_divideComplex, imaginary part

Percentage Accurate: 61.5% → 92.4%

Time: 10.4s

Precision: binary64

Cost: 39816

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right) \cdot \sqrt{\frac{1}{y.im}}\right)}^{2}}\right)\\ \end{array} \]

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 (hypot y.re y.im))) (t_1 (/ x.im (hypot y.re y.im))))
   (if (<= y.im -1.35e+154)
     (/ (* y.im (/ x.re (hypot y.re y.im))) (- (hypot y.re y.im)))
     (if (<= y.im 2e-309)
       (fma t_0 t_1 (/ (- x.re) (/ (pow (hypot y.re y.im) 2.0) y.im)))
       (fma
        t_0
        t_1
        (/ (- x.re) (pow (* (hypot y.re y.im) (sqrt (/ 1.0 y.im))) 2.0)))))))

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 / hypot(y_46_re, y_46_im);
	double t_1 = x_46_im / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -1.35e+154) {
		tmp = (y_46_im * (x_46_re / hypot(y_46_re, y_46_im))) / -hypot(y_46_re, y_46_im);
	} else if (y_46_im <= 2e-309) {
		tmp = fma(t_0, t_1, (-x_46_re / (pow(hypot(y_46_re, y_46_im), 2.0) / y_46_im)));
	} else {
		tmp = fma(t_0, t_1, (-x_46_re / pow((hypot(y_46_re, y_46_im) * sqrt((1.0 / y_46_im))), 2.0)));
	}
	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(y_46_re / hypot(y_46_re, y_46_im))
	t_1 = Float64(x_46_im / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.35e+154)
		tmp = Float64(Float64(y_46_im * Float64(x_46_re / hypot(y_46_re, y_46_im))) / Float64(-hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 2e-309)
		tmp = fma(t_0, t_1, Float64(Float64(-x_46_re) / Float64((hypot(y_46_re, y_46_im) ^ 2.0) / y_46_im)));
	else
		tmp = fma(t_0, t_1, Float64(Float64(-x_46_re) / (Float64(hypot(y_46_re, y_46_im) * sqrt(Float64(1.0 / y_46_im))) ^ 2.0)));
	end
	return 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[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.35e+154], N[(N[(y$46$im * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision])), $MachinePrecision], If[LessEqual[y$46$im, 2e-309], N[(t$95$0 * t$95$1 + N[((-x$46$re) / N[(N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * t$95$1 + N[((-x$46$re) / N[Power[N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[Sqrt[N[(1.0 / y$46$im), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.35000000000000003e154

   1. Initial program 30.9%

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

   2. Applied egg-rr 51.2%

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

      [Start] 30.9%	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 30.9%	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 30.9%	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 30.9%	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 30.9%	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 51.2%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in x.im around 0 51.3%

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

   4. Simplified 51.3%

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

      [Start] 51.3%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{-1 \cdot \left(x.re \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      mul-1-neg [=>] 51.3%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{-x.re \cdot y.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      *-commutative [=>] 51.3%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{-\color{blue}{y.im \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      distribute-rgt-neg-in [=>] 51.3%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{y.im \cdot \left(-x.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)} \]

   5. Applied egg-rr 88.7%

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

      [Start] 51.3%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.im \cdot \left(-x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      frac-2neg [=>] 51.3%	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{-y.im \cdot \left(-x.re\right)}{-\mathsf{hypot}\left(y.re, y.im\right)}} \]
      associate-*r/ [=>] 51.3%	\[ \color{blue}{\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-y.im \cdot \left(-x.re\right)\right)}{-\mathsf{hypot}\left(y.re, y.im\right)}} \]
      *-commutative [=>] 51.3%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\color{blue}{\left(-x.re\right) \cdot y.im}\right)}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      add-sqr-sqrt [=>] 33.1%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\color{blue}{\left(\sqrt{-x.re} \cdot \sqrt{-x.re}\right)} \cdot y.im\right)}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      sqrt-unprod [=>] 45.3%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}} \cdot y.im\right)}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      sqr-neg [=>] 45.3%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\sqrt{\color{blue}{x.re \cdot x.re}} \cdot y.im\right)}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      sqrt-unprod [<=] 12.2%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\color{blue}{\left(\sqrt{x.re} \cdot \sqrt{x.re}\right)} \cdot y.im\right)}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      add-sqr-sqrt [<=] 31.2%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\color{blue}{x.re} \cdot y.im\right)}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      distribute-lft-neg-out [<=] 31.2%	\[ \frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(-x.re\right) \cdot y.im\right)}}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*r* [=>] 31.9%	\[ \frac{\color{blue}{\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-x.re\right)\right) \cdot y.im}}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      associate-*l/ [=>] 31.9%	\[ \frac{\color{blue}{\frac{1 \cdot \left(-x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      *-un-lft-identity [<=] 31.9%	\[ \frac{\frac{\color{blue}{-x.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      add-sqr-sqrt [=>] 19.5%	\[ \frac{\frac{\color{blue}{\sqrt{-x.re} \cdot \sqrt{-x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      sqrt-unprod [=>] 40.3%	\[ \frac{\frac{\color{blue}{\sqrt{\left(-x.re\right) \cdot \left(-x.re\right)}}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      sqr-neg [=>] 40.3%	\[ \frac{\frac{\sqrt{\color{blue}{x.re \cdot x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      sqrt-unprod [<=] 32.2%	\[ \frac{\frac{\color{blue}{\sqrt{x.re} \cdot \sqrt{x.re}}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]
      add-sqr-sqrt [<=] 88.7%	\[ \frac{\frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot y.im}{-\mathsf{hypot}\left(y.re, y.im\right)} \]

