_divideComplex, real part

Average Error: 26.3 → 10.8

Time: 21.2s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)\\ \mathbf{if}\;y.im \leq -2.0284987936154364 \cdot 10^{+92}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-250}:\\ \;\;\;\;\frac{t_0}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_1}}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.200908651103965 \cdot 10^{+56}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

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 (/ 1.0 (hypot y.re y.im))) (t_1 (fma x.re y.re (* y.im x.im))))
   (if (<= y.im -2.0284987936154364e+92)
     (* (+ x.im (* (/ y.re y.im) x.re)) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -1e-250)
       (/ t_0 (/ (hypot y.re y.im) t_1))
       (if (<= y.im 1.5e-193)
         (+ (/ x.re y.re) (/ x.im (* y.re (/ y.re y.im))))
         (if (<= y.im 7.200908651103965e+56)
           (* t_0 (/ t_1 (hypot y.re y.im)))
           (fma (/ (/ x.re y.im) y.im) y.re (/ x.im 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);
	double t_1 = fma(x_46_re, y_46_re, (y_46_im * x_46_im));
	double tmp;
	if (y_46_im <= -2.0284987936154364e+92) {
		tmp = (x_46_im + ((y_46_re / y_46_im) * x_46_re)) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1e-250) {
		tmp = t_0 / (hypot(y_46_re, y_46_im) / t_1);
	} else if (y_46_im <= 1.5e-193) {
		tmp = (x_46_re / y_46_re) + (x_46_im / (y_46_re * (y_46_re / y_46_im)));
	} else if (y_46_im <= 7.200908651103965e+56) {
		tmp = t_0 * (t_1 / hypot(y_46_re, y_46_im));
	} else {
		tmp = fma(((x_46_re / y_46_im) / y_46_im), y_46_re, (x_46_im / 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(1.0 / hypot(y_46_re, y_46_im))
	t_1 = fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_im <= -2.0284987936154364e+92)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re / y_46_im) * x_46_re)) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -1e-250)
		tmp = Float64(t_0 / Float64(hypot(y_46_re, y_46_im) / t_1));
	elseif (y_46_im <= 1.5e-193)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64(y_46_re * Float64(y_46_re / y_46_im))));
	elseif (y_46_im <= 7.200908651103965e+56)
		tmp = Float64(t_0 * Float64(t_1 / hypot(y_46_re, y_46_im)));
	else
		tmp = fma(Float64(Float64(x_46_re / y_46_im) / y_46_im), y_46_re, Float64(x_46_im / 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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.0284987936154364e+92], N[(N[(x$46$im + N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1e-250], N[(t$95$0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.5e-193], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(y$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.200908651103965e+56], N[(t$95$0 * N[(t$95$1 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] * y$46$re + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.02849879361543641e92

   1. Initial program 39.6

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

   2. Applied egg-rr 26.9

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

   3. Taylor expanded in y.im around -inf 14.0

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

   4. Simplified 11.0

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(-x.im\right) - \frac{y.re}{y.im} \cdot x.re\right)} \]
      Proof
      (-.f64 (neg.f64 x.im) (*.f64 (/.f64 y.re y.im) x.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x.im)) (*.f64 (/.f64 y.re y.im) x.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 x.im) (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 y.im x.re)))): 26 points increase in error, 25 points decrease in error
      (-.f64 (*.f64 -1 x.im) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) y.im))): 27 points increase in error, 19 points decrease in error
      (-.f64 (*.f64 -1 x.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) y.im)): 0 points increase in error, 0 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 -2.02849879361543641e92 < y.im < -1.0000000000000001e-250

   1. Initial program 18.1

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

   2. Applied egg-rr 11.4

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

   3. Applied egg-rr 11.5

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

   if -1.0000000000000001e-250 < y.im < 1.5e-193

   1. Initial program 24.3

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

   2. Applied egg-rr 13.3

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

   3. Taylor expanded in y.re around inf 9.4

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

   4. Simplified 5.8

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      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)))): 35 points increase in error, 9 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

   5. Applied egg-rr 4.3

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

   if 1.5e-193 < y.im < 7.2009086511039648e56

   1. Initial program 16.4

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

   2. Applied egg-rr 10.5

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

   if 7.2009086511039648e56 < y.im

   1. Initial program 35.2

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

   2. Taylor expanded in y.re around 0 18.3

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

   3. Simplified 14.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)} \]
      Proof
      (fma.f64 (/.f64 (/.f64 x.re y.im) y.im) y.re (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-/r*_binary64 (/.f64 x.re (*.f64 y.im y.im))) y.re (/.f64 x.im y.im)): 27 points increase in error, 7 points decrease in error
      (fma.f64 (/.f64 x.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) y.re (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 x.re (pow.f64 y.im 2)) y.re) (/.f64 x.im y.im))): 1 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2))) (/.f64 x.im y.im)): 20 points increase in error, 18 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.0284987936154364 \cdot 10^{+92}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-250}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-193}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.200908651103965 \cdot 10^{+56}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	11.2
Cost	20560

\[\begin{array}{l} t_0 := \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{if}\;y.im \leq -2.0284987936154364 \cdot 10^{+92}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-250}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-260}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 7.200908651103965 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 2
Error	14.1
Cost	20040

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.7462056822629743 \cdot 10^{+90}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 3
Error	14.2
Cost	7496

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.7462056822629743 \cdot 10^{+90}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 4
Error	14.2
Cost	7372

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.7462056822629743 \cdot 10^{+90}:\\ \;\;\;\;\left(x.im + \frac{y.re}{y.im} \cdot x.re\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-150}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 5
Error	13.2
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -2.3862211260766862 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-162}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.8475690085853758 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	12.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -2.3862211260766862 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-127}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot y.im} \cdot \left(x.re - \frac{y.re \cdot x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.8475690085853758 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	18.1
Cost	1364

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ t_1 := \frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9.752845205853692 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -0.061306424113414346:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.649780066403465 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-230}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	20.0
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.im \leq -9.752845205853692 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	18.2
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9.752845205853692 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	18.0
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -9.752845205853692 \cdot 10^{+107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	24.0
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.449340246179198 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3233920652177107 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12
Error	58.9
Cost	192

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

Alternative 13
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2022318 
(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))))