Average Error: 25.9 → 9.8
Time: 18.3s
Precision: binary64
Cost: 20432
\[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
↓
\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\left(x.im + \frac{y.re}{y.im} \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-170}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{\frac{y.im}{y.re}}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 8.284516120883395 \cdot 10^{+80}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]
(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 (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.im -1.6483411845380726e+80)
     (/ (- (+ x.im (* (/ y.re y.im) x.re))) (hypot y.re y.im))
     (if (<= y.im -1e-170)
       t_0
       (if (<= y.im 1e-240)
         (fma x.im (/ (/ y.im y.re) y.re) (/ x.re y.re))
         (if (<= y.im 8.284516120883395e+80)
           t_0
           (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))))))))
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, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	double tmp;
	if (y_46_im <= -1.6483411845380726e+80) {
		tmp = -(x_46_im + ((y_46_re / y_46_im) * x_46_re)) / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1e-170) {
		tmp = t_0;
	} else if (y_46_im <= 1e-240) {
		tmp = fma(x_46_im, ((y_46_im / y_46_re) / y_46_re), (x_46_re / y_46_re));
	} else if (y_46_im <= 8.284516120883395e+80) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	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(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.6483411845380726e+80)
		tmp = Float64(Float64(-Float64(x_46_im + Float64(Float64(y_46_re / y_46_im) * x_46_re))) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1e-170)
		tmp = t_0;
	elseif (y_46_im <= 1e-240)
		tmp = fma(x_46_im, Float64(Float64(y_46_im / y_46_re) / y_46_re), Float64(x_46_re / y_46_re));
	elseif (y_46_im <= 8.284516120883395e+80)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	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[(y$46$im * x$46$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]}, If[LessEqual[y$46$im, -1.6483411845380726e+80], N[((-N[(x$46$im + N[(N[(y$46$re / y$46$im), $MachinePrecision] * x$46$re), $MachinePrecision]), $MachinePrecision]) / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1e-170], t$95$0, If[LessEqual[y$46$im, 1e-240], N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.284516120883395e+80], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

\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, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\
\mathbf{if}\;y.im \leq -1.6483411845380726 \cdot 10^{+80}:\\
\;\;\;\;\frac{-\left(x.im + \frac{y.re}{y.im} \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\

\mathbf{elif}\;y.im \leq -1 \cdot 10^{-170}:\\
\;\;\;\;t_0\\

\mathbf{elif}\;y.im \leq 10^{-240}:\\
\;\;\;\;\mathsf{fma}\left(x.im, \frac{\frac{y.im}{y.re}}{y.re}, \frac{x.re}{y.re}\right)\\

\mathbf{elif}\;y.im \leq 8.284516120883395 \cdot 10^{+80}:\\
\;\;\;\;t_0\\

\mathbf{else}:\\
\;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\


\end{array}

Error

Derivation

  1. Split input into 4 regimes

  2. if y.im < -1.6483411845380726e80

    1. Initial program 37.5

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

    2. Applied egg-rr26.4

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

    3. Applied egg-rr26.3

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

    4. Taylor expanded in y.im around -inf 15.8

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

    5. Simplified11.8

      \[\leadsto \frac{\color{blue}{\left(-x.im\right) - \frac{y.re}{y.im} \cdot x.re}}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      Proof
      (-.f64 (neg.f64 x.im) (*.f64 (/.f64 y.re y.im) x.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x.im)) (*.f64 (/.f64 y.re y.im) x.re)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 x.im) (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 y.im x.re)))): 31 points increase in error, 18 points decrease in error
      (-.f64 (*.f64 -1 x.im) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) y.im))): 33 points increase in error, 24 points decrease in error
      (-.f64 (*.f64 -1 x.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 x.im) (neg.f64 (/.f64 (*.f64 x.re y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 x.im) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 x.re y.re) y.im)) (*.f64 -1 x.im))): 0 points increase in error, 0 points decrease in error

    if -1.6483411845380726e80 < y.im < -9.99999999999999983e-171 or 9.9999999999999997e-241 < y.im < 8.28451612088339485e80

