_divideComplex, real part

Average Accuracy: 55.3% → 76.4%

Time: 16.0s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im \cdot y.im}{x.re}}\\ \end{array} \]

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)))))
   (if (<= y.im -2.7e+58)
     (* (/ y.im (hypot y.im y.re)) (/ x.im (hypot y.im y.re)))
     (if (<= y.im -9.5e-224)
       t_0
       (if (<= y.im -3.4e-299)
         (/ (+ x.re (/ x.im (/ y.re y.im))) y.re)
         (if (<= y.im 9.2e+76)
           t_0
           (+ (/ x.im y.im) (/ y.re (/ (* y.im y.im) x.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 = (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 tmp;
	if (y_46_im <= -2.7e+58) {
		tmp = (y_46_im / hypot(y_46_im, y_46_re)) * (x_46_im / hypot(y_46_im, y_46_re));
	} else if (y_46_im <= -9.5e-224) {
		tmp = t_0;
	} else if (y_46_im <= -3.4e-299) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_im <= 9.2e+76) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re / ((y_46_im * y_46_im) / x_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(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)))
	tmp = 0.0
	if (y_46_im <= -2.7e+58)
		tmp = Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) * Float64(x_46_im / hypot(y_46_im, y_46_re)));
	elseif (y_46_im <= -9.5e-224)
		tmp = t_0;
	elseif (y_46_im <= -3.4e-299)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_im <= 9.2e+76)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(Float64(y_46_im * y_46_im) / x_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[(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]}, If[LessEqual[y$46$im, -2.7e+58], N[(N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -9.5e-224], t$95$0, If[LessEqual[y$46$im, -3.4e-299], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 9.2e+76], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(N[(y$46$im * y$46$im), $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.7000000000000001e58

   1. Initial program 36.4%

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

   2. Taylor expanded in x.re around 0 35.0%

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

   3. Applied egg-rr 82.4%

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

      [Start] 35.0	\[ \frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 35.0	\[ \frac{y.im \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 48.6	\[ \color{blue}{\frac{y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      +-commutative [=>] 48.6	\[ \frac{y.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 48.6	\[ \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      +-commutative [=>] 48.6	\[ \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}} \]
      hypot-def [=>] 82.4	\[ \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}} \]

   if -2.7000000000000001e58 < y.im < -9.5000000000000003e-224 or -3.3999999999999998e-299 < y.im < 9.20000000000000005e76

   1. Initial program 73.5%

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

   2. Applied egg-rr 84.8%

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

      [Start] 73.5	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 73.5	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 73.5	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 73.5	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 73.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 73.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 84.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -9.5000000000000003e-224 < y.im < -3.3999999999999998e-299

   1. Initial program 55.0%

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

   2. Simplified 55.0%

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

      [Start] 55.0	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 55.0	\[ \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 55.0	\[ \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   3. Taylor expanded in y.re around inf 55.0%

      \[\leadsto \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{{y.re}^{2}}} \]

   4. Simplified 55.0%

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

      [Start] 55.0	\[ \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{{y.re}^{2}} \]
      unpow2 [=>] 55.0	\[ \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{y.re \cdot y.re}} \]

   5. Applied egg-rr 57.9%

      \[\leadsto \color{blue}{0 + \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot {y.re}^{-2}} \]
      Proof

      [Start] 55.0	\[ \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re} \]
      add-log-exp [=>] 6.1	\[ \color{blue}{\log \left(e^{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re}}\right)} \]
      *-un-lft-identity [=>] 6.1	\[ \log \color{blue}{\left(1 \cdot e^{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re}}\right)} \]
      log-prod [=>] 6.1	\[ \color{blue}{\log 1 + \log \left(e^{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re}}\right)} \]
      metadata-eval [=>] 6.1	\[ \color{blue}{0} + \log \left(e^{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re}}\right) \]
      add-log-exp [<=] 55.0	\[ 0 + \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re \cdot y.re}} \]
      div-inv [=>] 54.8	\[ 0 + \color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{y.re \cdot y.re}} \]
      pow2 [=>] 54.8	\[ 0 + \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\color{blue}{{y.re}^{2}}} \]
      pow-flip [=>] 57.9	\[ 0 + \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \color{blue}{{y.re}^{\left(-2\right)}} \]
      metadata-eval [=>] 57.9	\[ 0 + \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot {y.re}^{\color{blue}{-2}} \]

