_divideComplex, imaginary part

Average Accuracy: 55.8% → 79.0%

Time: 18.9s

Precision: binary64

Cost: 14424

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-277}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-220}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

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)))
        (t_1 (- (/ x.im y.re) (/ (* y.im x.re) (* y.re y.re)))))
   (if (<= y.im -7.6e+69)
     (* (/ 1.0 (hypot y.re y.im)) (- x.re (* (/ y.re y.im) x.im)))
     (if (<= y.im -1.4e-45)
       t_0
       (if (<= y.im -1.15e-277)
         t_1
         (if (<= y.im 1e-220)
           (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
           (if (<= y.im 3.2e-156)
             t_1
             (if (<= y.im 7.5e+68)
               t_0
               (/ (- (/ y.re (/ y.im x.im)) x.re) (hypot y.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 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 t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -7.6e+69) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	} else if (y_46_im <= -1.4e-45) {
		tmp = t_0;
	} else if (y_46_im <= -1.15e-277) {
		tmp = t_1;
	} else if (y_46_im <= 1e-220) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_im <= 3.2e-156) {
		tmp = t_1;
	} else if (y_46_im <= 7.5e+68) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / hypot(y_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 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 t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -7.6e+69) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	} else if (y_46_im <= -1.4e-45) {
		tmp = t_0;
	} else if (y_46_im <= -1.15e-277) {
		tmp = t_1;
	} else if (y_46_im <= 1e-220) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_im <= 3.2e-156) {
		tmp = t_1;
	} else if (y_46_im <= 7.5e+68) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / Math.hypot(y_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):
	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)
	t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / (y_46_re * y_46_re))
	tmp = 0
	if y_46_im <= -7.6e+69:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re / y_46_im) * x_46_im))
	elif y_46_im <= -1.4e-45:
		tmp = t_0
	elif y_46_im <= -1.15e-277:
		tmp = t_1
	elif y_46_im <= 1e-220:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_im <= 3.2e-156:
		tmp = t_1
	elif y_46_im <= 7.5e+68:
		tmp = t_0
	else:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / math.hypot(y_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)
	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))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im * x_46_re) / Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_im <= -7.6e+69)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_re - Float64(Float64(y_46_re / y_46_im) * x_46_im)));
	elseif (y_46_im <= -1.4e-45)
		tmp = t_0;
	elseif (y_46_im <= -1.15e-277)
		tmp = t_1;
	elseif (y_46_im <= 1e-220)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_im <= 3.2e-156)
		tmp = t_1;
	elseif (y_46_im <= 7.5e+68)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / hypot(y_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)
	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);
	t_1 = (x_46_im / y_46_re) - ((y_46_im * x_46_re) / (y_46_re * y_46_re));
	tmp = 0.0;
	if (y_46_im <= -7.6e+69)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	elseif (y_46_im <= -1.4e-45)
		tmp = t_0;
	elseif (y_46_im <= -1.15e-277)
		tmp = t_1;
	elseif (y_46_im <= 1e-220)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_im <= 3.2e-156)
		tmp = t_1;
	elseif (y_46_im <= 7.5e+68)
		tmp = t_0;
	else
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / hypot(y_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_] := 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]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im * x$46$re), $MachinePrecision] / N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7.6e+69], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re - N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.4e-45], t$95$0, If[LessEqual[y$46$im, -1.15e-277], t$95$1, If[LessEqual[y$46$im, 1e-220], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.2e-156], t$95$1, If[LessEqual[y$46$im, 7.5e+68], t$95$0, N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -7.60000000000000055e69

   1. Initial program 41.8%

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

   2. Applied egg-rr 59.4%

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

      [Start] 41.8	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 41.8	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 41.8	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 41.9	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 41.9	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 59.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.im around -inf 78.8%

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

   4. Simplified 83.2%

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

      [Start] 78.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \frac{y.re \cdot x.im}{y.im} + x.re\right) \]
      +-commutative [=>] 78.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re + -1 \cdot \frac{y.re \cdot x.im}{y.im}\right)} \]
      mul-1-neg [=>] 78.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \color{blue}{\left(-\frac{y.re \cdot x.im}{y.im}\right)}\right) \]
      unsub-neg [=>] 78.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re - \frac{y.re \cdot x.im}{y.im}\right)} \]
      associate-/l* [=>] 83.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \color{blue}{\frac{y.re}{\frac{y.im}{x.im}}}\right) \]
      associate-/r/ [=>] 83.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \color{blue}{\frac{y.re}{y.im} \cdot x.im}\right) \]

   if -7.60000000000000055e69 < y.im < -1.4000000000000001e-45 or 3.19999999999999982e-156 < y.im < 7.49999999999999959e68

