_divideComplex, imaginary part

Percentage Accurate: 61.3% → 97.2%

Time: 19.3s

Precision: binary64

Cost: 26944

Math

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

↓

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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.re (/ (hypot y.re y.im) x.im)))
  (* (/ y.im (hypot y.im y.re)) (/ x.re (hypot y.im y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * 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_re / (hypot(y_46_re, y_46_im) / x_46_im))) - ((y_46_im / hypot(y_46_im, y_46_re)) * (x_46_re / hypot(y_46_im, y_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_im * y_46_re) - (x_46_re * 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_re / (Math.hypot(y_46_re, y_46_im) / x_46_im))) - ((y_46_im / Math.hypot(y_46_im, y_46_re)) * (x_46_re / Math.hypot(y_46_im, y_46_re)));
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * 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(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im))) - Float64(Float64(y_46_im / hypot(y_46_im, y_46_re)) * Float64(x_46_re / hypot(y_46_im, y_46_re))))
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_im * y_46_re) - (x_46_re * 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_re / (hypot(y_46_re, y_46_im) / x_46_im))) - ((y_46_im / hypot(y_46_im, y_46_re)) * (x_46_re / hypot(y_46_im, y_46_re)));
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 63.3%

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

2. Applied egg-rr 72.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)} \]
   Step-by-step derivation

   [Start] 63.3	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   div-sub [=>] 61.2	\[ \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
   *-un-lft-identity [=>] 61.2	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   add-sqr-sqrt [=>] 61.2	\[ \frac{1 \cdot \left(x.im \cdot y.re\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   times-frac [=>] 61.3	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   fma-neg [=>] 61.3	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]
   hypot-def [=>] 61.3	\[ \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
   hypot-def [=>] 68.3	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
   associate-/l* [=>] 72.5	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}}\right) \]
   add-sqr-sqrt [=>] 72.5	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{y.im}}\right) \]
   pow2 [=>] 72.5	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}{y.im}}\right) \]
   hypot-def [=>] 72.5	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

3. Simplified 76.9%

   \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}} \]
   Step-by-step derivation

   [Start] 72.5	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right) \]
   fma-neg [<=] 72.5	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}} \]
   *-commutative [<=] 72.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{y.re \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \]
   associate-/l* [=>] 81.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \]
   associate-/l* [<=] 76.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{x.re \cdot y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]
   *-commutative [=>] 76.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{\color{blue}{y.im \cdot x.re}}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \]
   associate-/l* [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}} \]

4. Applied egg-rr 83.9%

   \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}} \]
   Step-by-step derivation

   [Start] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}} \]
   unpow2 [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}}{x.re}} \]
   *-un-lft-identity [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}{\color{blue}{1 \cdot x.re}}} \]
   times-frac [=>] 83.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\color{blue}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{1} \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}}} \]
   /-rgt-identity [=>] 83.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}} \]
   hypot-udef [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}} \]
   +-commutative [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}} \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}} \]
   hypot-def [=>] 83.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re}} \]
   hypot-udef [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{x.re}} \]
   +-commutative [=>] 76.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\sqrt{\color{blue}{y.im \cdot y.im + y.re \cdot y.re}}}{x.re}} \]
   hypot-def [=>] 83.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\color{blue}{\mathsf{hypot}\left(y.im, y.re\right)}}{x.re}} \]

5. Applied egg-rr 98.7%

   \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}} \]
   Step-by-step derivation

   [Start] 83.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}} \]
   associate-/r* [=>] 98.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}} \]
   div-inv [=>] 98.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{1}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}} \]
   clear-num [<=] 98.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}} \]

6. Final simplification 98.7%

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

Alternatives

Alternative 1
Accuracy	92.7%
Cost	27344

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.im}} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right) \cdot \frac{\mathsf{hypot}\left(y.im, y.re\right)}{x.re}}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-176}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 7.4 \cdot 10^{+124}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	89.6%
Cost	27080

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}\\ t_2 := t_1 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+110}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -9.5 \cdot 10^{-106}:\\ \;\;\;\;t_1 - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ \mathbf{elif}\;y.im \leq 10^{-148}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+96}:\\ \;\;\;\;t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Accuracy	89.1%
Cost	14788

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_1}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+263}:\\ \;\;\;\;t_0 \cdot \frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4
Accuracy	86.2%
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.re \leq -10200:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -8 \cdot 10^{-213}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{-156}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	84.5%
Cost	14160

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{if}\;y.re \leq -7 \cdot 10^{+19}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -5.7 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-165}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	83.0%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{+95}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Accuracy	71.3%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.2 \cdot 10^{+64} \lor \neg \left(y.im \leq 2.4 \cdot 10^{-31} \lor \neg \left(y.im \leq 9.5 \cdot 10^{+37}\right) \land y.im \leq 1.65 \cdot 10^{+75}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Accuracy	72.0%
Cost	1105

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.75 \cdot 10^{+38} \lor \neg \left(y.im \leq 5.8 \cdot 10^{+72}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	78.0%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+43} \lor \neg \left(y.im \leq 3.7 \cdot 10^{-32}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	63.1%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{+60} \lor \neg \left(y.im \leq 1.35 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Accuracy	42.4%
Cost	192

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

Reproduce

herbie shell --seed 2023159 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))