_divideComplex, imaginary part

Percentage Accurate: 62.4% → 89.6%

Time: 21.7s

Precision: binary64

Cost: 46344

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\sqrt{y.im}\right)}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}\right)\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))

↓

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -4.5e+54)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (if (<= y.im 9.8e-281)
     (*
      (/ 1.0 (hypot y.re y.im))
      (/ (- (* x.im y.re) (* y.im x.re)) (hypot y.re y.im)))
     (fma
      (/ y.re (hypot y.re y.im))
      (/ x.im (hypot y.re y.im))
      (/
       (* (/ x.re (hypot y.re y.im)) (- (sqrt y.im)))
       (/ (hypot y.re y.im) (sqrt 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 tmp;
	if (y_46_im <= -4.5e+54) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 9.8e-281) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (((x_46_im * y_46_re) - (y_46_im * x_46_re)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), (((x_46_re / hypot(y_46_re, y_46_im)) * -sqrt(y_46_im)) / (hypot(y_46_re, y_46_im) / sqrt(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)
	tmp = 0.0
	if (y_46_im <= -4.5e+54)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 9.8e-281)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)));
	else
		tmp = fma(Float64(y_46_re / hypot(y_46_re, y_46_im)), Float64(x_46_im / hypot(y_46_re, y_46_im)), Float64(Float64(Float64(x_46_re / hypot(y_46_re, y_46_im)) * Float64(-sqrt(y_46_im))) / Float64(hypot(y_46_re, y_46_im) / sqrt(y_46_im))));
	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_] := If[LessEqual[y$46$im, -4.5e+54], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 9.8e-281], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] + N[(N[(N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[y$46$im], $MachinePrecision])), $MachinePrecision] / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[Sqrt[y$46$im], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.49999999999999984e54

   1. Initial program 48.6%

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

   2. Taylor expanded in y.re around 0 80.7%

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

   3. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}} \]
      Step-by-step derivation

      [Start] 80.7	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 80.7	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]
      mul-1-neg [=>] 80.7	\[ \frac{y.re \cdot x.im}{{y.im}^{2}} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      unsub-neg [=>] 80.7	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} - \frac{x.re}{y.im}} \]
      unpow2 [=>] 80.7	\[ \frac{y.re \cdot x.im}{\color{blue}{y.im \cdot y.im}} - \frac{x.re}{y.im} \]
      times-frac [=>] 91.2	\[ \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im}} - \frac{x.re}{y.im} \]

   4. Taylor expanded in y.re around 0 80.7%

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

   5. Simplified 91.2%

      \[\leadsto \color{blue}{\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}} \]
      Step-by-step derivation

      [Start] 80.7	\[ -1 \cdot \frac{x.re}{y.im} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      mul-1-neg [=>] 80.7	\[ \color{blue}{\left(-\frac{x.re}{y.im}\right)} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      distribute-frac-neg [<=] 80.7	\[ \color{blue}{\frac{-x.re}{y.im}} + \frac{y.re \cdot x.im}{{y.im}^{2}} \]
      +-commutative [=>] 80.7	\[ \color{blue}{\frac{y.re \cdot x.im}{{y.im}^{2}} + \frac{-x.re}{y.im}} \]
      unpow2 [=>] 80.7	\[ \frac{y.re \cdot x.im}{\color{blue}{y.im \cdot y.im}} + \frac{-x.re}{y.im} \]
      times-frac [=>] 91.2	\[ \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im}} + \frac{-x.re}{y.im} \]
      distribute-frac-neg [=>] 91.2	\[ \frac{y.re}{y.im} \cdot \frac{x.im}{y.im} + \color{blue}{\left(-\frac{x.re}{y.im}\right)} \]
      sub-neg [<=] 91.2	\[ \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}} \]
      associate-*l/ [=>] 91.3	\[ \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im}}{y.im}} - \frac{x.re}{y.im} \]
      div-sub [<=] 91.2	\[ \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}} \]
      associate-*r/ [=>] 81.0	\[ \frac{\color{blue}{\frac{y.re \cdot x.im}{y.im}} - x.re}{y.im} \]
      *-commutative [=>] 81.0	\[ \frac{\frac{\color{blue}{x.im \cdot y.re}}{y.im} - x.re}{y.im} \]
      associate-*r/ [<=] 91.2	\[ \frac{\color{blue}{x.im \cdot \frac{y.re}{y.im}} - x.re}{y.im} \]

   if -4.49999999999999984e54 < y.im < 9.7999999999999999e-281

   1. Initial program 80.0%

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

   2. Applied egg-rr 92.1%

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

      [Start] 80.0	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 80.0	\[ \frac{\color{blue}{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 80.0	\[ \frac{1 \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 80.1	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 80.1	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 92.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if 9.7999999999999999e-281 < y.im

