_divideComplex, real part

Average Error: 26.4 → 10.8

Time: 20.3s

Precision: binary64

Cost: 20828

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -5.174344703148754 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -90.43388479818566:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-121}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-134}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6166223276898808 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

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
         (/
          (/ (fma x.re y.re (* y.im x.im)) (hypot y.re y.im))
          (hypot y.re y.im))))
   (if (<= y.im -3.185189906959867e+122)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
     (if (<= y.im -7.577584446069664e+94)
       (+ (/ x.re y.re) (/ (/ y.im y.re) (/ y.re x.im)))
       (if (<= y.im -5.174344703148754e+29)
         t_0
         (if (<= y.im -90.43388479818566)
           (* (/ y.re (hypot y.re y.im)) (/ x.re (hypot y.re y.im)))
           (if (<= y.im -1e-121)
             t_0
             (if (<= y.im 1e-134)
               (+ (/ x.re y.re) (/ (* x.im (/ y.im y.re)) y.re))
               (if (<= y.im 1.6166223276898808e+114)
                 t_0
                 (fma (/ (/ x.re y.im) y.im) y.re (/ x.im 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 = (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 <= -3.185189906959867e+122) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else if (y_46_im <= -7.577584446069664e+94) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) / (y_46_re / x_46_im));
	} else if (y_46_im <= -5.174344703148754e+29) {
		tmp = t_0;
	} else if (y_46_im <= -90.43388479818566) {
		tmp = (y_46_re / hypot(y_46_re, y_46_im)) * (x_46_re / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -1e-121) {
		tmp = t_0;
	} else if (y_46_im <= 1e-134) {
		tmp = (x_46_re / y_46_re) + ((x_46_im * (y_46_im / y_46_re)) / y_46_re);
	} else if (y_46_im <= 1.6166223276898808e+114) {
		tmp = t_0;
	} else {
		tmp = fma(((x_46_re / y_46_im) / y_46_im), y_46_re, (x_46_im / 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(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 <= -3.185189906959867e+122)
		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 <= -7.577584446069664e+94)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) / Float64(y_46_re / x_46_im)));
	elseif (y_46_im <= -5.174344703148754e+29)
		tmp = t_0;
	elseif (y_46_im <= -90.43388479818566)
		tmp = Float64(Float64(y_46_re / hypot(y_46_re, y_46_im)) * Float64(x_46_re / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -1e-121)
		tmp = t_0;
	elseif (y_46_im <= 1e-134)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(x_46_im * Float64(y_46_im / y_46_re)) / y_46_re));
	elseif (y_46_im <= 1.6166223276898808e+114)
		tmp = t_0;
	else
		tmp = fma(Float64(Float64(x_46_re / y_46_im) / y_46_im), y_46_re, Float64(x_46_im / 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[(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, -3.185189906959867e+122], 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, -7.577584446069664e+94], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5.174344703148754e+29], t$95$0, If[LessEqual[y$46$im, -90.43388479818566], N[(N[(y$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1e-121], t$95$0, If[LessEqual[y$46$im, 1e-134], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6166223276898808e+114], t$95$0, N[(N[(N[(x$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] * y$46$re + N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.im < -3.1851899069598672e122

   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. Applied egg-rr 28.5

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

   4. Taylor expanded in y.re around 0 16.9

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

   5. Simplified 9.9

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      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)))): 40 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

   if -3.1851899069598672e122 < y.im < -7.5775844460696643e94

   1. Initial program 23.8

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

   2. Simplified 23.8

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 19.8

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

   4. Taylor expanded in y.re around inf 48.1

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

   5. Simplified 42.8

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      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)))): 35 points increase in error, 13 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

   6. Applied egg-rr 42.8

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

   if -7.5775844460696643e94 < y.im < -5.1743447031487538e29 or -90.433884798185659 < y.im < -9.9999999999999998e-122 or 1.00000000000000004e-134 < y.im < 1.61662232768988079e114

   1. Initial program 16.7

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

   2. Simplified 16.7

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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. 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. Applied egg-rr 11.3

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

   if -5.1743447031487538e29 < y.im < -90.433884798185659

   1. Initial program 11.0

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

   2. Taylor expanded in x.re around inf 38.7

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

   3. Simplified 38.7

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

   4. Applied egg-rr 31.1

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

   if -9.9999999999999998e-122 < y.im < 1.00000000000000004e-134

   1. Initial program 22.1

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

   2. Simplified 22.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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. Applied egg-rr 12.5

