_divideComplex, imaginary part

Average Error: 26.3 → 11.3

Time: 13.2s

Precision: binary64

Cost: 14552

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := t_0 \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -3.15 \cdot 10^{+47}:\\ \;\;\;\;t_0 \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-165}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 5.9 \cdot 10^{-179}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-28}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re \cdot \frac{y.re}{-y.im} - y.im}\\ \end{array} \]

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
 (let* ((t_0 (/ 1.0 (hypot y.re y.im)))
        (t_1 (* t_0 (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im)))))
   (if (<= y.im -3.15e+47)
     (* t_0 (- x.re (* (/ y.re y.im) x.im)))
     (if (<= y.im -1.65e-165)
       t_1
       (if (<= y.im 5.9e-179)
         (/ (- x.im (* x.re (/ y.im y.re))) y.re)
         (if (<= y.im 6.2e-28)
           t_1
           (if (<= y.im 3.1e-16)
             (/ x.im y.re)
             (if (<= y.im 1.45e+68)
               t_1
               (/ x.re (- (* y.re (/ y.re (- y.im))) 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 = 1.0 / hypot(y_46_re, y_46_im);
	double t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -3.15e+47) {
		tmp = t_0 * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	} else if (y_46_im <= -1.65e-165) {
		tmp = t_1;
	} else if (y_46_im <= 5.9e-179) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 6.2e-28) {
		tmp = t_1;
	} else if (y_46_im <= 3.1e-16) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.45e+68) {
		tmp = t_1;
	} else {
		tmp = x_46_re / ((y_46_re * (y_46_re / -y_46_im)) - 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 = 1.0 / Math.hypot(y_46_re, y_46_im);
	double t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / Math.hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_im <= -3.15e+47) {
		tmp = t_0 * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	} else if (y_46_im <= -1.65e-165) {
		tmp = t_1;
	} else if (y_46_im <= 5.9e-179) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 6.2e-28) {
		tmp = t_1;
	} else if (y_46_im <= 3.1e-16) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_im <= 1.45e+68) {
		tmp = t_1;
	} else {
		tmp = x_46_re / ((y_46_re * (y_46_re / -y_46_im)) - 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 = 1.0 / math.hypot(y_46_re, y_46_im)
	t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / math.hypot(y_46_re, y_46_im))
	tmp = 0
	if y_46_im <= -3.15e+47:
		tmp = t_0 * (x_46_re - ((y_46_re / y_46_im) * x_46_im))
	elif y_46_im <= -1.65e-165:
		tmp = t_1
	elif y_46_im <= 5.9e-179:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 6.2e-28:
		tmp = t_1
	elif y_46_im <= 3.1e-16:
		tmp = x_46_im / y_46_re
	elif y_46_im <= 1.45e+68:
		tmp = t_1
	else:
		tmp = x_46_re / ((y_46_re * (y_46_re / -y_46_im)) - 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(1.0 / hypot(y_46_re, y_46_im))
	t_1 = Float64(t_0 * Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_im <= -3.15e+47)
		tmp = Float64(t_0 * Float64(x_46_re - Float64(Float64(y_46_re / y_46_im) * x_46_im)));
	elseif (y_46_im <= -1.65e-165)
		tmp = t_1;
	elseif (y_46_im <= 5.9e-179)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 6.2e-28)
		tmp = t_1;
	elseif (y_46_im <= 3.1e-16)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_im <= 1.45e+68)
		tmp = t_1;
	else
		tmp = Float64(x_46_re / Float64(Float64(y_46_re * Float64(y_46_re / Float64(-y_46_im))) - 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 = 1.0 / hypot(y_46_re, y_46_im);
	t_1 = t_0 * (((y_46_re * x_46_im) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	tmp = 0.0;
	if (y_46_im <= -3.15e+47)
		tmp = t_0 * (x_46_re - ((y_46_re / y_46_im) * x_46_im));
	elseif (y_46_im <= -1.65e-165)
		tmp = t_1;
	elseif (y_46_im <= 5.9e-179)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 6.2e-28)
		tmp = t_1;
	elseif (y_46_im <= 3.1e-16)
		tmp = x_46_im / y_46_re;
	elseif (y_46_im <= 1.45e+68)
		tmp = t_1;
	else
		tmp = x_46_re / ((y_46_re * (y_46_re / -y_46_im)) - 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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 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]}, If[LessEqual[y$46$im, -3.15e+47], N[(t$95$0 * N[(x$46$re - N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.65e-165], t$95$1, If[LessEqual[y$46$im, 5.9e-179], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 6.2e-28], t$95$1, If[LessEqual[y$46$im, 3.1e-16], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.45e+68], t$95$1, N[(x$46$re / N[(N[(y$46$re * N[(y$46$re / (-y$46$im)), $MachinePrecision]), $MachinePrecision] - y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.15000000000000002e47

