_divideComplex, real part

Average Error: 26.2 → 15.2

Time: 1.1min

Precision: binary64

Cost: 40648

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot -0.25 - x.re \cdot \frac{-1.25}{{y.im}^{2}}\right) \cdot y.re\\ t_1 := {y.im}^{2} + {y.re}^{2}\\ t_2 := y.re \cdot y.re + y.im \cdot y.im\\ t_3 := \frac{x.im}{t_2}\\ t_4 := \frac{x.re}{y.re} + y.im \cdot t_3\\ t_5 := \frac{y.re}{t_1}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+104}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im \cdot y.im}{t_1} + x.re \cdot \left(t_5 \cdot -0.25 - -1.25 \cdot t_5\right)\\ \mathbf{elif}\;y.re \leq -1.08 \cdot 10^{-191}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot \left(0.25 \cdot y.re - -1.25 \cdot y.re\right) + x.im \cdot y.im\right)}{t_2}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-270}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-150}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_2} - \left(-y.im\right) \cdot t_3\\ \mathbf{else}:\\ \;\;\;\;t_4\\ \end{array} \]

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
         (+
          (/ x.im y.im)
          (*
           (-
            (* (/ x.re (pow y.im 2.0)) -0.25)
            (* x.re (/ -1.25 (pow y.im 2.0))))
           y.re)))
        (t_1 (+ (pow y.im 2.0) (pow y.re 2.0)))
        (t_2 (+ (* y.re y.re) (* y.im y.im)))
        (t_3 (/ x.im t_2))
        (t_4 (+ (/ x.re y.re) (* y.im t_3)))
        (t_5 (/ y.re t_1)))
   (if (<= y.re -9e+104)
     t_4
     (if (<= y.re -6e-114)
       (+ (/ (* x.im y.im) t_1) (* x.re (- (* t_5 -0.25) (* -1.25 t_5))))
       (if (<= y.re -1.08e-191)
         t_0
         (if (<= y.re -7.8e-252)
           (/
            (+
             (/ (* y.re x.re) -2.0)
             (+ (* x.re (- (* 0.25 y.re) (* -1.25 y.re))) (* x.im y.im)))
            t_2)
           (if (<= y.re 4.2e-270)
             (/ x.im y.im)
             (if (<= y.re 9.5e-150)
               t_0
               (if (<= y.re 3e+98)
                 (- (/ (* y.re x.re) t_2) (* (- y.im) t_3))
                 t_4)))))))))

