_divideComplex, imaginary part

Average Error: 26.9 → 11.9

Time: 14.6s

Precision: binary64

Cost: 14420

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+33}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

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 (- (* (/ x.im y.im) (/ y.re y.im)) (/ x.re y.im)))
        (t_1
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (- (* x.im y.re) (* y.im x.re)) (hypot y.re y.im)))))
   (if (<= y.im -3.3e+33)
     t_0
     (if (<= y.im -2e-59)
       (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
       (if (<= y.im -1.8e-197)
         t_1
         (if (<= y.im 6.9e-152)
           (/ (- x.im (* y.im (/ x.re y.re))) y.re)
           (if (<= y.im 2.4e+137) t_1 t_0)))))))

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 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	double t_1 = (1.0 / hypot(y_46_re, y_46_im)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -3.3e+33) {
		tmp = t_0;
	} else if (y_46_im <= -2e-59) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= -1.8e-197) {
		tmp = t_1;
	} else if (y_46_im <= 6.9e-152) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.4e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	double t_1 = (1.0 / Math.hypot(y_46_re, y_46_im)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -3.3e+33) {
		tmp = t_0;
	} else if (y_46_im <= -2e-59) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= -1.8e-197) {
		tmp = t_1;
	} else if (y_46_im <= 6.9e-152) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_im <= 2.4e+137) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	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 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im)
	t_1 = (1.0 / math.hypot(y_46_re, y_46_im)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im))
	tmp = 0
	if y_46_im <= -3.3e+33:
		tmp = t_0
	elif y_46_im <= -2e-59:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	elif y_46_im <= -1.8e-197:
		tmp = t_1
	elif y_46_im <= 6.9e-152:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_im <= 2.4e+137:
		tmp = t_1
	else:
		tmp = t_0
	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(Float64(x_46_im / y_46_im) * Float64(y_46_re / y_46_im)) - Float64(x_46_re / y_46_im))
	t_1 = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_im <= -3.3e+33)
		tmp = t_0;
	elseif (y_46_im <= -2e-59)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)));
	elseif (y_46_im <= -1.8e-197)
		tmp = t_1;
	elseif (y_46_im <= 6.9e-152)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_im <= 2.4e+137)
		tmp = t_1;
	else
		tmp = t_0;
	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 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	t_1 = (1.0 / hypot(y_46_re, y_46_im)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	tmp = 0.0;
	if (y_46_im <= -3.3e+33)
		tmp = t_0;
	elseif (y_46_im <= -2e-59)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	elseif (y_46_im <= -1.8e-197)
		tmp = t_1;
	elseif (y_46_im <= 6.9e-152)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_im <= 2.4e+137)
		tmp = t_1;
	else
		tmp = t_0;
	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[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.3e+33], t$95$0, If[LessEqual[y$46$im, -2e-59], 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$46$im, -1.8e-197], t$95$1, If[LessEqual[y$46$im, 6.9e-152], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.4e+137], t$95$1, t$95$0]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.29999999999999976e33 or 2.39999999999999983e137 < y.im

   1. Initial program 38.9

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

   2. Taylor expanded in y.re around 0 16.8

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

   3. Simplified 11.1

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

      [Start] 16.8	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 16.8	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 16.8	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 16.8	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 16.8	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 16.8	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 11.1	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   if -3.29999999999999976e33 < y.im < -2.0000000000000001e-59

   1. Initial program 13.4

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

   2. Taylor expanded in y.re around inf 32.1

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

   3. Simplified 30.1

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

      [Start] 32.1	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 32.1	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 32.1	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 32.1	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 32.1	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 30.1	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   if -2.0000000000000001e-59 < y.im < -1.7999999999999999e-197 or 6.90000000000000039e-152 < y.im < 2.39999999999999983e137

   1. Initial program 17.2

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

   2. Applied egg-rr 11.6

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

   if -1.7999999999999999e-197 < y.im < 6.90000000000000039e-152

   1. Initial program 24.8

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

   2. Taylor expanded in y.re around inf 8.9

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

   3. Simplified 6.9

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

      [Start] 8.9	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 8.9	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 8.9	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 8.9	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 8.9	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 6.9	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   4. Taylor expanded in x.im around 0 8.9

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

   5. Simplified 6.4

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

      [Start] 8.9	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 8.9	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      *-commutative [<=] 8.9	\[ \frac{x.im}{y.re} + \left(-\frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}}\right) \]
      unpow2 [=>] 8.9	\[ \frac{x.im}{y.re} + \left(-\frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}}\right) \]
      associate-*l/ [<=] 9.8	\[ \frac{x.im}{y.re} + \left(-\color{blue}{\frac{y.im}{y.re \cdot y.re} \cdot x.re}\right) \]
      sub-neg [<=] 9.8	\[ \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot y.re} \cdot x.re} \]
      associate-*l/ [=>] 8.9	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im \cdot x.re}{y.re \cdot y.re}} \]
      associate-/r* [=>] 4.2	\[ \frac{x.im}{y.re} - \color{blue}{\frac{\frac{y.im \cdot x.re}{y.re}}{y.re}} \]
      div-sub [<=] 4.2	\[ \color{blue}{\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}} \]
      associate-*r/ [<=] 6.4	\[ \frac{x.im - \color{blue}{y.im \cdot \frac{x.re}{y.re}}}{y.re} \]

3. Recombined 4 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2 \cdot 10^{-59}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-197}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 6.9 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{+137}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.7
Cost	7628

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.6 \cdot 10^{-150}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{+137}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	13.6
Cost	1356

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.3 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-150}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+137}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	15.1
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.52 \cdot 10^{+27} \lor \neg \left(y.im \leq 5.8 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 4
Error	23.8
Cost	908

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.75 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 75000:\\ \;\;\;\;\frac{-x.re}{\frac{y.re \cdot y.re}{y.im}}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{+35}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	23.7
Cost	908

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.4 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 75000:\\ \;\;\;\;y.im \cdot \frac{\frac{x.re}{y.re}}{-y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	23.6
Cost	908

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.05 \cdot 10^{-89}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 75000:\\ \;\;\;\;\frac{y.im}{y.re} \cdot \frac{-x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 5.3 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	18.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+43} \lor \neg \left(y.im \leq 6.3 \cdot 10^{+32}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Error	22.6
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.3 \cdot 10^{+37} \lor \neg \left(y.im \leq 1.5 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9
Error	37.9
Cost	192

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

Reproduce

herbie shell --seed 2023060 
(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))))