_divideComplex, imaginary part

Average Error: 26.7 → 9.8

Time: 17.1s

Precision: binary64

Cost: 14160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.im}{-y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - {\left(\frac{y.re}{x.re \cdot \frac{y.im}{y.re}}\right)}^{-1}\\ \end{array} \]

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
         (/
          (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.re -4.6e+139)
     (/ (- x.im (* y.im (/ x.re y.re))) y.re)
     (if (<= y.re -1.7e-157)
       t_0
       (if (<= y.re 1.6e-58)
         (/ (- (/ (- x.im) (/ y.im (- y.re))) x.re) y.im)
         (if (<= y.re 1.9e+148)
           t_0
           (- (/ x.im y.re) (pow (/ y.re (* x.re (/ y.im y.re))) -1.0))))))))

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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -4.6e+139) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -1.7e-157) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-58) {
		tmp = ((-x_46_im / (y_46_im / -y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.9e+148) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - pow((y_46_re / (x_46_re * (y_46_im / y_46_re))), -1.0);
	}
	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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_re <= -4.6e+139) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -1.7e-157) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-58) {
		tmp = ((-x_46_im / (y_46_im / -y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.9e+148) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - Math.pow((y_46_re / (x_46_re * (y_46_im / y_46_re))), -1.0);
	}
	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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	tmp = 0
	if y_46_re <= -4.6e+139:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= -1.7e-157:
		tmp = t_0
	elif y_46_re <= 1.6e-58:
		tmp = ((-x_46_im / (y_46_im / -y_46_re)) - x_46_re) / y_46_im
	elif y_46_re <= 1.9e+148:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - math.pow((y_46_re / (x_46_re * (y_46_im / y_46_re))), -1.0)
	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(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_re <= -4.6e+139)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= -1.7e-157)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-58)
		tmp = Float64(Float64(Float64(Float64(-x_46_im) / Float64(y_46_im / Float64(-y_46_re))) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.9e+148)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - (Float64(y_46_re / Float64(x_46_re * Float64(y_46_im / y_46_re))) ^ -1.0));
	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 = (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	tmp = 0.0;
	if (y_46_re <= -4.6e+139)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= -1.7e-157)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-58)
		tmp = ((-x_46_im / (y_46_im / -y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.9e+148)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - ((y_46_re / (x_46_re * (y_46_im / y_46_re))) ^ -1.0);
	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[(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] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.6e+139], 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$46$re, -1.7e-157], t$95$0, If[LessEqual[y$46$re, 1.6e-58], N[(N[(N[((-x$46$im) / N[(y$46$im / (-y$46$re)), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+148], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[Power[N[(y$46$re / N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -4.6e139

   1. Initial program 43.6

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

   2. Taylor expanded in y.re around inf 13.4

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

   3. Simplified 6.5

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

      [Start] 13.4	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 13.4	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 13.4	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 13.4	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 13.4	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 6.5	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   4. Taylor expanded in x.im around 0 13.4

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

   5. Simplified 6.3

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

      [Start] 13.4	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 13.4	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      *-commutative [<=] 13.4	\[ \frac{x.im}{y.re} + \left(-\frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}}\right) \]
      unpow2 [=>] 13.4	\[ \frac{x.im}{y.re} + \left(-\frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}}\right) \]
      associate-*l/ [<=] 12.1	\[ \frac{x.im}{y.re} + \left(-\color{blue}{\frac{y.im}{y.re \cdot y.re} \cdot x.re}\right) \]
      sub-neg [<=] 12.1	\[ \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot y.re} \cdot x.re} \]
      associate-*l/ [=>] 13.4	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im \cdot x.re}{y.re \cdot y.re}} \]
      associate-/r* [=>] 11.1	\[ \frac{x.im}{y.re} - \color{blue}{\frac{\frac{y.im \cdot x.re}{y.re}}{y.re}} \]
      div-sub [<=] 11.1	\[ \color{blue}{\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}} \]
      associate-*r/ [<=] 6.3	\[ \frac{x.im - \color{blue}{y.im \cdot \frac{x.re}{y.re}}}{y.re} \]

   if -4.6e139 < y.re < -1.69999999999999989e-157 or 1.6e-58 < y.re < 1.8999999999999999e148

   1. Initial program 18.0

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

   2. Applied egg-rr 13.1

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

   3. Applied egg-rr 13.0

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

   if -1.69999999999999989e-157 < y.re < 1.6e-58

   1. Initial program 22.0

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

   2. Taylor expanded in y.re around 0 11.4

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

   3. Simplified 9.7

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

      [Start] 11.4	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 11.4	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 11.4	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 11.4	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      *-commutative [=>] 11.4	\[ \frac{\color{blue}{x.im \cdot y.re}}{{y.im}^{2}} - \frac{x.re}{y.im} \]
      unpow2 [=>] 11.4	\[ \frac{x.im \cdot y.re}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 9.7	\[ \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}} - \frac{x.re}{y.im} \]

   4. Applied egg-rr 8.1

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

   5. Applied egg-rr 7.9

      \[\leadsto \frac{\color{blue}{\left(-\frac{x.im}{\frac{y.im}{-y.re}}\right)} - x.re}{y.im} \]

   if 1.8999999999999999e148 < y.re

   1. Initial program 45.0

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

   2. Simplified 45.0

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

      [Start] 45.0	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-def [=>] 45.0	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]

   3. Taylor expanded in y.re around inf 15.6

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

   4. Simplified 14.1

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

      [Start] 15.6	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 15.6	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      associate-/l* [=>] 14.1	\[ \frac{x.im}{y.re} + \left(-\color{blue}{\frac{x.re}{\frac{{y.re}^{2}}{y.im}}}\right) \]
      unpow2 [=>] 14.1	\[ \frac{x.im}{y.re} + \left(-\frac{x.re}{\frac{\color{blue}{y.re \cdot y.re}}{y.im}}\right) \]

   5. Applied egg-rr 8.2

      \[\leadsto \frac{x.im}{y.re} + \left(-\color{blue}{{\left(\frac{y.re}{x.re \cdot \frac{y.im}{y.re}}\right)}^{-1}}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 9.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-157}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{-x.im}{\frac{y.im}{-y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+148}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - {\left(\frac{y.re}{x.re \cdot \frac{y.im}{y.re}}\right)}^{-1}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.1
Cost	7564

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.4 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{x.im}{y.re} - {\left(\frac{y.re}{x.re \cdot \frac{y.im}{y.re}}\right)}^{-1}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+146}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 2
Error	12.1
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -5.9 \cdot 10^{+121}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.85 \cdot 10^{-127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-152}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{+147}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 3
Error	16.0
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -320000000:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 860:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	16.2
Cost	1105

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -510000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-54} \lor \neg \left(y.im \leq 600\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5
Error	15.9
Cost	1104

\[\begin{array}{l} t_0 := \frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -25000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1050:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	18.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.95 \cdot 10^{-52} \lor \neg \left(y.re \leq 4.6 \cdot 10^{-26}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 7
Error	22.2
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 230:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 8
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2023018 
(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))))