_divideComplex, imaginary part

Percentage Accurate: 61.4% → 81.4%

Time: 10.4s

Alternatives: 8

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

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))))

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));
}

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

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));
}

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))

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

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

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]

Initial Program: 61.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \end{array} \]

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))))

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));
}

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

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));
}

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))

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

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

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]

Alternative 1: 81.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \frac{y.re}{y.im \cdot y.im}\\ t_1 := y.im \cdot y.im + y.re \cdot y.re\\ t_2 := x.im \cdot y.re - y.im \cdot x.re\\ \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot t\_0\right) + x.re \cdot \left(t\_0 + -1\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{t\_1} \cdot t\_2\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.82 \cdot 10^{+62}:\\ \;\;\;\;\frac{t\_2}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(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.im y.im) (* y.re y.re)))
        (t_2 (- (* x.im y.re) (* y.im x.re))))
   (if (<= y.im -2.55e+92)
     (/ (+ (* (/ x.im y.im) (- y.re (* y.re t_0))) (* x.re (+ t_0 -1.0))) y.im)
     (if (<= y.im -4.5e-118)
       (* (/ 1.0 t_1) t_2)
       (if (<= y.im 3.5e-57)
         (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
         (if (<= y.im 1.82e+62)
           (/ t_2 t_1)
           (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))))))

C

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_im * y_46_im) + (y_46_re * y_46_re);
	double t_2 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_im <= -2.55e+92) {
		tmp = (((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im;
	} else if (y_46_im <= -4.5e-118) {
		tmp = (1.0 / t_1) * t_2;
	} else if (y_46_im <= 3.5e-57) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= 1.82e+62) {
		tmp = t_2 / t_1;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	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
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = y_46re * (y_46re / (y_46im * y_46im))
    t_1 = (y_46im * y_46im) + (y_46re * y_46re)
    t_2 = (x_46im * y_46re) - (y_46im * x_46re)
    if (y_46im <= (-2.55d+92)) then
        tmp = (((x_46im / y_46im) * (y_46re - (y_46re * t_0))) + (x_46re * (t_0 + (-1.0d0)))) / y_46im
    else if (y_46im <= (-4.5d-118)) then
        tmp = (1.0d0 / t_1) * t_2
    else if (y_46im <= 3.5d-57) then
        tmp = (x_46im / y_46re) - ((y_46im / y_46re) * (x_46re / y_46re))
    else if (y_46im <= 1.82d+62) then
        tmp = t_2 / t_1
    else
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    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) {
	double t_0 = y_46_re * (y_46_re / (y_46_im * y_46_im));
	double t_1 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double t_2 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	double tmp;
	if (y_46_im <= -2.55e+92) {
		tmp = (((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im;
	} else if (y_46_im <= -4.5e-118) {
		tmp = (1.0 / t_1) * t_2;
	} else if (y_46_im <= 3.5e-57) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= 1.82e+62) {
		tmp = t_2 / t_1;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

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_im * y_46_im) + (y_46_re * y_46_re)
	t_2 = (x_46_im * y_46_re) - (y_46_im * x_46_re)
	tmp = 0
	if y_46_im <= -2.55e+92:
		tmp = (((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im
	elif y_46_im <= -4.5e-118:
		tmp = (1.0 / t_1) * t_2
	elif y_46_im <= 3.5e-57:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	elif y_46_im <= 1.82e+62:
		tmp = t_2 / t_1
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * Float64(y_46_re / Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))
	t_2 = Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re))
	tmp = 0.0
	if (y_46_im <= -2.55e+92)
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re - Float64(y_46_re * t_0))) + Float64(x_46_re * Float64(t_0 + -1.0))) / y_46_im);
	elseif (y_46_im <= -4.5e-118)
		tmp = Float64(Float64(1.0 / t_1) * t_2);
	elseif (y_46_im <= 3.5e-57)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)));
	elseif (y_46_im <= 1.82e+62)
		tmp = Float64(t_2 / t_1);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