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 = (x_46_im / y_46_im) + ((((x_46_re / pow(y_46_im, 2.0)) * -0.25) - (x_46_re * (-1.25 / pow(y_46_im, 2.0)))) * y_46_re);
	double t_1 = pow(y_46_im, 2.0) + pow(y_46_re, 2.0);
	double t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_3 = x_46_im / t_2;
	double t_4 = (x_46_re / y_46_re) + (y_46_im * t_3);
	double t_5 = y_46_re / t_1;
	double tmp;
	if (y_46_re <= -9e+104) {
		tmp = t_4;
	} else if (y_46_re <= -6e-114) {
		tmp = ((x_46_im * y_46_im) / t_1) + (x_46_re * ((t_5 * -0.25) - (-1.25 * t_5)));
	} else if (y_46_re <= -1.08e-191) {
		tmp = t_0;
	} else if (y_46_re <= -7.8e-252) {
		tmp = (((y_46_re * x_46_re) / -2.0) + ((x_46_re * ((0.25 * y_46_re) - (-1.25 * y_46_re))) + (x_46_im * y_46_im))) / t_2;
	} else if (y_46_re <= 4.2e-270) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 9.5e-150) {
		tmp = t_0;
	} else if (y_46_re <= 3e+98) {
		tmp = ((y_46_re * x_46_re) / t_2) - (-y_46_im * t_3);
	} else {
		tmp = t_4;
	}
	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) :: t_3
    real(8) :: t_4
    real(8) :: t_5
    real(8) :: tmp
    t_0 = (x_46im / y_46im) + ((((x_46re / (y_46im ** 2.0d0)) * (-0.25d0)) - (x_46re * ((-1.25d0) / (y_46im ** 2.0d0)))) * y_46re)
    t_1 = (y_46im ** 2.0d0) + (y_46re ** 2.0d0)
    t_2 = (y_46re * y_46re) + (y_46im * y_46im)
    t_3 = x_46im / t_2
    t_4 = (x_46re / y_46re) + (y_46im * t_3)
    t_5 = y_46re / t_1
    if (y_46re <= (-9d+104)) then
        tmp = t_4
    else if (y_46re <= (-6d-114)) then
        tmp = ((x_46im * y_46im) / t_1) + (x_46re * ((t_5 * (-0.25d0)) - ((-1.25d0) * t_5)))
    else if (y_46re <= (-1.08d-191)) then
        tmp = t_0
    else if (y_46re <= (-7.8d-252)) then
        tmp = (((y_46re * x_46re) / (-2.0d0)) + ((x_46re * ((0.25d0 * y_46re) - ((-1.25d0) * y_46re))) + (x_46im * y_46im))) / t_2
    else if (y_46re <= 4.2d-270) then
        tmp = x_46im / y_46im
    else if (y_46re <= 9.5d-150) then
        tmp = t_0
    else if (y_46re <= 3d+98) then
        tmp = ((y_46re * x_46re) / t_2) - (-y_46im * t_3)
    else
        tmp = t_4
    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 = (x_46_im / y_46_im) + ((((x_46_re / Math.pow(y_46_im, 2.0)) * -0.25) - (x_46_re * (-1.25 / Math.pow(y_46_im, 2.0)))) * y_46_re);
	double t_1 = Math.pow(y_46_im, 2.0) + Math.pow(y_46_re, 2.0);
	double t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_3 = x_46_im / t_2;
	double t_4 = (x_46_re / y_46_re) + (y_46_im * t_3);
	double t_5 = y_46_re / t_1;
	double tmp;
	if (y_46_re <= -9e+104) {
		tmp = t_4;
	} else if (y_46_re <= -6e-114) {
		tmp = ((x_46_im * y_46_im) / t_1) + (x_46_re * ((t_5 * -0.25) - (-1.25 * t_5)));
	} else if (y_46_re <= -1.08e-191) {
		tmp = t_0;
	} else if (y_46_re <= -7.8e-252) {
		tmp = (((y_46_re * x_46_re) / -2.0) + ((x_46_re * ((0.25 * y_46_re) - (-1.25 * y_46_re))) + (x_46_im * y_46_im))) / t_2;
	} else if (y_46_re <= 4.2e-270) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 9.5e-150) {
		tmp = t_0;
	} else if (y_46_re <= 3e+98) {
		tmp = ((y_46_re * x_46_re) / t_2) - (-y_46_im * t_3);
	} else {
		tmp = t_4;
	}
	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 = (x_46_im / y_46_im) + ((((x_46_re / math.pow(y_46_im, 2.0)) * -0.25) - (x_46_re * (-1.25 / math.pow(y_46_im, 2.0)))) * y_46_re)
	t_1 = math.pow(y_46_im, 2.0) + math.pow(y_46_re, 2.0)
	t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_3 = x_46_im / t_2
	t_4 = (x_46_re / y_46_re) + (y_46_im * t_3)
	t_5 = y_46_re / t_1
	tmp = 0
	if y_46_re <= -9e+104:
		tmp = t_4
	elif y_46_re <= -6e-114:
		tmp = ((x_46_im * y_46_im) / t_1) + (x_46_re * ((t_5 * -0.25) - (-1.25 * t_5)))
	elif y_46_re <= -1.08e-191:
		tmp = t_0
	elif y_46_re <= -7.8e-252:
