_divideComplex, imaginary part

Average Error: 26.2 → 12.5

Time: 19.2s

Precision: binary64

Cost: 2508

Math

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

↓

\[\begin{array}{l} t_0 := y.re \cdot x.im - y.im \cdot x.re\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{x.re}{-y.im}\\ t_3 := x.im \cdot \frac{2}{y.re}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+146}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\frac{t_0}{y.im}} + y.re \cdot \frac{y.re}{t_0}}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-77}:\\ \;\;\;\;t_3 + \left(t_3 - \left(\frac{x.im}{y.re} + \left(t_3 + \frac{x.re \cdot y.im}{t_1}\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_1} - x.re \cdot \frac{y.im}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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 x.im) (* y.im x.re)))
        (t_1 (+ (* y.re y.re) (* y.im y.im)))
        (t_2 (/ x.re (- y.im)))
        (t_3 (* x.im (/ 2.0 y.re))))
   (if (<= y.im -2e+146)
     t_2
     (if (<= y.im -2.2e-117)
       (/ 1.0 (+ (/ y.im (/ t_0 y.im)) (* y.re (/ y.re t_0))))
       (if (<= y.im 1.55e-77)
         (+ t_3 (- t_3 (+ (/ x.im y.re) (+ t_3 (/ (* x.re y.im) t_1)))))
         (if (<= y.im 1.35e+154)
           (- (* x.im (/ y.re t_1)) (* x.re (/ y.im t_1)))
           t_2))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * 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 * x_46_im) - (y_46_im * x_46_re);
	double t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_2 = x_46_re / -y_46_im;
	double t_3 = x_46_im * (2.0 / y_46_re);
	double tmp;
	if (y_46_im <= -2e+146) {
		tmp = t_2;
	} else if (y_46_im <= -2.2e-117) {
		tmp = 1.0 / ((y_46_im / (t_0 / y_46_im)) + (y_46_re * (y_46_re / t_0)));
	} else if (y_46_im <= 1.55e-77) {
		tmp = t_3 + (t_3 - ((x_46_im / y_46_re) + (t_3 + ((x_46_re * y_46_im) / t_1))));
	} else if (y_46_im <= 1.35e+154) {
		tmp = (x_46_im * (y_46_re / t_1)) - (x_46_re * (y_46_im / t_1));
	} 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_46im * y_46re) - (x_46re * 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 * x_46im) - (y_46im * x_46re)
    t_1 = (y_46re * y_46re) + (y_46im * y_46im)
    t_2 = x_46re / -y_46im
    t_3 = x_46im * (2.0d0 / y_46re)
    if (y_46im <= (-2d+146)) then
        tmp = t_2
    else if (y_46im <= (-2.2d-117)) then
        tmp = 1.0d0 / ((y_46im / (t_0 / y_46im)) + (y_46re * (y_46re / t_0)))
    else if (y_46im <= 1.55d-77) then
        tmp = t_3 + (t_3 - ((x_46im / y_46re) + (t_3 + ((x_46re * y_46im) / t_1))))
    else if (y_46im <= 1.35d+154) then
        tmp = (x_46im * (y_46re / t_1)) - (x_46re * (y_46im / t_1))
    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_im * y_46_re) - (x_46_re * 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 * x_46_im) - (y_46_im * x_46_re);
	double t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_2 = x_46_re / -y_46_im;
	double t_3 = x_46_im * (2.0 / y_46_re);
	double tmp;
	if (y_46_im <= -2e+146) {
		tmp = t_2;
	} else if (y_46_im <= -2.2e-117) {
		tmp = 1.0 / ((y_46_im / (t_0 / y_46_im)) + (y_46_re * (y_46_re / t_0)));
	} else if (y_46_im <= 1.55e-77) {
		tmp = t_3 + (t_3 - ((x_46_im / y_46_re) + (t_3 + ((x_46_re * y_46_im) / t_1))));
	} else if (y_46_im <= 1.35e+154) {
		tmp = (x_46_im * (y_46_re / t_1)) - (x_46_re * (y_46_im / t_1));
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return ((x_46_im * y_46_re) - (x_46_re * 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 * x_46_im) - (y_46_im * x_46_re)
	t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_2 = x_46_re / -y_46_im
	t_3 = x_46_im * (2.0 / y_46_re)
	tmp = 0
	if y_46_im <= -2e+146:
		tmp = t_2
	elif y_46_im <= -2.2e-117:
		tmp = 1.0 / ((y_46_im / (t_0 / y_46_im)) + (y_46_re * (y_46_re / t_0)))
	elif y_46_im <= 1.55e-77:
		tmp = t_3 + (t_3 - ((x_46_im / y_46_re) + (t_3 + ((x_46_re * y_46_im) / t_1))))
	elif y_46_im <= 1.35e+154:
		tmp = (x_46_im * (y_46_re / t_1)) - (x_46_re * (y_46_im / t_1))
	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_im * y_46_re) - Float64(x_46_re * 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 * x_46_im) - Float64(y_46_im * x_46_re))
	t_1 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_2 = Float64(x_46_re / Float64(-y_46_im))
	t_3 = Float64(x_46_im * Float64(2.0 / y_46_re))
	tmp = 0.0
	if (y_46_im <= -2e+146)
		tmp = t_2;
	elseif (y_46_im <= -2.2e-117)
		tmp = Float64(1.0 / Float64(Float64(y_46_im / Float64(t_0 / y_46_im)) + Float64(y_46_re * Float64(y_46_re / t_0))));
	elseif (y_46_im <= 1.55e-77)
		tmp = Float64(t_3 + Float64(t_3 - Float64(Float64(x_46_im / y_46_re) + Float64(t_3 + Float64(Float64(x_46_re * y_46_im) / t_1)))));
	elseif (y_46_im <= 1.35e+154)
		tmp = Float64(Float64(x_46_im * Float64(y_46_re / t_1)) - Float64(x_46_re * Float64(y_46_im / t_1)));
	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_im * y_46_re) - (x_46_re * 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 * x_46_im) - (y_46_im * x_46_re);
	t_1 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_2 = x_46_re / -y_46_im;
	t_3 = x_46_im * (2.0 / y_46_re);
	tmp = 0.0;
	if (y_46_im <= -2e+146)
		tmp = t_2;
	elseif (y_46_im <= -2.2e-117)
		tmp = 1.0 / ((y_46_im / (t_0 / y_46_im)) + (y_46_re * (y_46_re / t_0)));
	elseif (y_46_im <= 1.55e-77)
		tmp = t_3 + (t_3 - ((x_46_im / y_46_re) + (t_3 + ((x_46_re * y_46_im) / t_1))));
	elseif (y_46_im <= 1.35e+154)
		tmp = (x_46_im * (y_46_re / t_1)) - (x_46_re * (y_46_im / t_1));
	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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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 * x$46$im), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x$46$re / (-y$46$im)), $MachinePrecision]}, Block[{t$95$3 = N[(x$46$im * N[(2.0 / y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2e+146], t$95$2, If[LessEqual[y$46$im, -2.2e-117], N[(1.0 / N[(N[(y$46$im / N[(t$95$0 / y$46$im), $MachinePrecision]), $MachinePrecision] + N[(y$46$re * N[(y$46$re / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.55e-77], N[(t$95$3 + N[(t$95$3 - N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(t$95$3 + N[(N[(x$46$re * y$46$im), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.35e+154], N[(N[(x$46$im * N[(y$46$re / t$95$1), $MachinePrecision]), $MachinePrecision] - N[(x$46$re * N[(y$46$im / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.99999999999999987e146 or 1.35000000000000003e154 < y.im