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_im * y_46_im) + (y_46_re * y_46_re);
	t_2 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	tmp = 0.0;
	if (y_46_im <= -2.55e+92)
		tmp = (((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im;
	elseif (y_46_im <= -4.5e-118)
		tmp = (1.0 / t_1) * t_2;
	elseif (y_46_im <= 3.5e-57)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	elseif (y_46_im <= 1.82e+62)
		tmp = t_2 / t_1;
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(y$46$re * N[(y$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.55e+92], N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re - N[(y$46$re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.5e-118], N[(N[(1.0 / t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[y$46$im, 3.5e-57], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.82e+62], N[(t$95$2 / t$95$1), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.5500000000000001e92

   1. Initial program 30.0%

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

   3. Taylor expanded in y.im around inf

      \[\leadsto \color{blue}{\frac{\left(-1 \cdot x.re + \left(-1 \cdot \frac{x.im \cdot {y.re}^{3}}{{y.im}^{3}} + \frac{x.im \cdot y.re}{y.im}\right)\right) - -1 \cdot \frac{x.re \cdot {y.re}^{2}}{{y.im}^{2}}}{y.im}} \]

   5. Simplified 82.9%

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

   if -2.5500000000000001e92 < y.im < -4.5e-118

   1. Initial program 86.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 86.3%

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

   if -4.5e-118 < y.im < 3.49999999999999991e-57

   1. Initial program 73.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 73.8%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im}{y.re} + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im}{y.re} + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{\color{blue}{x.re \cdot y.im}}{{y.re}^{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), \color{blue}{\left({y.re}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left({\color{blue}{y.re}}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left(y.re \cdot \color{blue}{y.re}\right)\right)\right) \]

      9. *-lowering-*.f64 89.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \mathsf{*.f64}\left(y.re, \color{blue}{y.re}\right)\right)\right) \]

   7. Simplified 89.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{y.im \cdot x.re}{\color{blue}{y.re} \cdot y.re}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{y.im}{y.re} \cdot \color{blue}{\frac{x.re}{y.re}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(\left(\frac{y.im}{y.re}\right), \color{blue}{\left(\frac{x.re}{y.re}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \left(\frac{\color{blue}{x.re}}{y.re}\right)\right)\right) \]

      5. /-lowering-/.f64 94.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \mathsf{/.f64}\left(x.re, \color{blue}{y.re}\right)\right)\right) \]

   9. Applied egg-rr 94.6%

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

   if 3.49999999999999991e-57 < y.im < 1.81999999999999998e62

   1. Initial program 95.5%

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

   if 1.81999999999999998e62 < y.im

   1. Initial program 39.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(x.im \cdot y.re\right), y.im\right), x.re\right), y.im\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot x.im\right), y.im\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 81.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), y.im\right), x.re\right), y.im\right) \]

   5. Simplified 81.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 83.7%

      \[\leadsto \frac{\color{blue}{x.im \cdot \frac{y.re}{y.im}} - x.re}{y.im} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.55 \cdot 10^{+92}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im}\right)\right) + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{elif}\;y.im \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 1.82 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-118}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+62}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))))
   (if (<= y.im -1.25e+42)
     (/ (- (* (/ x.im y.im) y.re) x.re) y.im)
     (if (<= y.im -3e-118)
       t_0
       (if (<= y.im 6.5e-61)
         (- (/ x.im y.re) (* (/ y.im y.re) (/ x.re y.re)))
         (if (<= y.im 2.1e+62)
           t_0
           (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))))))

