_divideComplex, real part

Average Error: 25.8 → 13.3

Time: 17.2s

Precision: binary64

Cost: 2064

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{y.re \cdot x.re}{t_0}\\ t_2 := \frac{x.re}{y.re} - y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ t_3 := y.im \cdot y.im + y.re \cdot y.re\\ \mathbf{if}\;y.re \leq -2 \cdot 10^{+109}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im \cdot y.im}{t_3} - x.re \cdot \frac{-y.re}{t_3}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;t_1 - \left(-\frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;t_1 - x.im \cdot \frac{-y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

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 (/ (* y.re x.re) t_0))
        (t_2
         (-
          (/ x.re y.re)
          (* y.im (/ x.im (- (- (* y.im y.im)) (* y.re y.re))))))
        (t_3 (+ (* y.im y.im) (* y.re y.re))))
   (if (<= y.re -2e+109)
     t_2
     (if (<= y.re -5.2e-161)
       (- (/ (* x.im y.im) t_3) (* x.re (/ (- y.re) t_3)))
       (if (<= y.re 1.5e-157)
         (- t_1 (- (/ x.im y.im)))
         (if (<= y.re 1.55e+37) (- t_1 (* x.im (/ (- y.im) t_0))) t_2))))))

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 = (y_46_re * x_46_re) / t_0;
	double t_2 = (x_46_re / y_46_re) - (y_46_im * (x_46_im / (-(y_46_im * y_46_im) - (y_46_re * y_46_re))));
	double t_3 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -2e+109) {
		tmp = t_2;
	} else if (y_46_re <= -5.2e-161) {
		tmp = ((x_46_im * y_46_im) / t_3) - (x_46_re * (-y_46_re / t_3));
	} else if (y_46_re <= 1.5e-157) {
		tmp = t_1 - -(x_46_im / y_46_im);
	} else if (y_46_re <= 1.55e+37) {
		tmp = t_1 - (x_46_im * (-y_46_im / t_0));
	} else {
		tmp = t_2;
	}
	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) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (y_46re * x_46re) / t_0
    t_2 = (x_46re / y_46re) - (y_46im * (x_46im / (-(y_46im * y_46im) - (y_46re * y_46re))))
    t_3 = (y_46im * y_46im) + (y_46re * y_46re)
    if (y_46re <= (-2d+109)) then
        tmp = t_2
    else if (y_46re <= (-5.2d-161)) then
        tmp = ((x_46im * y_46im) / t_3) - (x_46re * (-y_46re / t_3))
    else if (y_46re <= 1.5d-157) then
        tmp = t_1 - -(x_46im / y_46im)
    else if (y_46re <= 1.55d+37) then
        tmp = t_1 - (x_46im * (-y_46im / t_0))
    else
        tmp = t_2
    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 = (y_46_re * x_46_re) / t_0;
	double t_2 = (x_46_re / y_46_re) - (y_46_im * (x_46_im / (-(y_46_im * y_46_im) - (y_46_re * y_46_re))));
	double t_3 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_re <= -2e+109) {
		tmp = t_2;
	} else if (y_46_re <= -5.2e-161) {
		tmp = ((x_46_im * y_46_im) / t_3) - (x_46_re * (-y_46_re / t_3));
	} else if (y_46_re <= 1.5e-157) {
		tmp = t_1 - -(x_46_im / y_46_im);
	} else if (y_46_re <= 1.55e+37) {
		tmp = t_1 - (x_46_im * (-y_46_im / t_0));
	} else {
		tmp = t_2;
	}
	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 = (y_46_re * x_46_re) / t_0
	t_2 = (x_46_re / y_46_re) - (y_46_im * (x_46_im / (-(y_46_im * y_46_im) - (y_46_re * y_46_re))))
	t_3 = (y_46_im * y_46_im) + (y_46_re * y_46_re)
	tmp = 0
	if y_46_re <= -2e+109:
		tmp = t_2
	elif y_46_re <= -5.2e-161:
		tmp = ((x_46_im * y_46_im) / t_3) - (x_46_re * (-y_46_re / t_3))
	elif y_46_re <= 1.5e-157:
		tmp = t_1 - -(x_46_im / y_46_im)
	elif y_46_re <= 1.55e+37:
		tmp = t_1 - (x_46_im * (-y_46_im / t_0))
	else:
		tmp = t_2
	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(y_46_re * x_46_re) / t_0)
	t_2 = Float64(Float64(x_46_re / y_46_re) - Float64(y_46_im * Float64(x_46_im / Float64(Float64(-Float64(y_46_im * y_46_im)) - Float64(y_46_re * y_46_re)))))
	t_3 = Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_re <= -2e+109)
		tmp = t_2;
	elseif (y_46_re <= -5.2e-161)
		tmp = Float64(Float64(Float64(x_46_im * y_46_im) / t_3) - Float64(x_46_re * Float64(Float64(-y_46_re) / t_3)));
	elseif (y_46_re <= 1.5e-157)
		tmp = Float64(t_1 - Float64(-Float64(x_46_im / y_46_im)));
	elseif (y_46_re <= 1.55e+37)
		tmp = Float64(t_1 - Float64(x_46_im * Float64(Float64(-y_46_im) / t_0)));
	else
		tmp = t_2;
	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 = (y_46_re * x_46_re) / t_0;
	t_2 = (x_46_re / y_46_re) - (y_46_im * (x_46_im / (-(y_46_im * y_46_im) - (y_46_re * y_46_re))));
	t_3 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	tmp = 0.0;
	if (y_46_re <= -2e+109)
		tmp = t_2;
	elseif (y_46_re <= -5.2e-161)
		tmp = ((x_46_im * y_46_im) / t_3) - (x_46_re * (-y_46_re / t_3));
	elseif (y_46_re <= 1.5e-157)
		tmp = t_1 - -(x_46_im / y_46_im);
	elseif (y_46_re <= 1.55e+37)
		tmp = t_1 - (x_46_im * (-y_46_im / t_0));
	else
		tmp = t_2;
	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[(y$46$re * x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$re / y$46$re), $MachinePrecision] - N[(y$46$im * N[(x$46$im / N[((-N[(y$46$im * y$46$im), $MachinePrecision]) - N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2e+109], t$95$2, If[LessEqual[y$46$re, -5.2e-161], N[(N[(N[(x$46$im * y$46$im), $MachinePrecision] / t$95$3), $MachinePrecision] - N[(x$46$re * N[((-y$46$re) / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.5e-157], N[(t$95$1 - (-N[(x$46$im / y$46$im), $MachinePrecision])), $MachinePrecision], If[LessEqual[y$46$re, 1.55e+37], N[(t$95$1 - N[(x$46$im * N[((-y$46$im) / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.99999999999999996e109 or 1.5500000000000001e37 < y.re

