_divideComplex, real part

Average Error: 26.1 → 16.9

Time: 4.1min

Precision: binary64

Cost: 28290

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.139893937144628 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -5711.206474923963:\\ \;\;\;\;\frac{y.im \cdot \left(x.im \cdot \frac{\frac{-y.re}{{y.im}^{2} + {y.re}^{2}}}{\frac{-1}{x.re}}\right)}{y.re \cdot x.re} - \frac{\frac{y.re}{{y.im}^{2} + {y.re}^{2}}}{\frac{-1}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x.im + \frac{y.re \cdot x.re}{y.im}\right) - 0.5 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im + 0.5 \cdot \frac{{y.re}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

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 -6.139893937144628e+141)
   (+
    (* x.re (/ 1.0 y.re))
    (/ (* y.im (* x.im (* x.re (/ 1.0 y.re)))) (* y.re x.re)))
   (if (<= y.re -5711.206474923963)
     (-
      (/
       (*
        y.im
        (*
         x.im
         (/ (/ (- y.re) (+ (pow y.im 2.0) (pow y.re 2.0))) (/ -1.0 x.re))))
       (* y.re x.re))
      (/ (/ y.re (+ (pow y.im 2.0) (pow y.re 2.0))) (/ -1.0 x.re)))
     (if (<= y.re 1.0936641726299874e-89)
       (/
        (-
         (+ x.im (/ (* y.re x.re) y.im))
         (* 0.5 (/ (* x.im (pow y.re 2.0)) (pow y.im 2.0))))
        (+ y.im (* 0.5 (/ (pow y.re 2.0) y.im))))
       (+
        (/ (* y.im x.im) (+ (pow y.im 2.0) (pow y.re 2.0)))
        (/ 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 <= -6.139893937144628e+141) {
		tmp = (x_46_re * (1.0 / y_46_re)) + ((y_46_im * (x_46_im * (x_46_re * (1.0 / y_46_re)))) / (y_46_re * x_46_re));
	} else if (y_46_re <= -5711.206474923963) {
		tmp = ((y_46_im * (x_46_im * ((-y_46_re / (pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / (-1.0 / x_46_re)))) / (y_46_re * x_46_re)) - ((y_46_re / (pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) / (-1.0 / x_46_re));
	} else if (y_46_re <= 1.0936641726299874e-89) {
		tmp = ((x_46_im + ((y_46_re * x_46_re) / y_46_im)) - (0.5 * ((x_46_im * pow(y_46_re, 2.0)) / pow(y_46_im, 2.0)))) / (y_46_im + (0.5 * (pow(y_46_re, 2.0) / y_46_im)));
	} else {
		tmp = ((y_46_im * x_46_im) / (pow(y_46_im, 2.0) + pow(y_46_re, 2.0))) + (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	16.4
Cost	27714

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.2850453642498544 \cdot 10^{+88}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -2.9987652470589066 \cdot 10^{-32}:\\ \;\;\;\;\frac{\left(y.re \cdot x.re + y.im \cdot x.im\right) \cdot \frac{1}{\sqrt{{y.im}^{2} + {y.re}^{2}}}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x.im + \frac{y.re \cdot x.re}{y.im}\right) - 0.5 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im + 0.5 \cdot \frac{{y.re}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

Alternative 2
Error	17.0
Cost	21827

\[\begin{array}{l} \mathbf{if}\;y.re \leq -7.846666014758605 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -5103.719554050937:\\ \;\;\;\;\frac{1}{\frac{\sqrt{{y.im}^{2} + {y.re}^{2}}}{\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x.im + \frac{y.re \cdot x.re}{y.im}\right) - 0.5 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im + 0.5 \cdot \frac{{y.re}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

Alternative 3
Error	17.2
Cost	20866

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.7445877215510606 \cdot 10^{+138}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -2473.655553969881:\\ \;\;\;\;\frac{1}{\frac{\sqrt{{y.im}^{2} + {y.re}^{2}}}{\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x.im + \frac{y.re \cdot x.re}{y.im}\right) - 0.5 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

Alternative 4
Error	17.1
Cost	14979

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.090490597274888 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -3992.372856152444:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x.im + \frac{y.re \cdot x.re}{y.im}\right) - 0.5 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

Alternative 5
Error	17.4
Cost	14595

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.0233979365331517 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -4951.847823832681:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{y.re \cdot x.re}{{y.im}^{2}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

Alternative 6
Error	18.0
Cost	8066

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.090490597274888 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -1957.084316548428:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im \cdot x.im}{y.re}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{y.re \cdot x.re}{{y.im}^{2}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]

Alternative 7
Error	17.9
Cost	8003

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.1551889227456226 \cdot 10^{+88}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -1957.084316548428:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{y.re \cdot x.re}{{y.im}^{2}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array}\]

Alternative 8
Error	18.0
Cost	8003

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.0996812048612831 \cdot 10^{+88}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -1957.084316548428:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{y.re \cdot x.re}{{y.im}^{2}} + \frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \end{array}\]

