Average Error: 26.6 → 11.9
Time: 25.8s
Precision: binary64
Cost: 20432
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ t_1 := \frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;x.im \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x.im \leq -1.36 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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 x.re y.re (* x.im y.im)) (hypot y.re y.im))
          (hypot y.re y.im)))
        (t_1 (/ (* x.im (/ y.im (hypot y.im y.re))) (hypot y.re y.im))))
   (if (<= x.im -4.8e+21)
     t_1
     (if (<= x.im -1.36e-161)
       t_0
       (if (<= x.im 2e-181)
         (/ (* y.re (/ x.re (hypot y.im y.re))) (hypot y.re y.im))
         (if (<= x.im 1.6e+110) t_0 t_1))))))
double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_re * y_46_re) + (x_46_im * 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(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double t_1 = (x_46_im * (y_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_re, y_46_im);
	double tmp;
	if (x_46_im <= -4.8e+21) {
		tmp = t_1;
	} else if (x_46_im <= -1.36e-161) {
		tmp = t_0;
	} else if (x_46_im <= 2e-181) {
		tmp = (y_46_re * (x_46_re / hypot(y_46_im, y_46_re))) / hypot(y_46_re, y_46_im);
	} else if (x_46_im <= 1.6e+110) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * 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(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	t_1 = Float64(Float64(x_46_im * Float64(y_46_im / hypot(y_46_im, y_46_re))) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (x_46_im <= -4.8e+21)
		tmp = t_1;
	elseif (x_46_im <= -1.36e-161)
		tmp = t_0;
	elseif (x_46_im <= 2e-181)
		tmp = Float64(Float64(y_46_re * Float64(x_46_re / hypot(y_46_im, y_46_re))) / hypot(y_46_re, y_46_im));
	elseif (x_46_im <= 1.6e+110)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $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]}, Block[{t$95$1 = N[(N[(x$46$im * N[(y$46$im / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$46$im, -4.8e+21], t$95$1, If[LessEqual[x$46$im, -1.36e-161], t$95$0, If[LessEqual[x$46$im, 2e-181], N[(N[(y$46$re * N[(x$46$re / N[Sqrt[y$46$im ^ 2 + y$46$re ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[x$46$im, 1.6e+110], t$95$0, t$95$1]]]]]]

\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}
↓
\begin{array}{l}
t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\
t_1 := \frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\
\mathbf{if}\;x.im \leq -4.8 \cdot 10^{+21}:\\
\;\;\;\;t_1\\

\mathbf{elif}\;x.im \leq -1.36 \cdot 10^{-161}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;x.im \leq 2 \cdot 10^{-181}:\\
\;\;\;\;\frac{y.re \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;x.im \leq 1.6 \cdot 10^{+110}:\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;t_1\\


\end{array}

Error

Derivation

  1. Split input into 3 regimes

  2. if x.im < -4.8e21 or 1.59999999999999997e110 < x.im

    1. Initial program 34.6

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

    2. Applied egg-rr27.6

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

    3. Applied egg-rr27.6

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

    4. Taylor expanded in x.re around 0 40.8

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

    5. Simplified17.1

      \[\leadsto \frac{\color{blue}{\frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot x.im}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Proof
      (*.f64 (/.f64 y.im (hypot.f64 y.im y.re)) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y.im (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 y.im y.im) (*.f64 y.re y.re))))) x.im): 39 points increase in error, 64 points decrease in error
      (*.f64 (/.f64 y.im (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) (*.f64 y.re y.re)))) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y.im (sqrt.f64 (+.f64 (pow.f64 y.im 2) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))))) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y.im (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))))) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y.im (sqrt.f64 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 y.re y.re)) (pow.f64 y.im 2)))) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y.im (sqrt.f64 (+.f64 (*.f64 y.re y.re) (Rewrite=> unpow2_binary64 (*.f64 y.im y.im))))) x.im): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 y.im (Rewrite=> hypot-def_binary64 (hypot.f64 y.re y.im))) x.im): 64 points increase in error, 39 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 y.im (/.f64 (hypot.f64 y.re y.im) x.im))): 28 points increase in error, 60 points decrease in error
      (/.f64 y.im (/.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))) x.im)): 36 points increase in error, 40 points decrease in error
      (/.f64 y.im (/.f64 (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) (*.f64 y.im y.im))) x.im)): 0 points increase in error, 0 points decrease in error
      (/.f64 y.im (/.f64 (sqrt.f64 (+.f64 (pow.f64 y.re 2) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))) x.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.im x.im) (sqrt.f64 (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))))): 71 points increase in error, 9 points decrease in error

    if -4.8e21 < x.im < -1.36e-161 or 2.00000000000000009e-181 < x.im < 1.59999999999999997e110

