_divideComplex, real part

Average Error: 26.9 → 10.2

Time: 17.7s

Precision: binary64

Cost: 20432

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -1.1921764556915153 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-190}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7060144739495957 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

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
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.re -1.1921764556915153e+97)
     (fma (/ y.im y.re) (/ x.im y.re) (/ x.re y.re))
     (if (<= y.re -1e-115)
       t_0
       (if (<= y.re 1e-190)
         (+
          (/ x.im y.im)
          (/ (* (/ y.re y.im) (- x.re (* y.re (/ x.im y.im)))) y.im))
         (if (<= y.re 1.7060144739495957e+142)
           t_0
           (* (/ y.re (hypot y.re y.im)) (/ x.re (hypot y.re 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 = (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -1.1921764556915153e+97) {
		tmp = fma((y_46_im / y_46_re), (x_46_im / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_re <= -1e-115) {
		tmp = t_0;
	} else if (y_46_re <= 1e-190) {
		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 <= 1.7060144739495957e+142) {
		tmp = t_0;
	} else {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, 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(Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -1.1921764556915153e+97)
		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_re <= -1e-115)
		tmp = t_0;
	elseif (y_46_re <= 1e-190)
		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 <= 1.7060144739495957e+142)
		tmp = t_0;
	else
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_re / hypot(y_46_re, 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[(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] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.1921764556915153e+97], 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$46$re, -1e-115], t$95$0, If[LessEqual[y$46$re, 1e-190], 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, 1.7060144739495957e+142], t$95$0, N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.1921764556915153e97

   1. Initial program 40.2

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

   2. Simplified 40.2

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 26.4

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

   4. Applied egg-rr 26.3

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

   5. Taylor expanded in y.re around inf 16.5

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

   6. Simplified 9.7

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)} \]
      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))): 1 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)): 41 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

   if -1.1921764556915153e97 < y.re < -1.0000000000000001e-115 or 1e-190 < y.re < 1.70601447394959573e142

   1. Initial program 17.6

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

   2. Simplified 17.6

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 x.re y.re (*.f64 x.im y.im)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 1 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 0 points increase in error, 0 points decrease in error

   3. Applied egg-rr 12.0

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

   4. Applied egg-rr 11.8

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

   if -1.0000000000000001e-115 < y.re < 1e-190

   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. Taylor expanded in y.re around 0 14.3

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

   3. Simplified 11.8

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)} \]
      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, 0 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)))): 7 points increase in error, 6 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)))): 10 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))))): 21 points increase in error, 4 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))))): 3 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 6.3

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

   if 1.70601447394959573e142 < y.re

   1. Initial program 44.1

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

   2. Taylor expanded in x.re around inf 44.3

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

   3. Simplified 44.3

      \[\leadsto \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} \]
      Proof
      (*.f64 y.re x.re): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 11.5

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

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.1921764556915153 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-190}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7060144739495957 \cdot 10^{+142}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.5
Cost	14036

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.1921764556915153 \cdot 10^{+97}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{t_0}\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Error	13.0
Cost	7636

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.1921764556915153 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	13.0
Cost	7636

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.1921764556915153 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	13.0
Cost	7636

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \mathsf{fma}\left(\frac{y.im}{y.re}, \frac{x.im}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{if}\;y.re \leq -1.1921764556915153 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.re, y.im \cdot x.im\right)}{t_0}\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.8
Cost	7444

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7692606865815425 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;x.re \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6
Error	16.2
Cost	1748

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7692606865815425 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)\\ \end{array} \]

Alternative 7
Error	16.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7692606865815425 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	15.9
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.7692606865815425 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-115}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-115}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot \left(x.re - y.re \cdot \frac{x.im}{y.im}\right)}{y.im}\\ \mathbf{elif}\;y.re \leq 5.5753156676145865 \cdot 10^{-40}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Error	20.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.re \leq -5.6351158376466 \cdot 10^{+126}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.7127129223089227 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.2825900732172963 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Error	18.8
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.7692606865815425 \cdot 10^{+97}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -7.536488317619283 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Error	23.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.2825900732172963 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.514784016688724 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12
Error	58.3
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.im \leq 5.608439046130346 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 13
Error	58.8
Cost	192

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

Alternative 14
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2022297 
(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))))