_divideComplex, real part

Average Error: 26.2 → 10.3

Time: 15.6s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.6221103327553225 \cdot 10^{+66}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-159}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.109495027883992 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

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 -1.6221103327553225e+66)
     (* (+ x.im (* y.re (/ x.re y.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -1e-110)
       t_0
       (if (<= y.im 1e-159)
         (* (/ 1.0 y.re) (+ x.re (* y.im (/ x.im y.re))))
         (if (<= y.im 9.109495027883992e+52)
           t_0
           (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.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)) * (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 <= -1.6221103327553225e+66) {
		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 <= -1e-110) {
		tmp = t_0;
	} else if (y_46_im <= 1e-159) {
		tmp = (1.0 / y_46_re) * (x_46_re + (y_46_im * (x_46_im / y_46_re)));
	} else if (y_46_im <= 9.109495027883992e+52) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_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(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 <= -1.6221103327553225e+66)
		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 <= -1e-110)
		tmp = t_0;
	elseif (y_46_im <= 1e-159)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))));
	elseif (y_46_im <= 9.109495027883992e+52)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_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[(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, -1.6221103327553225e+66], 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, -1e-110], t$95$0, If[LessEqual[y$46$im, 1e-159], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.109495027883992e+52], t$95$0, 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.62211033275532248e66

   1. Initial program 38.1

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

   2. Applied egg-rr 26.4

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

   3. Taylor expanded in y.im around -inf 15.0

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

   4. Simplified 11.2

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(-x.im\right) - \frac{x.re}{y.im} \cdot y.re\right)} \]
      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-*l/_binary64 (/.f64 (*.f64 x.re y.re) y.im))): 22 points increase in error, 10 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 -1.62211033275532248e66 < y.im < -1.0000000000000001e-110 or 9.99999999999999989e-160 < y.im < 9.10949502788399177e52

   1. Initial program 14.4

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

   2. Applied egg-rr 9.4

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

   if -1.0000000000000001e-110 < y.im < 9.99999999999999989e-160

   1. Initial program 23.1

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

   2. Applied egg-rr 11.8

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

   3. Taylor expanded in y.re around inf 29.6

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

   4. Simplified 30.5

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re + \frac{x.im}{y.re} \cdot y.im\right)} \]
      Proof
      (+.f64 x.re (*.f64 (/.f64 x.im y.re) y.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 x.re (Rewrite<= associate-/r/_binary64 (/.f64 x.im (/.f64 y.re y.im)))): 24 points increase in error, 20 points decrease in error
      (+.f64 x.re (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.im) y.re))): 27 points increase in error, 20 points decrease in error
      (+.f64 x.re (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) y.re)): 0 points increase in error, 0 points decrease in error

   5. Taylor expanded in y.re around inf 8.3

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

   if 9.10949502788399177e52 < y.im

   1. Initial program 36.2

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

   2. Applied egg-rr 24.7

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

   3. Taylor expanded in y.re around 0 17.1

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

   4. Simplified 15.1

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{y.im} \cdot x.re} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 (/.f64 y.re y.im) y.im) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (*.f64 (Rewrite<= associate-/r*_binary64 (/.f64 y.re (*.f64 y.im y.im))) x.re)): 22 points increase in error, 12 points decrease in error
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) x.re)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 (pow.f64 y.im 2) x.re)))): 6 points increase in error, 20 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) (pow.f64 y.im 2)))): 23 points increase in error, 11 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (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

   5. Taylor expanded in y.re around 0 17.1

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

   6. Simplified 13.0

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im \cdot \frac{y.im}{x.re}}} \]
      Proof
      (/.f64 y.re (*.f64 y.im (/.f64 y.im x.re))): 0 points increase in error, 0 points decrease in error
      (/.f64 y.re (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 y.im y.im) x.re))): 32 points increase in error, 23 points decrease in error
      (/.f64 y.re (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) x.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) (pow.f64 y.im 2))): 34 points increase in error, 21 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (pow.f64 y.im 2)): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6221103327553225 \cdot 10^{+66}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-110}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 10^{-159}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.109495027883992 \cdot 10^{+52}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	20560

\[\begin{array}{l} t_0 := \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{if}\;y.im \leq -1.6221103327553225 \cdot 10^{+66}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-159}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.109495027883992 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 2
Error	11.9
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.303166916027757 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-159}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 4.843721839438302 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im + y.re \cdot \frac{x.re}{y.im}\right)\\ \end{array} \]

Alternative 3
Error	11.7
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.im + y.re \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.303166916027757 \cdot 10^{+49}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-159}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 4.843721839438302 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_1\\ \end{array} \]

Alternative 4
Error	16.5
Cost	1496

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{if}\;y.im \leq -6.000877851733397 \cdot 10^{+40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.2539367517734567 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-63}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.554156736768857 \cdot 10^{-55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 6.458230857498665 \cdot 10^{+31}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.109495027883992 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 5
Error	12.3
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -4.303166916027757 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-159}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.109495027883992 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 6
Error	19.0
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.367277341472856 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.7459503277662954 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	15.5
Cost	968

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.367277341472856 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 15054256.082764637:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	15.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.367277341472856 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 15054256.082764637:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im}}{\frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + y.im \cdot \frac{x.im}{y.re}\right)\\ \end{array} \]

Alternative 9
Error	57.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.725526912866907 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.0545691943241164 \cdot 10^{-31}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Error	23.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.367277341472856 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 15054256.082764637:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Error	58.9
Cost	192

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

Alternative 12
Error	37.6
Cost	192

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

Reproduce

herbie shell --seed 2022301 
(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))))