   1. Initial program 35.8

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

   2. Applied egg-rr 24.3

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

   3. Taylor expanded in y.im around -inf 15.8

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

   4. Simplified 13.0

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(x.re - \frac{y.re}{y.im} \cdot x.im\right)} \]
      Proof
      (-.f64 x.re (*.f64 (/.f64 y.re y.im) x.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 x.re (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 y.im x.im)))): 19 points increase in error, 27 points decrease in error
      (-.f64 x.re (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.im) y.im))): 27 points increase in error, 19 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 x.re (neg.f64 (/.f64 (*.f64 y.re x.im) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 x.re (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 y.re x.im) y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y.re x.im) y.im)) x.re)): 0 points increase in error, 0 points decrease in error

   if -3.15000000000000002e47 < y.im < -1.6499999999999999e-165 or 5.90000000000000029e-179 < y.im < 6.19999999999999984e-28 or 3.1000000000000001e-16 < y.im < 1.45000000000000006e68

   1. Initial program 15.8

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

   2. Applied egg-rr 10.5

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

   if -1.6499999999999999e-165 < y.im < 5.90000000000000029e-179

   1. Initial program 24.2

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

   2. Taylor expanded in y.re around inf 9.0

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

   3. Simplified 7.0

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}} \]
      Proof
      (-.f64 (/.f64 x.im y.re) (*.f64 (/.f64 x.re y.re) (/.f64 y.im y.re))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x.re y.im) (*.f64 y.re y.re)))): 37 points increase in error, 11 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (*.f64 x.re y.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 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, 0 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))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in x.im around 0 9.0

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

   5. Simplified 4.8

      \[\leadsto \color{blue}{\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}} \]
      Proof
      (/.f64 (-.f64 x.im (*.f64 x.re (/.f64 y.im y.re))) y.re): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 x.im y.re) (/.f64 (*.f64 x.re (/.f64 y.im y.re)) y.re))): 1 points increase in error, 2 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 x.re y.re) (/.f64 y.im y.re)))): 11 points increase in error, 11 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x.re y.im) (*.f64 y.re y.re)))): 37 points increase in error, 11 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 x.im y.re) (neg.f64 (/.f64 (*.f64 x.re y.im) (*.f64 y.re y.re))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.re) (neg.f64 (/.f64 (*.f64 x.re y.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))))): 0 points increase in error, 0 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))))): 0 points increase in error, 0 points decrease in error

   if 6.19999999999999984e-28 < y.im < 3.1000000000000001e-16

   1. Initial program 12.8

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

   2. Taylor expanded in y.re around inf 37.1

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

   if 1.45000000000000006e68 < y.im

   1. Initial program 36.2

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

   2. Taylor expanded in x.im around 0 40.2

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

   3. Simplified 36.8

      \[\leadsto \color{blue}{\frac{x.re}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{-y.im}}} \]
      Proof
      (/.f64 x.re (/.f64 (fma.f64 y.im y.im (*.f64 y.re y.re)) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 x.re (/.f64 (fma.f64 y.im y.im (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 x.re (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.im y.im) (pow.f64 y.re 2))) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 x.re (/.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) (pow.f64 y.re 2)) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 x.re (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.re (neg.f64 y.im)) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2)))): 54 points increase in error, 16 points decrease in error
      (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x.re y.im))) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (*.f64 x.re y.im))) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (*.f64 x.re y.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))))): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y.im around 0 19.7