C

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_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -1.25e+42) {
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3e-118) {
		tmp = t_0;
	} else if (y_46_im <= 6.5e-61) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= 2.1e+62) {
		tmp = t_0;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    if (y_46im <= (-1.25d+42)) then
        tmp = (((x_46im / y_46im) * y_46re) - x_46re) / y_46im
    else if (y_46im <= (-3d-118)) then
        tmp = t_0
    else if (y_46im <= 6.5d-61) then
        tmp = (x_46im / y_46re) - ((y_46im / y_46re) * (x_46re / y_46re))
    else if (y_46im <= 2.1d+62) then
        tmp = t_0
    else
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    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) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -1.25e+42) {
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	} else if (y_46_im <= -3e-118) {
		tmp = t_0;
	} else if (y_46_im <= 6.5e-61) {
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	} else if (y_46_im <= 2.1e+62) {
		tmp = t_0;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	tmp = 0
	if y_46_im <= -1.25e+42:
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im
	elif y_46_im <= -3e-118:
		tmp = t_0
	elif y_46_im <= 6.5e-61:
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re))
	elif y_46_im <= 2.1e+62:
		tmp = t_0
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_im <= -1.25e+42)
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im);
	elseif (y_46_im <= -3e-118)
		tmp = t_0;
	elseif (y_46_im <= 6.5e-61)
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(Float64(y_46_im / y_46_re) * Float64(x_46_re / y_46_re)));
	elseif (y_46_im <= 2.1e+62)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	tmp = 0.0;
	if (y_46_im <= -1.25e+42)
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	elseif (y_46_im <= -3e-118)
		tmp = t_0;
	elseif (y_46_im <= 6.5e-61)
		tmp = (x_46_im / y_46_re) - ((y_46_im / y_46_re) * (x_46_re / y_46_re));
	elseif (y_46_im <= 2.1e+62)
		tmp = t_0;
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.25e+42], N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -3e-118], t$95$0, If[LessEqual[y$46$im, 6.5e-61], N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.1e+62], t$95$0, N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.25000000000000002e42

   1. Initial program 38.6%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(x.im \cdot y.re\right), y.im\right), x.re\right), y.im\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot x.im\right), y.im\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 68.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), y.im\right), x.re\right), y.im\right) \]

   5. Simplified 68.0%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im}{y.im} \cdot y.re\right), x.re\right), y.im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{x.im}{y.im}\right), y.re\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 82.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x.im, y.im\right), y.re\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 82.7%

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

   if -1.25000000000000002e42 < y.im < -3.00000000000000018e-118 or 6.4999999999999994e-61 < y.im < 2.1e62

   1. Initial program 90.9%

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

   if -3.00000000000000018e-118 < y.im < 6.4999999999999994e-61

   1. Initial program 73.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 73.8%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im}{y.re} + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im}{y.re} + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{\color{blue}{x.re \cdot y.im}}{{y.re}^{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), \color{blue}{\left({y.re}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left({\color{blue}{y.re}}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left(y.re \cdot \color{blue}{y.re}\right)\right)\right) \]

      9. *-lowering-*.f64 89.7%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \mathsf{*.f64}\left(y.re, \color{blue}{y.re}\right)\right)\right) \]

   7. Simplified 89.7%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{y.im \cdot x.re}{\color{blue}{y.re} \cdot y.re}\right)\right) \]

      2. times-frac N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{y.im}{y.re} \cdot \color{blue}{\frac{x.re}{y.re}}\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(\left(\frac{y.im}{y.re}\right), \color{blue}{\left(\frac{x.re}{y.re}\right)}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \left(\frac{\color{blue}{x.re}}{y.re}\right)\right)\right) \]

      5. /-lowering-/.f64 94.6%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), \mathsf{/.f64}\left(x.re, \color{blue}{y.re}\right)\right)\right) \]

   9. Applied egg-rr 94.6%

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

   if 2.1e62 < y.im

   1. Initial program 39.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(x.im \cdot y.re\right), y.im\right), x.re\right), y.im\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot x.im\right), y.im\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 81.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), y.im\right), x.re\right), y.im\right) \]