   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. Applied egg-rr 36.7

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

   3. Simplified 37.4

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

      [Start] 36.7	\[ \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 36.7	\[ \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 37.4	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{x.im \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

   4. Applied egg-rr 36.7

      \[\leadsto \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{\left(y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re} + 0\right)} \]

   5. Simplified 36.7

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

      [Start] 36.7	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re} + 0\right) \]
      rational_best-simplify-3 [<=] 36.7	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{\left(0 + y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\right)} \]
      rational_best-simplify-6 [=>] 36.7	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}} \]

   6. Taylor expanded in y.re around inf 15.6

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

   if -1.99999999999999996e109 < y.re < -5.19999999999999991e-161

   1. Initial program 17.4

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

   2. Applied egg-rr 32.1

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

   3. Applied egg-rr 17.0

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

   4. Simplified 15.4

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

      [Start] 17.0	\[ \frac{y.im \cdot x.im}{y.im \cdot y.im + y.re \cdot y.re} - \left(-y.re\right) \cdot \frac{x.re}{y.im \cdot y.im + y.re \cdot y.re} \]
      rational_best-simplify-1 [=>] 17.0	\[ \frac{\color{blue}{x.im \cdot y.im}}{y.im \cdot y.im + y.re \cdot y.re} - \left(-y.re\right) \cdot \frac{x.re}{y.im \cdot y.im + y.re \cdot y.re} \]
      rational_best-simplify-55 [=>] 15.4	\[ \frac{x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re} - \color{blue}{x.re \cdot \frac{-y.re}{y.im \cdot y.im + y.re \cdot y.re}} \]

   if -5.19999999999999991e-161 < y.re < 1.5e-157

   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 24.1

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

   3. Simplified 19.7

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

      [Start] 24.1	\[ \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 24.1	\[ \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 19.7	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{x.im \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

