_divideComplex, imaginary part

Average Error: 26.6 → 7.4

Time: 24.0s

Precision: binary64

Cost: 33552

Math

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

↓

\[\begin{array}{l} t_0 := \frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \mathsf{fma}\left(t_0, t_1, \frac{-x.re}{y.im}\right)\\ \mathbf{if}\;y.im \leq -1.1469635874565826 \cdot 10^{+68}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 10^{-192}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.673336263870845 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(t_0, t_1, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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)))
        (t_2 (fma t_0 t_1 (/ (- x.re) y.im))))
   (if (<= y.im -1.1469635874565826e+68)
     t_2
     (if (<= y.im -1e-230)
       (/
        (/ (- (* y.re x.im) (* y.im x.re)) (hypot y.re y.im))
        (hypot y.re y.im))
       (if (<= y.im 1e-192)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 5.673336263870845e+150)
           (fma t_0 t_1 (/ (* y.im (- x.re)) (pow (hypot y.re y.im) 2.0)))
           t_2))))))

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 t_2 = fma(t_0, t_1, (-x_46_re / y_46_im));
	double tmp;
	if (y_46_im <= -1.1469635874565826e+68) {
		tmp = t_2;
	} else if (y_46_im <= -1e-230) {
		tmp = (((y_46_re * x_46_im) - (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 <= 1e-192) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 5.673336263870845e+150) {
		tmp = fma(t_0, t_1, ((y_46_im * -x_46_re) / pow(hypot(y_46_re, y_46_im), 2.0)));
	} else {
		tmp = t_2;
	}
	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))
	t_2 = fma(t_0, t_1, Float64(Float64(-x_46_re) / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.1469635874565826e+68)
		tmp = t_2;
	elseif (y_46_im <= -1e-230)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) - Float64(y_46_im * x_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 1e-192)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 5.673336263870845e+150)
		tmp = fma(t_0, t_1, Float64(Float64(y_46_im * Float64(-x_46_re)) / (hypot(y_46_re, y_46_im) ^ 2.0)));
	else
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(t$95$0 * t$95$1 + N[((-x$46$re) / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.1469635874565826e+68], t$95$2, If[LessEqual[y$46$im, -1e-230], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1e-192], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 5.673336263870845e+150], N[(t$95$0 * t$95$1 + 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], t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.14696358745658257e68 or 5.67333626387084466e150 < y.im

   1. Initial program 40.0

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

   2. Applied egg-rr 31.4

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

   3. Taylor expanded in y.im around inf 5.8

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

   if -1.14696358745658257e68 < y.im < -1.00000000000000005e-230

   1. Initial program 17.4

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

   2. Applied egg-rr 18.1

      \[\leadsto \frac{\color{blue}{{\left(\sqrt[3]{x.im \cdot y.re - x.re \cdot y.im}\right)}^{3}}}{y.re \cdot y.re + y.im \cdot y.im} \]

   3. Applied egg-rr 10.5

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

   4. Applied egg-rr 10.4

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

   if -1.00000000000000005e-230 < y.im < 1.0000000000000001e-192

   1. Initial program 24.8

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

   2. Simplified 24.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y.im, -x.re, x.im \cdot y.re\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      Proof
      (/.f64 (fma.f64 y.im (neg.f64 x.re) (*.f64 x.im y.re)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.im (neg.f64 x.re)) (*.f64 x.im y.re))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 x.re) y.im)) (*.f64 x.im y.re)) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 x.im y.re) (*.f64 (neg.f64 x.re) y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im))) (fma.f64 y.re y.re (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))): 1 points increase in error, 0 points decrease in error

   3. Taylor expanded in y.im around 0 9.4

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

   4. Simplified 10.2

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re \cdot y.re} \cdot x.re} \]
      Proof
      (-.f64 (/.f64 x.im y.re) (*.f64 (/.f64 y.im (*.f64 y.re y.re)) x.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.re) (*.f64 (/.f64 y.im (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) x.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 x.im y.re) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.re) (pow.f64 y.re 2)))): 16 points increase in error, 15 points decrease in error
      (-.f64 (/.f64 x.im y.re) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.im)) (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 4.3

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

   6. Applied egg-rr 3.7

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

   7. Applied egg-rr 4.3

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

   if 1.0000000000000001e-192 < y.im < 5.67333626387084466e150

   1. Initial program 19.0

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

   2. Applied egg-rr 8.6

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, -\frac{x.re \cdot y.im}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 7.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.1469635874565826 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-230}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 10^{-192}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 5.673336263870845 \cdot 10^{+150}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	7.5
Cost	20996

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-x.re}{y.im}\right)\\ \end{array} \]

Alternative 2
Error	13.2
Cost	14492

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.2761489133332196 \cdot 10^{-8}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.0415742887118027 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.148732733672411 \cdot 10^{+97}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y.im, -x.re, y.re \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 3
Error	10.2
Cost	14340

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\mathsf{fma}\left(0.5, x.im \cdot \left(\frac{y.im}{y.re} \cdot \frac{y.im}{y.re}\right), x.re \cdot \frac{y.im}{y.re}\right) - x.im\right)\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.0415742887118027 \cdot 10^{-65}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.148732733672411 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 4
Error	10.1
Cost	14160

\[\begin{array}{l} t_0 := \frac{\frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -4.490116382232771 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.0415742887118027 \cdot 10^{-65}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.148732733672411 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 5
Error	13.2
Cost	1884

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -2.2761489133332196 \cdot 10^{-8}:\\ \;\;\;\;\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.0415742887118027 \cdot 10^{-65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.148732733672411 \cdot 10^{+97}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \end{array} \]

Alternative 6
Error	17.7
Cost	1632

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.5659726218618048 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 317500939.7906411:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1675900956826687 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.2168147847791367 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	16.4
Cost	1632

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ t_2 := \frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.5659726218618048 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 317500939.7906411:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1675900956826687 \cdot 10^{+39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 2.2168147847791367 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	16.4
Cost	1632

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.5659726218618048 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 317500939.7906411:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1675900956826687 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2168147847791367 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	16.5
Cost	1632

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ t_1 := \frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.5659726218618048 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 317500939.7906411:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1675900956826687 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.2168147847791367 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	19.1
Cost	1104

\[\begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.re \leq -1.9673740714989925 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.544508827741849 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.0142518985666746 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 11
Error	24.1
Cost	784

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.re \leq -4.567322487019083 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.0450242346714168 \cdot 10^{-79}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.0142518985666746 \cdot 10^{+67}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]

Alternative 12
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2022300 
(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))))