   1. Initial program 73.0%

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

   2. Applied egg-rr 81.2%

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

      [Start] 73.0	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 73.0	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 73.0	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 73.0	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 73.0	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 81.4%

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

      [Start] 81.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 81.4	\[ \color{blue}{\frac{1 \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      *-un-lft-identity [<=] 81.4	\[ \frac{\color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -1.4000000000000001e-45 < y.im < -1.15e-277 or 9.99999999999999992e-221 < y.im < 3.19999999999999982e-156

   1. Initial program 59.6%

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

   2. Taylor expanded in y.re around inf 69.9%

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

   3. Simplified 69.9%

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

      [Start] 69.9	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 69.9	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 69.9	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      unpow2 [=>] 69.9	\[ \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]

   if -1.15e-277 < y.im < 9.99999999999999992e-221

   1. Initial program 52.8%

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

   2. Taylor expanded in y.re around inf 74.9%

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

   3. Simplified 74.9%

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

      [Start] 74.9	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 74.9	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 74.9	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      unpow2 [=>] 74.9	\[ \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]

   4. Applied egg-rr 80.2%

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

      [Start] 74.9	\[ \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{y.re \cdot y.re} \]
      associate-/r* [=>] 82.1	\[ \frac{x.im}{y.re} - \color{blue}{\frac{\frac{x.re \cdot y.im}{y.re}}{y.re}} \]
      sub-div [=>] 82.1	\[ \color{blue}{\frac{x.im - \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
      *-commutative [=>] 82.1	\[ \frac{x.im - \frac{\color{blue}{y.im \cdot x.re}}{y.re}}{y.re} \]
      associate-/l* [=>] 80.2	\[ \frac{x.im - \color{blue}{\frac{y.im}{\frac{y.re}{x.re}}}}{y.re} \]

   if 7.49999999999999959e68 < y.im

   1. Initial program 41.4%

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

   2. Applied egg-rr 59.2%

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

      [Start] 41.4	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 41.4	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 41.4	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 41.4	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 41.4	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 59.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 59.3%

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

      [Start] 59.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 59.3	\[ \color{blue}{\frac{1 \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      *-un-lft-identity [<=] 59.3	\[ \frac{\color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   4. Taylor expanded in y.re around 0 76.8%

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

   5. Simplified 82.6%

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

      [Start] 76.8	\[ \frac{\frac{y.re \cdot x.im}{y.im} + -1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      +-commutative [=>] 76.8	\[ \frac{\color{blue}{-1 \cdot x.re + \frac{y.re \cdot x.im}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      mul-1-neg [=>] 76.8	\[ \frac{\color{blue}{\left(-x.re\right)} + \frac{y.re \cdot x.im}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-/l* [=>] 82.6	\[ \frac{\left(-x.re\right) + \color{blue}{\frac{y.re}{\frac{y.im}{x.im}}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

3. Recombined 5 regimes into one program.

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.6 \cdot 10^{+69}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 10^{-220}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+68}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	76.3%
Cost	7700

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+68}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.6 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Accuracy	76.4%
Cost	7700

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -1.45 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-277}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{-156}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+42}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 3
Accuracy	70.9%
Cost	1629

\[\begin{array}{l} t_0 := \frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 3.7 \cdot 10^{-73} \lor \neg \left(y.im \leq 4.2 \cdot 10^{-46}\right) \land y.im \leq 46000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Accuracy	70.9%
Cost	1628

\[\begin{array}{l} t_0 := \frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.9 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-44}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.15 \cdot 10^{-277}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.2 \cdot 10^{-45}:\\ \;\;\;\;\frac{y.re}{\frac{y.im \cdot y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 82000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Accuracy	76.2%
Cost	1620

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.4 \cdot 10^{-45}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.1 \cdot 10^{-277}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im \cdot x.re}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-155}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Accuracy	71.6%
Cost	1234

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+37} \lor \neg \left(y.im \leq 3.5 \cdot 10^{-73} \lor \neg \left(y.im \leq 6.3 \cdot 10^{-46}\right) \land y.im \leq 48000000\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \end{array} \]

Alternative 7
Accuracy	67.3%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+65} \lor \neg \left(y.im \leq 7.6 \cdot 10^{+65}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \end{array} \]

Alternative 8
Accuracy	60.5%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{+64} \lor \neg \left(y.im \leq 2.8 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	41.4%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.im \leq 3 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 10
Accuracy	40.2%
Cost	192

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

Reproduce

herbie shell --seed 2023153 
(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))))