   5. Simplified 81.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 83.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 83.7%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.25 \cdot 10^{+42}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -3 \cdot 10^{-118}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{x.im}{y.re} - \frac{y.im}{y.re} \cdot \frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6e-15)
   (/ (- (* (/ x.im y.im) y.re) x.re) y.im)
   (if (<= y.im 2.6e-53)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re)
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6e-15) {
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	} else if (y_46_im <= 2.6e-53) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	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
    real(8) :: tmp
    if (y_46im <= (-6d-15)) then
        tmp = (((x_46im / y_46im) * y_46re) - x_46re) / y_46im
    else if (y_46im <= 2.6d-53) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    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) {
	double tmp;
	if (y_46_im <= -6e-15) {
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	} else if (y_46_im <= 2.6e-53) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -6e-15:
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im
	elif y_46_im <= 2.6e-53:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -6e-15)
		tmp = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im);
	elseif (y_46_im <= 2.6e-53)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -6e-15)
		tmp = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	elseif (y_46_im <= 2.6e-53)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -6e-15], N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.6e-53], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6e-15

   1. Initial program 48.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(x.im \cdot y.re\right), y.im\right), x.re\right), y.im\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot x.im\right), y.im\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 66.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), y.im\right), x.re\right), y.im\right) \]

   5. Simplified 66.4%

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

      1. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(y.re \cdot \frac{x.im}{y.im}\right), x.re\right), y.im\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im}{y.im} \cdot y.re\right), x.re\right), y.im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\left(\frac{x.im}{y.im}\right), y.re\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 78.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(x.im, y.im\right), y.re\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 78.7%

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

   if -6e-15 < y.im < 2.59999999999999996e-53

   1. Initial program 75.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 75.2%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im}{y.re} + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im}{y.re} + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{\color{blue}{x.re \cdot y.im}}{{y.re}^{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), \color{blue}{\left({y.re}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left({\color{blue}{y.re}}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left(y.re \cdot \color{blue}{y.re}\right)\right)\right) \]

      9. *-lowering-*.f64 85.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \mathsf{*.f64}\left(y.re, \color{blue}{y.re}\right)\right)\right) \]

   7. Simplified 85.3%

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

      1. associate-/r* N/A

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

      2. sub-div N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), y.re\right)\right), y.re\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\left(y.im \cdot x.re\right), y.re\right)\right), y.re\right) \]

      7. *-lowering-*.f64 88.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, x.re\right), y.re\right)\right), y.re\right) \]

   9. Applied egg-rr 88.8%

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

       1. div-sub N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

       7. associate-*l/ N/A

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

       8. sub-div N/A

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

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{y.im}{y.re} \cdot x.re\right), \color{blue}{y.re}\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{y.im}{y.re} \cdot x.re\right)\right), y.re\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\left(\frac{y.im}{y.re}\right), x.re\right)\right), y.re\right) \]

       12. /-lowering-/.f64 89.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), x.re\right)\right), y.re\right) \]

   11. Applied egg-rr 89.5%

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

   if 2.59999999999999996e-53 < y.im

   1. Initial program 60.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(x.im \cdot y.re\right), y.im\right), x.re\right), y.im\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot x.im\right), y.im\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 75.6%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), y.im\right), x.re\right), y.im\right) \]

   5. Simplified 75.6%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 76.9%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 76.9%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-14}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.im -1.05e-14)
     t_0
     (if (<= y.im 2.6e-53) (/ (- x.im (* x.re (/ y.im y.re))) y.re) t_0))))

C

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_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.05e-14) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-53) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-1.05d-14)) then
        tmp = t_0
    else if (y_46im <= 2.6d-53) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = t_0
    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) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.05e-14) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e-53) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -1.05e-14:
		tmp = t_0
	elif y_46_im <= 2.6e-53:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.05e-14)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-53)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.05e-14)
		tmp = t_0;
	elseif (y_46_im <= 2.6e-53)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.05e-14], t$95$0, If[LessEqual[y$46$im, 2.6e-53], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.0499999999999999e-14 or 2.59999999999999996e-53 < y.im