Alternative 9
Error	14.7
Cost	8003

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.977462035739202 \cdot 10^{+38}:\\ \;\;\;\;x.im \cdot \frac{1}{y.im} + \frac{y.re \cdot \left(x.re \cdot \left(x.im \cdot \frac{1}{y.im}\right)\right)}{y.im \cdot x.im}\\ \mathbf{elif}\;y.im \leq -1.865044624534711 \cdot 10^{-126}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8.609066475272341 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.im \leq 8.680010860334413 \cdot 10^{+95}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{1}{y.im} + \frac{y.re \cdot \left(x.re \cdot \left(x.im \cdot \frac{1}{y.im}\right)\right)}{y.im \cdot x.im}\\ \end{array}\]

Alternative 10
Error	16.0
Cost	2500

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.977462035739202 \cdot 10^{+38}:\\ \;\;\;\;x.im \cdot \frac{1}{y.im} + \frac{y.re \cdot \left(x.re \cdot \left(x.im \cdot \frac{1}{y.im}\right)\right)}{y.im \cdot x.im}\\ \mathbf{elif}\;y.im \leq -1.1335800448969252 \cdot 10^{-133}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.516852061872786 \cdot 10^{-155}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.im \leq 3.8739115826785715 \cdot 10^{+95}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{1}{y.im} + \frac{y.re \cdot \left(x.re \cdot \left(x.im \cdot \frac{1}{y.im}\right)\right)}{y.im \cdot x.im}\\ \end{array}\]

Alternative 11
Error	16.0
Cost	2244

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.564022337473754 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.3572811904925184 \cdot 10^{-132}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.832868878772282 \cdot 10^{-156}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.im \leq 1.295224182584382 \cdot 10^{+96}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

Alternative 12
Error	15.9
Cost	2244

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.9002421296373794 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.3485016802441929 \cdot 10^{-133}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.325647280634228 \cdot 10^{-157}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 8.941350663866893 \cdot 10^{+95}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

Alternative 13
Error	22.4
Cost	1346

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2195710950727558 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -5.311819209579214 \cdot 10^{-62}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 40678255396950.66:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

Alternative 14
Error	23.3
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -33499306029128.684 \lor \neg \left(y.re \leq 9.070872177309683 \cdot 10^{-90}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array}\]

Alternative 15
Error	35.7
Cost	513

\[\begin{array}{l} \mathbf{if}\;y.re \leq 2.3553215216684362 \cdot 10^{+216}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Alternative 16
Error	51.9
Cost	64

\[0\]

Alternative 17
Error	61.7
Cost	64

\[1\]

Derivation

1. Split input into 4 regimes

2. if y.re < -6.1398939371446278e141

   1. Initial program 45.0

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

   3. Applied add-exp-log_binary64_798 45.2

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

   4. Applied add-exp-log_binary64_798 54.6

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

   5. Applied div-exp_binary64_811 54.6

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

   6. Simplified 45.9

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

   7. Taylor expanded around -inf 40.7

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

   8. Simplified 14.4

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

   9. Simplified 14.4

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

   if -6.1398939371446278e141 < y.re < -5711.20647492396256

   1. Initial program 19.6

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

   3. Applied add-exp-log_binary64_798 22.8

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

   4. Applied add-exp-log_binary64_798 44.0

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

   5. Applied div-exp_binary64_811 44.1

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

   6. Simplified 40.7

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

   7. Taylor expanded around -inf 44.2

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

   8. Simplified 19.7

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

   9. Simplified 19.7

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

   if -5711.20647492396256 < y.re < 1.09366417262998742e-89

   1. Initial program 20.1

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

   3. Applied add-sqr-sqrt_binary64_782 20.1

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

   4. Applied associate-/r*_binary64_704 20.0

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

   5. Simplified 20.0

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

   6. Taylor expanded around 0 40.4

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

   7. Simplified 40.4

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

   8. Taylor expanded around 0 14.1

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

   9. Simplified 14.1

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

   if 1.09366417262998742e-89 < y.re

   1. Initial program 28.3

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

   3. Applied add-exp-log_binary64_798 30.2

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

   4. Applied add-exp-log_binary64_798 47.0

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

   5. Applied div-exp_binary64_811 47.0

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

   6. Simplified 41.1

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

   7. Taylor expanded around 0 46.3

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

   8. Simplified 38.1

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

   9. Taylor expanded around 0 20.4

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

   10. Simplified 20.4

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

4. Final simplification 16.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.139893937144628 \cdot 10^{+141}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re} + \frac{y.im \cdot \left(x.im \cdot \left(x.re \cdot \frac{1}{y.re}\right)\right)}{y.re \cdot x.re}\\ \mathbf{elif}\;y.re \leq -5711.206474923963:\\ \;\;\;\;\frac{y.im \cdot \left(x.im \cdot \frac{\frac{-y.re}{{y.im}^{2} + {y.re}^{2}}}{\frac{-1}{x.re}}\right)}{y.re \cdot x.re} - \frac{\frac{y.re}{{y.im}^{2} + {y.re}^{2}}}{\frac{-1}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.0936641726299874 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(x.im + \frac{y.re \cdot x.re}{y.im}\right) - 0.5 \cdot \frac{x.im \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im + 0.5 \cdot \frac{{y.re}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.im \cdot x.im}{{y.im}^{2} + {y.re}^{2}} + \frac{x.re}{y.re}\\ \end{array}\]

Reproduce

herbie shell --seed 2021040 
(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))))