_divideComplex, imaginary part

Average Error: 26.1 → 10.0

Time: 15.6s

Precision: binary64

Cost: 14288

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.62 \cdot 10^{-230}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.6 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
 (let* ((t_0
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im))))
        (t_1 (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))))
   (if (<= y.re -2.5e+143)
     t_1
     (if (<= y.re -1.62e-230)
       t_0
       (if (<= y.re 1.15e-160)
         (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)
         (if (<= y.re 7.6e+123) t_0 t_1))))))

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) {
	double t_0 = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -2.5e+143) {
		tmp = t_1;
	} else if (y_46_re <= -1.62e-230) {
		tmp = t_0;
	} else if (y_46_re <= 1.15e-160) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7.6e+123) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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) {
	double t_0 = (1.0 / Math.hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im));
	double t_1 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	double tmp;
	if (y_46_re <= -2.5e+143) {
		tmp = t_1;
	} else if (y_46_re <= -1.62e-230) {
		tmp = t_0;
	} else if (y_46_re <= 1.15e-160) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7.6e+123) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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):
	t_0 = (1.0 / math.hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im))
	t_1 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	tmp = 0
	if y_46_re <= -2.5e+143:
		tmp = t_1
	elif y_46_re <= -1.62e-230:
		tmp = t_0
	elif y_46_re <= 1.15e-160:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	elif y_46_re <= 7.6e+123:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

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)
	t_0 = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)))
	tmp = 0.0
	if (y_46_re <= -2.5e+143)
		tmp = t_1;
	elseif (y_46_re <= -1.62e-230)
		tmp = t_0;
	elseif (y_46_re <= 1.15e-160)
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 7.6e+123)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
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_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (1.0 / hypot(y_46_re, y_46_im)) * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	t_1 = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	tmp = 0.0;
	if (y_46_re <= -2.5e+143)
		tmp = t_1;
	elseif (y_46_re <= -1.62e-230)
		tmp = t_0;
	elseif (y_46_re <= 1.15e-160)
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 7.6e+123)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
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_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e+143], t$95$1, If[LessEqual[y$46$re, -1.62e-230], t$95$0, If[LessEqual[y$46$re, 1.15e-160], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.6e+123], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.50000000000000006e143 or 7.59999999999999989e123 < y.re

   1. Initial program 43.0

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

   2. Taylor expanded in y.re around inf 15.2

      \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

   3. Simplified 8.3

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

      [Start] 15.2	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 15.2	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 15.2	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 15.2	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 15.2	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 8.3	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   if -2.50000000000000006e143 < y.re < -1.62000000000000002e-230 or 1.14999999999999992e-160 < y.re < 7.59999999999999989e123

   1. Initial program 17.5

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

   2. Applied egg-rr 12.1

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

   if -1.62000000000000002e-230 < y.re < 1.14999999999999992e-160

   1. Initial program 23.2

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

   2. Taylor expanded in y.re around 0 9.2

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}}} \]

   3. Simplified 7.0

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

      [Start] 9.2	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 9.2	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 9.2	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 9.2	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 9.2	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 9.2	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 7.0	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   4. Taylor expanded in x.im around 0 9.2

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}}} \]

   5. Simplified 5.9

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

      [Start] 9.2	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 9.2	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      *-commutative [=>] 9.2	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im} \]
      unpow2 [=>] 9.2	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} + -1 \cdot \frac{x.re}{y.im} \]
      associate-/l* [=>] 10.3	\[ \color{blue}{\frac{x.im}{\frac{y.im \cdot y.im}{y.re}}} + -1 \cdot \frac{x.re}{y.im} \]
      mul-1-neg [=>] 10.3	\[ \frac{x.im}{\frac{y.im \cdot y.im}{y.re}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      sub-neg [<=] 10.3	\[ \color{blue}{\frac{x.im}{\frac{y.im \cdot y.im}{y.re}} - \frac{x.re}{y.im}} \]
      associate-/l* [<=] 9.2	\[ \color{blue}{\frac{x.im \cdot y.re}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      associate-/r* [=>] 4.2	\[ \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im}}{y.im}} - \frac{x.re}{y.im} \]
      div-sub [<=] 4.2	\[ \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}} \]
      *-commutative [<=] 4.2	\[ \frac{\frac{\color{blue}{y.re \cdot x.im}}{y.im} - x.re}{y.im} \]
      associate-/l* [=>] 5.9	\[ \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}}} - x.re}{y.im} \]

3. Recombined 3 regimes into one program.

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.62 \cdot 10^{-230}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-160}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.9
Cost	14556

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ t_2 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{+125}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.95 \cdot 10^{+34}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_2}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y.im \cdot x.re}{-\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -6.4:\\ \;\;\;\;\frac{t_1}{t_2}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 8.8 \cdot 10^{+123}:\\ \;\;\;\;t_1 \cdot \frac{1}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.9
Cost	13968

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{y.re \cdot x.im - y.im \cdot x.re}{t_1}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{+33}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_1}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\frac{\frac{y.im \cdot x.re}{-\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -6.4:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+126}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	13.9
Cost	1884

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_2 := y.re \cdot y.re + y.im \cdot y.im\\ t_3 := \frac{y.re \cdot x.im - y.im \cdot x.re}{t_2}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -4.85 \cdot 10^{+123}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{+48}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_2}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{+24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -6.4:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.re \leq 7.2 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.56 \cdot 10^{+124}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	17.4
Cost	1761

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+150}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{+125}:\\ \;\;\;\;\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -510000:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.3 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+39} \lor \neg \left(y.re \leq 1.36 \cdot 10^{+91}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5
Error	16.3
Cost	1497

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{if}\;y.re \leq -480000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-71}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 9.6 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{+40} \lor \neg \left(y.re \leq 5.5 \cdot 10^{+92}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	1497

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -118000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-104}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-19}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.02 \cdot 10^{+40} \lor \neg \left(y.re \leq 6.6 \cdot 10^{+97}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7
Error	20.1
Cost	1371

\[\begin{array}{l} \mathbf{if}\;y.re \leq -250000 \lor \neg \left(y.re \leq 1.15 \cdot 10^{-104} \lor \neg \left(y.re \leq 1.55 \cdot 10^{-70}\right) \land \left(y.re \leq 1.9 \cdot 10^{-19} \lor \neg \left(y.re \leq 2 \cdot 10^{+27}\right) \land y.re \leq 8 \cdot 10^{+90}\right)\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 8
Error	16.3
Cost	1371

\[\begin{array}{l} \mathbf{if}\;y.re \leq -150000 \lor \neg \left(y.re \leq 1.15 \cdot 10^{-104}\right) \land \left(y.re \leq 9.5 \cdot 10^{-71} \lor \neg \left(y.re \leq 4.9 \cdot 10^{-20}\right) \land \left(y.re \leq 3.6 \cdot 10^{+37} \lor \neg \left(y.re \leq 2.4 \cdot 10^{+91}\right)\right)\right):\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 9
Error	23.3
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -240000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+91}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Error	34.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.4 \cdot 10^{+177}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+251}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 11
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2023045 
(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))))