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

   5. Simplified 19.7

      \[\leadsto \frac{x.re}{\color{blue}{\frac{y.re \cdot \left(-y.re\right)}{y.im} - y.im}} \]
      Proof
      (-.f64 (/.f64 (*.f64 y.re (neg.f64 y.re)) y.im) y.im): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 y.re y.re))) y.im) y.im): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (neg.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) y.im) y.im): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 (pow.f64 y.re 2) y.im))) y.im): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 y.re 2) y.im))) y.im): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 y.re 2) y.im)) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 (/.f64 (pow.f64 y.re 2) y.im)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 y.im))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in y.re around 0 19.7

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

   7. Simplified 15.5

      \[\leadsto \frac{x.re}{\color{blue}{y.re \cdot \frac{y.re}{-y.im}} - y.im} \]
      Proof
      (*.f64 y.re (/.f64 y.re (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (*.f64 y.re (/.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y.re 1)) (neg.f64 y.im))): 0 points increase in error, 0 points decrease in error
      (*.f64 y.re (Rewrite<= associate-*r/_binary64 (*.f64 y.re (/.f64 1 (neg.f64 y.im))))): 15 points increase in error, 9 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y.re y.re) (/.f64 1 (neg.f64 y.im)))): 31 points increase in error, 22 points decrease in error
      (Rewrite=> *-commutative_binary64 (*.f64 (/.f64 1 (neg.f64 y.im)) (*.f64 y.re y.re))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 1 (*.f64 y.re y.re)) (neg.f64 y.im))): 9 points increase in error, 10 points decrease in error
      (/.f64 (*.f64 1 (*.f64 y.re y.re)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> times-frac_binary64 (*.f64 (/.f64 1 -1) (/.f64 (*.f64 y.re y.re) y.im))): 0 points increase in error, 0 points decrease in error
      (*.f64 (Rewrite=> metadata-eval -1) (/.f64 (*.f64 y.re y.re) y.im)): 0 points increase in error, 0 points decrease in error
      (*.f64 -1 (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) y.im)): 0 points increase in error, 0 points decrease in error
3. Recombined 5 regimes into one program.

4. Final simplification 11.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.15 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-165}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 5.9 \cdot 10^{-179}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+68}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re \cdot \frac{y.re}{-y.im} - y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.9
Cost	7824

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{t_0}\\ \mathbf{if}\;y.im \leq -3.15 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-134}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.02 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{-33}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.re, x.im, y.im \cdot \left(-x.re\right)\right)}{t_0}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+63}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re \cdot \frac{y.re}{-y.im} - y.im}\\ \end{array} \]

Alternative 2
Error	12.9
Cost	7300

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.65 \cdot 10^{+47}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re - \frac{y.re}{y.im} \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq -2.8 \cdot 10^{-133}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-28}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re \cdot \frac{y.re}{-y.im} - y.im}\\ \end{array} \]

Alternative 3
Error	14.0
Cost	1356

\[\begin{array}{l} \mathbf{if}\;y.re \leq -7 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+106}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 4
Error	16.7
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -5.4 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{+43}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	16.8
Cost	1104

\[\begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.95 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.4 \cdot 10^{+64}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 6
Error	19.4
Cost	840

\[\begin{array}{l} t_0 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.85 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{-82}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.85 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.52 \cdot 10^{-81}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]

Alternative 8
Error	23.5
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-26}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9
Error	37.2
Cost	192

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

Reproduce

herbie shell --seed 2022328 
(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))))