   1. Initial program 55.6%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im \cdot y.re}{{y.im}^{2}} + \left(\mathsf{neg}\left(\frac{x.re}{y.im}\right)\right) \]

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x.im \cdot y.re}{y.im} - x.re\right), \color{blue}{y.im}\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(x.im \cdot y.re\right), y.im\right), x.re\right), y.im\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\left(y.re \cdot x.im\right), y.im\right), x.re\right), y.im\right) \]

      11. *-lowering-*.f64 71.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(y.re, x.im\right), y.im\right), x.re\right), y.im\right) \]

   5. Simplified 71.8%

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

      1. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(\frac{x.im \cdot y.re}{y.im}\right), x.re\right), y.im\right) \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\left(x.im \cdot \frac{y.re}{y.im}\right), x.re\right), y.im\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \left(\frac{y.re}{y.im}\right)\right), x.re\right), y.im\right) \]

      4. /-lowering-/.f64 75.4%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(\mathsf{*.f64}\left(x.im, \mathsf{/.f64}\left(y.re, y.im\right)\right), x.re\right), y.im\right) \]

   7. Applied egg-rr 75.4%

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

   if -1.0499999999999999e-14 < y.im < 2.59999999999999996e-53

   1. Initial program 75.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 75.2%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im}{y.re} + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im}{y.re} + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{\color{blue}{x.re \cdot y.im}}{{y.re}^{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), \color{blue}{\left({y.re}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left({\color{blue}{y.re}}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left(y.re \cdot \color{blue}{y.re}\right)\right)\right) \]

      9. *-lowering-*.f64 85.3%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \mathsf{*.f64}\left(y.re, \color{blue}{y.re}\right)\right)\right) \]

   7. Simplified 85.3%

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

      1. associate-/r* N/A

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

      2. sub-div N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), y.re\right)\right), y.re\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\left(y.im \cdot x.re\right), y.re\right)\right), y.re\right) \]

      7. *-lowering-*.f64 88.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, x.re\right), y.re\right)\right), y.re\right) \]

   9. Applied egg-rr 88.8%

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

       1. div-sub N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

       7. associate-*l/ N/A

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

       8. sub-div N/A

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

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{y.im}{y.re} \cdot x.re\right), \color{blue}{y.re}\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{y.im}{y.re} \cdot x.re\right)\right), y.re\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\left(\frac{y.im}{y.re}\right), x.re\right)\right), y.re\right) \]

       12. /-lowering-/.f64 89.5%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), x.re\right)\right), y.re\right) \]

   11. Applied egg-rr 89.5%

       \[\leadsto \color{blue}{\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.05 \cdot 10^{-14}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{-53}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6200000000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 x.re) y.im)))
   (if (<= y.im -3.1e+31)
     t_0
     (if (<= y.im 6200000000.0)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (0.0 - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -3.1e+31) {
		tmp = t_0;
	} else if (y_46_im <= 6200000000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.0d0 - x_46re) / y_46im
    if (y_46im <= (-3.1d+31)) then
        tmp = t_0
    else if (y_46im <= 6200000000.0d0) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = t_0
    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) {
	double t_0 = (0.0 - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -3.1e+31) {
		tmp = t_0;
	} else if (y_46_im <= 6200000000.0) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (0.0 - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -3.1e+31:
		tmp = t_0
	elif y_46_im <= 6200000000.0:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(0.0 - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.1e+31)
		tmp = t_0;
	elseif (y_46_im <= 6200000000.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (0.0 - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -3.1e+31)
		tmp = t_0;
	elseif (y_46_im <= 6200000000.0)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(0.0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.1e+31], t$95$0, If[LessEqual[y$46$im, 6200000000.0], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.1000000000000002e31 or 6.2e9 < y.im

   1. Initial program 45.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)} \cdot \frac{\color{blue}{1}}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\color{blue}{1}}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(y.re \cdot y.re + y.im \cdot y.im\right)}\right)} \]