		tmp = (((y_46_re * x_46_re) / -2.0) + ((x_46_re * ((0.25 * y_46_re) - (-1.25 * y_46_re))) + (x_46_im * y_46_im))) / t_2
	elif y_46_re <= 4.2e-270:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 9.5e-150:
		tmp = t_0
	elif y_46_re <= 3e+98:
		tmp = ((y_46_re * x_46_re) / t_2) - (-y_46_im * t_3)
	else:
		tmp = t_4
	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(x_46_im / y_46_im) + Float64(Float64(Float64(Float64(x_46_re / (y_46_im ^ 2.0)) * -0.25) - Float64(x_46_re * Float64(-1.25 / (y_46_im ^ 2.0)))) * y_46_re))
	t_1 = Float64((y_46_im ^ 2.0) + (y_46_re ^ 2.0))
	t_2 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_3 = Float64(x_46_im / t_2)
	t_4 = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * t_3))
	t_5 = Float64(y_46_re / t_1)
	tmp = 0.0
	if (y_46_re <= -9e+104)
		tmp = t_4;
	elseif (y_46_re <= -6e-114)
		tmp = Float64(Float64(Float64(x_46_im * y_46_im) / t_1) + Float64(x_46_re * Float64(Float64(t_5 * -0.25) - Float64(-1.25 * t_5))));
	elseif (y_46_re <= -1.08e-191)
		tmp = t_0;
	elseif (y_46_re <= -7.8e-252)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_re) / -2.0) + Float64(Float64(x_46_re * Float64(Float64(0.25 * y_46_re) - Float64(-1.25 * y_46_re))) + Float64(x_46_im * y_46_im))) / t_2);
	elseif (y_46_re <= 4.2e-270)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 9.5e-150)
		tmp = t_0;
	elseif (y_46_re <= 3e+98)
		tmp = Float64(Float64(Float64(y_46_re * x_46_re) / t_2) - Float64(Float64(-y_46_im) * t_3));
	else
		tmp = t_4;
	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 = (x_46_im / y_46_im) + ((((x_46_re / (y_46_im ^ 2.0)) * -0.25) - (x_46_re * (-1.25 / (y_46_im ^ 2.0)))) * y_46_re);
	t_1 = (y_46_im ^ 2.0) + (y_46_re ^ 2.0);
	t_2 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_3 = x_46_im / t_2;
	t_4 = (x_46_re / y_46_re) + (y_46_im * t_3);
	t_5 = y_46_re / t_1;
	tmp = 0.0;
	if (y_46_re <= -9e+104)
		tmp = t_4;
	elseif (y_46_re <= -6e-114)
		tmp = ((x_46_im * y_46_im) / t_1) + (x_46_re * ((t_5 * -0.25) - (-1.25 * t_5)));
	elseif (y_46_re <= -1.08e-191)
		tmp = t_0;
	elseif (y_46_re <= -7.8e-252)
		tmp = (((y_46_re * x_46_re) / -2.0) + ((x_46_re * ((0.25 * y_46_re) - (-1.25 * y_46_re))) + (x_46_im * y_46_im))) / t_2;
	elseif (y_46_re <= 4.2e-270)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 9.5e-150)
		tmp = t_0;
	elseif (y_46_re <= 3e+98)
		tmp = ((y_46_re * x_46_re) / t_2) - (-y_46_im * t_3);
	else
		tmp = t_4;
	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[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(N[(N[(x$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] - N[(x$46$re * N[(-1.25 / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[y$46$im, 2.0], $MachinePrecision] + N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(x$46$im / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * t$95$3), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(y$46$re / t$95$1), $MachinePrecision]}, If[LessEqual[y$46$re, -9e+104], t$95$4, If[LessEqual[y$46$re, -6e-114], N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / t$95$1), $MachinePrecision] + N[(x$46$re * N[(N[(t$95$5 * -0.25), $MachinePrecision] - N[(-1.25 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.08e-191], t$95$0, If[LessEqual[y$46$re, -7.8e-252], N[(N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] / -2.0), $MachinePrecision] + N[(N[(x$46$re * N[(N[(0.25 * y$46$re), $MachinePrecision] - N[(-1.25 * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[y$46$re, 4.2e-270], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-150], t$95$0, If[LessEqual[y$46$re, 3e+98], N[(N[(N[(y$46$re * x$46$re), $MachinePrecision] / t$95$2), $MachinePrecision] - N[((-y$46$im) * t$95$3), $MachinePrecision]), $MachinePrecision], t$95$4]]]]]]]]]]]]]

