_divideComplex, real part

Average Error: 26.4 → 14.1

Time: 19.4s

Precision: binary64

Cost: 33296

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2736672553024105 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-70}:\\ \;\;\;\;{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{-2} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

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
 (if (<= y.im -2.2232753094407857e+130)
   (+ (/ x.im y.im) (/ (/ y.re (/ y.im x.re)) y.im))
   (if (<= y.im -4.8551994884230834e+94)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (if (<= y.im -1.2736672553024105e+76)
       (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
       (if (<= y.im -1e-70)
         (*
          (pow (sqrt (hypot y.re y.im)) -2.0)
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im)))
         (if (<= y.im 1.5953427575134446e-25)
           (* (+ x.re (* y.im (/ x.im y.re))) (/ 1.0 y.re))
           (+ (/ x.im y.im) (/ y.re (/ y.im (/ x.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 tmp;
	if (y_46_im <= -2.2232753094407857e+130) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / (y_46_im / x_46_re)) / y_46_im);
	} else if (y_46_im <= -4.8551994884230834e+94) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else if (y_46_im <= -1.2736672553024105e+76) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_im <= -1e-70) {
		tmp = pow(sqrt(hypot(y_46_re, y_46_im)), -2.0) * (fma(x_46_re, y_46_re, (y_46_im * x_46_im)) / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 1.5953427575134446e-25) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) * (1.0 / y_46_re);
	} else {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im / (x_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)
	tmp = 0.0
	if (y_46_im <= -2.2232753094407857e+130)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / Float64(y_46_im / x_46_re)) / y_46_im));
	elseif (y_46_im <= -4.8551994884230834e+94)
		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_im <= -1.2736672553024105e+76)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	elseif (y_46_im <= -1e-70)
		tmp = Float64((sqrt(hypot(y_46_re, y_46_im)) ^ -2.0) * Float64(fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im)) / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 1.5953427575134446e-25)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) * Float64(1.0 / y_46_re));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im / Float64(x_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_] := If[LessEqual[y$46$im, -2.2232753094407857e+130], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.8551994884230834e+94], 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$46$im, -1.2736672553024105e+76], 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], If[LessEqual[y$46$im, -1e-70], N[(N[Power[N[Sqrt[N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]], $MachinePrecision], -2.0], $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$46$im, 1.5953427575134446e-25], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im / N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.im < -2.22327530944078571e130

   1. Initial program 42.9

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

   2. Simplified 42.9

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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. Taylor expanded in y.re around 0 24.7

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

   4. Simplified 19.4

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot y.im} \cdot \left(x.re - \frac{x.im \cdot y.re}{y.im}\right)} \]
      Proof
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re (*.f64 y.im y.im)) (-.f64 x.re (/.f64 (*.f64 x.im y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (*.f64 (/.f64 y.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) (-.f64 x.re (/.f64 (*.f64 x.im y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= distribute-lft-out--_binary64 (-.f64 (*.f64 (/.f64 y.re (pow.f64 y.im 2)) x.re) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 y.re (/.f64 (pow.f64 y.im 2) x.re))) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im)))): 7 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y.re x.re) (pow.f64 y.im 2))) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im)))): 14 points increase in error, 6 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 x.re y.re)) (pow.f64 y.im 2)) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (*.f64 x.im y.re) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (*.f64 (/.f64 y.re (pow.f64 y.im 2)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 y.re x.im)) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.re (*.f64 y.re x.im)) (*.f64 (pow.f64 y.im 2) y.im))))): 17 points increase in error, 4 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 y.re y.re) x.im)) (*.f64 (pow.f64 y.im 2) y.im)))): 4 points increase in error, 1 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)) x.im) (*.f64 (pow.f64 y.im 2) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (*.f64 (Rewrite=> unpow2_binary64 (*.f64 y.im y.im)) y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (-.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (Rewrite<= unpow3_binary64 (pow.f64 y.im 3))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (neg.f64 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.im y.im) (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))))): 0 points increase in error, 0 points decrease in error
      (Rewrite=> +-commutative_binary64 (+.f64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-+r+_binary64 (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (+.f64 (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3))) (/.f64 x.im y.im)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2)) (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 x.im y.im) (*.f64 -1 (/.f64 (*.f64 (pow.f64 y.re 2) x.im) (pow.f64 y.im 3)))))): 0 points increase in error, 0 points decrease in error

   5. Applied egg-rr 15.1

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

   6. Taylor expanded in y.re around 0 13.2

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

   7. Simplified 8.7

      \[\leadsto \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot \frac{x.re}{y.im}}}{y.im} \]
      Proof
      (*.f64 y.re (/.f64 x.re y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 x.re y.im) y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x.re y.re) y.im)): 52 points increase in error, 40 points decrease in error

