_divideComplex, real part

Average Error: 26.0 → 10.4

Time: 17.4s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := x.re + \frac{y.im}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -2.0068511289131843 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{y.re} \cdot t_2\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-200}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.6926259780825635 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.482515939733231 \cdot 10^{+75}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot t_0\\ \end{array} \]

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 -2.0068511289131843e+124)
     (* (/ 1.0 y.re) t_2)
     (if (<= y.re -1e-200)
       t_1
       (if (<= y.re 4.6926259780825635e-76)
         (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
         (if (<= y.re 3.482515939733231e+75) t_1 (* t_2 t_0)))))))

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 <= -2.0068511289131843e+124) {
		tmp = (1.0 / y_46_re) * t_2;
	} else if (y_46_re <= -1e-200) {
		tmp = t_1;
	} else if (y_46_re <= 4.6926259780825635e-76) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_re <= 3.482515939733231e+75) {
		tmp = t_1;
	} else {
		tmp = t_2 * t_0;
	}
	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 <= -2.0068511289131843e+124)
		tmp = Float64(Float64(1.0 / y_46_re) * t_2);
	elseif (y_46_re <= -1e-200)
		tmp = t_1;
	elseif (y_46_re <= 4.6926259780825635e-76)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_re <= 3.482515939733231e+75)
		tmp = t_1;
	else
		tmp = Float64(t_2 * t_0);
	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, -2.0068511289131843e+124], N[(N[(1.0 / y$46$re), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[y$46$re, -1e-200], t$95$1, If[LessEqual[y$46$re, 4.6926259780825635e-76], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.482515939733231e+75], t$95$1, N[(t$95$2 * t$95$0), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.00685112891318433e124

   1. Initial program 41.7

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

   2. Applied egg-rr 28.0

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

   3. Taylor expanded in y.re around inf 47.1

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

   4. Simplified 46.9

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re + \frac{y.im}{y.re} \cdot x.im\right)} \]
      Proof
      (+.f64 x.re (*.f64 (/.f64 y.im y.re) x.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 x.re (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.im) y.re))): 21 points increase in error, 23 points decrease in error

   5. Taylor expanded in y.re around inf 9.5

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

   6. Applied egg-rr 8.9

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

   if -2.00685112891318433e124 < y.re < -9.9999999999999998e-201 or 4.6926259780825635e-76 < y.re < 3.48251593973323099e75

   1. Initial program 17.0

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

   2. Applied egg-rr 11.6

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

   if -9.9999999999999998e-201 < y.re < 4.6926259780825635e-76

   1. Initial program 22.2

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

   2. Taylor expanded in y.re around 0 14.9

      \[\leadsto \color{blue}{\frac{x.re \cdot y.re}{{y.im}^{2}} + \left(\frac{x.im}{y.im} + -1 \cdot \frac{{y.re}^{2} \cdot x.im}{{y.im}^{3}}\right)} \]

   3. Simplified 12.6

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re (*.f64 y.im y.im)) (-.f64 x.re (/.f64 (*.f64 x.im y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) (-.f64 x.re (/.f64 (*.f64 x.im y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 y.re (pow.f64 y.im 2)) x.re) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im))))): 1 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 (pow.f64 y.im 2) x.re))) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im)))): 9 points increase in error, 3 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) (pow.f64 y.im 2))) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im)))): 22 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (pow.f64 y.im 2)) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 y.re x.im)) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re (*.f64 y.re x.im)) (*.f64 (pow.f64 y.im 2) y.im))))): 15 points increase in error, 3 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y.re y.re) x.im)) (*.f64 (pow.f64 y.im 2) y.im)))): 12 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) x.im) (*.f64 (pow.f64 y.im 2) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 y.im y.im)) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (Rewrite<= unpow3_binary64 (pow.f64 y.im 3))))): 1 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (neg.f64 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3))) (/.f64 x.im y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x.im y.im) (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y.re around 0 11.6

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

   5. Simplified 9.7

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im))): 63 points increase in error, 27 points decrease in error
      (/.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.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))): 0 points increase in error, 0 points decrease in error

   if 3.48251593973323099e75 < y.re

   1. Initial program 36.0

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

   2. Applied egg-rr 23.7

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

   3. Taylor expanded in y.re around inf 13.8

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

   4. Simplified 10.6

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re + \frac{y.im}{y.re} \cdot x.im\right)} \]
      Proof
      (+.f64 x.re (*.f64 (/.f64 y.im y.re) x.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 x.re (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.im) y.re))): 21 points increase in error, 23 points decrease in error

   5. Applied egg-rr 10.3

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \color{blue}{\frac{y.im}{\frac{y.re}{x.im}}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.0068511289131843 \cdot 10^{+124}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-200}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 4.6926259780825635 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.482515939733231 \cdot 10^{+75}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	7240

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.083455114343713 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{1}{\frac{y.re}{\mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)}}\\ \mathbf{elif}\;y.im \leq 1.2320390717440114 \cdot 10^{+132}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	20.3
Cost	1496

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{if}\;y.re \leq -2.0068511289131843 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -8.630568195952071 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -9.630404667672953 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7.080983159766232 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.245367067430426 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	17.7
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ t_1 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{if}\;y.re \leq -2.0068511289131843 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -8.630568195952071 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.2298125846985904 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -7.080983159766232 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.293590831195469 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.7
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\\ t_1 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{if}\;y.re \leq -2.0068511289131843 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -8.630568195952071 \cdot 10^{+90}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.2298125846985904 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -7.080983159766232 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.293590831195469 \cdot 10^{-10}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.9
Cost	1356

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.083455114343713 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-121}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.2320390717440114 \cdot 10^{+132}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	24.6
Cost	984

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.0068511289131843 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.353915732253755 \cdot 10^{+90}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -9.915289969042209 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -7.080983159766232 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 7.245367067430426 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Error	15.2
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.083455114343713 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.001975080593576 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	58.8
Cost	192

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

Alternative 9
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(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))))