_divideComplex, real part

Average Accuracy: 58.5% → 98.4%

Time: 22.9s

Precision: binary64

Cost: 20352

Math

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

↓

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

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
 (*
  (/ 1.0 (hypot y.re y.im))
  (+ (* y.im (/ x.im (hypot y.re y.im))) (* (/ y.re (hypot y.re 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) {
	return (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_im / hypot(y_46_re, y_46_im))) + ((y_46_re / hypot(y_46_re, y_46_im)) * x_46_re));
}

Java

public static 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));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_im / Math.hypot(y_46_re, y_46_im))) + ((y_46_re / Math.hypot(y_46_re, y_46_im)) * x_46_re));
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_im / math.hypot(y_46_re, y_46_im))) + ((y_46_re / math.hypot(y_46_re, y_46_im)) * x_46_re))

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(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_im * Float64(x_46_im / hypot(y_46_re, y_46_im))) + Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * x_46_re)))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_im * (x_46_im / hypot(y_46_re, y_46_im))) + ((y_46_re / hypot(y_46_re, y_46_im)) * x_46_re));
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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$im * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.5%

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

2. Applied egg-rr 73.0%

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

   [Start] 58.5	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   *-un-lft-identity [=>] 58.5	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
   add-sqr-sqrt [=>] 58.5	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
   times-frac [=>] 58.5	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
   hypot-def [=>] 58.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
   fma-def [=>] 58.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
   hypot-def [=>] 73.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

3. Applied egg-rr 73.0%

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

   [Start] 73.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
   div-inv [=>] 72.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right) \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   *-commutative [<=] 72.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)\right)} \]
   fma-udef [=>] 72.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re \cdot y.re + x.im \cdot y.im\right)}\right) \]
   distribute-lft-in [=>] 72.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re \cdot y.re\right) + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right)} \]
   associate-*l/ [=>] 72.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{1 \cdot \left(x.re \cdot y.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
   *-un-lft-identity [<=] 72.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{\color{blue}{x.re \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im \cdot y.im\right)\right) \]
   associate-*l/ [=>] 73.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \color{blue}{\frac{1 \cdot \left(x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}\right) \]
   *-un-lft-identity [<=] 73.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{\color{blue}{x.im \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

4. Simplified 85.7%

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

   [Start] 73.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   +-commutative [=>] 73.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\frac{x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \]
   associate-/l* [=>] 85.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}} + \frac{x.re \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   *-commutative [=>] 85.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{\color{blue}{y.re \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]

5. Applied egg-rr 99.0%

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

   [Start] 85.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right) \]
   associate-/l* [=>] 98.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \color{blue}{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}}\right) \]
   associate-/r/ [=>] 99.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \color{blue}{\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re}\right) \]

6. Applied egg-rr 98.4%

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

   [Start] 99.0	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}} + \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right) \]
   associate-/r/ [=>] 98.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im} + \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right) \]

7. Final simplification 98.4%

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

Alternatives

Alternative 1
Accuracy	88.9%
Cost	22089

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 10^{+287}\right):\\ \;\;\;\;t_1 \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Accuracy	79.2%
Cost	14424

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+102}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -7500000000:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-29}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 3
Accuracy	77.4%
Cost	14164

\[\begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -2.1 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -3300000000:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.12 \cdot 10^{-132}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 4
Accuracy	82.3%
Cost	14028

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.8 \cdot 10^{+103}:\\ \;\;\;\;t_0 \cdot \left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re - \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \mathbf{elif}\;y.re \leq -8.2 \cdot 10^{-133}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 3.05 \cdot 10^{-27}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.re + \frac{x.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{y.im}}\right)\\ \end{array} \]

Alternative 5
Accuracy	77.2%
Cost	13772

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1900000000:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -4.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{-134}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.8 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+75}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 6
Accuracy	77.1%
Cost	8092

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.6 \cdot 10^{+46}:\\ \;\;\;\;\frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.re}}\\ \mathbf{elif}\;y.re \leq -95:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 7
Accuracy	79.5%
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+103}:\\ \;\;\;\;\frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq -1.16 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{y.re}{x.im}}\right)\\ \end{array} \]

Alternative 8
Accuracy	76.5%
Cost	7372

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.4 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{+46}:\\ \;\;\;\;\frac{y.re}{\frac{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}{x.re}}\\ \mathbf{elif}\;y.re \leq -3:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot \left(y.re \cdot \frac{1}{x.im}\right)}\\ \end{array} \]

Alternative 9
Accuracy	76.2%
Cost	7180

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7.6 \cdot 10^{+46}:\\ \;\;\;\;x.re \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -49:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.26 \cdot 10^{-133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot \left(y.re \cdot \frac{1}{x.im}\right)}\\ \end{array} \]

Alternative 10
Accuracy	76.3%
Cost	1884

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.95:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-21}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{\frac{y.im \cdot y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot \left(y.re \cdot \frac{1}{x.im}\right)}\\ \end{array} \]

Alternative 11
Accuracy	73.6%
Cost	1360

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -7.6 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{+49}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot \left(y.re \cdot \frac{1}{x.im}\right)}\\ \end{array} \]

Alternative 12
Accuracy	69.7%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{+48}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{+79}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 13
Accuracy	74.1%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -6.4 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.45 \cdot 10^{+69}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{+50}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Accuracy	73.6%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -4.4 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{+47}:\\ \;\;\;\;x.re \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq 3.05 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot \frac{y.re}{x.im}}\\ \end{array} \]

Alternative 15
Accuracy	63.6%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+28}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 16
Accuracy	41.9%
Cost	192

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

Reproduce

herbie shell --seed 2023137 
(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))))