_divideComplex, real part

Average Accuracy: 58.8% → 99.2%

Time: 19.2s

Precision: binary64

Cost: 26496

Math

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

↓

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

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
 (/
  (fma x.im (/ y.im (hypot y.re y.im)) (/ x.re (/ (hypot y.re y.im) y.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) {
	return fma(x_46_im, (y_46_im / hypot(y_46_re, y_46_im)), (x_46_re / (hypot(y_46_re, y_46_im) / y_46_re))) / hypot(y_46_re, y_46_im);
}

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)
	return Float64(fma(x_46_im, Float64(y_46_im / hypot(y_46_re, y_46_im)), Float64(x_46_re / Float64(hypot(y_46_re, y_46_im) / y_46_re))) / hypot(y_46_re, y_46_im))
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_] := N[(N[(x$46$im * N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(x$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.8%

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

2. Applied egg-rr 73.4%

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

   [Start] 58.8	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   *-un-lft-identity [=>] 58.8	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
   add-sqr-sqrt [=>] 58.8	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
   times-frac [=>] 58.8	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
   hypot-def [=>] 58.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
   fma-def [=>] 58.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
   hypot-def [=>] 73.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

3. Applied egg-rr 73.4%

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

   [Start] 73.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   div-inv [=>] 73.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   *-commutative [<=] 73.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
   fma-udef [=>] 73.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
   distribute-lft-in [=>] 73.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
   associate-*l/ [=>] 73.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
   *-un-lft-identity [<=] 73.3	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
   associate-*l/ [=>] 73.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
   *-un-lft-identity [<=] 73.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

4. Simplified 85.0%

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

   [Start] 73.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   +-commutative [=>] 73.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   associate-/l* [=>] 85.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   *-commutative [=>] 85.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

5. Applied egg-rr 99.2%

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

   [Start] 85.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   associate-/r/ [<=] 84.4	\[ \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}}} \]
   clear-num [<=] 85.2	\[ \color{blue}{\frac{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
   div-inv [=>] 85.2	\[ \frac{\color{blue}{x.im \cdot \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   fma-def [=>] 85.2	\[ \frac{\color{blue}{\mathsf{fma}\left(x.im, \frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   clear-num [<=] 85.3	\[ \frac{\mathsf{fma}\left(x.im, \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   *-commutative [=>] 85.3	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   associate-/l* [=>] 99.2	\[ \frac{\mathsf{fma}\left(x.im, \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]

6. Final simplification 99.2%

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

Alternatives

Alternative 1
Accuracy	88.3%
Cost	22988

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.re + x.im \cdot y.im\\ t_2 := \frac{t_1}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_2 \leq -2 \cdot 10^{+294}:\\ \;\;\;\;t_0 \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \mathbf{elif}\;t_2 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ \mathbf{elif}\;t_2 \leq 2 \cdot 10^{+262}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.im, 1, \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.re}}\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Accuracy	87.8%
Cost	21832

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ t_2 := y.re \cdot x.re + x.im \cdot y.im\\ t_3 := \frac{t_2}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_3 \leq -2 \cdot 10^{+294}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+262}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	98.9%
Cost	20352

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

Alternative 4
Accuracy	87.9%
Cost	16844

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ t_2 := y.re \cdot x.re + x.im \cdot y.im\\ t_3 := \frac{t_2}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_3 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_3 \leq -1 \cdot 10^{-102}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{+262}:\\ \;\;\;\;t_0 \cdot \frac{t_2}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	82.9%
Cost	14028

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+38}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-131}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 6
Accuracy	79.1%
Cost	7436

\[\begin{array}{l} \mathbf{if}\;y.im \leq -0.65:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.06 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 7
Accuracy	79.4%
Cost	7436

\[\begin{array}{l} \mathbf{if}\;y.im \leq -0.65:\\ \;\;\;\;\frac{\frac{x.re}{y.im} \cdot \left(-y.re\right) - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 8
Accuracy	78.9%
Cost	1356

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -0.65:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.9 \cdot 10^{-133}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+55}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Accuracy	76.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -0.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{y.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq 5.9 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Accuracy	76.0%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -0.6:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Accuracy	70.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -0.6 \lor \neg \left(y.im \leq 1.5 \cdot 10^{+45}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12
Accuracy	76.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -0.65 \lor \neg \left(y.im \leq 5.3 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 13
Accuracy	76.2%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -0.56 \lor \neg \left(y.im \leq 5.3 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 14
Accuracy	76.5%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -0.62 \lor \neg \left(y.im \leq 2.3 \cdot 10^{+44}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{\frac{y.re}{x.im}}}{y.re}\\ \end{array} \]

Alternative 15
Accuracy	64.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 16
Accuracy	41.3%
Cost	192

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

Reproduce

herbie shell --seed 2023138 
(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))))