_divideComplex, imaginary part

Average Error: 26.2 → 12.6

Time: 16.3s

Precision: binary64

Cost: 14552

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+144}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))
          (/ (- (* x.im y.re) (* y.im x.re)) (hypot y.re y.im))))
        (t_1 (- (* (/ x.im y.im) (/ y.re y.im)) (/ x.re y.im))))
   (if (<= y.im -4e+152)
     t_1
     (if (<= y.im -1.7e-25)
       t_0
       (if (<= y.im 1.4e-78)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 1.7e+37)
           t_0
           (if (<= y.im 2.5e+107)
             (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
             (if (<= y.im 1.1e+144) 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)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	double t_1 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4e+152) {
		tmp = t_1;
	} else if (y_46_im <= -1.7e-25) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-78) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.7e+37) {
		tmp = t_0;
	} else if (y_46_im <= 2.5e+107) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= 1.1e+144) {
		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)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im));
	double t_1 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4e+152) {
		tmp = t_1;
	} else if (y_46_im <= -1.7e-25) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e-78) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.7e+37) {
		tmp = t_0;
	} else if (y_46_im <= 2.5e+107) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= 1.1e+144) {
		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)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im))
	t_1 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -4e+152:
		tmp = t_1
	elif y_46_im <= -1.7e-25:
		tmp = t_0
	elif y_46_im <= 1.4e-78:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 1.7e+37:
		tmp = t_0
	elif y_46_im <= 2.5e+107:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	elif y_46_im <= 1.1e+144:
		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(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)))
	t_1 = Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re / y_46_im)) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4e+152)
		tmp = t_1;
	elseif (y_46_im <= -1.7e-25)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-78)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.7e+37)
		tmp = t_0;
	elseif (y_46_im <= 2.5e+107)
		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 <= 1.1e+144)
		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)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	t_1 = ((x_46_im / y_46_im) * (y_46_re / y_46_im)) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4e+152)
		tmp = t_1;
	elseif (y_46_im <= -1.7e-25)
		tmp = t_0;
	elseif (y_46_im <= 1.4e-78)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 1.7e+37)
		tmp = t_0;
	elseif (y_46_im <= 2.5e+107)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	elseif (y_46_im <= 1.1e+144)
		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[(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]}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$im, -4e+152], t$95$1, If[LessEqual[y$46$im, -1.7e-25], t$95$0, If[LessEqual[y$46$im, 1.4e-78], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.7e+37], t$95$0, If[LessEqual[y$46$im, 2.5e+107], 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, 1.1e+144], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -4.0000000000000002e152 or 1.09999999999999994e144 < y.im

   1. Initial program 44.9

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

   2. Applied egg-rr 44.9

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

   3. Taylor expanded in y.re around 0 15.0

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

   4. Simplified 6.9

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

      [Start] 15.0	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 15.0	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 15.0	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 15.0	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 15.0	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 15.0	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 6.9	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   if -4.0000000000000002e152 < y.im < -1.70000000000000001e-25 or 1.40000000000000012e-78 < y.im < 1.70000000000000003e37 or 2.5000000000000001e107 < y.im < 1.09999999999999994e144

   1. Initial program 17.8

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

   2. Applied egg-rr 13.0

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

   if -1.70000000000000001e-25 < y.im < 1.40000000000000012e-78

   1. Initial program 19.7

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

   2. Taylor expanded in y.re around inf 14.9

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

   3. Simplified 13.4

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

      [Start] 14.9	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 14.9	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 14.9	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 14.9	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 14.9	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 13.4	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   4. Applied egg-rr 11.8

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

   if 1.70000000000000003e37 < y.im < 2.5000000000000001e107

   1. Initial program 18.4

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

   2. Taylor expanded in y.re around inf 45.5

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

   3. Simplified 41.3

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

      [Start] 45.5	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 45.5	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 45.5	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 45.5	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 45.5	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 41.3	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 12.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+152}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.8
Cost	14296

\[\begin{array}{l} t_0 := x.im \cdot y.re - y.im \cdot x.re\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.7 \cdot 10^{+121}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{t_1}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7.4 \cdot 10^{+141}:\\ \;\;\;\;t_0 \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	13.7
Cost	7560

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im \cdot y.re - y.im \cdot x.re}{t_0}\\ t_2 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+117}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{t_0}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 5.4 \cdot 10^{+139}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	13.7
Cost	1752

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.36 \cdot 10^{+116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.95 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+140}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.5
Cost	1233

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3 \cdot 10^{-23}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{+87} \lor \neg \left(y.im \leq 2.55 \cdot 10^{+107}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5
Error	15.8
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+89}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.im \cdot y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	15.8
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+88}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.im \cdot y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	20.0
Cost	1106

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-21} \lor \neg \left(y.im \leq 1.32 \cdot 10^{+16}\right) \land \left(y.im \leq 7 \cdot 10^{+69} \lor \neg \left(y.im \leq 2.3 \cdot 10^{+108}\right)\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Error	19.7
Cost	1105

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{+14}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.6 \cdot 10^{+65} \lor \neg \left(y.im \leq 9 \cdot 10^{+108}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 9
Error	23.7
Cost	786

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.7 \cdot 10^{-22} \lor \neg \left(y.im \leq 2.9 \cdot 10^{-46} \lor \neg \left(y.im \leq 10^{+89}\right) \land y.im \leq 3.8 \cdot 10^{+108}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Error	38.1
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(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))))