      6. frac-times N/A

         \[\leadsto \frac{1 \cdot -1}{\color{blue}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)\right)}\right) \]

   4. Applied egg-rr 45.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{x.im \cdot y.re - x.re \cdot y.im} \cdot \left(y.im \cdot \left(0 - y.im\right) - y.re \cdot y.re\right)}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{y.im}{x.re}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 67.5%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(y.im, \color{blue}{x.re}\right)\right) \]

   7. Simplified 67.5%

      \[\leadsto \frac{-1}{\color{blue}{\frac{y.im}{x.re}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x.re}{y.im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x.re}{y.im}\right)\right) \]

      5. /-lowering-/.f64 67.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x.re, y.im\right)\right) \]

   9. Applied egg-rr 67.8%

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

   if -3.1000000000000002e31 < y.im < 6.2e9

   1. Initial program 78.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im}{y.re} + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im}{y.re} + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{\color{blue}{x.re \cdot y.im}}{{y.re}^{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), \color{blue}{\left({y.re}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left({\color{blue}{y.re}}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left(y.re \cdot \color{blue}{y.re}\right)\right)\right) \]

      9. *-lowering-*.f64 78.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \mathsf{*.f64}\left(y.re, \color{blue}{y.re}\right)\right)\right) \]

   7. Simplified 78.2%

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

      1. associate-/r* N/A

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

      2. sub-div N/A

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

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re \cdot y.im}{y.re}\right), \color{blue}{y.re}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re \cdot y.im}{y.re}\right)\right), y.re\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), y.re\right)\right), y.re\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\left(y.im \cdot x.re\right), y.re\right)\right), y.re\right) \]

      7. *-lowering-*.f64 81.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y.im, x.re\right), y.re\right)\right), y.re\right) \]

   9. Applied egg-rr 81.1%

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

       1. div-sub N/A

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

       2. *-commutative N/A

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

       3. associate-/r* N/A

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

       4. associate-*r/ N/A

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

       5. *-commutative N/A

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

       6. associate-/r* N/A

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

       7. associate-*l/ N/A

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

       8. sub-div N/A

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

       9. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{y.im}{y.re} \cdot x.re\right), \color{blue}{y.re}\right) \]

       10. --lowering--.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{y.im}{y.re} \cdot x.re\right)\right), y.re\right) \]

       11. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\left(\frac{y.im}{y.re}\right), x.re\right)\right), y.re\right) \]

       12. /-lowering-/.f64 81.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(y.im, y.re\right), x.re\right)\right), y.re\right) \]

   11. Applied egg-rr 81.7%

       \[\leadsto \color{blue}{\frac{x.im - \frac{y.im}{y.re} \cdot x.re}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 6200000000:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 10500000000:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 x.re) y.im)))
   (if (<= y.im -1.2e+39)
     t_0
     (if (<= y.im 10500000000.0)
       (/ (- x.im (* y.im (/ x.re y.re))) y.re)
       t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (0.0 - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e+39) {
		tmp = t_0;
	} else if (y_46_im <= 10500000000.0) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	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
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.0d0 - x_46re) / y_46im
    if (y_46im <= (-1.2d+39)) then
        tmp = t_0
    else if (y_46im <= 10500000000.0d0) then
        tmp = (x_46im - (y_46im * (x_46re / y_46re))) / y_46re
    else
        tmp = t_0
    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) {
	double t_0 = (0.0 - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e+39) {
		tmp = t_0;
	} else if (y_46_im <= 10500000000.0) {
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (0.0 - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -1.2e+39:
		tmp = t_0
	elif y_46_im <= 10500000000.0:
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(0.0 - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+39)
		tmp = t_0;
	elseif (y_46_im <= 10500000000.0)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im * Float64(x_46_re / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (0.0 - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.2e+39)
		tmp = t_0;
	elseif (y_46_im <= 10500000000.0)
		tmp = (x_46_im - (y_46_im * (x_46_re / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(0.0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e+39], t$95$0, If[LessEqual[y$46$im, 10500000000.0], N[(N[(x$46$im - N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.2e39 or 1.05e10 < y.im

