_divideComplex, real part

Average Accuracy: 56.1% → 78.3%

Time: 18.4s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_0\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_0}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

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))))
   (if (<= y.im -7.4e+118)
     (+ (/ x.im y.im) (/ x.re (/ (* y.im y.im) y.re)))
     (if (<= y.im -6.2e-155)
       (* (/ 1.0 (hypot y.re y.im)) t_0)
       (if (<= y.im 8.5e-155)
         (+ (/ x.re y.re) (/ (/ (* y.im x.im) y.re) y.re))
         (if (<= y.im 2.5e+43)
           (/ 1.0 (/ (hypot y.re y.im) t_0))
           (+ (/ x.im y.im) (* y.re (/ (/ x.re y.im) 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);
	double tmp;
	if (y_46_im <= -7.4e+118) {
		tmp = (x_46_im / y_46_im) + (x_46_re / ((y_46_im * y_46_im) / y_46_re));
	} else if (y_46_im <= -6.2e-155) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * t_0;
	} else if (y_46_im <= 8.5e-155) {
		tmp = (x_46_re / y_46_re) + (((y_46_im * x_46_im) / y_46_re) / y_46_re);
	} else if (y_46_im <= 2.5e+43) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) / t_0);
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re * ((x_46_re / y_46_im) / 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(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 <= -7.4e+118)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re / Float64(Float64(y_46_im * y_46_im) / y_46_re)));
	elseif (y_46_im <= -6.2e-155)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * t_0);
	elseif (y_46_im <= 8.5e-155)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(Float64(y_46_im * x_46_im) / y_46_re) / y_46_re));
	elseif (y_46_im <= 2.5e+43)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) / t_0));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(Float64(x_46_re / y_46_im) / 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[(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]}, If[LessEqual[y$46$im, -7.4e+118], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re / N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -6.2e-155], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision], If[LessEqual[y$46$im, 8.5e-155], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.5e+43], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -7.39999999999999973e118

   1. Initial program 35.5%

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

   2. Applied egg-rr 18.1%

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

      [Start] 35.5	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-commutative [=>] 35.5	\[ \frac{\color{blue}{y.re \cdot x.re} + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 18.1	\[ \frac{y.re \cdot \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      associate-*r* [=>] 18.1	\[ \frac{\color{blue}{\left(y.re \cdot \sqrt{x.re}\right) \cdot \sqrt{x.re}} + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 18.1	\[ \frac{\color{blue}{\mathsf{fma}\left(y.re \cdot \sqrt{x.re}, \sqrt{x.re}, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

   3. Taylor expanded in y.re around 0 76.3%

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

   4. Simplified 77.8%

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

      [Start] 76.3	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 76.3	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      associate-/l* [=>] 77.8	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]
      unpow2 [=>] 77.8	\[ \frac{x.im}{y.im} + \frac{x.re}{\frac{\color{blue}{y.im \cdot y.im}}{y.re}} \]

   if -7.39999999999999973e118 < y.im < -6.2e-155

   1. Initial program 71.6%

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

   2. Applied egg-rr 78.5%

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

      [Start] 71.6	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 71.6	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 71.5	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 71.5	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 71.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 71.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 78.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if -6.2e-155 < y.im < 8.4999999999999996e-155

   1. Initial program 53.6%

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

   2. Taylor expanded in y.re around inf 73.4%

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

   3. Simplified 72.1%

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

      [Start] 73.4	\[ \frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}} \]
      associate-/l* [=>] 69.9	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{\frac{{y.re}^{2}}{x.im}}} \]
      associate-/r/ [=>] 72.1	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im}{{y.re}^{2}} \cdot x.im} \]
      unpow2 [=>] 72.1	\[ \frac{x.re}{y.re} + \frac{y.im}{\color{blue}{y.re \cdot y.re}} \cdot x.im \]

   4. Applied egg-rr 78.1%

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

      [Start] 72.1	\[ \frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot x.im \]
      associate-*l/ [=>] 73.4	\[ \frac{x.re}{y.re} + \color{blue}{\frac{y.im \cdot x.im}{y.re \cdot y.re}} \]
      associate-/r* [=>] 78.1	\[ \frac{x.re}{y.re} + \color{blue}{\frac{\frac{y.im \cdot x.im}{y.re}}{y.re}} \]

   if 8.4999999999999996e-155 < y.im < 2.5000000000000002e43

   1. Initial program 72.8%

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

   2. Applied egg-rr 80.4%

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

      [Start] 72.8	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 72.8	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 72.8	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 72.7	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 72.7	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 72.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 80.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 80.0%

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

      [Start] 80.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 80.6	\[ \color{blue}{\frac{1 \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      associate-/l* [=>] 80.0	\[ \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}} \]

   if 2.5000000000000002e43 < y.im

   1. Initial program 44.4%

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

   2. Applied egg-rr 23.2%

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

      [Start] 44.4	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-commutative [=>] 44.4	\[ \frac{\color{blue}{y.re \cdot x.re} + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 23.2	\[ \frac{y.re \cdot \color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      associate-*r* [=>] 23.2	\[ \frac{\color{blue}{\left(y.re \cdot \sqrt{x.re}\right) \cdot \sqrt{x.re}} + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 23.2	\[ \frac{\color{blue}{\mathsf{fma}\left(y.re \cdot \sqrt{x.re}, \sqrt{x.re}, x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

   3. Taylor expanded in y.re around 0 70.9%

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

   4. Simplified 77.5%

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

      [Start] 70.9	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 70.9	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      associate-/l* [=>] 71.4	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{\frac{{y.im}^{2}}{y.re}}} \]
      associate-/r/ [=>] 73.0	\[ \frac{x.im}{y.im} + \color{blue}{\frac{x.re}{{y.im}^{2}} \cdot y.re} \]
      unpow2 [=>] 73.0	\[ \frac{x.im}{y.im} + \frac{x.re}{\color{blue}{y.im \cdot y.im}} \cdot y.re \]
      associate-/r* [=>] 77.5	\[ \frac{x.im}{y.im} + \color{blue}{\frac{\frac{x.re}{y.im}}{y.im}} \cdot y.re \]

3. Recombined 5 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{-155}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+43}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	78.3%
Cost	20560

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -2.6 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+40}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 2
Accuracy	67.1%
Cost	1496

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ t_1 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -3.2 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.65 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.8 \cdot 10^{-270}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 55000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	75.6%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + x.re \cdot y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq -3.6 \cdot 10^{-158}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{-154}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 4
Accuracy	68.2%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-147}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 90000000:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Accuracy	69.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ t_1 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-147}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.66 \cdot 10^{-44}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.im \leq 54000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	71.7%
Cost	1228

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.4 \cdot 10^{+37}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{+66}:\\ \;\;\;\;\frac{y.im}{\frac{y.im \cdot y.im + y.re \cdot y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{x.re}} \cdot \frac{1}{y.im}\\ \end{array} \]

Alternative 7
Accuracy	71.3%
Cost	1100

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+66}:\\ \;\;\;\;\frac{y.im}{\frac{y.im \cdot y.im + y.re \cdot y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	67.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+53} \lor \neg \left(y.re \leq 2.1 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 9
Accuracy	66.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+48} \lor \neg \left(y.re \leq 4.6 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{y.im \cdot y.im}\\ \end{array} \]

Alternative 10
Accuracy	60.4%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+70}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{+21}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -7 \cdot 10^{+17}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 11
Accuracy	61.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12
Accuracy	38.8%
Cost	192

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

Reproduce

herbie shell --seed 2023153 
(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))))