_divideComplex, real part

Average Error: 26.2 → 10.4

Time: 22.2s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.re \leq -3.4074423324263793 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-186}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 10^{-180}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 6.8607979028319614 \cdot 10^{+128}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

FPCore

(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
         (*
          (/ 1.0 (hypot y.re y.im))
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))))
   (if (<= y.re -3.4074423324263793e+59)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (if (<= y.re -1e-186)
       t_0
       (if (<= y.re 1e-180)
         (+ (/ x.im y.im) (/ (/ x.re y.im) (/ y.im y.re)))
         (if (<= y.re 6.8607979028319614e+128)
           t_0
           (+ (/ x.re y.re) (/ (/ x.im y.re) (/ y.re y.im)))))))))

C

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 = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	double tmp;
	if (y_46_re <= -3.4074423324263793e+59) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_re <= -1e-186) {
		tmp = t_0;
	} else if (y_46_re <= 1e-180) {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) / (y_46_im / y_46_re));
	} else if (y_46_re <= 6.8607979028319614e+128) {
		tmp = t_0;
	} else {
		tmp = (x_46_re / y_46_re) + ((x_46_im / y_46_re) / (y_46_re / y_46_im));
	}
	return tmp;
}

Julia

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(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.4074423324263793e+59)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	elseif (y_46_re <= -1e-186)
		tmp = t_0;
	elseif (y_46_re <= 1e-180)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) / Float64(y_46_im / y_46_re)));
	elseif (y_46_re <= 6.8607979028319614e+128)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im / y_46_re) / Float64(y_46_re / y_46_im)));
	end
	return tmp
end

Wolfram

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[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * 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]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.4074423324263793e+59], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1e-186], t$95$0, If[LessEqual[y$46$re, 1e-180], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 6.8607979028319614e+128], t$95$0, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.40744233242637929e59

   1. Initial program 37.1

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

   2. Simplified 37.1

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

   3. Taylor expanded in y.re around inf 24.8

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{3}} + \left(\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\right)} \]

   4. Simplified 20.3

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (*.f64 y.re y.re)) (-.f64 x.im (/.f64 (*.f64 y.im x.re) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) (-.f64 x.im (/.f64 (*.f64 y.im x.re) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (pow.f64 y.re 2)) (-.f64 x.im (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x.re y.im)) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x.im (/.f64 y.im (pow.f64 y.re 2))) (*.f64 (/.f64 (*.f64 x.re y.im) y.re) (/.f64 y.im (pow.f64 y.re 2)))))): 0 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y.im (pow.f64 y.re 2)) x.im)) (*.f64 (/.f64 (*.f64 x.re y.im) y.re) (/.f64 y.im (pow.f64 y.re 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (*.f64 (/.f64 (*.f64 x.re y.im) y.re) (/.f64 y.im (pow.f64 y.re 2))))): 19 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 x.re y.im) y.im) (*.f64 y.re (pow.f64 y.re 2)))))): 17 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 x.re (*.f64 y.im y.im))) (*.f64 y.re (pow.f64 y.re 2))))): 8 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (*.f64 x.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) (*.f64 y.re (pow.f64 y.re 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (*.f64 y.re (Rewrite=> unpow2_binary64 (*.f64 y.re y.re)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (Rewrite<= cube-mult_binary64 (pow.f64 y.re 3))))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (neg.f64 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3))) (+.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

   5. Taylor expanded in y.im around 0 18.2

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

   6. Simplified 16.0

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}} \]
      Proof
      (/.f64 x.im (/.f64 y.re (/.f64 y.im y.re))): 0 points increase in error, 0 points decrease in error
      (/.f64 x.im (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re y.re) y.im))): 31 points increase in error, 12 points decrease in error
      (/.f64 x.im (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.im) (pow.f64 y.re 2))): 40 points increase in error, 25 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) (pow.f64 y.re 2)): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 12.2

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

   if -3.40744233242637929e59 < y.re < -9.9999999999999991e-187 or 1e-180 < y.re < 6.86079790283196141e128

   1. Initial program 16.9

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

   2. Simplified 16.9

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

   3. Applied egg-rr 11.3

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

   if -9.9999999999999991e-187 < y.re < 1e-180

   1. Initial program 24.5

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

   2. Simplified 24.5

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

   3. Applied egg-rr 13.8

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

   4. Taylor expanded in y.re around 0 9.1

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

   5. Simplified 7.1

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y.re}{y.im}, \frac{x.re}{y.im}, \frac{x.im}{y.im}\right)} \]
      Proof
      (fma.f64 (/.f64 y.re y.im) (/.f64 x.re y.im) (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 y.re y.im) (/.f64 x.re y.im)) (/.f64 x.im y.im))): 0 points increase in error, 2 points decrease in error
      (+.f64 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re x.re) (*.f64 y.im y.im))) (/.f64 x.im y.im)): 43 points increase in error, 9 points decrease in error
      (+.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (*.f64 y.im y.im)) (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 x.re y.re) (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error

