_divideComplex, imaginary part

Average Error: 26.5 → 8.3

Time: 17.8s

Precision: binary64

Cost: 33816

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-y.im \cdot x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.5729780386512144 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-172}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.2241809059768097 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.926004778278525 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (- x.im (* y.im (/ x.re y.re))) y.re))
        (t_1
         (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_2 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.im -2.5355883399872014e+249)
     t_2
     (if (<= y.im -9.766740365443902e+235)
       t_0
       (if (<= y.im -3.5729780386512144e+121)
         (- (/ 1.0 (* (/ y.im y.re) (/ y.im x.im))) (/ x.re y.im))
         (if (<= y.im -1e-172)
           t_1
           (if (<= y.im 3.2241809059768097e-149)
             t_0
             (if (<= y.im 2.926004778278525e+141) t_1 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 = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	double t_1 = 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_2 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.5355883399872014e+249) {
		tmp = t_2;
	} else if (y_46_im <= -9.766740365443902e+235) {
		tmp = t_0;
	} else if (y_46_im <= -3.5729780386512144e+121) {
		tmp = (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_im))) - (x_46_re / y_46_im);
	} else if (y_46_im <= -1e-172) {
		tmp = t_1;
	} else if (y_46_im <= 3.2241809059768097e-149) {
		tmp = t_0;
	} else if (y_46_im <= 2.926004778278525e+141) {
		tmp = t_1;
	} 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(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re)
	t_1 = 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(y_46_im * x_46_re)) / (hypot(y_46_re, y_46_im) ^ 2.0)))
	t_2 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.5355883399872014e+249)
		tmp = t_2;
	elseif (y_46_im <= -9.766740365443902e+235)
		tmp = t_0;
	elseif (y_46_im <= -3.5729780386512144e+121)
		tmp = Float64(Float64(1.0 / Float64(Float64(y_46_im / y_46_re) * Float64(y_46_im / x_46_im))) - Float64(x_46_re / y_46_im));
	elseif (y_46_im <= -1e-172)
		tmp = t_1;
	elseif (y_46_im <= 3.2241809059768097e-149)
		tmp = t_0;
	elseif (y_46_im <= 2.926004778278525e+141)
		tmp = t_1;
	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[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$1 = 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$2 = 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, -2.5355883399872014e+249], t$95$2, If[LessEqual[y$46$im, -9.766740365443902e+235], t$95$0, If[LessEqual[y$46$im, -3.5729780386512144e+121], N[(N[(1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] * 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-172], t$95$1, If[LessEqual[y$46$im, 3.2241809059768097e-149], t$95$0, If[LessEqual[y$46$im, 2.926004778278525e+141], t$95$1, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.5355883399872014e249 or 2.9260047782785248e141 < y.im

   1. Initial program 42.5

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

   2. Applied egg-rr 34.7

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

   3. Taylor expanded in y.re around 0 15.2

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

   4. Simplified 7.6

      \[\leadsto \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}} \]
      Proof
      (-.f64 (*.f64 (/.f64 x.im y.im) (/.f64 y.re y.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x.im y.re) (*.f64 y.im y.im))) (/.f64 x.re y.im)): 52 points increase in error, 9 points decrease in error
      (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.re x.im)) (*.f64 y.im y.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 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

   5. Applied egg-rr 7.7

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

   if -2.5355883399872014e249 < y.im < -9.76674036544390209e235 or -1e-172 < y.im < 3.2241809059768097e-149

   1. Initial program 25.4

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

   2. Applied egg-rr 11.0

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

   3. Taylor expanded in y.re around inf 12.5

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

   4. Simplified 10.5

      \[\leadsto \color{blue}{\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}} \]
      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)))): 51 points increase in error, 12 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 9.8

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

   if -9.76674036544390209e235 < y.im < -3.5729780386512144e121

   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 31.3

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

   3. Taylor expanded in y.re around 0 20.1

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

   4. Simplified 14.4

      \[\leadsto \color{blue}{\frac{x.im}{y.im} \cdot \frac{y.re}{y.im} - \frac{x.re}{y.im}} \]
      Proof
      (-.f64 (*.f64 (/.f64 x.im y.im) (/.f64 y.re y.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 x.im y.re) (*.f64 y.im y.im))) (/.f64 x.re y.im)): 52 points increase in error, 9 points decrease in error
      (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.re x.im)) (*.f64 y.im y.im)) (/.f64 x.re y.im)): 0 points increase in error, 0 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

   5. Applied egg-rr 14.5

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

   if -3.5729780386512144e121 < y.im < -1e-172 or 3.2241809059768097e-149 < y.im < 2.9260047782785248e141

   1. Initial program 17.2

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

   2. Applied egg-rr 6.6

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

4. Final simplification 8.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq -3.5729780386512144 \cdot 10^{+121}:\\ \;\;\;\;\frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-172}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-y.im \cdot x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{elif}\;y.im \leq 3.2241809059768097 \cdot 10^{-149}:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.926004778278525 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{x.im}{\mathsf{hypot}\left(y.re, y.im\right)}, \frac{-y.im \cdot x.re}{{\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	12.6
Cost	14424

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{\frac{x.im \cdot y.re - y.im \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_2 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.939221699676908 \cdot 10^{+146}:\\ \;\;\;\;\frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -76127214.12311965:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.9168056692032655 \cdot 10^{-159}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.424138041514158 \cdot 10^{+124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 2
Error	14.7
Cost	1620

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.516151575779809 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9704135634555555 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.424138041514158 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	16.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{\frac{x.im}{y.im}}{\frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.516151575779809 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.001975080593576 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.2
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.516151575779809 \cdot 10^{+29}:\\ \;\;\;\;\frac{y.re}{y.im \cdot \frac{y.im}{x.im}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.001975080593576 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.im}}{\frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]

Alternative 5
Error	19.8
Cost	1104

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ t_1 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.7248893549882116 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.970989366873508 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.4
Cost	1104

\[\begin{array}{l} t_0 := \frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ t_1 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.5355883399872014 \cdot 10^{+249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -9.766740365443902 \cdot 10^{+235}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.516151575779809 \cdot 10^{+29}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.001975080593576 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	23.1
Cost	520

\[\begin{array}{l} t_0 := \frac{-x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.7248893549882116 \cdot 10^{+32}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.001975080593576 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	37.1
Cost	192

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

Reproduce

herbie shell --seed 2022308 
(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))))