_divideComplex, imaginary part

Average Accuracy: 56.2% → 84.8%

Time: 19.5s

Precision: binary64

Cost: 27344

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}\\ t_1 := t_0 - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+160}:\\ \;\;\;\;t_0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.42 \cdot 10^{-92}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+143}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

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
 (let* ((t_0 (* (/ 1.0 (hypot y.re y.im)) (/ y.re (/ (hypot y.re y.im) x.im))))
        (t_1 (- t_0 (/ y.im (/ (pow (hypot y.re y.im) 2.0) x.re)))))
   (if (<= y.im -1e+160)
     (- t_0 (/ x.re y.im))
     (if (<= y.im -1.42e-92)
       t_1
       (if (<= y.im 5.1e-138)
         (- (/ x.im y.re) (/ (/ (* y.im x.re) y.re) y.re))
         (if (<= y.im 9.5e+143)
           t_1
           (- (/ (/ y.re y.im) (/ y.im x.im)) (/ 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 t_0 = (1.0 / hypot(y_46_re, y_46_im)) * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im));
	double t_1 = t_0 - (y_46_im / (pow(hypot(y_46_re, y_46_im), 2.0) / x_46_re));
	double tmp;
	if (y_46_im <= -1e+160) {
		tmp = t_0 - (x_46_re / y_46_im);
	} else if (y_46_im <= -1.42e-92) {
		tmp = t_1;
	} else if (y_46_im <= 5.1e-138) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 9.5e+143) {
		tmp = t_1;
	} else {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (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 t_0 = (1.0 / Math.hypot(y_46_re, y_46_im)) * (y_46_re / (Math.hypot(y_46_re, y_46_im) / x_46_im));
	double t_1 = t_0 - (y_46_im / (Math.pow(Math.hypot(y_46_re, y_46_im), 2.0) / x_46_re));
	double tmp;
	if (y_46_im <= -1e+160) {
		tmp = t_0 - (x_46_re / y_46_im);
	} else if (y_46_im <= -1.42e-92) {
		tmp = t_1;
	} else if (y_46_im <= 5.1e-138) {
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	} else if (y_46_im <= 9.5e+143) {
		tmp = t_1;
	} else {
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (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):
	t_0 = (1.0 / math.hypot(y_46_re, y_46_im)) * (y_46_re / (math.hypot(y_46_re, y_46_im) / x_46_im))
	t_1 = t_0 - (y_46_im / (math.pow(math.hypot(y_46_re, y_46_im), 2.0) / x_46_re))
	tmp = 0
	if y_46_im <= -1e+160:
		tmp = t_0 - (x_46_re / y_46_im)
	elif y_46_im <= -1.42e-92:
		tmp = t_1
	elif y_46_im <= 5.1e-138:
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re)
	elif y_46_im <= 9.5e+143:
		tmp = t_1
	else:
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (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)
	t_0 = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(y_46_re / Float64(hypot(y_46_re, y_46_im) / x_46_im)))
	t_1 = Float64(t_0 - Float64(y_46_im / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / x_46_re)))
	tmp = 0.0
	if (y_46_im <= -1e+160)
		tmp = Float64(t_0 - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -1.42e-92)
		tmp = t_1;
	elseif (y_46_im <= 5.1e-138)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(Float64(y_46_im * x_46_re) / y_46_re) / y_46_re));
	elseif (y_46_im <= 9.5e+143)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) / Float64(y_46_im / x_46_im)) - 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)
	t_0 = (1.0 / hypot(y_46_re, y_46_im)) * (y_46_re / (hypot(y_46_re, y_46_im) / x_46_im));
	t_1 = t_0 - (y_46_im / ((hypot(y_46_re, y_46_im) ^ 2.0) / x_46_re));
	tmp = 0.0;
	if (y_46_im <= -1e+160)
		tmp = t_0 - (x_46_re / y_46_im);
	elseif (y_46_im <= -1.42e-92)
		tmp = t_1;
	elseif (y_46_im <= 5.1e-138)
		tmp = (x_46_im / y_46_re) - (((y_46_im * x_46_re) / y_46_re) / y_46_re);
	elseif (y_46_im <= 9.5e+143)
		tmp = t_1;
	else
		tmp = ((y_46_re / y_46_im) / (y_46_im / x_46_im)) - (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_] := Block[{t$95$0 = N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[(y$46$im / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1e+160], N[(t$95$0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.42e-92], t$95$1, If[LessEqual[y$46$im, 5.1e-138], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.5e+143], t$95$1, N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.00000000000000001e160

   1. Initial program 15.6%

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

   2. Applied egg-rr 20.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)} \]
      Proof

      [Start] 15.6	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      div-sub [=>] 15.6	\[ \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
      *-un-lft-identity [=>] 15.6	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 15.6	\[ \frac{1 \cdot \left(x.im \cdot y.re\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      times-frac [=>] 15.6	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-neg [=>] 15.6	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]
      hypot-def [=>] 15.6	\[ \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      hypot-def [=>] 18.7	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      associate-/l* [=>] 20.2	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}}\right) \]
      add-sqr-sqrt [=>] 20.2	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{y.im}}\right) \]
      pow2 [=>] 20.2	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}{y.im}}\right) \]
      hypot-def [=>] 20.2	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

   3. Simplified 34.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}} \]
      Proof

      [Start] 20.2	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right) \]
      fma-neg [<=] 20.2	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}} \]
      *-commutative [<=] 20.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{y.re \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \]
      associate-/l* [=>] 34.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \]
      associate-/l* [<=] 28.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{x.re \cdot y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]
      *-commutative [=>] 28.9	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{\color{blue}{y.im \cdot x.re}}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \]
      associate-/l* [=>] 34.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}} \]

