_divideComplex, real part

Average Error: 25.8 → 9.4

Time: 13.5s

Precision: binary64

Cost: 20432

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.1618754408931414 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{y.re} \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.5643768901090682 \cdot 10^{-111}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-172}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.091556390258079 \cdot 10^{+56}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]

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
         (/
          (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.re -4.1618754408931414e+66)
     (/ (- (- x.re) (* (/ x.im y.re) y.im)) (hypot y.re y.im))
     (if (<= y.re -1.5643768901090682e-111)
       t_0
       (if (<= y.re 1e-172)
         (+
          (/ x.im y.im)
          (/ (* (/ y.re y.im) (- x.re (* y.re (/ x.im y.im)))) y.im))
         (if (<= y.re 5.091556390258079e+56)
           t_0
           (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.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 = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -4.1618754408931414e+66) {
		tmp = (-x_46_re - ((x_46_im / y_46_re) * y_46_im)) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.5643768901090682e-111) {
		tmp = t_0;
	} else if (y_46_re <= 1e-172) {
		tmp = (x_46_im / y_46_im) + (((y_46_re / y_46_im) * (x_46_re - (y_46_re * (x_46_im / y_46_im)))) / y_46_im);
	} else if (y_46_re <= 5.091556390258079e+56) {
		tmp = t_0;
	} else {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_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(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -4.1618754408931414e+66)
		tmp = Float64(Float64(Float64(-x_46_re) - Float64(Float64(x_46_im / y_46_re) * y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.5643768901090682e-111)
		tmp = t_0;
	elseif (y_46_re <= 1e-172)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re - Float64(y_46_re * Float64(x_46_im / y_46_im)))) / y_46_im));
	elseif (y_46_re <= 5.091556390258079e+56)
		tmp = t_0;
	else
		tmp = fma(Float64(y_46_im / y_46_re), Float64(x_46_im / y_46_re), Float64(x_46_re / y_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[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.1618754408931414e+66], N[(N[((-x$46$re) - N[(N[(x$46$im / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.5643768901090682e-111], t$95$0, If[LessEqual[y$46$re, 1e-172], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re - N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 5.091556390258079e+56], t$95$0, 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]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.1618754408931414e66

   1. Initial program 36.6

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

   2. Applied egg-rr 25.1

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

   3. Applied egg-rr 25.0

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

   4. Taylor expanded in y.re around -inf 13.3

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

   5. Simplified 10.1

      \[\leadsto \frac{\color{blue}{\left(-x.re\right) - \frac{x.im}{y.re} \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Proof
      (-.f64 (neg.f64 x.re) (*.f64 (/.f64 x.im y.re) y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x.re)) (*.f64 (/.f64 x.im y.re) y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 x.re) (Rewrite<= associate-/r/_binary64 (/.f64 x.im (/.f64 y.re y.im)))): 21 points increase in error, 9 points decrease in error
      (-.f64 (*.f64 -1 x.re) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.im) y.re))): 14 points increase in error, 22 points decrease in error
      (-.f64 (*.f64 -1 x.re) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 x.re) (neg.f64 (/.f64 (*.f64 y.im x.im) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 x.re) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 y.im x.im) y.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y.im x.im) y.re)) (*.f64 -1 x.re))): 0 points increase in error, 0 points decrease in error

   if -4.1618754408931414e66 < y.re < -1.56437689010906819e-111 or 1e-172 < y.re < 5.0915563902580793e56

   1. Initial program 14.5

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

   2. Applied egg-rr 9.7

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

   3. Applied egg-rr 9.5

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

   if -1.56437689010906819e-111 < y.re < 1e-172

   1. Initial program 23.0

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

   2. Taylor expanded in y.re around 0 14.2

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

   3. Simplified 10.8

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)} \]
      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, 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, 7 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)))): 11 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))))): 17 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)))): 7 points increase in error, 2 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))))): 2 points increase in error, 2 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2))) (/.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 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))) (/.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=> associate--l+_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (-.f64 (/.f64 x.im y.im) (/.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 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 x.im y.im) (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 (*.f64 x.re y.re) (pow.f64 y.im 2)) (+.f64 (/.f64 x.im y.im) (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

   4. Applied egg-rr 6.4

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

   if 5.0915563902580793e56 < y.re

   1. Initial program 35.7

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

   2. Taylor expanded in y.re around inf 17.0

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

   3. Simplified 11.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)} \]
      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, 0 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)): 32 points increase in error, 13 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 4 regimes into one program.

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.1618754408931414 \cdot 10^{+66}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{y.re} \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.5643768901090682 \cdot 10^{-111}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-172}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.091556390258079 \cdot 10^{+56}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.9
Cost	7372

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -3.5339124098640167 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.091556390258079 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	12.7
Cost	7372

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.5339124098640167 \cdot 10^{+27}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{y.re} \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.091556390258079 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \end{array} \]

Alternative 3
Error	12.7
Cost	1356

\[\begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.5339124098640167 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.091556390258079 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	12.7
Cost	1356

\[\begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.5339124098640167 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.091556390258079 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.3
Cost	1104

\[\begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.1695883990742602 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6788228432867958 \cdot 10^{-78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.3925395676626935 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	14.6
Cost	968

\[\begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -3.5339124098640167 \cdot 10^{+27}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.3925395676626935 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	23.3
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{-15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1695883990742602 \cdot 10^{-111}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6788228432867958 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5669758865422497 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	58.7
Cost	192

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

Alternative 9
Error	37.0
Cost	192

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

Reproduce

herbie shell --seed 2022296 
(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))))