   4. Taylor expanded in y.im around inf 8.4

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

   5. Simplified 8.4

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

      [Start] 8.4	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - -1 \cdot \frac{x.im}{y.im} \]
      rational_best-simplify-1 [=>] 8.4	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{\frac{x.im}{y.im} \cdot -1} \]
      rational_best-simplify-10 [=>] 8.4	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{\left(-\frac{x.im}{y.im}\right)} \]

   if 1.5e-157 < y.re < 1.5500000000000001e37

   1. Initial program 13.7

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

   2. Applied egg-rr 16.6

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

   3. Simplified 11.4

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

      [Start] 16.6	\[ \frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-1 [=>] 16.6	\[ \frac{\color{blue}{y.re \cdot x.re}}{y.re \cdot y.re + y.im \cdot y.im} - \left(-y.im\right) \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im} \]
      rational_best-simplify-55 [=>] 11.4	\[ \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \color{blue}{x.im \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]

3. Recombined 4 regimes into one program.

4. Final simplification 13.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2 \cdot 10^{+109}:\\ \;\;\;\;\frac{x.re}{y.re} - y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.im \cdot y.im + y.re \cdot y.re} - x.re \cdot \frac{-y.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.5 \cdot 10^{-157}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-\frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - x.im \cdot \frac{-y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} - y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \end{array} \]

Alternatives

Alternative 1
Error	13.6
Cost	2064

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} - y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ t_1 := y.im \cdot y.im + y.re \cdot y.re\\ t_2 := \frac{x.im \cdot y.im}{t_1} - x.re \cdot \frac{-y.re}{t_1}\\ \mathbf{if}\;y.re \leq -4.15 \cdot 10^{+108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-167}:\\ \;\;\;\;\frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im} - \left(-\frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 6.8 \cdot 10^{+55}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	15.3
Cost	1552

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} - y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ \mathbf{if}\;y.re \leq -1.2 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-161}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{-236}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	14.2
Cost	1552

\[\begin{array}{l} t_0 := \frac{x.re}{y.re} - y.im \cdot \frac{x.im}{\left(-y.im \cdot y.im\right) - y.re \cdot y.re}\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{x.re \cdot y.re + x.im \cdot y.im}{t_1}\\ \mathbf{if}\;y.re \leq -1.16 \cdot 10^{+58}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-161}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{-167}:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_1} - \left(-\frac{x.im}{y.im}\right)\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+37}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	23.9
Cost	1496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -0.00092:\\ \;\;\;\;\frac{y.re \cdot x.re}{t_0}\\ \mathbf{elif}\;y.im \leq -9.2 \cdot 10^{-52}:\\ \;\;\;\;y.im \cdot \frac{x.im}{t_0}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-56}:\\ \;\;\;\;\frac{y.im \cdot x.im}{t_0}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+14}:\\ \;\;\;\;x.re \cdot \frac{y.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 5
Error	23.9
Cost	1496

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.im \leq -6.5 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -1.35 \cdot 10^{-5}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(y.re \cdot x.re\right)\\ \mathbf{elif}\;y.im \leq -7.8 \cdot 10^{-52}:\\ \;\;\;\;y.im \cdot \frac{x.im}{t_0}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.2 \cdot 10^{-55}:\\ \;\;\;\;\frac{y.im \cdot x.im}{t_0}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+16}:\\ \;\;\;\;x.re \cdot \frac{y.re}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 6
Error	16.5
Cost	1488

\[\begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.8 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.3 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-235}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 5.6 \cdot 10^{+159}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7
Error	23.5
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.re \leq -3.7 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{-97}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+26}:\\ \;\;\;\;y.im \cdot \frac{x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8
Error	23.3
Cost	1100

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.2 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{-148}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{+26}:\\ \;\;\;\;\frac{y.im}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 9
Error	23.2
Cost	456

\[\begin{array}{l} \mathbf{if}\;y.re \leq -2.55 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 10
Error	36.9
Cost	192

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

Reproduce

herbie shell --seed 2023100 
(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))))