    1. Initial program 24.1

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

    2. Applied egg-rr11.0

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

    3. Applied egg-rr10.8

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

    if -1.36e-161 < x.im < 2.00000000000000009e-181

    1. Initial program 17.9

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

    2. Applied egg-rr10.4

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

    3. Applied egg-rr10.3

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

    4. Taylor expanded in x.re around inf 21.1

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

    5. Simplified5.4

      \[\leadsto \frac{\color{blue}{\frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)} \cdot y.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Proof
      (*.f64 (/.f64 x.re (hypot.f64 y.im y.re)) y.re): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x.re (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 y.im y.im) (*.f64 y.re y.re))))) y.re): 34 points increase in error, 45 points decrease in error
      (*.f64 (/.f64 x.re (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)) (*.f64 y.re y.re)))) y.re): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x.re (sqrt.f64 (+.f64 (pow.f64 y.im 2) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))))) y.re): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x.re (sqrt.f64 (Rewrite<= +-commutative_binary64 (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))))) y.re): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x.re (sqrt.f64 (+.f64 (Rewrite=> unpow2_binary64 (*.f64 y.re y.re)) (pow.f64 y.im 2)))) y.re): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x.re (sqrt.f64 (+.f64 (*.f64 y.re y.re) (Rewrite=> unpow2_binary64 (*.f64 y.im y.im))))) y.re): 0 points increase in error, 0 points decrease in error
      (*.f64 (/.f64 x.re (Rewrite=> hypot-def_binary64 (hypot.f64 y.re y.im))) y.re): 45 points increase in error, 34 points decrease in error
      (Rewrite<= associate-/r/_binary64 (/.f64 x.re (/.f64 (hypot.f64 y.re y.im) y.re))): 51 points increase in error, 47 points decrease in error
      (/.f64 x.re (/.f64 (Rewrite<= hypot-def_binary64 (sqrt.f64 (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))) y.re)): 33 points increase in error, 63 points decrease in error
      (/.f64 x.re (/.f64 (sqrt.f64 (+.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) (*.f64 y.im y.im))) y.re)): 0 points increase in error, 0 points decrease in error
      (/.f64 x.re (/.f64 (sqrt.f64 (+.f64 (pow.f64 y.re 2) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))) y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.re y.re) (sqrt.f64 (+.f64 (pow.f64 y.re 2) (pow.f64 y.im 2))))): 47 points increase in error, 20 points decrease in error
  3. Recombined 3 regimes into one program.

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;x.im \leq -4.8 \cdot 10^{+21}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;x.im \leq -1.36 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;x.im \leq 2 \cdot 10^{-181}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.re}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;x.im \leq 1.6 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error13.9
Cost13904
\[\begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.8 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re \cdot \frac{\frac{1}{y.im}}{y.im}, \frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq -1.15 \cdot 10^{-234}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.im}{\mathsf{hypot}\left(y.im, y.re\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.62 \cdot 10^{-71}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 39:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{y.im}{\frac{y.re}{x.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
Alternative 2
Error12.8
Cost7700
\[\begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1 \cdot 10^{+145}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.im \leq -2.55 \cdot 10^{-135}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
Alternative 3
Error12.9
Cost7700
\[\begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.15 \cdot 10^{+145}:\\ \;\;\;\;\frac{\left(-x.im\right) - \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.08 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-104}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-16}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + y.re \cdot \frac{x.re}{y.im}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
Alternative 4
Error12.5
Cost7700
\[\begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := x.im + y.re \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.32 \cdot 10^{+76}:\\ \;\;\;\;t_1 \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]
Alternative 5
Error19.1
Cost1496
\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.3 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-128}:\\ \;\;\;\;\left(y.re \cdot x.re\right) \cdot \frac{\frac{-1}{y.im}}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-72}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.95 \cdot 10^{-44}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 50:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 6
Error13.0
Cost1488
\[\begin{array}{l} t_0 := \frac{x.im \cdot y.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+144}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -6.9 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-9}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error19.4
Cost1232
\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
Alternative 8
Error19.6
Cost1232
\[\begin{array}{l} \mathbf{if}\;y.im \leq -1 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq -5.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + x.im \cdot \frac{y.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
Alternative 9
Error19.5
Cost1232
\[\begin{array}{l} \mathbf{if}\;y.im \leq -9 \cdot 10^{+52}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
Alternative 10
Error16.1
Cost1232
\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{+52}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im}{y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 11
Error15.4
Cost968
\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 100:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 12
Error23.4
Cost720
\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-39}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.25 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{+19}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
Alternative 13
Error37.1
Cost192
\[\frac{x.im}{y.im} \]

Error

Reproduce

herbie shell --seed 2022325 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))