_divideComplex, imaginary part

Average Error: 26.2 → 7.9

Time: 20.1s

Precision: binary64

Cost: 33552

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{if}\;y.im \leq -1.1833245676509435 \cdot 10^{+130}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-155}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-190}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re \cdot \frac{y.im}{y.re}}{-y.re}\\ \mathbf{elif}\;y.im \leq 9.160572549621019 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

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
         (fma
          (/ y.re (hypot y.re y.im))
          (/ x.im (hypot y.re y.im))
          (/ (* y.im (- x.re)) (pow (hypot y.re y.im) 2.0)))))
   (if (<= y.im -1.1833245676509435e+130)
     (- (/ y.re (* y.im (/ y.im x.im))) (/ x.re y.im))
     (if (<= y.im -1e-155)
       t_0
       (if (<= y.im 1e-190)
         (+ (/ x.im y.re) (/ (* x.re (/ y.im y.re)) (- y.re)))
         (if (<= y.im 9.160572549621019e+66)
           t_0
           (- (* (/ y.re y.im) (/ x.im y.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 = fma((y_46_re / hypot(y_46_re, y_46_im)), (x_46_im / hypot(y_46_re, y_46_im)), ((y_46_im * -x_46_re) / pow(hypot(y_46_re, y_46_im), 2.0)));
	double tmp;
	if (y_46_im <= -1.1833245676509435e+130) {
		tmp = (y_46_re / (y_46_im * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= -1e-155) {
		tmp = t_0;
	} else if (y_46_im <= 1e-190) {
		tmp = (x_46_im / y_46_re) + ((x_46_re * (y_46_im / y_46_re)) / -y_46_re);
	} else if (y_46_im <= 9.160572549621019e+66) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re / y_46_im) * (x_46_im / y_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 = 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(y_46_im * Float64(-x_46_re)) / (hypot(y_46_re, y_46_im) ^ 2.0)))
	tmp = 0.0
	if (y_46_im <= -1.1833245676509435e+130)
		tmp = Float64(Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -1e-155)
		tmp = t_0;
	elseif (y_46_im <= 1e-190)
		tmp = Float64(Float64(x_46_im / y_46_re) + Float64(Float64(x_46_re * Float64(y_46_im / y_46_re)) / Float64(-y_46_re)));
	elseif (y_46_im <= 9.160572549621019e+66)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re / y_46_im) * Float64(x_46_im / y_46_im)) - Float64(x_46_re / 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_] := Block[{t$95$0 = 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[(y$46$im * (-x$46$re)), $MachinePrecision] / N[Power[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.1833245676509435e+130], N[(N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1e-155], t$95$0, If[LessEqual[y$46$im, 1e-190], N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.160572549621019e+66], t$95$0, N[(N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.18332456765094353e130

   1. Initial program 42.7

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

   2. Applied egg-rr 63.3

      \[\leadsto \frac{x.im \cdot y.re - x.re \cdot y.im}{\color{blue}{\frac{{y.re}^{4} - {y.im}^{4}}{y.re \cdot y.re - y.im \cdot y.im}}} \]

   3. Taylor expanded in y.re around 0 16.0

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

   4. Simplified 14.7

      \[\leadsto \color{blue}{\frac{y.re}{\frac{y.im \cdot y.im}{x.im}} - \frac{x.re}{y.im}} \]
      Proof
      (-.f64 (/.f64 y.re (/.f64 (*.f64 y.im y.im) x.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 y.re (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) x.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2))) (/.f64 x.re y.im)): 21 points increase in error, 11 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)) (neg.f64 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x.re y.im)) (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 10.8

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

   if -1.18332456765094353e130 < y.im < -1.00000000000000001e-155 or 1e-190 < y.im < 9.16057254962101886e66

   1. Initial program 16.3

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

   2. Applied egg-rr 6.1

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

   if -1.00000000000000001e-155 < y.im < 1e-190

   1. Initial program 24.4

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

   2. Applied egg-rr 9.3

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

   3. Taylor expanded in y.re around inf 9.5

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

   4. Simplified 7.4

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]
      Proof
      (-.f64 (/.f64 x.im y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.re y.re))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.re) (*.f64 y.re y.re)))): 47 points increase in error, 15 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (Rewrite<= *-commutative_binary64 (*.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) (/.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

   5. Applied egg-rr 4.8

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

   if 9.16057254962101886e66 < y.im

   1. Initial program 37.1

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

   2. Applied egg-rr 27.4

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

   3. Taylor expanded in y.re around 0 18.5

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

   4. Simplified 12.8

      \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}} \]
      Proof
      (-.f64 (*.f64 (/.f64 y.re y.im) (/.f64 x.im y.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.im) (*.f64 y.im y.im))) (/.f64 x.re y.im)): 43 points increase in error, 7 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)) (neg.f64 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 x.re y.im)) (/.f64 (*.f64 y.re x.im) (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
3. Recombined 4 regimes into one program.

4. Final simplification 7.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1833245676509435 \cdot 10^{+130}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-155}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y.im \leq 10^{-190}:\\ \;\;\;\;\frac{x.im}{y.re} + \frac{x.re \cdot \frac{y.im}{y.re}}{-y.re}\\ \mathbf{elif}\;y.im \leq 9.160572549621019 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.0
Cost	14160

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -6.849960252831045 \cdot 10^{+169}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.6435499227547897 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-214}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 2
Error	13.9
Cost	7172

\[\begin{array}{l} t_0 := y.im \cdot \frac{x.re}{y.re}\\ \mathbf{if}\;y.re \leq -2.0478690137712028 \cdot 10^{+110}:\\ \;\;\;\;\frac{t_0 - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.6435499227547897 \cdot 10^{-108}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - t_0}{y.re}\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 3
Error	21.0
Cost	1628

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{if}\;y.re \leq -7.65555750962981 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.4925377672796953 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.991267105011886 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-143}:\\ \;\;\;\;y.re \cdot \frac{x.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4457440215525287 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	21.1
Cost	1628

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{if}\;y.re \leq -7.65555750962981 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.4925377672796953 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.991267105011886 \cdot 10^{-81}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-143}:\\ \;\;\;\;y.re \cdot \frac{x.im}{y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4457440215525287 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 5
Error	17.1
Cost	1496

\[\begin{array}{l} t_0 := \frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -7.65555750962981 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -2.4925377672796953 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.3241526844745973 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 6
Error	16.2
Cost	1496

\[\begin{array}{l} t_0 := \frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -7.65555750962981 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -2.4925377672796953 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -3.3241526844745973 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 7
Error	14.1
Cost	1364

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.0478690137712028 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.6435499227547897 \cdot 10^{-108}:\\ \;\;\;\;\frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 8
Error	16.6
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.re \leq -7.65555750962981 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.086470413219434 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{x.re}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 9
Error	19.5
Cost	1104

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -7.65555750962981 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.4457440215525287 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	24.3
Cost	1048

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.re \leq -6.849960252831045 \cdot 10^{+169}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -9.184134998734538 \cdot 10^{+115}:\\ \;\;\;\;\frac{\frac{x.re}{y.re}}{y.re} \cdot \left(-y.im\right)\\ \mathbf{elif}\;y.re \leq -18.10085919335649:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4457440215525287 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Error	23.5
Cost	784

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.re \leq -18.10085919335649:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.6308962564249265 \cdot 10^{-26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.0626847866249067 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 1.4457440215525287 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12
Error	36.4
Cost	324

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.923259067017494 \cdot 10^{+134}:\\ \;\;\;\;\frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 13
Error	57.0
Cost	192

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

Reproduce

herbie shell --seed 2022311 
(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))))