   if -1.35000000000000003e154 < y.im < 1.9999999999999988e-309

   1. Initial program 69.8%

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

   2. Applied egg-rr 91.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)} \]
      Step-by-step derivation

      [Start] 69.8%	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      div-sub [=>] 66.6%	\[ \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
      *-commutative [=>] 66.6%	\[ \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 66.6%	\[ \frac{y.re \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      times-frac [=>] 68.9%	\[ \color{blue}{\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-neg [=>] 68.9%	\[ \color{blue}{\mathsf{fma}\left(\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]
      hypot-def [=>] 68.9%	\[ \mathsf{fma}\left(\frac{y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      hypot-def [=>] 90.3%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      associate-/l* [=>] 91.6%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}}\right) \]
      add-sqr-sqrt [=>] 91.6%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{y.im}}\right) \]
      pow2 [=>] 91.6%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}{y.im}}\right) \]
      hypot-def [=>] 91.6%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

   if 1.9999999999999988e-309 < y.im

   1. Initial program 62.4%

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

   2. Applied egg-rr 81.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)} \]
      Step-by-step derivation

      [Start] 62.4%	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      div-sub [=>] 57.6%	\[ \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
      *-commutative [=>] 57.6%	\[ \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 57.6%	\[ \frac{y.re \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      times-frac [=>] 60.4%	\[ \color{blue}{\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-neg [=>] 60.4%	\[ \color{blue}{\mathsf{fma}\left(\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]
      hypot-def [=>] 60.4%	\[ \mathsf{fma}\left(\frac{y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      hypot-def [=>] 80.2%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      associate-/l* [=>] 81.4%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}}\right) \]
      add-sqr-sqrt [=>] 81.4%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{y.im}}\right) \]
      pow2 [=>] 81.4%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}{y.im}}\right) \]
      hypot-def [=>] 81.4%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

   3. Applied egg-rr 96.6%

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

      [Start] 81.4%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right) \]
      add-sqr-sqrt [=>] 81.3%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}}\right) \]
      pow2 [=>] 81.3%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\color{blue}{{\left(\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)}^{2}}}\right) \]
      div-inv [=>] 81.3%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{{\left(\sqrt{\color{blue}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2} \cdot \frac{1}{y.im}}}\right)}^{2}}\right) \]
      sqrt-prod [=>] 81.3%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{{\color{blue}{\left(\sqrt{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}} \cdot \sqrt{\frac{1}{y.im}}\right)}}^{2}}\right) \]
      unpow2 [=>] 81.3%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{{\left(\sqrt{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}} \cdot \sqrt{\frac{1}{y.im}}\right)}^{2}}\right) \]
      sqrt-prod [=>] 96.5%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{{\left(\color{blue}{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)} \cdot \sqrt{\frac{1}{y.im}}\right)}^{2}}\right) \]
      add-sqr-sqrt [<=] 96.6%	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{{\left(\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{\frac{1}{y.im}}\right)}^{2}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 93.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right) \cdot \sqrt{\frac{1}{y.im}}\right)}^{2}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	92.4%
Cost	39816

\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{-\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{-x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right) \cdot \sqrt{\frac{1}{y.im}}\right)}^{2}}\right)\\ \end{array} \]

Alternative 2
Accuracy	84.4%
Cost	14788

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+243}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \end{array} \]

Alternative 3
Accuracy	81.7%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-97}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{+83}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im \cdot \frac{x.re}{y.re}}}\\ \end{array} \]

Alternative 4
Accuracy	77.7%
Cost	1096

\[\begin{array}{l} \mathbf{if}\;y.re \leq -8.8 \cdot 10^{-5}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-11}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{-1}{\frac{y.re}{y.im \cdot \frac{x.re}{y.re}}}\\ \end{array} \]

Alternative 5
Accuracy	71.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.85 \lor \neg \left(y.re \leq 5.8 \cdot 10^{-60}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 6
Accuracy	77.9%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.015 \lor \neg \left(y.re \leq 1.15 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re} \cdot \frac{y.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 7
Accuracy	64.9%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.3 \cdot 10^{+53} \lor \neg \left(y.im \leq 2.9 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 8
Accuracy	47.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+205}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.3 \cdot 10^{+165}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \end{array} \]

Alternative 9
Accuracy	43.2%
Cost	192

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

Reproduce

herbie shell --seed 2023165 
(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))))