_divideComplex, real part

Average Accuracy: 58.1% → 83.4%

Time: 15.8s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

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
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1 (* t_0 (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))))
   (if (<= y.re -4.6e+73)
     (+ (/ x.re y.re) (/ (* y.im (/ x.im y.re)) y.re))
     (if (<= y.re -8.6e-160)
       t_1
       (if (<= y.re 4.1e-262)
         (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
         (if (<= y.re 5.8e+97)
           t_1
           (* t_0 (+ x.re (* x.im (/ y.im 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 t_0 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = t_0 * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_re <= -4.6e+73) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (x_46_im / y_46_re)) / y_46_re);
	} else if (y_46_re <= -8.6e-160) {
		tmp = t_1;
	} else if (y_46_re <= 4.1e-262) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_re <= 5.8e+97) {
		tmp = t_1;
	} else {
		tmp = t_0 * (x_46_re + (x_46_im * (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
end

↓

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(t_0 * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_re <= -4.6e+73)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(x_46_im / y_46_re)) / y_46_re));
	elseif (y_46_re <= -8.6e-160)
		tmp = t_1;
	elseif (y_46_re <= 4.1e-262)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(y_46_im * Float64(y_46_im / y_46_re))));
	elseif (y_46_re <= 5.8e+97)
		tmp = t_1;
	else
		tmp = Float64(t_0 * Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * N[(N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+73], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -8.6e-160], t$95$1, If[LessEqual[y$46$re, 4.1e-262], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.8e+97], t$95$1, N[(t$95$0 * N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.6e73

   1. Initial program 40.3%

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

   2. Taylor expanded in y.re around inf 72.9%

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

   3. Simplified 73.0%

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

      [Start] 72.9	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      associate-/l* [=>] 74.9	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{\frac{{y.re}^{2}}{x.im}}} \]
      associate-/r/ [=>] 73.0	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{{y.re}^{2}} \cdot x.im} \]
      unpow2 [=>] 73.0	\[ \frac{x.re}{y.re} + \frac{y.im}{\color{blue}{y.re \cdot y.re}} \cdot x.im \]

   4. Applied egg-rr 81.6%

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

      [Start] 73.0	\[ \frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im \]
      associate-/r* [=>] 75.8	\[ \frac{x.re}{y.re} + \color{blue}{\frac{\frac{y.im}{y.re}}{y.re}} \cdot x.im \]
      associate-*l/ [=>] 81.6	\[ \frac{x.re}{y.re} + \color{blue}{\frac{\frac{y.im}{y.re} \cdot x.im}{y.re}} \]

   5. Taylor expanded in y.im around 0 76.8%

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

   6. Simplified 82.0%

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

      [Start] 76.8	\[ \frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re} \]
      associate-*r/ [<=] 82.0	\[ \frac{x.re}{y.re} + \frac{\color{blue}{y.im \cdot \frac{x.im}{y.re}}}{y.re} \]

   if -4.6e73 < y.re < -8.60000000000000028e-160 or 4.10000000000000026e-262 < y.re < 5.79999999999999974e97

   1. Initial program 72.3%

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

   2. Applied egg-rr 81.2%

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

      [Start] 72.3	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 72.3	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 72.3	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 72.3	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 72.3	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 72.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -8.60000000000000028e-160 < y.re < 4.10000000000000026e-262

   1. Initial program 62.2%

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

   2. Taylor expanded in y.re around 0 85.5%

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

   3. Simplified 89.8%

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

      [Start] 85.5	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 85.5	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 85.5	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 85.5	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 89.8	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

   4. Applied egg-rr 92.4%

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

      [Start] 89.8	\[ \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im} \]
      clear-num [=>] 89.6	\[ \frac{x.im}{y.im} + \color{blue}{\frac{1}{\frac{y.im}{y.re}}} \cdot \frac{x.re}{y.im} \]
      frac-times [=>] 92.4	\[ \frac{x.im}{y.im} + \color{blue}{\frac{1 \cdot x.re}{\frac{y.im}{y.re} \cdot y.im}} \]
      *-un-lft-identity [<=] 92.4	\[ \frac{x.im}{y.im} + \frac{\color{blue}{x.re}}{\frac{y.im}{y.re} \cdot y.im} \]

   if 5.79999999999999974e97 < y.re

   1. Initial program 39.5%

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

   2. Applied egg-rr 59.2%

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

      [Start] 39.5	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 39.5	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 39.5	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 39.5	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 39.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 39.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 59.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.re around inf 77.9%

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

   4. Simplified 83.6%

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

      [Start] 77.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right) \]
      associate-/l* [=>] 84.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}\right) \]
      associate-/r/ [=>] 83.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \color{blue}{\frac{y.im}{y.re} \cdot x.im}\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -8.6 \cdot 10^{-160}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{-262}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+97}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	80.5%
Cost	7828

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+78}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{x.re \cdot \frac{y.re}{y.im}}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \end{array} \]

Alternative 2
Accuracy	80.5%
Cost	7636

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+82}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{x.re \cdot \frac{y.re}{y.im}}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.65 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]

Alternative 3
Accuracy	80.6%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{x.re \cdot \frac{y.re}{y.im}}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	69.0%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{+53} \lor \neg \left(y.re \leq -29000000\right) \land y.re \leq 1.15 \cdot 10^{+103}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 5
Accuracy	75.9%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.8 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{+60}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	73.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.9 \cdot 10^{-51} \lor \neg \left(y.im \leq 8.2 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \end{array} \]

Alternative 7
Accuracy	74.8%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-43} \lor \neg \left(y.im \leq 3.9 \cdot 10^{+59}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \]

Alternative 8
Accuracy	76.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.25 \cdot 10^{-43} \lor \neg \left(y.im \leq 4.9 \cdot 10^{+58}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	63.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.012:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	41.8%
Cost	192

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

Reproduce

herbie shell --seed 2023151 
(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))))