Derivation

1. Split input into 6 regimes

2. if y.re < -8.9999999999999997e104 or 3.0000000000000001e98 < y.re

   1. Initial program 39.7

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

   2. Applied egg-rr 39.3

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

   3. Simplified 39.3

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

      [Start] 39.3	\[ \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 39.3	\[ \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]

   4. Taylor expanded in y.re around inf 13.9

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

   5. Applied egg-rr 13.9

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

   if -8.9999999999999997e104 < y.re < -6.0000000000000003e-114

   1. Initial program 15.9

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

   2. Applied egg-rr 15.9

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

   3. Applied egg-rr 16.1

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

   4. Simplified 16.1

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

      [Start] 16.1	\[ \frac{\left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right) + \frac{x.re \cdot y.re}{-2}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 16.1	\[ \frac{\color{blue}{\frac{x.re \cdot y.re}{-2} + \left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 16.1	\[ \frac{\frac{\color{blue}{y.re \cdot x.re}}{-2} + \left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-59 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(x.re \cdot y.re - \left(-\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-14 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(0 - \left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(\frac{x.re \cdot y.re}{-2} - \left(x.im \cdot y.im - 0\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-9 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \left(\frac{x.re \cdot y.re}{-2} - \color{blue}{x.im \cdot y.im}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-6 [<=] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(0 + \left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(x.re \cdot y.re - 0\right) + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-9 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{x.re \cdot y.re} + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{y.re \cdot x.re} + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-14 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \color{blue}{\left(0 - \left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \color{blue}{\left(x.im \cdot y.im - \left(\frac{x.re \cdot y.re}{-2} - 0\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(\color{blue}{y.im \cdot x.im} - \left(\frac{x.re \cdot y.re}{-2} - 0\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [<=] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\frac{x.re \cdot y.re}{-2} - \color{blue}{\left(1 + -1\right)}\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [<=] 22.0	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\left(\frac{x.re \cdot y.re}{-2} - -1\right) - 1\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-20 [=>] 22.0	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\color{blue}{\left(\frac{x.re \cdot y.re}{-2} + 1\right)} - 1\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [<=] 22.0	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\color{blue}{\left(1 + \frac{x.re \cdot y.re}{-2}\right)} - 1\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-19 [<=] 22.0	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\left(1 + \frac{x.re \cdot y.re}{-2}\right) + -1\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [<=] 22.0	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(-1 + \left(1 + \frac{x.re \cdot y.re}{-2}\right)\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-47 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\frac{x.re \cdot y.re}{-2} + \left(1 + -1\right)\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\frac{x.re \cdot y.re}{-2} + \color{blue}{0}\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(0 + \frac{x.re \cdot y.re}{-2}\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-6 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\frac{x.re \cdot y.re}{-2}}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \frac{\color{blue}{y.re \cdot x.re}}{-2}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   5. Applied egg-rr 16.1

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right) - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

   6. Simplified 16.1

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      Proof

      [Start] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right) - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(y.re \cdot x.re - \left(\frac{y.re \cdot x.re}{-4} - \left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re - \left(\frac{y.re \cdot x.re}{-4} - \color{blue}{\left(\frac{y.re \cdot x.re}{4} + y.im \cdot x.im\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [<=] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re - \color{blue}{\left(\left(\frac{y.re \cdot x.re}{-4} - y.im \cdot x.im\right) - \frac{y.re \cdot x.re}{4}\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [<=] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\frac{y.re \cdot x.re}{4} - \left(\left(\frac{y.re \cdot x.re}{-4} - y.im \cdot x.im\right) - y.re \cdot x.re\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\frac{y.re \cdot x.re}{4} - \color{blue}{\left(\frac{y.re \cdot x.re}{-4} - \left(y.re \cdot x.re + y.im \cdot x.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\frac{y.re \cdot x.re}{4} - \color{blue}{\left(\left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right) + \left(-y.im \cdot x.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(\frac{y.re \cdot x.re}{4} - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{\color{blue}{x.re \cdot y.re}}{4} - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-112 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(1 - -4\right) \cdot \frac{y.re \cdot x.re}{-4}}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(y.re \cdot x.re\right) \cdot \frac{1 - -4}{-4}}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(x.re \cdot y.re\right)} \cdot \frac{1 - -4}{-4}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot \frac{\color{blue}{5}}{-4}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot \color{blue}{-1.25}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-13 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + \color{blue}{\frac{-y.im \cdot x.im}{-1}}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-13 [=>] 16.1	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + \frac{\color{blue}{\frac{y.im \cdot x.im}{-1}}}{-1}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   7. Taylor expanded in x.re around 0 13.7

      \[\leadsto \color{blue}{\left(\left(0.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}} + -0.5 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \cdot x.re + \frac{y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}}} \]

   8. Simplified 13.7

      \[\leadsto \color{blue}{\frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + x.re \cdot \left(\frac{y.re}{{y.im}^{2} + {y.re}^{2}} \cdot -0.25 - -1.25 \cdot \frac{y.re}{{y.im}^{2} + {y.re}^{2}}\right)} \]
      Proof

