_divideComplex, imaginary part

Average Error: 26.1 → 10.3

Time: 12.8s

Precision: binary64

Cost: 14288

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+88}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 -5.5e+88)
     t_1
     (if (<= y.re -6.2e-71)
       t_0
       (if (<= y.re 2.2e-60)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 1.55e+130) 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 <= -5.5e+88) {
		tmp = t_1;
	} else if (y_46_re <= -6.2e-71) {
		tmp = t_0;
	} else if (y_46_re <= 2.2e-60) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.55e+130) {
		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 <= -5.5e+88) {
		tmp = t_1;
	} else if (y_46_re <= -6.2e-71) {
		tmp = t_0;
	} else if (y_46_re <= 2.2e-60) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.55e+130) {
		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 <= -5.5e+88:
		tmp = t_1
	elif y_46_re <= -6.2e-71:
		tmp = t_0
	elif y_46_re <= 2.2e-60:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 1.55e+130:
		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 <= -5.5e+88)
		tmp = t_1;
	elseif (y_46_re <= -6.2e-71)
		tmp = t_0;
	elseif (y_46_re <= 2.2e-60)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.55e+130)
		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 <= -5.5e+88)
		tmp = t_1;
	elseif (y_46_re <= -6.2e-71)
		tmp = t_0;
	elseif (y_46_re <= 2.2e-60)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.55e+130)
		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, -5.5e+88], t$95$1, If[LessEqual[y$46$re, -6.2e-71], t$95$0, If[LessEqual[y$46$re, 2.2e-60], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.55e+130], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -5.5e88 or 1.55e130 < y.re

   1. Initial program 39.5

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

   2. Taylor expanded in y.re around inf 15.5

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

   3. Simplified 9.3

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]
      Proof
      (-.f64 (/.f64 x.im y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.re y.re))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.re) (*.f64 y.re y.re)))): 0 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.im)) (*.f64 y.re y.re))): 0 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (*.f64 x.re y.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 6 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 x.im y.re) (neg.f64 (/.f64 (*.f64 x.re y.im) (pow.f64 y.re 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 x.im y.re) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re y.im) (pow.f64 y.re 2))))): 1 points increase in error, 0 points decrease in error

   if -5.5e88 < y.re < -6.20000000000000004e-71 or 2.1999999999999999e-60 < y.re < 1.55e130

   1. Initial program 16.2

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

   2. Applied egg-rr 11.9

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

   if -6.20000000000000004e-71 < y.re < 2.1999999999999999e-60

   1. Initial program 21.4

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

   2. Taylor expanded in y.re around 0 12.6

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

   3. Simplified 11.7

      \[\leadsto \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}} \]
      Proof
      (-.f64 (/.f64 x.im y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.re y.re))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.re) (*.f64 y.re y.re)))): 0 points increase in error, 1 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.im)) (*.f64 y.re y.re))): 0 points increase in error, 6 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (*.f64 x.re y.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 6 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 x.im y.re) (neg.f64 (/.f64 (*.f64 x.re y.im) (pow.f64 y.re 2))))): 0 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 x.im y.re) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re y.im) (pow.f64 y.re 2))))): 1 points increase in error, 0 points decrease in error

   4. Applied egg-rr 10.0

      \[\leadsto \color{blue}{\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 10.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.5 \cdot 10^{+88}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -6.2 \cdot 10^{-71}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{-60}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+130}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.5
Cost	7760

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.3 \cdot 10^{-71}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-63}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.35 \cdot 10^{+33}:\\ \;\;\;\;\frac{t_0}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 2
Error	12.4
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.6 \cdot 10^{-70}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 3
Error	15.1
Cost	1233

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -980000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+30} \lor \neg \left(y.re \leq 6.8 \cdot 10^{+80}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 4
Error	15.1
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -1360000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 5
Error	15.2
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1120000:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-59}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 10^{+31}:\\ \;\;\;\;\frac{y.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.re \leq 3.55 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 6
Error	23.4
Cost	1108

\[\begin{array}{l} t_0 := -\frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -7.2 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{-51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 7
Error	15.4
Cost	1106

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1750000 \lor \neg \left(y.re \leq 1.4 \cdot 10^{-52}\right) \land \left(y.re \leq 3.9 \cdot 10^{+31} \lor \neg \left(y.re \leq 3.5 \cdot 10^{+78}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 8
Error	15.1
Cost	1105

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -520000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+31} \lor \neg \left(y.re \leq 1.6 \cdot 10^{+82}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \end{array} \]

Alternative 9
Error	18.1
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.2 \cdot 10^{-8} \lor \neg \left(y.re \leq 1.95 \cdot 10^{-53}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 10
Error	22.4
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{-51}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Error	36.0
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.im \leq 2.1 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 12
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2022343 
(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))))