   6. Simplified 89.9%

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

      [Start] 57.9	\[ 0 + \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot {y.re}^{-2} \]
      +-lft-identity [=>] 57.9	\[ \color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot {y.re}^{-2}} \]
      *-commutative [<=] 57.9	\[ \color{blue}{{y.re}^{-2} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)} \]
      metadata-eval [<=] 57.9	\[ {y.re}^{\color{blue}{\left(2 \cdot -1\right)}} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \]
      pow-sqr [<=] 57.7	\[ \color{blue}{\left({y.re}^{-1} \cdot {y.re}^{-1}\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \]
      unpow-1 [=>] 57.7	\[ \left(\color{blue}{\frac{1}{y.re}} \cdot {y.re}^{-1}\right) \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \]
      unpow-1 [=>] 57.7	\[ \left(\frac{1}{y.re} \cdot \color{blue}{\frac{1}{y.re}}\right) \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \]
      associate-*l* [=>] 64.5	\[ \color{blue}{\frac{1}{y.re} \cdot \left(\frac{1}{y.re} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
      associate-*l/ [=>] 64.7	\[ \frac{1}{y.re} \cdot \color{blue}{\frac{1 \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re}} \]
      *-lft-identity [=>] 64.7	\[ \frac{1}{y.re} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re} \]
      associate-*l/ [=>] 65.1	\[ \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re}}{y.re}} \]
      associate-*r/ [=>] 65.1	\[ \frac{\color{blue}{\frac{1 \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{y.re}}}{y.re} \]
      associate-*l/ [<=] 65.0	\[ \frac{\color{blue}{\frac{1}{y.re} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{y.re} \]
      fma-udef [=>] 65.0	\[ \frac{\frac{1}{y.re} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re} \]
      *-commutative [<=] 65.0	\[ \frac{\frac{1}{y.re} \cdot \left(x.re \cdot y.re + \color{blue}{y.im \cdot x.im}\right)}{y.re} \]
      distribute-rgt-in [=>] 65.0	\[ \frac{\color{blue}{\left(x.re \cdot y.re\right) \cdot \frac{1}{y.re} + \left(y.im \cdot x.im\right) \cdot \frac{1}{y.re}}}{y.re} \]
      associate-*l* [=>] 89.8	\[ \frac{\color{blue}{x.re \cdot \left(y.re \cdot \frac{1}{y.re}\right)} + \left(y.im \cdot x.im\right) \cdot \frac{1}{y.re}}{y.re} \]
      rgt-mult-inverse [=>] 89.9	\[ \frac{x.re \cdot \color{blue}{1} + \left(y.im \cdot x.im\right) \cdot \frac{1}{y.re}}{y.re} \]
      *-rgt-identity [=>] 89.9	\[ \frac{\color{blue}{x.re} + \left(y.im \cdot x.im\right) \cdot \frac{1}{y.re}}{y.re} \]
      associate-*r/ [=>] 89.9	\[ \frac{x.re + \color{blue}{\frac{\left(y.im \cdot x.im\right) \cdot 1}{y.re}}}{y.re} \]
      *-rgt-identity [=>] 89.9	\[ \frac{x.re + \frac{\color{blue}{y.im \cdot x.im}}{y.re}}{y.re} \]
      *-commutative [=>] 89.9	\[ \frac{x.re + \frac{\color{blue}{x.im \cdot y.im}}{y.re}}{y.re} \]
      associate-/l* [=>] 89.9	\[ \frac{x.re + \color{blue}{\frac{x.im}{\frac{y.re}{y.im}}}}{y.re} \]

   if 9.20000000000000005e76 < y.im

   1. Initial program 46.7%

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

   2. Taylor expanded in y.re around 0 79.6%

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

   3. Simplified 85.1%

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

      [Start] 79.6	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 79.6	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 79.6	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 79.6	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      associate-/l* [=>] 85.1	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{\frac{y.im \cdot y.im}{x.re}}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+58}:\\ \;\;\;\;\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -3.4 \cdot 10^{-299}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im \cdot y.im}{x.re}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	72.7%
Cost	20700

\[\begin{array}{l} t_0 := \mathsf{fma}\left(y.im, x.im, y.re \cdot x.re\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ t_1 := \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ t_2 := \frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.im \leq -3.5 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-66}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im \cdot y.im}{x.re}}\\ \end{array} \]

Alternative 2
Accuracy	72.6%
Cost	14168

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ t_1 := \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.im, y.re\right)}\\ t_2 := \frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.56 \cdot 10^{-139}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.15 \cdot 10^{-13}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+76}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im \cdot y.im}{x.re}}\\ \end{array} \]

Alternative 3
Accuracy	76.0%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+46}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 4
Accuracy	70.9%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \end{array} \]

Alternative 5
Accuracy	70.7%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+67} \lor \neg \left(y.re \leq -2.8 \cdot 10^{+21} \lor \neg \left(y.re \leq -2.7 \cdot 10^{-47}\right) \land y.re \leq 4.7 \cdot 10^{+55}\right):\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]

Alternative 6
Accuracy	65.9%
Cost	1105

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+68}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{+18} \lor \neg \left(y.re \leq -8.2 \cdot 10^{-38}\right) \land y.re \leq 5.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Accuracy	70.8%
Cost	1105

\[\begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.7 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{-44} \lor \neg \left(y.re \leq 7.2 \cdot 10^{+58}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]

Alternative 8
Accuracy	60.7%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Accuracy	40.0%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+147}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Accuracy	39.4%
Cost	192

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

Reproduce

herbie shell --seed 2023157 
(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))))