   8. Applied egg-rr 8.7

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

   if -2.22327530944078571e130 < y.im < -4.8551994884230834e94

   1. Initial program 21.8

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

   2. Simplified 21.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 16.2

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

   4. Taylor expanded in y.re around inf 47.9

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

   5. Simplified 41.7

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      Proof
      (+.f64 (/.f64 x.re y.re) (*.f64 (/.f64 y.im y.re) (/.f64 x.im y.re))): 0 points increase in error, 0 points decrease in error
      (+.f64 (/.f64 x.re y.re) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 y.im x.im) (*.f64 y.re y.re)))): 38 points increase in error, 10 points decrease in error
      (+.f64 (/.f64 x.re y.re) (/.f64 (*.f64 y.im x.im) (Rewrite<= unpow2_binary64 (pow.f64 y.re 2)))): 0 points increase in error, 0 points decrease in error

   if -4.8551994884230834e94 < y.im < -1.273667255302411e76

   1. Initial program 20.0

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

   2. Simplified 20.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 17.2

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

   4. Taylor expanded in y.re around 0 21.6

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

   5. Simplified 20.5

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      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)))): 36 points increase in error, 8 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

   if -1.273667255302411e76 < y.im < -9.99999999999999996e-71

   1. Initial program 16.8

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

   2. Simplified 16.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 11.2

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

   4. Applied egg-rr 11.4

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

   if -9.99999999999999996e-71 < y.im < 1.5953427575134446e-25

   1. Initial program 20.0

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

   2. Simplified 20.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 11.4

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

   4. Taylor expanded in y.re around -inf 34.5

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

   5. Simplified 35.1

      \[\leadsto \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(\left(-x.re\right) - \frac{x.im}{y.re} \cdot y.im\right)} \]
      Proof
      (-.f64 (neg.f64 x.re) (*.f64 (/.f64 x.im y.re) y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= mul-1-neg_binary64 (*.f64 -1 x.re)) (*.f64 (/.f64 x.im y.re) y.im)): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 -1 x.re) (Rewrite<= associate-/r/_binary64 (/.f64 x.im (/.f64 y.re y.im)))): 22 points increase in error, 18 points decrease in error
      (-.f64 (*.f64 -1 x.re) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x.im y.im) y.re))): 20 points increase in error, 21 points decrease in error
      (-.f64 (*.f64 -1 x.re) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y.im x.im)) y.re)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= unsub-neg_binary64 (+.f64 (*.f64 -1 x.re) (neg.f64 (/.f64 (*.f64 y.im x.im) y.re)))): 0 points increase in error, 0 points decrease in error
      (+.f64 (*.f64 -1 x.re) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (*.f64 y.im x.im) y.re)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (*.f64 y.im x.im) y.re)) (*.f64 -1 x.re))): 0 points increase in error, 0 points decrease in error

   6. Taylor expanded in y.re around -inf 12.7

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

   if 1.5953427575134446e-25 < y.im

   1. Initial program 31.0

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

   2. Simplified 31.0

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 20.6

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

   4. Taylor expanded in y.re around 0 20.4

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

   5. Simplified 16.1

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      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)))): 36 points increase in error, 8 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

   6. Applied egg-rr 16.8

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}} \]
3. Recombined 6 regimes into one program.

4. Final simplification 14.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2736672553024105 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-70}:\\ \;\;\;\;{\left(\sqrt{\mathsf{hypot}\left(y.re, y.im\right)}\right)}^{-2} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	20560

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2736672553024105 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 2
Error	14.7
Cost	20172

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right) \cdot {\left(\mathsf{hypot}\left(y.re, y.im\right)\right)}^{-2}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 3
Error	14.7
Cost	1356

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-70}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 4
Error	19.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -7.888878529761534 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2126397624409436 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	19.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{if}\;y.im \leq -7.888878529761534 \cdot 10^{+116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2126397624409436 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re \cdot \frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 6
Error	19.6
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \mathbf{if}\;y.im \leq -7.888878529761534 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2126397624409436 \cdot 10^{-60}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	19.6
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -7.888878529761534 \cdot 10^{+116}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2126397624409436 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 8
Error	16.5
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4.442316394251276 \cdot 10^{-27}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 9
Error	16.3
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq -4.442316394251276 \cdot 10^{-27}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\left(x.re + y.im \cdot \frac{x.im}{y.re}\right) \cdot \frac{1}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{\frac{y.im}{\frac{x.re}{y.im}}}\\ \end{array} \]

Alternative 10
Error	24.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -2.2232753094407857 \cdot 10^{+130}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8551994884230834 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -1.2126397624409436 \cdot 10^{-60}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.5953427575134446 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Error	58.8
Cost	192

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

Alternative 12
Error	37.5
Cost	192

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

Reproduce

herbie shell --seed 2022306 
(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))))