_divideComplex, real part

Average Error: 26.1 → 10.9

Time: 21.3s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -3.1292067878106438 \cdot 10^{+112}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -7.554869422178032 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.554695517200356 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))))
        (t_1
         (+
          (/ x.im y.im)
          (/ (* (/ y.re y.im) (- x.re (* (/ x.im y.im) y.re))) y.im))))
   (if (<= y.im -3.1292067878106438e+112)
     t_1
     (if (<= y.im -7.554869422178032e-105)
       t_0
       (if (<= y.im 1e-150)
         (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
         (if (<= y.im 1.554695517200356e+65) t_0 t_1))))))

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 t_1 = (x_46_im / y_46_im) + (((y_46_re / y_46_im) * (x_46_re - ((x_46_im / y_46_im) * y_46_re))) / y_46_im);
	double tmp;
	if (y_46_im <= -3.1292067878106438e+112) {
		tmp = t_1;
	} else if (y_46_im <= -7.554869422178032e-105) {
		tmp = t_0;
	} else if (y_46_im <= 1e-150) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_im <= 1.554695517200356e+65) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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)))
	t_1 = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re - Float64(Float64(x_46_im / y_46_im) * y_46_re))) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -3.1292067878106438e+112)
		tmp = t_1;
	elseif (y_46_im <= -7.554869422178032e-105)
		tmp = t_0;
	elseif (y_46_im <= 1e-150)
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_46_re));
	elseif (y_46_im <= 1.554695517200356e+65)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re - N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.1292067878106438e+112], t$95$1, If[LessEqual[y$46$im, -7.554869422178032e-105], t$95$0, If[LessEqual[y$46$im, 1e-150], N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.554695517200356e+65], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.12920678781064375e112 or 1.55469551720035598e65 < y.im

   1. Initial program 38.2

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

   2. Taylor expanded in y.re around 0 25.3

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

   3. Simplified 20.4

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)} \]
      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))))): 0 points increase in error, 2 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)))): 6 points increase in error, 10 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)))): 19 points increase in error, 8 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))))): 19 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)))): 9 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 (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. Applied egg-rr 12.3

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

   if -3.12920678781064375e112 < y.im < -7.5548694221780317e-105 or 1.00000000000000001e-150 < y.im < 1.55469551720035598e65

   1. Initial program 15.8

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

   2. Applied egg-rr 10.8

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

   if -7.5548694221780317e-105 < y.im < 1.00000000000000001e-150

   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 12.4

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

   3. Taylor expanded in y.re around inf 11.0

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

   4. Simplified 9.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)} \]
      Proof
      (fma.f64 (/.f64 y.im y.re) (/.f64 x.im y.re) (/.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re)) (/.f64 x.re y.re))): 0 points increase in error, 1 points decrease in error
      (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re))) (/.f64 x.re y.re)): 31 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) (/.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 10.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1292067878106438 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -7.554869422178032 \cdot 10^{-105}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.554695517200356 \cdot 10^{+65}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.1
Cost	7636

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ \mathbf{if}\;y.re \leq -2.781374860782003 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-160}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 199404161345.77786:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.2735514438597952 \cdot 10^{+113}:\\ \;\;\;\;x.re \cdot \frac{y.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.4962737985278269 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7504

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ t_2 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.521174256902956 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -488396607700958.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -7.554869422178032 \cdot 10^{-105}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.554695517200356 \cdot 10^{+65}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.7
Cost	1752

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re - \frac{x.im}{y.im} \cdot y.re}{y.im \cdot \frac{y.im}{y.re}}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.1409161625501377 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7175025798514832 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.4193883045570712 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 4
Error	14.3
Cost	1752

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.1409161625501377 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-160}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7175025798514832 \cdot 10^{+25}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.4193883045570712 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 5
Error	13.6
Cost	1616

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - \frac{x.im}{y.im} \cdot y.re\right)}{y.im}\\ \mathbf{if}\;y.im \leq -7.956850444945925 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -7.554869422178032 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-150}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 1.554695517200356 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.9
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.1409161625501377 \cdot 10^{+156}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{-166}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-164}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.4193883045570712 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Error	23.1
Cost	1364

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.521174256902956 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.041088000471931 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2765558034807314 \cdot 10^{+49}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.160572549621019 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.396640388738088 \cdot 10^{+113}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Error	22.6
Cost	1360

\[\begin{array}{l} t_0 := x.im \cdot \frac{-1}{-0.5 \cdot \left(y.re \cdot \frac{y.re}{y.im}\right) - y.im}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -2.521174256902956 \cdot 10^{+52}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.041088000471931 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2765558034807314 \cdot 10^{+49}:\\ \;\;\;\;\frac{y.im \cdot x.im}{t_1}\\ \mathbf{elif}\;y.im \leq 2.1236915110350935 \cdot 10^{+72}:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	23.2
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.521174256902956 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.041088000471931 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2765558034807314 \cdot 10^{+49}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.160572549621019 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Error	57.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -135799796856.87495:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.634856843409071 \cdot 10^{-122}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 11
Error	23.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.521174256902956 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 9.160572549621019 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12
Error	58.5
Cost	192

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

Alternative 13
Error	36.9
Cost	192

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

Reproduce

herbie shell --seed 2022311 
(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))))