_divideComplex, real part

Percentage Accurate: 61.9% → 86.6%

Time: 22.8s

Precision: binary64

Cost: 20560

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)\\ \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+106}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{t_0}}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

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 (fma x.re y.re (* y.im x.im))))
   (if (<= y.im -1.65e+106)
     (* (+ x.im (* y.re (/ x.re y.im))) (/ -1.0 (hypot y.re y.im)))
     (if (<= y.im -7.8e-106)
       (/ 1.0 (* (hypot y.re y.im) (/ (hypot y.re y.im) t_0)))
       (if (<= y.im 4.6e-190)
         (* (/ 1.0 y.re) (+ x.re (/ (* y.im x.im) y.re)))
         (if (<= y.im 2.65e+98)
           (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
           (+ (/ x.im y.im) (* (/ x.re y.im) (/ 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 = fma(x_46_re, y_46_re, (y_46_im * x_46_im));
	double tmp;
	if (y_46_im <= -1.65e+106) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= -7.8e-106) {
		tmp = 1.0 / (hypot(y_46_re, y_46_im) * (hypot(y_46_re, y_46_im) / t_0));
	} else if (y_46_im <= 4.6e-190) {
		tmp = (1.0 / y_46_re) * (x_46_re + ((y_46_im * x_46_im) / y_46_re));
	} else if (y_46_im <= 2.65e+98) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re / y_46_im) * (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 = fma(x_46_re, y_46_re, Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_im <= -1.65e+106)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= -7.8e-106)
		tmp = Float64(1.0 / Float64(hypot(y_46_re, y_46_im) * Float64(hypot(y_46_re, y_46_im) / t_0)));
	elseif (y_46_im <= 4.6e-190)
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)));
	elseif (y_46_im <= 2.65e+98)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re / y_46_im) * 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[(x$46$re * y$46$re + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.65e+106], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7.8e-106], N[(1.0 / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] * N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 4.6e-190], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.65e+98], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re / y$46$im), $MachinePrecision] * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -1.65000000000000004e106

   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. Applied egg-rr 64.2%

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

      [Start] 42.9	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 42.9	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 42.9	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 42.8	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 42.8	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 42.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 64.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.im around -inf 80.6%

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

   4. Simplified 88.2%

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

      [Start] 80.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot \frac{x.re \cdot y.re}{y.im} + -1 \cdot x.im\right) \]
      +-commutative [=>] 80.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.im + -1 \cdot \frac{x.re \cdot y.re}{y.im}\right)} \]
      mul-1-neg [=>] 80.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(-1 \cdot x.im + \color{blue}{\left(-\frac{x.re \cdot y.re}{y.im}\right)}\right) \]
      unsub-neg [=>] 80.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\left(-1 \cdot x.im - \frac{x.re \cdot y.re}{y.im}\right)} \]
      mul-1-neg [=>] 80.6	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\color{blue}{\left(-x.im\right)} - \frac{x.re \cdot y.re}{y.im}\right) \]
      associate-/l* [=>] 88.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(-x.im\right) - \color{blue}{\frac{x.re}{\frac{y.im}{y.re}}}\right) \]
      associate-/r/ [=>] 88.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(\left(-x.im\right) - \color{blue}{\frac{x.re}{y.im} \cdot y.re}\right) \]

   if -1.65000000000000004e106 < y.im < -7.80000000000000019e-106

   1. Initial program 70.3%

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

   2. Applied egg-rr 87.1%

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

      [Start] 70.3	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 70.3	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 70.3	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 70.2	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 70.2	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 70.2	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 87.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Applied egg-rr 87.3%

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

      [Start] 87.1	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)} \]
      clear-num [=>] 87.4	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \color{blue}{\frac{1}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}} \]
      frac-times [=>] 87.3	\[ \color{blue}{\frac{1 \cdot 1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}} \]
      metadata-eval [=>] 87.3	\[ \frac{\color{blue}{1}}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}} \]

   if -7.80000000000000019e-106 < y.im < 4.59999999999999984e-190

   1. Initial program 70.7%

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

   2. Applied egg-rr 82.8%

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

      [Start] 70.7	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 70.7	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 70.7	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 70.7	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 70.7	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 70.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 82.8	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   3. Taylor expanded in y.re around inf 56.7%

