac{x.re}{y.re}\]

Alternative 42
Error	37.4
Cost	192

\[\frac{x.im}{y.im}\]

Alternative 43
Error	61.6
Cost	64

\[1\]

Alternative 44
Error	52.0
Cost	64

\[0\]

Alternative 45
Error	61.7
Cost	64

\[-1\]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.2664477750268591e143 or 1.41635508708514771e142 < y.re

   1. Initial program 42.6

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

   2. Taylor expanded around inf 14.5

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

   3. Simplified 14.5

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

   if -1.2664477750268591e143 < y.re < -4.7218745403061054e135 or -2.2148262768226444e22 < y.re < -3084.18200427928195 or -9.8394525987137508e-91 < y.re < 1.9690354844540234e-126

   1. Initial program 22.4

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

   2. Taylor expanded around 0 12.9

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

   3. Simplified 12.9

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

   4. Simplified 12.9

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

   if -4.7218745403061054e135 < y.re < -2.2148262768226444e22

   1. Initial program 22.1

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

   3. Applied add-sqr-sqrt_binary64_1092 22.1

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

   4. Applied *-un-lft-identity_binary64_1070 22.1

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

   5. Applied times-frac_binary64_1076 22.1

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

   6. Simplified 22.1

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

   7. Simplified 22.1

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

   8. Simplified 22.1

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

   if -3084.18200427928195 < y.re < -9.8394525987137508e-91 or 1.9690354844540234e-126 < y.re < 1.41635508708514771e142

   1. Initial program 16.3

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

   3. Applied add-sqr-sqrt_binary64_1092 16.3

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

   4. Applied associate-/r*_binary64_1014 16.2

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

   5. Simplified 16.2

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

   6. Simplified 16.2

      \[\leadsto \color{blue}{\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{{y.re}^{2} + {y.im}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}\]
3. Recombined 4 regimes into one program.

4. Final simplification 15.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.266447775026859 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.7218745403061054 \cdot 10^{+135}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq -2.2148262768226444 \cdot 10^{+22}:\\ \;\;\;\;\frac{1}{\sqrt{{y.im}^{2} + {y.re}^{2}}} \cdot \frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}\\ \mathbf{elif}\;y.re \leq -3084.182004279282:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq -9.83945259871375 \cdot 10^{-91}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 1.9690354844540234 \cdot 10^{-126}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.4163550870851477 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re + x.im \cdot y.im}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))))