   1. Initial program 45.2

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

   2. Taylor expanded in y.re around 0 14.1

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

   3. Simplified 14.1

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

      [Start] 14.1	\[ -1 \cdot \frac{x.re}{y.im} \]
      rational.json-simplify-18 [=>] 14.4	\[ \color{blue}{\frac{-1}{\frac{y.im}{x.re}}} \]
      rational.json-simplify-1 [=>] 14.1	\[ \color{blue}{\frac{x.re}{\frac{y.im}{-1}}} \]
      rational.json-simplify-69 [<=] 14.1	\[ \frac{x.re}{\color{blue}{-y.im}} \]

   if -1.99999999999999987e146 < y.im < -2.2000000000000001e-117

   1. Initial program 16.9

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

   2. Applied egg-rr 17.0

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

   3. Applied egg-rr 17.1

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

   4. Applied egg-rr 12.8

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

   if -2.2000000000000001e-117 < y.im < 1.55000000000000004e-77

   1. Initial program 21.4

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

   2. Applied egg-rr 19.0

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

   3. Taylor expanded in y.re around inf 12.1

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

   4. Applied egg-rr 11.4

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

   if 1.55000000000000004e-77 < y.im < 1.35000000000000003e154

   1. Initial program 16.7

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

   2. Applied egg-rr 11.9

      \[\leadsto \color{blue}{x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} - x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2 \cdot 10^{+146}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \mathbf{elif}\;y.im \leq -2.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\frac{y.re \cdot x.im - y.im \cdot x.re}{y.im}} + y.re \cdot \frac{y.re}{y.re \cdot x.im - y.im \cdot x.re}}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-77}:\\ \;\;\;\;x.im \cdot \frac{2}{y.re} + \left(x.im \cdot \frac{2}{y.re} - \left(\frac{x.im}{y.re} + \left(x.im \cdot \frac{2}{y.re} + \frac{x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\right)\right)\right)\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} - x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{-y.im}\\ \end{array} \]

Alternatives

Alternative 1
Error	14.0
Cost	2000

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ t_2 := x.re \cdot \frac{y.im}{t_0}\\ \mathbf{if}\;y.im \leq -9.5 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\left(x.im \cdot \left(y.re + y.re\right) - x.re \cdot y.im\right) - x.im \cdot y.re}{t_0}\\ \mathbf{elif}\;y.im \leq 3.55 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im}{y.re} - t_2\\ \mathbf{elif}\;y.im \leq 10^{+154}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_0} - t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	13.3
Cost	2000

