_divideComplex, imaginary part

Average Error: 26.0 → 6.6

Time: 13.7s

Precision: binary64

Cost: 33552

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+79}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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))))
        (t_1 (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))
   (if (<= y.im -1.45e+79)
     t_1
     (if (<= y.im -1.65e-130)
       t_0
       (if (<= y.im 5e-198)
         (- (/ x.im y.re) (/ x.re (+ y.im (/ y.re (/ y.im y.re)))))
         (if (<= y.im 1.2e+135) t_0 t_1))))))

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 t_1 = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.45e+79) {
		tmp = t_1;
	} else if (y_46_im <= -1.65e-130) {
		tmp = t_0;
	} else if (y_46_im <= 5e-198) {
		tmp = (x_46_im / y_46_re) - (x_46_re / (y_46_im + (y_46_re / (y_46_im / y_46_re))));
	} else if (y_46_im <= 1.2e+135) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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)))
	t_1 = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.45e+79)
		tmp = t_1;
	elseif (y_46_im <= -1.65e-130)
		tmp = t_0;
	elseif (y_46_im <= 5e-198)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re / Float64(y_46_im + Float64(y_46_re / Float64(y_46_im / y_46_re)))));
	elseif (y_46_im <= 1.2e+135)
		tmp = t_0;
	else
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.45e+79], t$95$1, If[LessEqual[y$46$im, -1.65e-130], t$95$0, If[LessEqual[y$46$im, 5e-198], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re / N[(y$46$im + N[(y$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.2e+135], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.44999999999999996e79 or 1.19999999999999999e135 < y.im

   1. Initial program 40.2

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

   2. Taylor expanded in y.re around 0 16.2

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

   3. Simplified 9.1

      \[\leadsto \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.im}{y.im} - \frac{x.re}{y.im}} \]
      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)): 34 points increase in error, 9 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

   4. Taylor expanded in y.re around 0 16.2

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

   5. Simplified 8.9

      \[\leadsto \color{blue}{\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}} \]
      Proof
      (/.f64 (-.f64 (*.f64 y.re (/.f64 x.im y.im)) x.re) y.im): 0 points increase in error, 0 points decrease in error
      (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (*.f64 y.re (/.f64 x.im y.im)) y.im) (/.f64 x.re y.im))): 0 points increase in error, 3 points decrease in error
      (-.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 y.re y.im) (/.f64 x.im y.im))) (/.f64 x.re y.im)): 10 points increase in error, 11 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 (/.f64 y.re y.im) (/.f64 x.im y.im)) (neg.f64 (/.f64 x.re y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 (/.f64 y.re y.im) (/.f64 x.im y.im)) (Rewrite<= distribute-frac-neg_binary64 (/.f64 (neg.f64 x.re) y.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (/.f64 (neg.f64 x.re) y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.im y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (neg.f64 x.re) y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.im) (*.f64 y.im y.im)))): 34 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 (neg.f64 x.re) y.im) (/.f64 (*.f64 y.re x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> distribute-frac-neg_binary64 (neg.f64 (/.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
      (+.f64 (Rewrite<= mul-1-neg_binary64 (*.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

   if -1.44999999999999996e79 < y.im < -1.6499999999999999e-130 or 4.9999999999999999e-198 < y.im < 1.19999999999999999e135

   1. Initial program 16.1

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

   2. Applied egg-rr 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 -1.6499999999999999e-130 < y.im < 4.9999999999999999e-198

   1. Initial program 23.4

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

   2. Taylor expanded in x.im around 0 23.4

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

   3. Simplified 24.1

      \[\leadsto \color{blue}{\frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{\frac{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)}{y.im}}} \]
      Proof
      (-.f64 (/.f64 (*.f64 y.re x.im) (fma.f64 y.im y.im (*.f64 y.re y.re))) (/.f64 x.re (/.f64 (fma.f64 y.im y.im (*.f64 y.re y.re)) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 y.im y.im) (*.f64 y.re y.re)))) (/.f64 x.re (/.f64 (fma.f64 y.im y.im (*.f64 y.re y.re)) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))) (/.f64 x.re (/.f64 (fma.f64 y.im y.im (*.f64 y.re y.re)) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) (*.f64 y.im y.im))) (/.f64 x.re (/.f64 (fma.f64 y.im y.im (*.f64 y.re y.re)) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))) (/.f64 x.re (/.f64 (fma.f64 y.im y.im (*.f64 y.re y.re)) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (/.f64 x.re (/.f64 (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 y.im y.im) (*.f64 y.re y.re))) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (/.f64 x.re (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (/.f64 x.re (/.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) (*.f64 y.im y.im)) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (/.f64 x.re (/.f64 (+.f64 (pow.f64 y.re 2) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) y.im))): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.re y.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))))): 42 points increase in error, 7 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (neg.f64 (/.f64 (*.f64 x.re y.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.re x.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re y.im) (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2)))))): 0 points increase in error, 0 points decrease in error

   4. Applied egg-rr 23.9

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

   5. Taylor expanded in y.re around 0 23.9

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

   6. Simplified 23.9

      \[\leadsto \frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - \frac{x.re}{\color{blue}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}} \]
      Proof
      (+.f64 y.im (/.f64 y.re (/.f64 y.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 y.im (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re y.re) y.im))): 19 points increase in error, 14 points decrease in error
      (+.f64 y.im (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (pow.f64 y.re 2) y.im) y.im)): 0 points increase in error, 0 points decrease in error

   7. Taylor expanded in y.re around inf 4.0

      \[\leadsto \color{blue}{\frac{x.im}{y.re}} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 6.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{+79}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.65 \cdot 10^{-130}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-198}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{y.im \cdot \left(-x.re\right)}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.3
Cost	33292

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ t_2 := {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}\\ \mathbf{if}\;y.im \leq -1.6 \cdot 10^{+101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.05 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-43}:\\ \;\;\;\;\mathsf{fma}\left(y.re, \frac{x.im}{t_2}, \frac{y.im \cdot \left(-x.re\right)}{t_2}\right)\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.56 \cdot 10^{+131}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{y.re \cdot x.im - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	10.3
Cost	14420

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ t_2 := y.re \cdot x.im - y.im \cdot x.re\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+99}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -5.8 \cdot 10^{+43}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -2.65 \cdot 10^{-42}:\\ \;\;\;\;\frac{t_2}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.4 \cdot 10^{+133}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_2}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	7.8
Cost	7880

\[\begin{array}{l} t_0 := \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ t_1 := \frac{x.im}{y.re} - t_0\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.9 \cdot 10^{+69}:\\ \;\;\;\;\frac{y.re \cdot x.im}{\mathsf{fma}\left(y.im, y.im, y.re \cdot y.re\right)} - t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	10.7
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.re \cdot x.im - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+78}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-42}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{elif}\;y.im \leq 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	13.5
Cost	1096

\[\begin{array}{l} t_0 := \frac{x.im}{y.re} - \frac{x.re}{y.im + \frac{y.re}{\frac{y.im}{y.re}}}\\ \mathbf{if}\;y.re \leq -1.42 \cdot 10^{-61}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.3
Cost	840

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+77}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	16.6
Cost	840

\[\begin{array}{l} t_0 := \frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.85 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	24.2
Cost	520

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -5.6 \cdot 10^{+53}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	37.7
Cost	192

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

Reproduce

herbie shell --seed 2022330 
(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))))