_divideComplex, imaginary part

Average Accuracy: 59.8% → 84.8%

Time: 18.2s

Precision: binary64

Cost: 14160

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.68 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+143}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

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 -2.8e+63)
     (/ (- x.im (* y.im (/ x.re y.re))) y.re)
     (if (<= y.re -9.2e-170)
       t_0
       (if (<= y.re 1.68e-99)
         (/ (- (/ x.im (/ y.im y.re)) x.re) y.im)
         (if (<= y.re 6.8e+143)
           t_0
           (- (/ x.im y.re) (* (/ x.re y.re) (/ y.im y.re)))))))))

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 <= -2.8e+63) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -9.2e-170) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-99) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6.8e+143) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	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 <= -2.8e+63) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else if (y_46_re <= -9.2e-170) {
		tmp = t_0;
	} else if (y_46_re <= 1.68e-99) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_re <= 6.8e+143) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	}
	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 <= -2.8e+63:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	elif y_46_re <= -9.2e-170:
		tmp = t_0
	elif y_46_re <= 1.68e-99:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im
	elif y_46_re <= 6.8e+143:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re))
	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 <= -2.8e+63)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	elseif (y_46_re <= -9.2e-170)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-99)
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) / y_46_im);
	elseif (y_46_re <= 6.8e+143)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(x_46_re / y_46_re) * Float64(y_46_im / y_46_re)));
	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 <= -2.8e+63)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	elseif (y_46_re <= -9.2e-170)
		tmp = t_0;
	elseif (y_46_re <= 1.68e-99)
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_re <= 6.8e+143)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - ((x_46_re / y_46_re) * (y_46_im / y_46_re));
	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, -2.8e+63], 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, -9.2e-170], t$95$0, If[LessEqual[y$46$re, 1.68e-99], 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, 6.8e+143], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(x$46$re / y$46$re), $MachinePrecision] * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.79999999999999987e63

   1. Initial program 43.3%

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

   2. Taylor expanded in y.re around inf 72.6%

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

   3. Simplified 81.1%

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

      [Start] 72.6	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 72.6	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 72.6	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 72.6	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 72.6	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 81.1	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

   4. Taylor expanded in x.im around 0 72.6%

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

   5. Simplified 81.5%

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

      [Start] 72.6	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 72.6	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      *-commutative [<=] 72.6	\[ \frac{x.im}{y.re} + \left(-\frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}}\right) \]
      unpow2 [=>] 72.6	\[ \frac{x.im}{y.re} + \left(-\frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}}\right) \]
      associate-*l/ [<=] 73.3	\[ \frac{x.im}{y.re} + \left(-\color{blue}{\frac{y.im}{y.re \cdot y.re} \cdot x.re}\right) \]
      sub-neg [<=] 73.3	\[ \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot y.re} \cdot x.re} \]
      associate-*l/ [=>] 72.6	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im \cdot x.re}{y.re \cdot y.re}} \]
      associate-/r* [=>] 76.2	\[ \frac{x.im}{y.re} - \color{blue}{\frac{\frac{y.im \cdot x.re}{y.re}}{y.re}} \]
      div-sub [<=] 76.2	\[ \color{blue}{\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}} \]
      associate-*r/ [<=] 81.5	\[ \frac{x.im - \color{blue}{y.im \cdot \frac{x.re}{y.re}}}{y.re} \]

   if -2.79999999999999987e63 < y.re < -9.19999999999999948e-170 or 1.68000000000000007e-99 < y.re < 6.79999999999999964e143

   1. Initial program 74.3%

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

   2. Applied egg-rr 82.2%

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

      [Start] 74.3	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 74.3	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 74.3	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 74.3	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 74.3	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 82.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 82.4%

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

      [Start] 82.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 82.4	\[ \color{blue}{\frac{1 \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}} \]
      *-un-lft-identity [<=] 82.4	\[ \frac{\color{blue}{\frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -9.19999999999999948e-170 < y.re < 1.68000000000000007e-99

   1. Initial program 65.9%

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

   2. Applied egg-rr 80.9%

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

      [Start] 65.9	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 65.9	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 65.9	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 65.9	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 65.9	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 80.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.re around 0 84.4%

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

   4. Simplified 82.4%

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

      [Start] 84.4	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      fma-def [=>] 84.4	\[ \color{blue}{\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re \cdot x.im}{{y.im}^{2}}\right)} \]
      associate-/l* [=>] 79.1	\[ \mathsf{fma}\left(-1, \frac{x.re}{y.im}, \color{blue}{\frac{y.re}{\frac{{y.im}^{2}}{x.im}}}\right) \]
      unpow2 [=>] 79.1	\[ \mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re}{\frac{\color{blue}{y.im \cdot y.im}}{x.im}}\right) \]
      associate-/l* [=>] 82.4	\[ \mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re}{\color{blue}{\frac{y.im}{\frac{x.im}{y.im}}}}\right) \]

