_divideComplex, real part

Percentage Accurate: 61.7% → 83.0%

Time: 14.8s

Precision: binary64

Cost: 7568

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

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
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ y.im (/ y.re x.im))))
   (if (<= y.re -8e+86)
     (/ (- (- x.re) t_1) (hypot y.re y.im))
     (if (<= y.re -9.5e-129)
       t_0
       (if (<= y.re 4e-94)
         (+ (/ x.im y.im) (* (/ (* x.re y.re) y.im) (/ 1.0 y.im)))
         (if (<= y.re 6.2e+54) t_0 (/ (+ x.re t_1) (hypot y.re y.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 t_0 = ((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 t_1 = y_46_im / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -8e+86) {
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9.5e-129) {
		tmp = t_0;
	} else if (y_46_re <= 4e-94) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) * (1.0 / y_46_im));
	} else if (y_46_re <= 6.2e+54) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + t_1) / hypot(y_46_re, y_46_im);
	}
	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 = ((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 t_1 = y_46_im / (y_46_re / x_46_im);
	double tmp;
	if (y_46_re <= -8e+86) {
		tmp = (-x_46_re - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -9.5e-129) {
		tmp = t_0;
	} else if (y_46_re <= 4e-94) {
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) * (1.0 / y_46_im));
	} else if (y_46_re <= 6.2e+54) {
		tmp = t_0;
	} else {
		tmp = (x_46_re + t_1) / Math.hypot(y_46_re, y_46_im);
	}
	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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = y_46_im / (y_46_re / x_46_im)
	tmp = 0
	if y_46_re <= -8e+86:
		tmp = (-x_46_re - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -9.5e-129:
		tmp = t_0
	elif y_46_re <= 4e-94:
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) * (1.0 / y_46_im))
	elif y_46_re <= 6.2e+54:
		tmp = t_0
	else:
		tmp = (x_46_re + t_1) / math.hypot(y_46_re, y_46_im)
	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(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)))
	t_1 = Float64(y_46_im / Float64(y_46_re / x_46_im))
	tmp = 0.0
	if (y_46_re <= -8e+86)
		tmp = Float64(Float64(Float64(-x_46_re) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -9.5e-129)
		tmp = t_0;
	elseif (y_46_re <= 4e-94)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(x_46_re * y_46_re) / y_46_im) * Float64(1.0 / y_46_im)));
	elseif (y_46_re <= 6.2e+54)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re + t_1) / hypot(y_46_re, y_46_im));
	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 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = y_46_im / (y_46_re / x_46_im);
	tmp = 0.0;
	if (y_46_re <= -8e+86)
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -9.5e-129)
		tmp = t_0;
	elseif (y_46_re <= 4e-94)
		tmp = (x_46_im / y_46_im) + (((x_46_re * y_46_re) / y_46_im) * (1.0 / y_46_im));
	elseif (y_46_re <= 6.2e+54)
		tmp = t_0;
	else
		tmp = (x_46_re + t_1) / hypot(y_46_re, y_46_im);
	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[(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]}, Block[{t$95$1 = N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -8e+86], N[(N[((-x$46$re) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9.5e-129], t$95$0, If[LessEqual[y$46$re, 4e-94], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+54], t$95$0, N[(N[(x$46$re + t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -8.0000000000000001e86

   1. Initial program 29.6%

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

   2. Applied egg-rr 51.5%

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

      [Start] 29.6%	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 29.6%	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 29.6%	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 29.7%	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 29.7%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 29.7%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 51.5%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 51.6%

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

      [Start] 51.5%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 51.6%	\[ \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      *-un-lft-identity [<=] 51.6%	\[ \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   4. Taylor expanded in y.re around -inf 83.9%

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

   5. Simplified 89.8%

      \[\leadsto \frac{\color{blue}{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Step-by-step derivation

      [Start] 83.9%	\[ \frac{-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      +-commutative [=>] 83.9%	\[ \frac{\color{blue}{-1 \cdot x.re + -1 \cdot \frac{y.im \cdot x.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      mul-1-neg [=>] 83.9%	\[ \frac{-1 \cdot x.re + \color{blue}{\left(-\frac{y.im \cdot x.im}{y.re}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      unsub-neg [=>] 83.9%	\[ \frac{\color{blue}{-1 \cdot x.re - \frac{y.im \cdot x.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      neg-mul-1 [<=] 83.9%	\[ \frac{\color{blue}{\left(-x.re\right)} - \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-/l* [=>] 89.8%	\[ \frac{\left(-x.re\right) - \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -8.0000000000000001e86 < y.re < -9.5000000000000006e-129 or 3.9999999999999998e-94 < y.re < 6.1999999999999999e54

   1. Initial program 83.7%

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

   if -9.5000000000000006e-129 < y.re < 3.9999999999999998e-94

   1. Initial program 63.7%

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

   2. Taylor expanded in y.re around 0 77.7%

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

   3. Simplified 77.7%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}} \]
      Step-by-step derivation

      [Start] 77.7%	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 77.7%	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 77.7%	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 77.7%	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]

   4. Applied egg-rr 87.4%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}} \]
      Step-by-step derivation

      [Start] 77.7%	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im} \]
      associate-/r* [=>] 87.4%	\[ \frac{x.im}{y.im} + \color{blue}{\frac{\frac{y.re \cdot x.re}{y.im}}{y.im}} \]
      div-inv [=>] 87.4%	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re \cdot x.re}{y.im} \cdot \frac{1}{y.im}} \]
      *-commutative [=>] 87.4%	\[ \frac{x.im}{y.im} + \frac{\color{blue}{x.re \cdot y.re}}{y.im} \cdot \frac{1}{y.im} \]

   if 6.1999999999999999e54 < y.re

   1. Initial program 44.0%

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

   2. Applied egg-rr 57.5%

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

      [Start] 44.0%	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 44.0%	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 44.0%	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 44.0%	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 44.0%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 44.0%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 57.5%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 57.6%

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

      [Start] 57.5%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 57.6%	\[ \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      *-un-lft-identity [<=] 57.6%	\[ \frac{\color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   4. Taylor expanded in y.re around inf 76.7%

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

   5. Simplified 85.0%

      \[\leadsto \frac{\color{blue}{x.re + \frac{y.im}{\frac{y.re}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Step-by-step derivation

      [Start] 76.7%	\[ \frac{x.re + \frac{y.im \cdot x.im}{y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-/l* [=>] 85.0%	\[ \frac{x.re + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

3. Recombined 4 regimes into one program.

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	83.0%
Cost	7568

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+86}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -9.5 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Accuracy	85.2%
Cost	20932

\[\begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 3
Accuracy	82.8%
Cost	7568

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{+59}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4
Accuracy	82.4%
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{-128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 3.9 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	67.8%
Cost	1100

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-175}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-94}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Accuracy	76.7%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{-35} \lor \neg \left(y.re \leq 3.6 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im} \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 7
Accuracy	76.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-31} \lor \neg \left(y.re \leq 4 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 8
Accuracy	76.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.85 \cdot 10^{-35} \lor \neg \left(y.re \leq 3.4 \cdot 10^{-65}\right):\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 9
Accuracy	76.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.88 \cdot 10^{-34} \lor \neg \left(y.re \leq 3.5 \cdot 10^{-66}\right):\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \end{array} \]

Alternative 10
Accuracy	62.0%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.6 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 6.4 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Accuracy	42.2%
Cost	192

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

Reproduce

herbie shell --seed 2023263 
(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))))