   1. Initial program 45.5%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)} \cdot \frac{\color{blue}{1}}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\color{blue}{1}}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(y.re \cdot y.re + y.im \cdot y.im\right)}\right)} \]

      6. frac-times N/A

         \[\leadsto \frac{1 \cdot -1}{\color{blue}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)\right)}\right) \]

   4. Applied egg-rr 45.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{x.im \cdot y.re - x.re \cdot y.im} \cdot \left(y.im \cdot \left(0 - y.im\right) - y.re \cdot y.re\right)}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{y.im}{x.re}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 67.5%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(y.im, \color{blue}{x.re}\right)\right) \]

   7. Simplified 67.5%

      \[\leadsto \frac{-1}{\color{blue}{\frac{y.im}{x.re}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x.re}{y.im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x.re}{y.im}\right)\right) \]

      5. /-lowering-/.f64 67.8%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x.re, y.im\right)\right) \]

   9. Applied egg-rr 67.8%

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

   if -1.2e39 < y.im < 1.05e10

   1. Initial program 78.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. flip3-+ N/A

         \[\leadsto \frac{1}{\frac{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}} \cdot \left(x.im \cdot \color{blue}{y.re} - x.re \cdot y.im\right) \]

      4. clear-num N/A

         \[\leadsto \frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}} \cdot \left(\color{blue}{x.im \cdot y.re} - x.re \cdot y.im\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{\left(y.re \cdot y.re\right) \cdot \left(y.re \cdot y.re\right) + \left(\left(y.im \cdot y.im\right) \cdot \left(y.im \cdot y.im\right) - \left(y.re \cdot y.re\right) \cdot \left(y.im \cdot y.im\right)\right)}{{\left(y.re \cdot y.re\right)}^{3} + {\left(y.im \cdot y.im\right)}^{3}}\right), \color{blue}{\left(x.im \cdot y.re - x.re \cdot y.im\right)}\right) \]

   4. Applied egg-rr 78.6%

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

   5. Taylor expanded in y.im around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}} + \frac{x.im}{y.re}} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{x.im}{y.re} + \color{blue}{-1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg N/A

         \[\leadsto \frac{x.im}{y.re} + \left(\mathsf{neg}\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(\frac{x.re \cdot y.im}{{y.re}^{2}}\right)}\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\frac{\color{blue}{x.re \cdot y.im}}{{y.re}^{2}}\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\left(x.re \cdot y.im\right), \color{blue}{\left({y.re}^{2}\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left({\color{blue}{y.re}}^{2}\right)\right)\right) \]

      8. unpow2 N/A

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \left(y.re \cdot \color{blue}{y.re}\right)\right)\right) \]

      9. *-lowering-*.f64 78.2%

         \[\leadsto \mathsf{\_.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(x.re, y.im\right), \mathsf{*.f64}\left(y.re, \color{blue}{y.re}\right)\right)\right) \]

   7. Simplified 78.2%

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

      1. times-frac N/A

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

      2. associate-*r/ N/A

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

      3. sub-div N/A

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

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x.im - \frac{x.re}{y.re} \cdot y.im\right), \color{blue}{y.re}\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(\frac{x.re}{y.re} \cdot y.im\right)\right), y.re\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\left(\frac{x.re}{y.re}\right), y.im\right)\right), y.re\right) \]

      7. /-lowering-/.f64 81.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \mathsf{*.f64}\left(\mathsf{/.f64}\left(x.re, y.re\right), y.im\right)\right), y.re\right) \]