   1. Initial program 62.5%

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

   2. Applied egg-rr 79.2%

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

      [Start] 62.5	\[ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      div-sub [=>] 60.7	\[ \color{blue}{\frac{x.im \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
      *-commutative [=>] 60.7	\[ \frac{\color{blue}{y.re \cdot x.im}}{y.re \cdot y.re + y.im \cdot y.im} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 60.7	\[ \frac{y.re \cdot x.im}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      times-frac [=>] 62.0	\[ \color{blue}{\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} - \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      fma-neg [=>] 62.0	\[ \color{blue}{\mathsf{fma}\left(\frac{y.re}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)} \]
      hypot-def [=>] 62.0	\[ \mathsf{fma}\left(\frac{y.re}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, \frac{x.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      hypot-def [=>] 78.0	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}, -\frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right) \]
      associate-/l* [=>] 79.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}}\right) \]
      add-sqr-sqrt [=>] 79.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}}{y.im}}\right) \]
      pow2 [=>] 79.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{\color{blue}{{\left(\sqrt{y.re \cdot y.re + y.im \cdot y.im}\right)}^{2}}}{y.im}}\right) \]
      hypot-def [=>] 79.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\color{blue}{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}}^{2}}{y.im}}\right) \]

   3. Applied egg-rr 98.8%

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

      [Start] 79.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right) \]
      *-un-lft-identity [=>] 79.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{\color{blue}{1 \cdot x.re}}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right) \]
      add-sqr-sqrt [=>] 79.1	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1 \cdot x.re}{\color{blue}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}} \cdot \sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}}\right) \]
      times-frac [=>] 79.1	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{1}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}} \cdot \frac{x.re}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}}\right) \]
      sqrt-div [=>] 79.1	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\color{blue}{\frac{\sqrt{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}}{\sqrt{y.im}}}} \cdot \frac{x.re}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}\right) \]
      unpow2 [=>] 79.1	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\sqrt{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{y.im}}} \cdot \frac{x.re}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}\right) \]
      sqrt-prod [=>] 79.1	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\color{blue}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{y.im}}} \cdot \frac{x.re}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}\right) \]
      add-sqr-sqrt [<=] 79.1	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}{\sqrt{y.im}}} \cdot \frac{x.re}{\sqrt{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}}\right) \]
      sqrt-div [=>] 79.4	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}} \cdot \frac{x.re}{\color{blue}{\frac{\sqrt{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}}{\sqrt{y.im}}}}\right) \]
      unpow2 [=>] 79.4	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}} \cdot \frac{x.re}{\frac{\sqrt{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{y.im}}}\right) \]
      sqrt-prod [=>] 98.9	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}} \cdot \frac{x.re}{\frac{\color{blue}{\sqrt{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{\mathsf{hypot}\left(y.re, y.im\right)}}}{\sqrt{y.im}}}\right) \]
      add-sqr-sqrt [<=] 98.8	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}} \cdot \frac{x.re}{\frac{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}}{\sqrt{y.im}}}\right) \]

   4. Simplified 98.2%

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

      [Start] 98.8	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}} \cdot \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}\right) \]
      associate-*l/ [=>] 98.8	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\color{blue}{\frac{1 \cdot \frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}}\right) \]
      *-lft-identity [=>] 98.8	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{\color{blue}{\frac{x.re}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}\right) \]
      associate-/r/ [=>] 98.2	\[ \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{\color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \sqrt{y.im}}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}\right) \]

3. Recombined 3 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-281}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-\sqrt{y.im}\right)}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\sqrt{y.im}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.1%
Cost	33684

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{-280}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot t_0\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{-205}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_0}}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{\frac{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}{y.im}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Accuracy	83.8%
Cost	14288

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.7 \cdot 10^{+54}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9.8 \cdot 10^{-281}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-224}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	82.2%
Cost	13904

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{y.im}{\frac{y.re}{x.re}} \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.re \leq -3.6 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.3 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{+103}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4
Accuracy	82.9%
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.im}{\frac{y.re}{x.re}}\\ \mathbf{if}\;y.re \leq -2.75 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im}{y.re} + t_1 \cdot \frac{-1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 7.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+104}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - t_1}{y.re}\\ \end{array} \]

Alternative 5
Accuracy	75.5%
Cost	1233

\[\begin{array}{l} \mathbf{if}\;y.im \leq -8.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{+43}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{-88} \lor \neg \left(y.im \leq 4.7 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \end{array} \]

Alternative 6
Accuracy	70.0%
Cost	1106

\[\begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+53} \lor \neg \left(y.im \leq -2 \cdot 10^{+41}\right) \land \left(y.im \leq -4 \cdot 10^{-55} \lor \neg \left(y.im \leq 1.5 \cdot 10^{+120}\right)\right):\\ \;\;\;\;\frac{-x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \end{array} \]

Alternative 7
Accuracy	70.4%
Cost	1105

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-52} \lor \neg \left(y.im \leq 1.4 \cdot 10^{+122}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \end{array} \]

Alternative 8
Accuracy	75.6%
Cost	1105

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.3 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -6.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -7.2 \cdot 10^{-87} \lor \neg \left(y.im \leq 8.5 \cdot 10^{+26}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \end{array} \]

Alternative 9
Accuracy	64.0%
Cost	521

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+29} \lor \neg \left(y.re \leq 0.106\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]

Alternative 10
Accuracy	41.9%
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Accuracy	9.7%
Cost	192

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

Reproduce

herbie shell --seed 2023160 
(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))))