_divideComplex, real part

Average Error: 26.0 → 10.2

Time: 14.5s

Precision: binary64

Cost: 20824

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := x.re + \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -9.8 \cdot 10^{+122}:\\ \;\;\;\;t_2 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-193}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.72 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot t_2\\ \end{array} \]

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))))
        (t_2 (+ x.re (/ y.im (/ y.re x.im)))))
   (if (<= y.re -9.8e+122)
     (* t_2 (/ -1.0 (hypot y.re y.im)))
     (if (<= y.re -5.2e-193)
       t_1
       (if (<= y.re 1.3e-218)
         (+ (/ x.im y.im) (/ x.re (* y.im (/ y.im y.re))))
         (if (<= y.re 1.02e-29)
           t_1
           (if (<= y.re 3e-11)
             (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))
             (if (<= y.re 1.72e+36) t_1 (* t_0 t_2)))))))))

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 t_2 = x_46_re + (y_46_im / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -9.8e+122) {
		tmp = t_2 * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -5.2e-193) {
		tmp = t_1;
	} else if (y_46_re <= 1.3e-218) {
		tmp = (x_46_im / y_46_im) + (x_46_re / (y_46_im * (y_46_im / y_46_re)));
	} else if (y_46_re <= 1.02e-29) {
		tmp = t_1;
	} else if (y_46_re <= 3e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_re <= 1.72e+36) {
		tmp = t_1;
	} else {
		tmp = t_0 * t_2;
	}
	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)))
	t_2 = Float64(x_46_re + Float64(y_46_im / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -9.8e+122)
		tmp = Float64(t_2 * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -5.2e-193)
		tmp = t_1;
	elseif (y_46_re <= 1.3e-218)
		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 <= 1.02e-29)
		tmp = t_1;
	elseif (y_46_re <= 3e-11)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	elseif (y_46_re <= 1.72e+36)
		tmp = t_1;
	else
		tmp = Float64(t_0 * t_2);
	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]}, Block[{t$95$2 = N[(x$46$re + N[(y$46$im / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.8e+122], N[(t$95$2 * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -5.2e-193], t$95$1, If[LessEqual[y$46$re, 1.3e-218], 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, 1.02e-29], t$95$1, If[LessEqual[y$46$re, 3e-11], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.72e+36], t$95$1, N[(t$95$0 * t$95$2), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -9.7999999999999995e122

   1. Initial program 42.1

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

   2. Applied egg-rr 27.7

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

   3. Taylor expanded in y.re around -inf 11.9

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

   4. Simplified 7.6

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

      [Start] 11.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \frac{y.im \cdot x.im}{y.re} + -1 \cdot x.re\right) \]
      distribute-lft-out [=>] 11.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot \left(\frac{y.im \cdot x.im}{y.re} + x.re\right)\right)} \]
      +-commutative [<=] 11.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \color{blue}{\left(x.re + \frac{y.im \cdot x.im}{y.re}\right)}\right) \]
      associate-/l* [=>] 7.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \left(x.re + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}\right)\right) \]

   if -9.7999999999999995e122 < y.re < -5.20000000000000015e-193 or 1.29999999999999992e-218 < y.re < 1.01999999999999994e-29 or 3e-11 < y.re < 1.7199999999999999e36

   1. Initial program 16.9

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

   2. Applied egg-rr 10.9

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

   if -5.20000000000000015e-193 < y.re < 1.29999999999999992e-218

   1. Initial program 24.8

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

   2. Taylor expanded in y.re around 0 10.3

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

   3. Simplified 11.4

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

      [Start] 10.3	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 10.3	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      associate-/l* [=>] 11.4	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]
      unpow2 [=>] 11.4	\[ \frac{x.im}{y.im} + \frac{x.re}{\frac{\color{blue}{y.im \cdot y.im}}{y.re}} \]

   4. Applied egg-rr 5.0

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

   if 1.01999999999999994e-29 < y.re < 3e-11

   1. Initial program 9.8

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

   2. Applied egg-rr 28.4

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

   3. Taylor expanded in y.re around 0 32.0

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

   4. Simplified 30.7

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

      [Start] 32.0	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 32.0	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      associate-/l* [=>] 32.6	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]
      associate-/r/ [=>] 32.1	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]
      unpow2 [=>] 32.1	\[ \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]
      associate-/r* [=>] 30.7	\[ \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re}{y.im}}{y.im}} \cdot y.re \]

   5. Applied egg-rr 30.8

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

   if 1.7199999999999999e36 < y.re

   1. Initial program 34.1

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

   2. Applied egg-rr 23.1

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

   3. Taylor expanded in y.re around inf 16.1

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

   4. Simplified 12.8

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

      [Start] 16.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right) \]
      associate-/l* [=>] 12.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}\right) \]

3. Recombined 5 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.8 \cdot 10^{+122}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-193}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-218}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.72 \cdot 10^{+36}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	13.8
Cost	7828

\[\begin{array}{l} t_0 := x.re + \frac{y.im}{\frac{y.re}{x.im}}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.16 \cdot 10^{-27}:\\ \;\;\;\;t_0 \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_0\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7828

\[\begin{array}{l} t_0 := x.re + \frac{y.im}{\frac{y.re}{x.im}}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-28}:\\ \;\;\;\;t_0 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-161}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-30}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.72 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_0\\ \end{array} \]

Alternative 3
Error	13.9
Cost	1620

\[\begin{array}{l} t_0 := \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-159}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{-29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-9}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	19.0
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+52} \lor \neg \left(y.re \leq 1.32 \cdot 10^{+36}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 5
Error	18.6
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{+55} \lor \neg \left(y.re \leq 3.5 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 6
Error	14.9
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4 \cdot 10^{-24} \lor \neg \left(y.re \leq 9.8 \cdot 10^{-8}\right):\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + x.re \cdot \frac{y.re}{y.im}\right)\\ \end{array} \]

Alternative 7
Error	23.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	37.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+151}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	37.4
Cost	192

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

Reproduce

herbie shell --seed 2023039 
(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))))