_divideComplex, real part

Average Error: 26.2 → 9.5

Time: 13.6s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{1}{y.re} \cdot \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \end{array} \]

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))
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))))
   (if (<= y.im -5e+114)
     (* (+ x.im (* y.re (/ x.re y.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -4.8e-115)
       t_0
       (if (<= y.im 1.25e-135)
         (fma x.im (* (/ 1.0 y.re) (/ y.im y.re)) (/ x.re y.re))
         (if (<= y.im 8e+76)
           t_0
           (+ (/ x.im y.im) (* (/ y.re (/ y.im x.re)) (/ 1.0 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 = (1.0 / hypot(y_46_re, y_46_im)) * (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_im <= -5e+114) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -4.8e-115) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e-135) {
		tmp = fma(x_46_im, ((1.0 / y_46_re) * (y_46_im / y_46_re)), (x_46_re / y_46_re));
	} else if (y_46_im <= 8e+76) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / (y_46_im / x_46_re)) * (1.0 / 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(1.0 / hypot(y_46_re, y_46_im)) * 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_im <= -5e+114)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -4.8e-115)
		tmp = t_0;
	elseif (y_46_im <= 1.25e-135)
		tmp = fma(x_46_im, Float64(Float64(1.0 / y_46_re) * Float64(y_46_im / y_46_re)), Float64(x_46_re / y_46_re));
	elseif (y_46_im <= 8e+76)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / Float64(y_46_im / x_46_re)) * Float64(1.0 / y_46_im)));
	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[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * 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$46$im, -5e+114], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.8e-115], t$95$0, If[LessEqual[y$46$im, 1.25e-135], N[(x$46$im * N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8e+76], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -5.0000000000000001e114

   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 26.7

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

   3. Taylor expanded in y.im around -inf 11.6

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

   4. Simplified 7.8

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(-x.im\right) - \frac{x.re}{y.im} \cdot y.re\right)} \]
      Proof
      (-.f64 (neg.f64 x.im) (*.f64 (/.f64 x.re y.im) y.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x.im)) (*.f64 (/.f64 x.re y.im) y.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 x.im) (Rewrite<= associate-/r/_binary64 (/.f64 x.re (/.f64 y.im y.re)))): 24 points increase in error, 24 points decrease in error
      (-.f64 (*.f64 -1 x.im) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.re y.re) y.im))): 34 points increase in error, 27 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 x.im) (neg.f64 (/.f64 (*.f64 x.re y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 x.im) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 x.re y.re) y.im)) (*.f64 -1 x.im))): 0 points increase in error, 0 points decrease in error

   if -5.0000000000000001e114 < y.im < -4.80000000000000042e-115 or 1.25000000000000005e-135 < y.im < 8.0000000000000004e76

   1. Initial program 16.0

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

   2. Applied egg-rr 11.3

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

   if -4.80000000000000042e-115 < y.im < 1.25000000000000005e-135

   1. Initial program 23.6

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

   2. Taylor expanded in y.re around inf 10.7

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

   3. Simplified 8.9

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 42 points increase in error, 11 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 7.4

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

   if 8.0000000000000004e76 < y.im

   1. Initial program 37.1

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

   2. Taylor expanded in y.re around 0 17.2

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

   3. Simplified 17.2

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 10.2

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 9.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5 \cdot 10^{+114}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-115}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{1}{y.re} \cdot \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	11.2
Cost	7500

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.6 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-135}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{1}{y.re} \cdot \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 2
Error	11.2
Cost	7500

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+116}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.4 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.95 \cdot 10^{-134}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{1}{y.re} \cdot \frac{y.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 3
Error	11.3
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.22 \cdot 10^{-136}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.35 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 4
Error	18.9
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.65 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.08 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 8.4 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.6 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -8 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.3 \cdot 10^{+38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.7 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ t_1 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.48 \cdot 10^{+41}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -8.8 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.08 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -8 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-17}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	15.2
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.42 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1.08 \cdot 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.08 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	23.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Error	37.6
Cost	192

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

Reproduce

herbie shell --seed 2022329 
(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))))