    1. Initial program 15.8

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

    2. Applied egg-rr9.9

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

    3. Applied egg-rr9.8

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

    if -9.99999999999999983e-171 < y.im < 9.9999999999999997e-241

    1. Initial program 23.9

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

    2. Applied egg-rr13.2

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

    3. Applied egg-rr13.1

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

    4. Taylor expanded in y.re around inf 30.0

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

    5. Taylor expanded in y.im around 0 8.7

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

    6. Simplified4.2

      \[\leadsto \color{blue}{\mathsf{fma}\left(x.im, \frac{\frac{y.im}{y.re}}{y.re}, \frac{x.re}{y.re}\right)} \]
      Proof
      (fma.f64 x.im (/.f64 (/.f64 y.im y.re) y.re) (/.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.im (Rewrite<= associate-/r*_binary64 (/.f64 y.im (*.f64 y.re y.re))) (/.f64 x.re y.re)): 19 points increase in error, 9 points decrease in error
      (fma.f64 x.im (/.f64 y.im (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) (/.f64 x.re y.re)): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.im (/.f64 y.im (pow.f64 y.re 2)) (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 x.re y.re))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 x.im (/.f64 y.im (pow.f64 y.re 2)) (neg.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 x.re y.re))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-udef_binary64 (+.f64 (*.f64 x.im (/.f64 y.im (pow.f64 y.re 2))) (neg.f64 (*.f64 -1 (/.f64 x.re y.re))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y.im (pow.f64 y.re 2)) x.im)) (neg.f64 (*.f64 -1 (/.f64 x.re y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (neg.f64 (*.f64 -1 (/.f64 x.re y.re)))): 27 points increase in error, 12 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (neg.f64 (Rewrite=> mul-1-neg_binary64 (neg.f64 (/.f64 x.re y.re))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (Rewrite=> remove-double-neg_binary64 (/.f64 x.re y.re))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

    if 8.28451612088339485e80 < y.im

    1. Initial program 39.1

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

    2. Applied egg-rr26.7

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

    3. Taylor expanded in y.re around 0 18.1

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

    4. Simplified11.7

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im)))): 49 points increase in error, 11 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
  3. Recombined 4 regimes into one program.

  4. Final simplification9.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\left(x.im + \frac{y.re}{y.im} \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-170}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 10^{-240}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{\frac{y.im}{y.re}}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 8.284516120883395 \cdot 10^{+80}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error11.9
Cost7372
\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0576790863064915 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{\frac{y.im}{y.re}}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error11.8
Cost7372
\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;\frac{-\left(x.im + \frac{y.re}{y.im} \cdot x.re\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0576790863064915 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(x.im, \frac{\frac{y.im}{y.re}}{y.re}, \frac{x.re}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]
Alternative 3
Error12.3
Cost1488
\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.6483411845380726 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.0576790863064915 \cdot 10^{-132}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.5751383169809345 \cdot 10^{+78}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error16.4
Cost1232
\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.7819103948735785 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.5108845785376295 \cdot 10^{-12}:\\ \;\;\;\;\frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \end{array} \]
Alternative 5
Error24.5
Cost1108
\[\begin{array}{l} t_0 := y.im \cdot \frac{x.im}{y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.765362979944965 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 4.395899517888614 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
Alternative 6
Error24.5
Cost1108
\[\begin{array}{l} t_0 := y.im \cdot \frac{\frac{x.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.0955419659833187 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.474451949234269 \cdot 10^{+55}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3.337487325098142 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.765362979944965 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 4.395899517888614 \cdot 10^{+31}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
Alternative 7
Error20.5
Cost968
\[\begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(x.im + \frac{y.re \cdot x.re}{y.im}\right)\\ \mathbf{if}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 8
Error18.9
Cost968
\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]
Alternative 9
Error18.9
Cost968
\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \end{array} \]
Alternative 10
Error16.5
Cost968
\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{y.im} \cdot x.re}{y.im}\\ \end{array} \]
Alternative 11
Error23.7
Cost456
\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.041012913590577 \cdot 10^{-80}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 7.089272260876135 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
Alternative 12
Error58.5
Cost192
\[\frac{x.im}{y.re} \]
Alternative 13
Error37.4
Cost192
\[\frac{x.re}{y.re} \]

Error

Reproduce

herbie shell --seed 2022294 
(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))))