_divideComplex, real part

Average Error: 25.6 → 15.7

Time: 29.2s

Precision: binary64

Cost: 7832

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.re \cdot y.re + x.im \cdot y.im\\ t_2 := \frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -0.21:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{\frac{t_0}{y.re \cdot x.re + y.im \cdot x.im}}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{-136}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

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 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (+ (* x.re y.re) (* x.im y.im)))
        (t_2 (+ (/ x.im y.im) (* x.re (/ y.re (pow y.im 2.0))))))
   (if (<= y.re -2.45e+123)
     (/ x.re y.re)
     (if (<= y.re -5.1e+54)
       (* (/ 1.0 t_0) t_1)
       (if (<= y.re -1.85e+53)
         (/ x.re y.re)
         (if (<= y.re -0.21)
           t_2
           (if (<= y.re -2.3e-136)
             (/ 1.0 (/ t_0 (+ (* y.re x.re) (* y.im x.im))))
             (if (<= y.re 2.9e-136)
               t_2
               (if (<= y.re 1.6e+139) (/ t_1 t_0) (/ x.re y.re))))))))))

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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double t_2 = (x_46_im / y_46_im) + (x_46_re * (y_46_re / pow(y_46_im, 2.0)));
	double tmp;
	if (y_46_re <= -2.45e+123) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.1e+54) {
		tmp = (1.0 / t_0) * t_1;
	} else if (y_46_re <= -1.85e+53) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -0.21) {
		tmp = t_2;
	} else if (y_46_re <= -2.3e-136) {
		tmp = 1.0 / (t_0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im)));
	} else if (y_46_re <= 2.9e-136) {
		tmp = t_2;
	} else if (y_46_re <= 1.6e+139) {
		tmp = t_1 / t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

↓

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46re * y_46re) + (x_46im * y_46im)
    t_2 = (x_46im / y_46im) + (x_46re * (y_46re / (y_46im ** 2.0d0)))
    if (y_46re <= (-2.45d+123)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-5.1d+54)) then
        tmp = (1.0d0 / t_0) * t_1
    else if (y_46re <= (-1.85d+53)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-0.21d0)) then
        tmp = t_2
    else if (y_46re <= (-2.3d-136)) then
        tmp = 1.0d0 / (t_0 / ((y_46re * x_46re) + (y_46im * x_46im)))
    else if (y_46re <= 2.9d-136) then
        tmp = t_2
    else if (y_46re <= 1.6d+139) then
        tmp = t_1 / t_0
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static 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));
}

↓

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double t_2 = (x_46_im / y_46_im) + (x_46_re * (y_46_re / Math.pow(y_46_im, 2.0)));
	double tmp;
	if (y_46_re <= -2.45e+123) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.1e+54) {
		tmp = (1.0 / t_0) * t_1;
	} else if (y_46_re <= -1.85e+53) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -0.21) {
		tmp = t_2;
	} else if (y_46_re <= -2.3e-136) {
		tmp = 1.0 / (t_0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im)));
	} else if (y_46_re <= 2.9e-136) {
		tmp = t_2;
	} else if (y_46_re <= 1.6e+139) {
		tmp = t_1 / t_0;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, 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))

↓

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	t_2 = (x_46_im / y_46_im) + (x_46_re * (y_46_re / math.pow(y_46_im, 2.0)))
	tmp = 0
	if y_46_re <= -2.45e+123:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -5.1e+54:
		tmp = (1.0 / t_0) * t_1
	elif y_46_re <= -1.85e+53:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -0.21:
		tmp = t_2
	elif y_46_re <= -2.3e-136:
		tmp = 1.0 / (t_0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im)))
	elif y_46_re <= 2.9e-136:
		tmp = t_2
	elif y_46_re <= 1.6e+139:
		tmp = t_1 / t_0
	else:
		tmp = x_46_re / y_46_re
	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(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im))
	t_2 = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(y_46_re / (y_46_im ^ 2.0))))
	tmp = 0.0
	if (y_46_re <= -2.45e+123)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -5.1e+54)
		tmp = Float64(Float64(1.0 / t_0) * t_1);
	elseif (y_46_re <= -1.85e+53)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -0.21)
		tmp = t_2;
	elseif (y_46_re <= -2.3e-136)
		tmp = Float64(1.0 / Float64(t_0 / Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im))));
	elseif (y_46_re <= 2.9e-136)
		tmp = t_2;
	elseif (y_46_re <= 1.6e+139)
		tmp = Float64(t_1 / t_0);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
end