      [Start] 13.7	\[ \left(\left(0.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}} + -0.5 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \cdot x.re + \frac{y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}} \]
      rational_best-simplify-3 [=>] 13.7	\[ \color{blue}{\frac{y.im \cdot x.im}{{y.re}^{2} + {y.im}^{2}} + \left(\left(0.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}} + -0.5 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \cdot x.re} \]
      rational_best-simplify-1 [=>] 13.7	\[ \frac{\color{blue}{x.im \cdot y.im}}{{y.re}^{2} + {y.im}^{2}} + \left(\left(0.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}} + -0.5 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \cdot x.re \]
      rational_best-simplify-3 [=>] 13.7	\[ \frac{x.im \cdot y.im}{\color{blue}{{y.im}^{2} + {y.re}^{2}}} + \left(\left(0.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}} + -0.5 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \cdot x.re \]
      rational_best-simplify-1 [=>] 13.7	\[ \frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + \color{blue}{x.re \cdot \left(\left(0.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}} + -0.5 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right)} \]
      rational_best-simplify-63 [=>] 13.7	\[ \frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + x.re \cdot \left(\color{blue}{\frac{y.re}{{y.re}^{2} + {y.im}^{2}} \cdot \left(0.25 + -0.5\right)} - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \]
      rational_best-simplify-3 [=>] 13.7	\[ \frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + x.re \cdot \left(\frac{y.re}{\color{blue}{{y.im}^{2} + {y.re}^{2}}} \cdot \left(0.25 + -0.5\right) - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \]
      metadata-eval [=>] 13.7	\[ \frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + x.re \cdot \left(\frac{y.re}{{y.im}^{2} + {y.re}^{2}} \cdot \color{blue}{-0.25} - -1.25 \cdot \frac{y.re}{{y.re}^{2} + {y.im}^{2}}\right) \]
      rational_best-simplify-3 [=>] 13.7	\[ \frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + x.re \cdot \left(\frac{y.re}{{y.im}^{2} + {y.re}^{2}} \cdot -0.25 - -1.25 \cdot \frac{y.re}{\color{blue}{{y.im}^{2} + {y.re}^{2}}}\right) \]

   if -6.0000000000000003e-114 < y.re < -1.07999999999999996e-191 or 4.19999999999999992e-270 < y.re < 9.50000000000000013e-150

   1. Initial program 21.9

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

   2. Applied egg-rr 21.9

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

   3. Applied egg-rr 21.9

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

   4. Simplified 21.9

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

      [Start] 21.9	\[ \frac{\left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right) + \frac{x.re \cdot y.re}{-2}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 21.9	\[ \frac{\color{blue}{\frac{x.re \cdot y.re}{-2} + \left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 21.9	\[ \frac{\frac{\color{blue}{y.re \cdot x.re}}{-2} + \left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-59 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(x.re \cdot y.re - \left(-\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-14 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(0 - \left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(\frac{x.re \cdot y.re}{-2} - \left(x.im \cdot y.im - 0\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-9 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \left(\frac{x.re \cdot y.re}{-2} - \color{blue}{x.im \cdot y.im}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-6 [<=] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(0 + \left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(x.re \cdot y.re - 0\right) + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-9 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{x.re \cdot y.re} + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{y.re \cdot x.re} + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-14 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \color{blue}{\left(0 - \left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \color{blue}{\left(x.im \cdot y.im - \left(\frac{x.re \cdot y.re}{-2} - 0\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(\color{blue}{y.im \cdot x.im} - \left(\frac{x.re \cdot y.re}{-2} - 0\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [<=] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\frac{x.re \cdot y.re}{-2} - \color{blue}{\left(1 + -1\right)}\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [<=] 26.7	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\left(\frac{x.re \cdot y.re}{-2} - -1\right) - 1\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-20 [=>] 26.7	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\color{blue}{\left(\frac{x.re \cdot y.re}{-2} + 1\right)} - 1\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [<=] 26.7	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\color{blue}{\left(1 + \frac{x.re \cdot y.re}{-2}\right)} - 1\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-19 [<=] 26.7	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\left(1 + \frac{x.re \cdot y.re}{-2}\right) + -1\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [<=] 26.7	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(-1 + \left(1 + \frac{x.re \cdot y.re}{-2}\right)\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-47 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\frac{x.re \cdot y.re}{-2} + \left(1 + -1\right)\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\frac{x.re \cdot y.re}{-2} + \color{blue}{0}\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(0 + \frac{x.re \cdot y.re}{-2}\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-6 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\frac{x.re \cdot y.re}{-2}}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \frac{\color{blue}{y.re \cdot x.re}}{-2}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   5. Applied egg-rr 21.9

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right) - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