   4. Taylor expanded in y.im around inf 95.8%

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

   if -1.00000000000000001e160 < y.im < -1.42e-92 or 5.1000000000000002e-138 < y.im < 9.50000000000000066e143

   1. Initial program 70.8%

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

   2. Applied egg-rr 80.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)} \]
      Proof

      [Start] 70.8	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      div-sub [=>] 70.8	\[ \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
      *-un-lft-identity [=>] 70.8	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 70.8	\[ \frac{1 \cdot \left(x.im \cdot y.re\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      times-frac [=>] 70.7	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-neg [=>] 70.7	\[ \color{blue}{\mathsf{fma}\left(\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]
      hypot-def [=>] 70.7	\[ \mathsf{fma}\left(\frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im \cdot y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      hypot-def [=>] 73.4	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      associate-/l* [=>] 80.4	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}}\right) \]
      add-sqr-sqrt [=>] 80.4	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{y.im}}\right) \]
      pow2 [=>] 80.4	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}{y.im}}\right) \]
      hypot-def [=>] 80.4	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

   3. Simplified 93.2%

      \[\leadsto \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}} \]
      Proof

      [Start] 80.4	\[ \mathsf{fma}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right) \]
      fma-neg [<=] 80.4	\[ \color{blue}{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}} \]
      *-commutative [<=] 80.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{y.re \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \]
      associate-/l* [=>] 93.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}}} - \frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \]
      associate-/l* [<=] 85.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{x.re \cdot y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}} \]
      *-commutative [=>] 85.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{\color{blue}{y.im \cdot x.re}}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \]
      associate-/l* [=>] 93.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \color{blue}{\frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}} \]

   if -1.42e-92 < y.im < 5.1000000000000002e-138

   1. Initial program 59.1%

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

   2. Taylor expanded in y.re around inf 74.7%

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

   3. Simplified 76.0%

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

      [Start] 74.7	\[ \frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} \]
      mul-1-neg [=>] 74.7	\[ \frac{x.im}{y.re} + \color{blue}{\left(-\frac{x.re \cdot y.im}{{y.re}^{2}}\right)} \]
      unsub-neg [=>] 74.7	\[ \color{blue}{\frac{x.im}{y.re} - \frac{x.re \cdot y.im}{{y.re}^{2}}} \]
      unpow2 [=>] 74.7	\[ \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{\color{blue}{y.re \cdot y.re}} \]
      associate-/r* [=>] 76.0	\[ \frac{x.im}{y.re} - \color{blue}{\frac{\frac{x.re \cdot y.im}{y.re}}{y.re}} \]

   if 9.50000000000000066e143 < y.im

   1. Initial program 30.4%

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

   2. Taylor expanded in y.re around 0 79.3%

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

   3. Simplified 99.9%

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

      [Start] 79.3	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 79.3	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 79.3	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 79.3	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      unpow2 [=>] 79.3	\[ \frac{y.re \cdot x.im}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 99.9	\[ \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im}} - \frac{x.re}{y.im} \]

   4. Applied egg-rr 100.0%

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

      [Start] 99.9	\[ \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im} \]
      clear-num [=>] 99.9	\[ \frac{y.re}{y.im} \cdot \color{blue}{\frac{1}{\frac{y.im}{x.im}}} - \frac{x.re}{y.im} \]
      un-div-inv [=>] 100.0	\[ \color{blue}{\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}}} - \frac{x.re}{y.im} \]

3. Recombined 4 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+160}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.42 \cdot 10^{-92}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ \mathbf{elif}\;y.im \leq 5.1 \cdot 10^{-138}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+143}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{y.im}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	84.8%
Cost	15817

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ t_2 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+215}\right):\\ \;\;\;\;t_2 \cdot \frac{y.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_2 \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 2
Accuracy	79.2%
Cost	14288

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{-171}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-60}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 650000000000:\\ \;\;\;\;\frac{x.im}{y.re} - y.im \cdot \frac{x.re}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 9.6 \cdot 10^{+76}:\\ \;\;\;\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternative 3
Accuracy	74.7%
Cost	1620

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{+27}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 3.3 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	70.4%
Cost	1233

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{+18}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-39} \lor \neg \left(y.re \leq 1.1 \cdot 10^{+56}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5
Accuracy	65.5%
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -6000000000:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+57}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 6
Accuracy	70.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.05 \cdot 10^{+20}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+57}:\\ \;\;\;\;x.im \cdot \frac{\frac{y.re}{y.im}}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Accuracy	70.4%
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.1 \cdot 10^{+20}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-36}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{+58}:\\ \;\;\;\;\frac{\frac{y.re}{y.im}}{\frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Accuracy	65.7%
Cost	1105

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1 \cdot 10^{+69}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9000000000 \lor \neg \left(y.re \leq -2.95 \cdot 10^{-36}\right) \land y.re \leq 8.8 \cdot 10^{+58}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	60.1%
Cost	785

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.6 \cdot 10^{+84}:\\ \;\;\;\;\frac{y.re}{\frac{y.im \cdot y.im}{x.im}}\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-29} \lor \neg \left(y.im \leq 10^{-13}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 10
Accuracy	60.2%
Cost	520

\[\begin{array}{l} \mathbf{if}\;y.re \leq -6.3 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{+55}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Accuracy	39.2%
Cost	192

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

Reproduce

herbie shell --seed 2023157 
(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))))