   9. Applied egg-rr 81.1%

      \[\leadsto \color{blue}{\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 10500000000:\\ \;\;\;\;\frac{x.im - y.im \cdot \frac{x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 21500000000000:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8e-108)
   (/ x.im y.re)
   (if (<= y.re 21500000000000.0) (/ (- 0.0 x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8e-108) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 21500000000000.0) {
		tmp = (0.0 - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-8d-108)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 21500000000000.0d0) then
        tmp = (0.0d0 - x_46re) / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8e-108) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 21500000000000.0) {
		tmp = (0.0 - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8e-108:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 21500000000000.0:
		tmp = (0.0 - x_46_re) / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8e-108)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 21500000000000.0)
		tmp = Float64(Float64(0.0 - x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -8e-108)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 21500000000000.0)
		tmp = (0.0 - x_46_re) / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8e-108], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 21500000000000.0], N[(N[(0.0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -8.00000000000000032e-108 or 2.15e13 < y.re

   1. Initial program 58.4%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 61.3%

         \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.re}\right) \]

   5. Simplified 61.3%

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

   if -8.00000000000000032e-108 < y.re < 2.15e13

   1. Initial program 73.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

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

      2. flip3-- N/A

         \[\leadsto \frac{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)} \cdot \frac{\color{blue}{1}}{y.re \cdot y.re + y.im \cdot y.im} \]

      3. clear-num N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\color{blue}{1}}{y.re \cdot y.re + y.im \cdot y.im} \]

      4. frac-2neg N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{\mathsf{neg}\left(1\right)}{\color{blue}{\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)}} \]

      5. metadata-eval N/A

         \[\leadsto \frac{1}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \frac{-1}{\mathsf{neg}\left(\color{blue}{\left(y.re \cdot y.re + y.im \cdot y.im\right)}\right)} \]

      6. frac-times N/A

         \[\leadsto \frac{1 \cdot -1}{\color{blue}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)}} \]

      7. metadata-eval N/A

         \[\leadsto \frac{-1}{\color{blue}{\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)} \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{\left(x.im \cdot y.re\right) \cdot \left(x.im \cdot y.re\right) + \left(\left(x.re \cdot y.im\right) \cdot \left(x.re \cdot y.im\right) + \left(x.im \cdot y.re\right) \cdot \left(x.re \cdot y.im\right)\right)}{{\left(x.im \cdot y.re\right)}^{3} - {\left(x.re \cdot y.im\right)}^{3}} \cdot \left(\mathsf{neg}\left(\left(y.re \cdot y.re + y.im \cdot y.im\right)\right)\right)\right)}\right) \]

   4. Applied egg-rr 73.2%

      \[\leadsto \color{blue}{\frac{-1}{\frac{1}{x.im \cdot y.re - x.re \cdot y.im} \cdot \left(y.im \cdot \left(0 - y.im\right) - y.re \cdot y.re\right)}} \]

   5. Taylor expanded in y.re around 0

      \[\leadsto \mathsf{/.f64}\left(-1, \color{blue}{\left(\frac{y.im}{x.re}\right)}\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 64.7%

         \[\leadsto \mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(y.im, \color{blue}{x.re}\right)\right) \]

   7. Simplified 64.7%

      \[\leadsto \frac{-1}{\color{blue}{\frac{y.im}{x.re}}} \]
   8. Step-by-step derivation

      1. div-inv N/A

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

      2. clear-num N/A

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

      3. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{x.re}{y.im}\right) \]

      4. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{x.re}{y.im}\right)\right) \]

      5. /-lowering-/.f64 65.0%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(x.re, y.im\right)\right) \]

   9. Applied egg-rr 65.0%

      \[\leadsto \color{blue}{-\frac{x.re}{y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 21500000000000:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{x.im}{y.re} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.im y.re))

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;
}

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
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;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	return x_46_im / y_46_re

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_46_im / y_46_re)
end

MATLAB

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

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := N[(x$46$im / y$46$re), $MachinePrecision]

Derivation

1. Initial program 65.3%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 40.4%

      \[\leadsto \mathsf{/.f64}\left(x.im, \color{blue}{y.re}\right) \]

5. Simplified 40.4%

   \[\leadsto \color{blue}{\frac{x.im}{y.re}} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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))))