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := y.re \cdot x.im - y.im \cdot x.re\\ t_2 := \frac{x.re}{-y.im}\\ t_3 := x.re \cdot \frac{y.im}{t_0}\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+151}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{1}{\frac{y.im}{\frac{t_1}{y.im}} + y.re \cdot \frac{y.re}{t_1}}\\ \mathbf{elif}\;y.im \leq 3.55 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im}{y.re} - t_3\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;x.im \cdot \frac{y.re}{t_0} - t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	15.3
Cost	1880

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ t_2 := x.im \cdot y.re - x.re \cdot y.im\\ t_3 := \frac{t_2}{t_0}\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -3.8 \cdot 10^{-25}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 1.36 \cdot 10^{-77}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{+33}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	15.2
Cost	1880

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ t_2 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -5.8 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.75 \cdot 10^{-23}:\\ \;\;\;\;\frac{t_2}{t_0}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{t_0}{y.re \cdot x.im - y.im \cdot x.re}}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+78}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.im \leq 1.1 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	15.2
Cost	1880

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ t_2 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;y.im \leq -2.75 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -4.4 \cdot 10^{-24}:\\ \;\;\;\;\frac{t_2}{t_0}\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{-1}{y.im \cdot x.re - y.re \cdot x.im} \cdot t_0}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t_0} \cdot t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	15.2
Cost	1880

\[\begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ t_1 := y.re \cdot y.re + y.im \cdot y.im\\ t_2 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+142}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-24}:\\ \;\;\;\;\frac{\frac{1}{\frac{1}{t_0}}}{t_1}\\ \mathbf{elif}\;y.im \leq 4.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{t_1}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{-1}{y.im \cdot x.re - y.re \cdot x.im} \cdot t_1}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.im \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t_1} \cdot t_0\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	15.2
Cost	1880

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{+151}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -1.7 \cdot 10^{-25}:\\ \;\;\;\;\frac{\left(x.im \cdot \left(y.re + y.re\right) - x.re \cdot y.im\right) - x.im \cdot y.re}{t_0}\\ \mathbf{elif}\;y.im \leq 2.06 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+35}:\\ \;\;\;\;\frac{1}{\frac{-1}{y.im \cdot x.re - y.re \cdot x.im} \cdot t_0}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+77}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.im \leq 6.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{1}{t_0} \cdot \left(x.im \cdot y.re - x.re \cdot y.im\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	15.3
Cost	1752

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ t_2 := \frac{x.im \cdot y.re - x.re \cdot y.im}{t_0}\\ \mathbf{if}\;y.im \leq -1.16 \cdot 10^{+152}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -2.5 \cdot 10^{-24}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{-76}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{t_0}\\ \mathbf{elif}\;y.im \leq 2.65 \cdot 10^{+35}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+82}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{elif}\;y.im \leq 6 \cdot 10^{+153}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	21.3
Cost	1424

\[\begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+127}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1.9 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 10
Error	24.4
Cost	1364

\[\begin{array}{l} t_0 := \frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ t_1 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -7.5 \cdot 10^{-21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.4 \cdot 10^{-75}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{+15}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	22.9
Cost	1364

\[\begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ t_1 := \frac{x.re}{\left(-y.re \cdot y.re\right) - y.im \cdot y.im} \cdot y.im\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -4 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 2.2 \cdot 10^{-48}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 4.2 \cdot 10^{+153}:\\ \;\;\;\;\frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 12
Error	22.6
Cost	1364

\[\begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ t_1 := \frac{y.im}{\left(-y.re \cdot y.re\right) - y.im \cdot y.im} \cdot x.re\\ \mathbf{if}\;y.im \leq -5 \cdot 10^{+126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -1 \cdot 10^{-23}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 3 \cdot 10^{-51}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+86}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq 4.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{y.re}{y.re \cdot y.re + y.im \cdot y.im} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 13
Error	22.6
Cost	1364

\[\begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -2 \cdot 10^{+125}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.im \leq -5.2 \cdot 10^{-23}:\\ \;\;\;\;\frac{-x.re}{\frac{t_0}{y.im}}\\ \mathbf{elif}\;y.im \leq 1.05 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 2.46 \cdot 10^{+86}:\\ \;\;\;\;\frac{y.im}{\left(-y.re \cdot y.re\right) - y.im \cdot y.im} \cdot x.re\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{+153}:\\ \;\;\;\;\frac{y.re}{t_0} \cdot x.im\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	18.1
Cost	1292

\[\begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -1.35 \cdot 10^{+71}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.8 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+147}:\\ \;\;\;\;\frac{y.im}{y.im \cdot \frac{y.im}{-x.re} - y.re \cdot \frac{y.re}{x.re}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 15
Error	23.7
Cost	784

\[\begin{array}{l} t_0 := \frac{x.re}{-y.im}\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{-21}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.9 \cdot 10^{-46}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.im \leq 1.9 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 16
Error	38.1
Cost	192

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

Reproduce

herbie shell --seed 2023066 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))