   5. Applied egg-rr 87.7%

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

      [Start] 82.4	\[ \mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}}\right) \]
      fma-udef [=>] 82.4	\[ \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}}} \]
      +-commutative [=>] 82.4	\[ \color{blue}{\frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}} + -1 \cdot \frac{x.re}{y.im}} \]
      div-inv [=>] 82.4	\[ \frac{y.re}{\frac{y.im}{\color{blue}{x.im \cdot \frac{1}{y.im}}}} + -1 \cdot \frac{x.re}{y.im} \]
      associate-/r* [=>] 82.4	\[ \frac{y.re}{\color{blue}{\frac{\frac{y.im}{x.im}}{\frac{1}{y.im}}}} + -1 \cdot \frac{x.re}{y.im} \]
      associate-/r/ [=>] 87.9	\[ \color{blue}{\frac{y.re}{\frac{y.im}{x.im}} \cdot \frac{1}{y.im}} + -1 \cdot \frac{x.re}{y.im} \]
      mul-1-neg [=>] 87.9	\[ \frac{y.re}{\frac{y.im}{x.im}} \cdot \frac{1}{y.im} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      div-inv [=>] 87.7	\[ \frac{y.re}{\frac{y.im}{x.im}} \cdot \frac{1}{y.im} + \left(-\color{blue}{x.re \cdot \frac{1}{y.im}}\right) \]
      distribute-lft-neg-in [=>] 87.7	\[ \frac{y.re}{\frac{y.im}{x.im}} \cdot \frac{1}{y.im} + \color{blue}{\left(-x.re\right) \cdot \frac{1}{y.im}} \]
      distribute-rgt-out [=>] 87.7	\[ \color{blue}{\frac{1}{y.im} \cdot \left(\frac{y.re}{\frac{y.im}{x.im}} + \left(-x.re\right)\right)} \]

   6. Simplified 89.5%

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

      [Start] 87.7	\[ \frac{1}{y.im} \cdot \left(\frac{y.re}{\frac{y.im}{x.im}} + \left(-x.re\right)\right) \]
      +-commutative [<=] 87.7	\[ \frac{1}{y.im} \cdot \color{blue}{\left(\left(-x.re\right) + \frac{y.re}{\frac{y.im}{x.im}}\right)} \]
      associate-*l/ [=>] 88.0	\[ \color{blue}{\frac{1 \cdot \left(\left(-x.re\right) + \frac{y.re}{\frac{y.im}{x.im}}\right)}{y.im}} \]
      *-lft-identity [=>] 88.0	\[ \frac{\color{blue}{\left(-x.re\right) + \frac{y.re}{\frac{y.im}{x.im}}}}{y.im} \]
      +-commutative [=>] 88.0	\[ \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}} + \left(-x.re\right)}}{y.im} \]
      unsub-neg [=>] 88.0	\[ \frac{\color{blue}{\frac{y.re}{\frac{y.im}{x.im}} - x.re}}{y.im} \]
      associate-/l* [<=] 90.2	\[ \frac{\color{blue}{\frac{y.re \cdot x.im}{y.im}} - x.re}{y.im} \]
      *-commutative [=>] 90.2	\[ \frac{\frac{\color{blue}{x.im \cdot y.re}}{y.im} - x.re}{y.im} \]
      associate-/l* [=>] 89.5	\[ \frac{\color{blue}{\frac{x.im}{\frac{y.im}{y.re}}} - x.re}{y.im} \]

   if 6.79999999999999964e143 < y.re

   1. Initial program 30.7%

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

   2. Taylor expanded in y.re around inf 75.0%

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

   3. Simplified 87.6%

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

      [Start] 75.0	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 75.0	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 75.0	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      *-commutative [=>] 75.0	\[ \frac{x.im}{y.re} - \frac{\color{blue}{y.im \cdot x.re}}{{y.re}^{2}} \]
      unpow2 [=>] 75.0	\[ \frac{x.im}{y.re} - \frac{y.im \cdot x.re}{\color{blue}{y.re \cdot y.re}} \]
      times-frac [=>] 87.6	\[ \frac{x.im}{y.re} - \color{blue}{\frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -9.2 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.68 \cdot 10^{-99}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+143}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	77.2%
Cost	14296

\[\begin{array}{l} t_0 := \left(y.re \cdot x.im - y.im \cdot x.re\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ t_1 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(-1, \frac{x.re}{y.im}, \frac{y.re}{\frac{y.im}{\frac{x.im}{y.im}}}\right)\\ \mathbf{elif}\;y.im \leq -7.5 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 8.2 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 330000000000:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - \frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \]

Alternative 2
Accuracy	82.1%
Cost	2000

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.45 \cdot 10^{-117}:\\ \;\;\;\;\frac{t_0}{t_1}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{-108}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+126}:\\ \;\;\;\;\frac{y.im \cdot x.re + \left(t_0 - y.im \cdot x.re\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 3
Accuracy	82.0%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.55 \cdot 10^{-117}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+123}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 4
Accuracy	73.7%
Cost	1369

\[\begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{+16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 850000000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+97} \lor \neg \left(y.im \leq 5.5 \cdot 10^{+114}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 5
Accuracy	73.4%
Cost	1168

\[\begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -2.45 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 150000000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - \frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \]

Alternative 6
Accuracy	73.3%
Cost	1168

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 560000000000:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\left(-y.im\right) - \frac{y.re}{\frac{y.im}{y.re}}}\\ \end{array} \]

Alternative 7
Accuracy	68.3%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.18 \cdot 10^{+173} \lor \neg \left(y.im \leq -1.6 \cdot 10^{+130}\right) \land \left(y.im \leq -6.8 \cdot 10^{+14} \lor \neg \left(y.im \leq 800000000000\right)\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Accuracy	68.8%
Cost	1105

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.18 \cdot 10^{+173}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -26000000000 \lor \neg \left(y.im \leq 67000000000\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	62.7%
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.08 \cdot 10^{-61}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	40.7%
Cost	192

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

Reproduce

herbie shell --seed 2023131 
(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))))