↓

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	t_2 = (x_46_im / y_46_im) + (x_46_re * (y_46_re / (y_46_im ^ 2.0)));
	tmp = 0.0;
	if (y_46_re <= -2.45e+123)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -5.1e+54)
		tmp = (1.0 / t_0) * t_1;
	elseif (y_46_re <= -1.85e+53)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -0.21)
		tmp = t_2;
	elseif (y_46_re <= -2.3e-136)
		tmp = 1.0 / (t_0 / ((y_46_re * x_46_re) + (y_46_im * x_46_im)));
	elseif (y_46_re <= 2.9e-136)
		tmp = t_2;
	elseif (y_46_re <= 1.6e+139)
		tmp = t_1 / t_0;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = 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[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(y$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.45e+123], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.1e+54], N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$re, -1.85e+53], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -0.21], t$95$2, If[LessEqual[y$46$re, -2.3e-136], N[(1.0 / N[(t$95$0 / N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 2.9e-136], t$95$2, If[LessEqual[y$46$re, 1.6e+139], N[(t$95$1 / t$95$0), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -2.44999999999999988e123 or -5.10000000000000009e54 < y.re < -1.85e53 or 1.6000000000000001e139 < y.re

   1. Initial program 41.4

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

   2. Taylor expanded in y.re around inf 14.3

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

   if -2.44999999999999988e123 < y.re < -5.10000000000000009e54

   1. Initial program 18.9

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

   2. Applied egg-rr 18.9

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

   if -1.85e53 < y.re < -0.209999999999999992 or -2.29999999999999998e-136 < y.re < 2.89999999999999995e-136

   1. Initial program 21.7

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

   2. Taylor expanded in y.re around 0 14.7

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

   3. Simplified 15.5

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

      [Start] 14.7	\[ \frac{x.re \cdot y.re}{{y.im}^{2}} + \frac{x.im}{y.im} \]
      rational.json-simplify-1 [=>] 14.7	\[ \color{blue}{\frac{x.im}{y.im} + \frac{x.re \cdot y.re}{{y.im}^{2}}} \]
      rational.json-simplify-2 [=>] 14.7	\[ \frac{x.im}{y.im} + \frac{\color{blue}{y.re \cdot x.re}}{{y.im}^{2}} \]
      rational.json-simplify-49 [=>] 15.5	\[ \frac{x.im}{y.im} + \color{blue}{x.re \cdot \frac{y.re}{{y.im}^{2}}} \]

   if -0.209999999999999992 < y.re < -2.29999999999999998e-136

   1. Initial program 15.4

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

   2. Applied egg-rr 15.5

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

   3. Applied egg-rr 15.6

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

   if 2.89999999999999995e-136 < y.re < 1.6000000000000001e139

   1. Initial program 16.8

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 15.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.45 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.1 \cdot 10^{+54}:\\ \;\;\;\;\frac{1}{y.re \cdot y.re + y.im \cdot y.im} \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -0.21:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq -2.3 \cdot 10^{-136}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.re + y.im \cdot x.im}}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{-136}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+139}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	15.5
Cost	7700

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + x.re \cdot \frac{y.re}{{y.im}^{2}}\\ t_1 := \frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ t_2 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -4.5:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -2.35 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{\frac{t_2}{y.re \cdot x.re + y.im \cdot x.im}}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{-132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{t_2}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.8
Cost	1752

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -4.5:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -2.6 \cdot 10^{-139}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-198}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 6.5 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 3
Error	16.9
Cost	1752

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.re \cdot y.re + x.im \cdot y.im\\ t_2 := \frac{t_1}{t_0}\\ \mathbf{if}\;y.re \leq -1.45 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;y.re \leq -4.4:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -5.8 \cdot 10^{-140}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 3.75 \cdot 10^{-199}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{+141}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 4
Error	16.9
Cost	1752

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+123}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.25 \cdot 10^{+51}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;y.re \leq -4.5:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -5.4 \cdot 10^{-140}:\\ \;\;\;\;\frac{1}{\frac{t_0}{y.re \cdot x.re + y.im \cdot x.im}}\\ \mathbf{elif}\;y.re \leq 2.25 \cdot 10^{-197}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{+143}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 5
Error	21.4
Cost	1496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re}{t_0} \cdot x.re\\ \mathbf{if}\;y.re \leq -2.85 \cdot 10^{+51}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -3.9:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -2.2 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -7.5 \cdot 10^{-133}:\\ \;\;\;\;\frac{y.im}{t_0} \cdot x.im\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 6
Error	21.4
Cost	1496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re}{t_0} \cdot x.re\\ \mathbf{if}\;y.re \leq -1.85 \cdot 10^{+53}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -0.057:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -2.7 \cdot 10^{-57}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.2 \cdot 10^{-134}:\\ \;\;\;\;\frac{x.im}{\frac{t_0}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+141}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Error	23.2
Cost	1232

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.55 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-134}:\\ \;\;\;\;\frac{y.im}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	22.6
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.im \leq -9.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	37.4
Cost	192

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

Reproduce

herbie shell --seed 2023074 
(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))))