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

   4. Taylor expanded in y.re around inf 9.7

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

   5. Simplified 8.8

      \[\leadsto \color{blue}{\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}} \]
      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)))): 35 points increase in error, 13 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

   6. Applied egg-rr 6.7

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

   if 1.61662232768988079e114 < y.im

   1. Initial program 41.1

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

   2. Simplified 41.1

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}} \]
      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 0 15.2

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

   4. Simplified 9.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)} \]
      Proof
      (fma.f64 (/.f64 (/.f64 x.re y.im) y.im) y.re (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= associate-/r*_binary64 (/.f64 x.re (*.f64 y.im y.im))) y.re (/.f64 x.im y.im)): 16 points increase in error, 5 points decrease in error
      (fma.f64 (/.f64 x.re (Rewrite<= unpow2_binary64 (pow.f64 y.im 2))) y.re (/.f64 x.im y.im)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (/.f64 x.re (pow.f64 y.im 2)) y.re) (/.f64 x.im y.im))): 0 points increase in error, 0 points decrease in error
      (+.f64 (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x.re y.re) (pow.f64 y.im 2))) (/.f64 x.im y.im)): 22 points increase in error, 11 points decrease in error
3. Recombined 6 regimes into one program.

4. Final simplification 10.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -5.174344703148754 \cdot 10^{+29}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -90.43388479818566:\\ \;\;\;\;\frac{y.re}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-121}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 10^{-134}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6166223276898808 \cdot 10^{+114}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	12.4
Cost	14164

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -1.11247770032971 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6166223276898808 \cdot 10^{+114}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 2
Error	12.4
Cost	7636

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -1.11247770032971 \cdot 10^{-67}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6166223276898808 \cdot 10^{+114}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\frac{x.re}{y.im}}{y.im}, y.re, \frac{x.im}{y.im}\right)\\ \end{array} \]

Alternative 3
Error	16.0
Cost	1760

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_2 := \frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -5.174344703148754 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -0.22406358666434653:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.445044566550207 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.3745397382678178 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.1331976565155514 \cdot 10^{-6}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 5.7761498226557776 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	16.0
Cost	1760

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ t_1 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_2 := \frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -5.174344703148754 \cdot 10^{+29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -0.22406358666434653:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.445044566550207 \cdot 10^{-38}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.3745397382678178 \cdot 10^{-63}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.1331976565155514 \cdot 10^{-6}:\\ \;\;\;\;\frac{y.im \cdot x.im + y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 5.7761498226557776 \cdot 10^{+26}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	12.2
Cost	1620

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ t_1 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -7.577584446069664 \cdot 10^{+94}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -1.11247770032971 \cdot 10^{-67}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 10^{-105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6166223276898808 \cdot 10^{+114}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	16.3
Cost	1496

\[\begin{array}{l} \mathbf{if}\;y.re \leq -4.0336545901129105 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -189435.4630368468:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq -8.370943940408767 \cdot 10^{-65}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.821178075742817 \cdot 10^{-63}:\\ \;\;\;\;\frac{x.im}{y.im} - \frac{\frac{y.re}{y.im} \cdot x.re}{-y.im}\\ \mathbf{elif}\;y.re \leq 1.3331030262292513 \cdot 10^{+37}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.1551020739359037 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 7
Error	20.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.22406358666434653:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -3.445044566550207 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9389250469649876 \cdot 10^{+148}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Error	20.3
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.22406358666434653:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -3.445044566550207 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.9389250469649876 \cdot 10^{+148}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	15.4
Cost	1232

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \frac{\frac{y.re \cdot x.re}{y.im}}{y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{\frac{y.im}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -4.0336545901129105 \cdot 10^{+32}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -189435.4630368468:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -8.370943940408767 \cdot 10^{-65}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.2382339467019525 \cdot 10^{+21}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	24.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.185189906959867 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.22406358666434653:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.445044566550207 \cdot 10^{-38}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3745397382678178 \cdot 10^{-63}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Error	58.8
Cost	192

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

Alternative 12
Error	37.0
Cost	192

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

Reproduce

herbie shell --seed 2022295 
(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))))