   6. Simplified 21.9

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      Proof

      [Start] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right) - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(y.re \cdot x.re - \left(\frac{y.re \cdot x.re}{-4} - \left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re - \left(\frac{y.re \cdot x.re}{-4} - \color{blue}{\left(\frac{y.re \cdot x.re}{4} + y.im \cdot x.im\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [<=] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re - \color{blue}{\left(\left(\frac{y.re \cdot x.re}{-4} - y.im \cdot x.im\right) - \frac{y.re \cdot x.re}{4}\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [<=] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\frac{y.re \cdot x.re}{4} - \left(\left(\frac{y.re \cdot x.re}{-4} - y.im \cdot x.im\right) - y.re \cdot x.re\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\frac{y.re \cdot x.re}{4} - \color{blue}{\left(\frac{y.re \cdot x.re}{-4} - \left(y.re \cdot x.re + y.im \cdot x.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\frac{y.re \cdot x.re}{4} - \color{blue}{\left(\left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right) + \left(-y.im \cdot x.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(\frac{y.re \cdot x.re}{4} - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{\color{blue}{x.re \cdot y.re}}{4} - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-112 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(1 - -4\right) \cdot \frac{y.re \cdot x.re}{-4}}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(y.re \cdot x.re\right) \cdot \frac{1 - -4}{-4}}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(x.re \cdot y.re\right)} \cdot \frac{1 - -4}{-4}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot \frac{\color{blue}{5}}{-4}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot \color{blue}{-1.25}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-13 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + \color{blue}{\frac{-y.im \cdot x.im}{-1}}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-13 [=>] 21.9	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + \frac{\color{blue}{\frac{y.im \cdot x.im}{-1}}}{-1}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   7. Taylor expanded in y.re around 0 15.6

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \left(\left(0.25 \cdot \frac{x.re}{{y.im}^{2}} + -0.5 \cdot \frac{x.re}{{y.im}^{2}}\right) - -1.25 \cdot \frac{x.re}{{y.im}^{2}}\right) \cdot y.re} \]

   8. Simplified 16.1

      \[\leadsto \color{blue}{\frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot -0.25 - x.re \cdot \frac{-1.25}{{y.im}^{2}}\right) \cdot y.re} \]
      Proof

      [Start] 15.6	\[ \frac{x.im}{y.im} + \left(\left(0.25 \cdot \frac{x.re}{{y.im}^{2}} + -0.5 \cdot \frac{x.re}{{y.im}^{2}}\right) - -1.25 \cdot \frac{x.re}{{y.im}^{2}}\right) \cdot y.re \]
      rational_best-simplify-63 [=>] 15.6	\[ \frac{x.im}{y.im} + \left(\color{blue}{\frac{x.re}{{y.im}^{2}} \cdot \left(0.25 + -0.5\right)} - -1.25 \cdot \frac{x.re}{{y.im}^{2}}\right) \cdot y.re \]
      metadata-eval [=>] 15.6	\[ \frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot \color{blue}{-0.25} - -1.25 \cdot \frac{x.re}{{y.im}^{2}}\right) \cdot y.re \]
      rational_best-simplify-55 [=>] 16.1	\[ \frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot -0.25 - \color{blue}{x.re \cdot \frac{-1.25}{{y.im}^{2}}}\right) \cdot y.re \]