   6. Applied egg-rr 7.1

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

   7. Applied egg-rr 7.3

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

   if 6.86079790283196141e128 < y.re

   1. Initial program 41.7

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

   2. Simplified 41.7

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

   3. Taylor expanded in y.re around inf 23.6

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot {y.im}^{2}}{{y.re}^{3}} + \left(\frac{x.re}{y.re} + \frac{y.im \cdot x.im}{{y.re}^{2}}\right)} \]

   4. Simplified 18.5

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re \cdot y.re} \cdot \left(x.im - \frac{y.im \cdot x.re}{y.re}\right)} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (*.f64 y.re y.re)) (-.f64 x.im (/.f64 (*.f64 y.im x.re) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (Rewrite<= unpow2_binary64 (pow.f64 y.re 2))) (-.f64 x.im (/.f64 (*.f64 y.im x.re) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im (pow.f64 y.re 2)) (-.f64 x.im (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x.re y.im)) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 x.im (/.f64 y.im (pow.f64 y.re 2))) (*.f64 (/.f64 (*.f64 x.re y.im) y.re) (/.f64 y.im (pow.f64 y.re 2)))))): 0 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 y.im (pow.f64 y.re 2)) x.im)) (*.f64 (/.f64 (*.f64 x.re y.im) y.re) (/.f64 y.im (pow.f64 y.re 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (*.f64 (/.f64 (*.f64 x.re y.im) y.re) (/.f64 y.im (pow.f64 y.re 2))))): 19 points increase in error, 7 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (*.f64 x.re y.im) y.im) (*.f64 y.re (pow.f64 y.re 2)))))): 17 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (Rewrite<= associate-*r*_binary64 (*.f64 x.re (*.f64 y.im y.im))) (*.f64 y.re (pow.f64 y.re 2))))): 8 points increase in error, 2 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (*.f64 x.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) (*.f64 y.re (pow.f64 y.re 2))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (*.f64 y.re (Rewrite=> unpow2_binary64 (*.f64 y.re y.re)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (-.f64 (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2)) (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (Rewrite<= cube-mult_binary64 (pow.f64 y.re 3))))): 0 points increase in error, 1 points decrease in error
      (Rewrite<= associate--l+_binary64 (-.f64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (neg.f64 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (pow.f64 y.re 2))) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3))))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 x.re (pow.f64 y.im 2)) (pow.f64 y.re 3))) (+.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

   5. Taylor expanded in y.im around 0 14.6

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

   6. Simplified 12.2

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{x.im}{\frac{y.re}{\frac{y.im}{y.re}}}} \]
      Proof
      (/.f64 x.im (/.f64 y.re (/.f64 y.im y.re))): 0 points increase in error, 0 points decrease in error
      (/.f64 x.im (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re y.re) y.im))): 31 points increase in error, 12 points decrease in error
      (/.f64 x.im (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.im) (pow.f64 y.re 2))): 40 points increase in error, 25 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) (pow.f64 y.re 2)): 0 points increase in error, 0 points decrease in error

   7. Applied egg-rr 9.0

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

   8. Applied egg-rr 8.8

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4074423324263793 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-186}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq 10^{-180}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 6.8607979028319614 \cdot 10^{+128}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.4
Cost	14100

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.re \leq -3.4074423324263793 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.376001427379716 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.5319451451925894 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \mathsf{fma}\left(\frac{y.im}{y.re}, x.im, x.re\right)\\ \end{array} \]

Alternative 2
Error	17.2
Cost	1756

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.0832892480691606 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.2502913451501095 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.6689945781376443 \cdot 10^{+81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.8607979028319614 \cdot 10^{+128}:\\ \;\;\;\;x.im \cdot \frac{-1}{-0.5 \cdot \frac{y.re \cdot y.re}{y.im} - y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 3
Error	13.5
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{if}\;y.re \leq -3.4074423324263793 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.376001427379716 \cdot 10^{-34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.5319451451925894 \cdot 10^{+25}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 4
Error	16.3
Cost	1364

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.0832892480691606 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.2502913451501095 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq -1 \cdot 10^{-298}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 5
Error	16.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.0832892480691606 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.2502913451501095 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.3
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.0832892480691606 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.re \leq -1.2502913451501095 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{y.re \cdot \frac{y.re}{y.im}}\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \end{array} \]

Alternative 7
Error	20.1
Cost	968

\[\begin{array}{l} \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 5.362293137817875 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	15.5
Cost	968

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{if}\;y.im \leq -9.834115852018582 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 4.484683706448499 \cdot 10^{-9}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 9
Error	16.7
Cost	968

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	15.9
Cost	968

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{\frac{x.im}{y.re}}{\frac{y.re}{y.im}}\\ \mathbf{if}\;y.re \leq -0.006765725800101391:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.191442538112069 \cdot 10^{+47}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{x.re}{y.im}}{\frac{y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 11
Error	22.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.714762643781653 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 700128388402989.4:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12
Error	58.8
Cost	192

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

Alternative 13
Error	37.3
Cost	192

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

Reproduce

herbie shell --seed 2022316 
(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))))