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

   4. Taylor expanded in y.re around inf 93.6%

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

   if 4.59999999999999984e-190 < y.im < 2.64999999999999999e98

   1. Initial program 73.4%

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

   2. Applied egg-rr 81.7%

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

      [Start] 73.4	\[ \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      *-un-lft-identity [=>] 73.4	\[ \frac{\color{blue}{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      add-sqr-sqrt [=>] 73.4	\[ \frac{1 \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)}{\color{blue}{\sqrt{y.re \cdot y.re + y.im \cdot y.im} \cdot \sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      times-frac [=>] 73.5	\[ \color{blue}{\frac{1}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}}} \]
      hypot-def [=>] 73.5	\[ \frac{1}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      fma-def [=>] 73.5	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\color{blue}{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}}{\sqrt{y.re \cdot y.re + y.im \cdot y.im}} \]
      hypot-def [=>] 81.7	\[ \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\color{blue}{\mathsf{hypot}\left(y.re, y.im\right)}} \]

   if 2.64999999999999999e98 < y.im

   1. Initial program 37.3%

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

   2. Taylor expanded in y.re around 0 72.8%

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

   3. Simplified 89.1%

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]
      Step-by-step derivation

      [Start] 72.8	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      +-commutative [=>] 72.8	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      *-commutative [=>] 72.8	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      unpow2 [=>] 72.8	\[ \frac{x.im}{y.im} + \frac{y.re \cdot x.re}{\color{blue}{y.im \cdot y.im}} \]
      times-frac [=>] 89.1	\[ \frac{x.im}{y.im} + \color{blue}{\frac{y.re}{y.im} \cdot \frac{x.re}{y.im}} \]

3. Recombined 5 regimes into one program.

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.65 \cdot 10^{+106}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-106}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right) \cdot \frac{\mathsf{hypot}\left(y.re, y.im\right)}{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}}\\ \mathbf{elif}\;y.im \leq 4.6 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+98}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	86.7%
Cost	20560

\[\begin{array}{l} t_0 := \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, y.im \cdot x.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{if}\;y.im \leq -5.2 \cdot 10^{+110}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-110}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-190}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+97}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \end{array} \]

Alternative 2
Accuracy	81.6%
Cost	7824

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -6 \cdot 10^{+102}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -2.55 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im} + {\left(\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}\right)}^{-1}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+110}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im}{\frac{-y.re}{-x.im}}\right)\\ \end{array} \]

Alternative 3
Accuracy	80.8%
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+106}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -1.95 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \end{array} \]

Alternative 4
Accuracy	80.8%
Cost	7696

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -5.1 \cdot 10^{-13}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im} + {\left(\frac{y.im}{x.re \cdot \frac{y.re}{y.im}}\right)}^{-1}\\ \mathbf{elif}\;y.re \leq 4.5 \cdot 10^{+111}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \end{array} \]

Alternative 5
Accuracy	71.2%
Cost	1497

\[\begin{array}{l} t_0 := \frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{if}\;y.im \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{-31}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 36000:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{+26} \lor \neg \left(y.im \leq 4.6 \cdot 10^{+38}\right) \land y.im \leq 6.5 \cdot 10^{+72}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 6
Accuracy	74.7%
Cost	1497

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.re \leq -2.75 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+31} \lor \neg \left(y.re \leq 4.7 \cdot 10^{+67}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 7
Accuracy	74.7%
Cost	1497

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+96}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{+63}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.re \leq -3.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{\frac{y.re \cdot y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im \cdot \frac{y.im}{y.re}}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{+31} \lor \neg \left(y.re \leq 10^{+67}\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 8
Accuracy	81.4%
Cost	1488

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im + y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{if}\;y.re \leq -4.1 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.95 \cdot 10^{-172}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.3 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Accuracy	60.6%
Cost	1300

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.3 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-47}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+128}:\\ \;\;\;\;\left(y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{--1}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Accuracy	61.7%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 6.6 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Accuracy	60.5%
Cost	1108

\[\begin{array}{l} \mathbf{if}\;y.im \leq -5.4 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+24}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 3.4 \cdot 10^{+72}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.35 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 12
Accuracy	77.3%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3.5 \cdot 10^{+43} \lor \neg \left(y.im \leq 4.1 \cdot 10^{-41}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{\frac{x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \end{array} \]

Alternative 13
Accuracy	78.1%
Cost	969

\[\begin{array}{l} \mathbf{if}\;y.im \leq -3 \cdot 10^{+43} \lor \neg \left(y.im \leq 1.25 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re}{y.im} \cdot \frac{y.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(x.re + \frac{y.im \cdot x.im}{y.re}\right)\\ \end{array} \]

Alternative 14
Accuracy	63.4%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -6.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 15
Accuracy	42.5%
Cost	192

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

Reproduce

herbie shell --seed 2023159 
(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))))