_divideComplex, real part

Average Accuracy: 60.2% → 93.7%

Time: 22.6s

Precision: binary64

Cost: 20616

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\\ \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{+94}:\\ \;\;\;\;t_0 \cdot \left(t_1 + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.re + t_1\right)\\ \end{array} \]

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 (/ x.im (/ (hypot y.re y.im) y.im))))
   (if (<= y.re -1.9e+125)
     (* (+ x.re (/ y.im (/ y.re x.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re 1e+94)
       (* t_0 (+ t_1 (/ (* y.re x.re) (hypot y.re y.im))))
       (* t_0 (+ x.re t_1))))))

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 = x_46_im / (hypot(y_46_re, y_46_im) / y_46_im);
	double tmp;
	if (y_46_re <= -1.9e+125) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 1e+94) {
		tmp = t_0 * (t_1 + ((y_46_re * x_46_re) / hypot(y_46_re, y_46_im)));
	} else {
		tmp = t_0 * (x_46_re + t_1);
	}
	return tmp;
}

Java

public static 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));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = x_46_im / (Math.hypot(y_46_re, y_46_im) / y_46_im);
	double tmp;
	if (y_46_re <= -1.9e+125) {
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= 1e+94) {
		tmp = t_0 * (t_1 + ((y_46_re * x_46_re) / Math.hypot(y_46_re, y_46_im)));
	} else {
		tmp = t_0 * (x_46_re + t_1);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = x_46_im / (math.hypot(y_46_re, y_46_im) / y_46_im)
	tmp = 0
	if y_46_re <= -1.9e+125:
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= 1e+94:
		tmp = t_0 * (t_1 + ((y_46_re * x_46_re) / math.hypot(y_46_re, y_46_im)))
	else:
		tmp = t_0 * (x_46_re + t_1)
	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(x_46_im / Float64(hypot(y_46_re, y_46_im) / y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.9e+125)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= 1e+94)
		tmp = Float64(t_0 * Float64(t_1 + Float64(Float64(y_46_re * x_46_re) / hypot(y_46_re, y_46_im))));
	else
		tmp = Float64(t_0 * Float64(x_46_re + t_1));
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = x_46_im / (hypot(y_46_re, y_46_im) / y_46_im);
	tmp = 0.0;
	if (y_46_re <= -1.9e+125)
		tmp = (x_46_re + (y_46_im / (y_46_re / x_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= 1e+94)
		tmp = t_0 * (t_1 + ((y_46_re * x_46_re) / hypot(y_46_re, y_46_im)));
	else
		tmp = t_0 * (x_46_re + t_1);
	end
	tmp_2 = 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[(x$46$im / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.9e+125], N[(N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1e+94], N[(t$95$0 * N[(t$95$1 + N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(x$46$re + t$95$1), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.90000000000000001e125

   1. Initial program 35.0%

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

   2. Applied egg-rr 55.9%

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

      [Start] 35.0	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 35.0	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 35.0	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 35.0	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 35.0	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 35.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 55.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.re around -inf 81.6%

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

   4. Simplified 88.4%

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

      [Start] 81.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re\right) \]
      distribute-lft-out [=>] 81.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot \left(\frac{y.im \cdot x.im}{y.re} + x.re\right)\right)} \]
      +-commutative [<=] 81.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \color{blue}{\left(x.re + \frac{y.im \cdot x.im}{y.re}\right)}\right) \]
      associate-/l* [=>] 88.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \left(x.re + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}\right)\right) \]

   if -1.90000000000000001e125 < y.re < 1e94

   1. Initial program 71.5%

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

   2. Applied egg-rr 81.4%

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

      [Start] 71.5	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 71.5	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 71.5	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 71.5	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 71.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 71.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 81.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 81.4%

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

      [Start] 81.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      div-inv [=>] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      *-commutative [<=] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
      fma-udef [=>] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
      distribute-lft-in [=>] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
      associate-*l/ [=>] 81.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      *-un-lft-identity [<=] 81.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      associate-*l/ [=>] 81.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
      *-un-lft-identity [<=] 81.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   4. Simplified 95.2%

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

      [Start] 81.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      +-commutative [=>] 81.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      associate-/l* [=>] 95.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      *-commutative [=>] 95.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   if 1e94 < y.re

   1. Initial program 40.8%

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

   2. Applied egg-rr 61.1%

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

      [Start] 40.8	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 40.8	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 40.8	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 40.8	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 40.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 40.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 61.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 61.1%

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

      [Start] 61.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      div-inv [=>] 61.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      *-commutative [<=] 61.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
      fma-udef [=>] 61.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
      distribute-lft-in [=>] 61.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
      associate-*l/ [=>] 61.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      *-un-lft-identity [<=] 61.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
      associate-*l/ [=>] 61.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
      *-un-lft-identity [<=] 61.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   4. Simplified 68.8%

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

      [Start] 61.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      +-commutative [=>] 61.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
      associate-/l* [=>] 68.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
      *-commutative [=>] 68.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

   5. Taylor expanded in y.re around inf 92.4%

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

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.9 \cdot 10^{+125}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{+94}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	85.3%
Cost	20168

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+121}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 2
Accuracy	82.9%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+106}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 3
Accuracy	80.3%
Cost	13904

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+106}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+113}:\\ \;\;\;\;\frac{y.im}{\frac{t_0}{x.im}} + \frac{y.re \cdot x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \end{array} \]

Alternative 4
Accuracy	81.2%
Cost	7300

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+106}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 1.52 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8.9 \cdot 10^{+98}:\\ \;\;\;\;\frac{y.im}{\frac{t_0}{x.im}} + \frac{y.re \cdot x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 5
Accuracy	81.0%
Cost	2000

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.7 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-63}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+98}:\\ \;\;\;\;\frac{y.im}{\frac{t_0}{x.im}} + \frac{y.re \cdot x.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 6
Accuracy	78.4%
Cost	1884

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -106000000000:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-147}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 40000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.1 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 7
Accuracy	71.7%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{-52} \lor \neg \left(y.re \leq 2.4 \cdot 10^{+33} \lor \neg \left(y.re \leq 4.3 \cdot 10^{+61}\right) \land y.re \leq 4.4 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 8
Accuracy	74.7%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{-12} \lor \neg \left(y.re \leq 4.2 \cdot 10^{+32}\right) \land \left(y.re \leq 2.2 \cdot 10^{+61} \lor \neg \left(y.re \leq 1.45 \cdot 10^{+94}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{\frac{y.re}{\frac{x.im}{y.re}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 9
Accuracy	68.6%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+34} \lor \neg \left(y.re \leq 2.05 \cdot 10^{+60}\right) \land y.re \leq 3.1 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	69.0%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.3 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+34} \lor \neg \left(y.re \leq 5.2 \cdot 10^{+60}\right) \land y.re \leq 5 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Accuracy	74.8%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{\frac{y.re}{\frac{x.im}{y.re}}}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Accuracy	75.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.re \leq -7.4 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+61}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{\frac{y.re}{\frac{x.im}{y.re}}}\\ \mathbf{elif}\;y.re \leq 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 13
Accuracy	41.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.18 \cdot 10^{-167}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 14
Accuracy	64.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 15
Accuracy	40.7%
Cost	192

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

Reproduce

herbie shell --seed 2023133 
(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))))