   if -1.07999999999999996e-191 < y.re < -7.7999999999999998e-252

   1. Initial program 22.4

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

   2. Applied egg-rr 22.4

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

   3. Applied egg-rr 22.4

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

   4. Simplified 22.4

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

      [Start] 22.4	\[ \frac{\left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right) + \frac{x.re \cdot y.re}{-2}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 22.4	\[ \frac{\color{blue}{\frac{x.re \cdot y.re}{-2} + \left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 22.4	\[ \frac{\frac{\color{blue}{y.re \cdot x.re}}{-2} + \left(\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right) + x.re \cdot y.re\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-59 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(x.re \cdot y.re - \left(-\left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-14 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(0 - \left(x.im \cdot y.im - \frac{x.re \cdot y.re}{-2}\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(\frac{x.re \cdot y.re}{-2} - \left(x.im \cdot y.im - 0\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-9 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \left(\frac{x.re \cdot y.re}{-2} - \color{blue}{x.im \cdot y.im}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-6 [<=] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot y.re - \color{blue}{\left(0 + \left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(x.re \cdot y.re - 0\right) + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-9 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{x.re \cdot y.re} + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{y.re \cdot x.re} + \left(-\left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-14 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \color{blue}{\left(0 - \left(\frac{x.re \cdot y.re}{-2} - x.im \cdot y.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \color{blue}{\left(x.im \cdot y.im - \left(\frac{x.re \cdot y.re}{-2} - 0\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(\color{blue}{y.im \cdot x.im} - \left(\frac{x.re \cdot y.re}{-2} - 0\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [<=] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\frac{x.re \cdot y.re}{-2} - \color{blue}{\left(1 + -1\right)}\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [<=] 26.2	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\left(\frac{x.re \cdot y.re}{-2} - -1\right) - 1\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-20 [=>] 26.2	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\color{blue}{\left(\frac{x.re \cdot y.re}{-2} + 1\right)} - 1\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [<=] 26.2	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\color{blue}{\left(1 + \frac{x.re \cdot y.re}{-2}\right)} - 1\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-19 [<=] 26.2	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\left(1 + \frac{x.re \cdot y.re}{-2}\right) + -1\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [<=] 26.2	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(-1 + \left(1 + \frac{x.re \cdot y.re}{-2}\right)\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-47 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(\frac{x.re \cdot y.re}{-2} + \left(1 + -1\right)\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \left(\frac{x.re \cdot y.re}{-2} + \color{blue}{0}\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\left(0 + \frac{x.re \cdot y.re}{-2}\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-6 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \color{blue}{\frac{x.re \cdot y.re}{-2}}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re + \left(y.im \cdot x.im - \frac{\color{blue}{y.re \cdot x.re}}{-2}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   5. Applied egg-rr 22.5

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right) - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]

   6. Simplified 22.5

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + x.im \cdot y.im\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      Proof

      [Start] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right) - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(y.re \cdot x.re - \left(\frac{y.re \cdot x.re}{-4} - \left(y.im \cdot x.im + \frac{y.re \cdot x.re}{4}\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-3 [=>] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re - \left(\frac{y.re \cdot x.re}{-4} - \color{blue}{\left(\frac{y.re \cdot x.re}{4} + y.im \cdot x.im\right)}\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [<=] 22.4	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(y.re \cdot x.re - \color{blue}{\left(\left(\frac{y.re \cdot x.re}{-4} - y.im \cdot x.im\right) - \frac{y.re \cdot x.re}{4}\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-51 [<=] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\frac{y.re \cdot x.re}{4} - \left(\left(\frac{y.re \cdot x.re}{-4} - y.im \cdot x.im\right) - y.re \cdot x.re\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-52 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\frac{y.re \cdot x.re}{4} - \color{blue}{\left(\frac{y.re \cdot x.re}{-4} - \left(y.re \cdot x.re + y.im \cdot x.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\frac{y.re \cdot x.re}{4} - \color{blue}{\left(\left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right) + \left(-y.im \cdot x.im\right)\right)}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-57 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \color{blue}{\left(\left(\frac{y.re \cdot x.re}{4} - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{\color{blue}{x.re \cdot y.re}}{4} - \left(\frac{y.re \cdot x.re}{-4} - y.re \cdot x.re\right)\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-112 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(1 - -4\right) \cdot \frac{y.re \cdot x.re}{-4}}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(y.re \cdot x.re\right) \cdot \frac{1 - -4}{-4}}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [<=] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \color{blue}{\left(x.re \cdot y.re\right)} \cdot \frac{1 - -4}{-4}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot \frac{\color{blue}{5}}{-4}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      metadata-eval [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot \color{blue}{-1.25}\right) + \left(-\left(-y.im \cdot x.im\right)\right)\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-13 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + \color{blue}{\frac{-y.im \cdot x.im}{-1}}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-13 [=>] 22.5	\[ \frac{\frac{y.re \cdot x.re}{-2} + \left(\left(\frac{x.re \cdot y.re}{4} - \left(x.re \cdot y.re\right) \cdot -1.25\right) + \frac{\color{blue}{\frac{y.im \cdot x.im}{-1}}}{-1}\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   7. Taylor expanded in x.re around 0 22.5

      \[\leadsto \frac{\frac{y.re \cdot x.re}{-2} + \left(\color{blue}{x.re \cdot \left(0.25 \cdot y.re - -1.25 \cdot y.re\right)} + x.im \cdot y.im\right)}{y.re \cdot y.re + y.im \cdot y.im} \]

   if -7.7999999999999998e-252 < y.re < 4.19999999999999992e-270

   1. Initial program 25.0

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

   2. Taylor expanded in y.re around 0 12.3

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

   if 9.50000000000000013e-150 < y.re < 3.0000000000000001e98

   1. Initial program 16.4

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

   2. Applied egg-rr 17.7

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

   3. Simplified 17.7

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

      [Start] 17.7	\[ \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 17.7	\[ \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]

