_divideComplex, imaginary part

Average Error: 58.8% → 97.5%

Time: 28.3s

Precision: binary64

Cost: 20617.00

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-208} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-145}\right):\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]

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
 (if (or (<= y.re -3.4e-208) (not (<= y.re 1.4e-145)))
   (*
    (/ 1.0 (hypot y.re y.im))
    (-
     (* y.re (/ x.im (hypot y.re y.im)))
     (* (/ y.im (hypot y.re y.im)) x.re)))
   (- (/ x.im (* y.im (/ y.im y.re))) (/ x.re y.im))))

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 tmp;
	if ((y_46_re <= -3.4e-208) || !(y_46_re <= 1.4e-145)) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - ((y_46_im / hypot(y_46_re, y_46_im)) * x_46_re));
	} else {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	}
	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 tmp;
	if ((y_46_re <= -3.4e-208) || !(y_46_re <= 1.4e-145)) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / Math.hypot(y_46_re, y_46_im))) - ((y_46_im / Math.hypot(y_46_re, y_46_im)) * x_46_re));
	} else {
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	}
	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):
	tmp = 0
	if (y_46_re <= -3.4e-208) or not (y_46_re <= 1.4e-145):
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / math.hypot(y_46_re, y_46_im))) - ((y_46_im / math.hypot(y_46_re, y_46_im)) * x_46_re))
	else:
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im)
	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)
	tmp = 0.0
	if ((y_46_re <= -3.4e-208) || !(y_46_re <= 1.4e-145))
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(y_46_re * Float64(x_46_im / hypot(y_46_re, y_46_im))) - Float64(Float64(y_46_im / hypot(y_46_re, y_46_im)) * x_46_re)));
	else
		tmp = Float64(Float64(x_46_im / Float64(y_46_im * Float64(y_46_im / y_46_re))) - Float64(x_46_re / y_46_im));
	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)
	tmp = 0.0;
	if ((y_46_re <= -3.4e-208) || ~((y_46_re <= 1.4e-145)))
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * ((y_46_re * (x_46_im / hypot(y_46_re, y_46_im))) - ((y_46_im / hypot(y_46_re, y_46_im)) * x_46_re));
	else
		tmp = (x_46_im / (y_46_im * (y_46_im / y_46_re))) - (x_46_re / y_46_im);
	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_] := If[Or[LessEqual[y$46$re, -3.4e-208], N[Not[LessEqual[y$46$re, 1.4e-145]], $MachinePrecision]], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(y$46$re * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(y$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.4e-208 or 1.4000000000000001e-145 < y.re

   1. Initial program 58.2

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

   2. Applied egg-rr 71.1

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

   3. Applied egg-rr 99.0

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

   if -3.4e-208 < y.re < 1.4000000000000001e-145

   1. Initial program 61.5

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

   2. Taylor expanded in y.re around 0 84.5

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

   3. Simplified 83.5

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

      [Start] 84.5	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 84.5	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 84.5	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 84.5	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      associate-/l* [=>] 79.1	\[ \color{blue}{\frac{y.re}{\frac{{y.im}^{2}}{x.im}}} - \frac{x.re}{y.im} \]
      associate-/r/ [=>] 83.5	\[ \color{blue}{\frac{y.re}{{y.im}^{2}} \cdot x.im} - \frac{x.re}{y.im} \]
      unpow2 [=>] 83.5	\[ \frac{y.re}{\color{blue}{y.im \cdot y.im}} \cdot x.im - \frac{x.re}{y.im} \]

   4. Applied egg-rr 91.0

      \[\leadsto \color{blue}{\frac{x.im}{\frac{y.im}{y.re} \cdot y.im}} - \frac{x.re}{y.im} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-208} \lor \neg \left(y.re \leq 1.4 \cdot 10^{-145}\right):\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	99.1%
Cost	20352.00

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

Alternative 2
Error	83.2%
Cost	17744.00

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-316}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \end{array} \]

Alternative 3
Error	89.2%
Cost	15688.00

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ t_2 := \mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_2} - \frac{x.re}{\frac{t_2}{y.im}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \end{array} \]

Alternative 4
Error	88.4%
Cost	14660.00

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \end{array} \]

Alternative 5
Error	83.2%
Cost	14292.00

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -9.2 \cdot 10^{+152}:\\ \;\;\;\;t_0 \cdot \mathsf{fma}\left(-1, x.im, \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-101}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-161}:\\ \;\;\;\;\frac{-y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}\\ \mathbf{elif}\;y.re \leq -2.05 \cdot 10^{-178}:\\ \;\;\;\;t_0 \cdot \left(y.re \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)} - x.re\right)\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(x.im - \frac{y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\\ \end{array} \]

Alternative 6
Error	80.2%
Cost	13700.00

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(-1, x.im, \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \mathbf{elif}\;y.re \leq -6.5 \cdot 10^{-107}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-98}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 0.88:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.95 \cdot 10^{+48}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 7
Error	80.1%
Cost	1488.00

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-105}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{-94}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 0.35:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+45}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 8
Error	74.2%
Cost	1234.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{-13} \lor \neg \left(y.re \leq 3.6 \cdot 10^{-92}\right) \land \left(y.re \leq 0.32 \lor \neg \left(y.re \leq 1.6 \cdot 10^{+45}\right)\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 9
Error	73.0%
Cost	1233.00

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -5 \cdot 10^{-14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.im \cdot y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 0.46 \lor \neg \left(y.re \leq 1.62 \cdot 10^{+40}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 10
Error	72.9%
Cost	1233.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.65 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.im \cdot y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 0.44 \lor \neg \left(y.re \leq 5.6 \cdot 10^{+41}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 11
Error	74.4%
Cost	1233.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im}{y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 0.5 \lor \neg \left(y.re \leq 1.3 \cdot 10^{+46}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 12
Error	75.0%
Cost	1233.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.6 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-92}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 0.205 \lor \neg \left(y.re \leq 7.6 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 13
Error	69.7%
Cost	1106.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{-18} \lor \neg \left(y.re \leq 6.2 \cdot 10^{-80} \lor \neg \left(y.re \leq 0.205\right) \land y.re \leq 1.15 \cdot 10^{+34}\right):\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \end{array} \]

Alternative 14
Error	63.9%
Cost	520.00

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{+82}:\\ \;\;\;\;-\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 15
Error	8.5%
Cost	192.00

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

Alternative 16
Error	41.8%
Cost	192.00

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

Reproduce

herbie shell --seed 2023093 
(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))))