3. Recombined 6 regimes into one program.

4. Final simplification 15.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{-114}:\\ \;\;\;\;\frac{x.im \cdot y.im}{{y.im}^{2} + {y.re}^{2}} + x.re \cdot \left(\frac{y.re}{{y.im}^{2} + {y.re}^{2}} \cdot -0.25 - -1.25 \cdot \frac{y.re}{{y.im}^{2} + {y.re}^{2}}\right)\\ \mathbf{elif}\;y.re \leq -1.08 \cdot 10^{-191}:\\ \;\;\;\;\frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot -0.25 - x.re \cdot \frac{-1.25}{{y.im}^{2}}\right) \cdot y.re\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-252}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.re}{-2} + \left(x.re \cdot \left(0.25 \cdot y.re - -1.25 \cdot y.re\right) + x.im \cdot y.im\right)}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 4.2 \cdot 10^{-270}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-150}:\\ \;\;\;\;\frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot -0.25 - x.re \cdot \frac{-1.25}{{y.im}^{2}}\right) \cdot y.re\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{+98}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.4
Cost	14544

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.1 \cdot 10^{-46}:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_0} - \left(-y.im\right) \cdot \frac{x.im}{t_0}\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{y.im \cdot x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+89}:\\ \;\;\;\;\frac{\frac{\left(x.re \cdot y.re + y.im \cdot \left(x.im + x.im\right)\right) + x.re \cdot y.re}{2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \left(\frac{x.re}{{y.im}^{2}} \cdot -0.25 - x.re \cdot \frac{-1.25}{{y.im}^{2}}\right) \cdot y.re\\ \end{array} \]

Alternative 2
Error	14.7
Cost	2064

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re \cdot x.re}{t_0} - \left(-y.im\right) \cdot \frac{x.im}{t_0}\\ \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.55 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{y.im \cdot x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.96 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 3
Error	14.6
Cost	2000

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-46}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x.re \cdot y.re + x.im \cdot y.im\right)\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{y.im \cdot x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\frac{\left(x.re \cdot y.re + y.im \cdot \left(x.im + x.im\right)\right) + x.re \cdot y.re}{2}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 4
Error	23.7
Cost	1496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.im \cdot x.im}{t_0}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+89}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.82 \cdot 10^{+64}:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -6.4 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.08 \cdot 10^{+36}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 5
Error	23.7
Cost	1496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.im \cdot x.im}{t_0}\\ \mathbf{if}\;y.im \leq -6.2 \cdot 10^{+87}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(y.re \cdot x.re\right)\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-157}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 10^{+37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+45}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 6
Error	19.4
Cost	1488

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -4.9 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.56 \cdot 10^{+64}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(y.re \cdot x.re\right)\\ \mathbf{elif}\;y.im \leq -4.1 \cdot 10^{+30}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 7
Error	15.6
Cost	1488

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re \cdot y.re + x.im \cdot y.im}{t_0}\\ \mathbf{if}\;y.im \leq -6.2 \cdot 10^{+117}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -6 \cdot 10^{-158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{t_0}\\ \mathbf{elif}\;y.im \leq 2.12 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 8
Error	14.6
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -1.1 \cdot 10^{+118}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -3.05 \cdot 10^{-46}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.5 \cdot 10^{-124}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{y.im \cdot x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+135}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9
Error	14.6
Cost	1488

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{+117}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.15 \cdot 10^{-45}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_1\\ \mathbf{elif}\;y.im \leq 9.4 \cdot 10^{-125}:\\ \;\;\;\;\frac{x.re}{y.re} - \frac{y.im \cdot x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+135}:\\ \;\;\;\;\frac{t_1}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10
Error	23.3
Cost	1232

\[\begin{array}{l} t_0 := \frac{y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -7.4 \cdot 10^{+117}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -2.7 \cdot 10^{-157}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.45 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11
Error	23.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -1.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 8.2 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12
Error	38.0
Cost	192

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

Reproduce

herbie shell --seed 2023099 
(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))))