_divideComplex, imaginary part

Percentage Accurate: 61.3% → 80.3%

Time: 10.4s

Alternatives: 10

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: 61.3% 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: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \frac{y.re}{y.im \cdot y.im}\\ t_1 := x.re \cdot \left(t\_0 + -1\right)\\ \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} + t\_1}{y.im}\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 + \frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot t\_0\right)}{y.im}\\ \end{array} \end{array} \]

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 (* x.re (+ t_0 -1.0))))
   (if (<= y.im -3.8e+85)
     (/ (+ (* x.im (/ y.re y.im)) t_1) y.im)
     (if (<= y.im -4.6e-106)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.im 7.7e+74)
         (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
         (/ (+ t_1 (* (/ x.im y.im) (- y.re (* y.re t_0)))) 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 = x_46_re * (t_0 + -1.0);
	double tmp;
	if (y_46_im <= -3.8e+85) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) + t_1) / y_46_im;
	} else if (y_46_im <= -4.6e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = (t_1 + ((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0)))) / 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) :: tmp
    t_0 = y_46re * (y_46re / (y_46im * y_46im))
    t_1 = x_46re * (t_0 + (-1.0d0))
    if (y_46im <= (-3.8d+85)) then
        tmp = ((x_46im * (y_46re / y_46im)) + t_1) / y_46im
    else if (y_46im <= (-4.6d-106)) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else if (y_46im <= 7.7d+74) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else
        tmp = (t_1 + ((x_46im / y_46im) * (y_46re - (y_46re * t_0)))) / 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 = x_46_re * (t_0 + -1.0);
	double tmp;
	if (y_46_im <= -3.8e+85) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) + t_1) / y_46_im;
	} else if (y_46_im <= -4.6e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = (t_1 + ((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0)))) / 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 = x_46_re * (t_0 + -1.0)
	tmp = 0
	if y_46_im <= -3.8e+85:
		tmp = ((x_46_im * (y_46_re / y_46_im)) + t_1) / y_46_im
	elif y_46_im <= -4.6e-106:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	elif y_46_im <= 7.7e+74:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	else:
		tmp = (t_1 + ((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0)))) / 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(x_46_re * Float64(t_0 + -1.0))
	tmp = 0.0
	if (y_46_im <= -3.8e+85)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) + t_1) / y_46_im);
	elseif (y_46_im <= -4.6e-106)
		tmp = 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)));
	elseif (y_46_im <= 7.7e+74)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	else
		tmp = Float64(Float64(t_1 + Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re - Float64(y_46_re * t_0)))) / 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 = x_46_re * (t_0 + -1.0);
	tmp = 0.0;
	if (y_46_im <= -3.8e+85)
		tmp = ((x_46_im * (y_46_re / y_46_im)) + t_1) / y_46_im;
	elseif (y_46_im <= -4.6e-106)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	elseif (y_46_im <= 7.7e+74)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	else
		tmp = (t_1 + ((x_46_im / y_46_im) * (y_46_re - (y_46_re * t_0)))) / 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[(x$46$re * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -3.8e+85], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.6e-106], 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, 7.7e+74], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(t$95$1 + N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re - N[(y$46$re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.79999999999999992e85

   1. Initial program 48.8%

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

   3. Taylor expanded in y.im around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot x.re + \left(\frac{x.im \cdot y.re}{y.im} + \frac{x.re \cdot {y.re}^{2}}{{y.im}^{2}}\right)\right), \color{blue}{y.im}\right) \]

   5. Simplified 86.6%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 90.1%

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

   7. Applied egg-rr 90.1%

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

   if -3.79999999999999992e85 < y.im < -4.6000000000000002e-106

   1. Initial program 93.5%

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

   if -4.6000000000000002e-106 < y.im < 7.70000000000000042e74

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 89.3%

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

   5. Simplified 89.3%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

   if 7.70000000000000042e74 < y.im

   1. Initial program 46.6%

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

   3. Taylor expanded in y.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 76.9%

      \[\leadsto \color{blue}{\frac{\frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im}\right)\right) + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.8 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.6 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right) + \frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im}\right)\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+85}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (+
           (* x.im (/ y.re y.im))
           (* x.re (+ (* y.re (/ y.re (* y.im y.im))) -1.0)))
          y.im)))
   (if (<= y.im -2.9e+85)
     t_0
     (if (<= y.im -4.5e-106)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.im 2.05e+133)
         (/ (- x.im (/ x.re (/ y.re y.im))) 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_re * (y_46_re / (y_46_im * y_46_im))) + -1.0))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.9e+85) {
		tmp = t_0;
	} else if (y_46_im <= -4.5e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 2.05e+133) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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_46re * (y_46re / (y_46im * y_46im))) + (-1.0d0)))) / y_46im
    if (y_46im <= (-2.9d+85)) then
        tmp = t_0
    else if (y_46im <= (-4.5d-106)) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else if (y_46im <= 2.05d+133) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / 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_re * (y_46_re / (y_46_im * y_46_im))) + -1.0))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.9e+85) {
		tmp = t_0;
	} else if (y_46_im <= -4.5e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 2.05e+133) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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_re * (y_46_re / (y_46_im * y_46_im))) + -1.0))) / y_46_im
	tmp = 0
	if y_46_im <= -2.9e+85:
		tmp = t_0
	elif y_46_im <= -4.5e-106:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	elif y_46_im <= 2.05e+133:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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)) + Float64(x_46_re * Float64(Float64(y_46_re * Float64(y_46_re / Float64(y_46_im * y_46_im))) + -1.0))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.9e+85)
		tmp = t_0;
	elseif (y_46_im <= -4.5e-106)
		tmp = 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)));
	elseif (y_46_im <= 2.05e+133)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / 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_re * (y_46_re / (y_46_im * y_46_im))) + -1.0))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.9e+85)
		tmp = t_0;
	elseif (y_46_im <= -4.5e-106)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	elseif (y_46_im <= 2.05e+133)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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] + N[(x$46$re * N[(N[(y$46$re * N[(y$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.9e+85], t$95$0, If[LessEqual[y$46$im, -4.5e-106], 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, 2.05e+133], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.89999999999999997e85 or 2.05000000000000002e133 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.im around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot x.re + \left(\frac{x.im \cdot y.re}{y.im} + \frac{x.re \cdot {y.re}^{2}}{{y.im}^{2}}\right)\right), \color{blue}{y.im}\right) \]

   5. Simplified 84.4%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 87.6%

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

   7. Applied egg-rr 87.6%

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

   if -2.89999999999999997e85 < y.im < -4.49999999999999955e-106

   1. Initial program 93.5%

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

   if -4.49999999999999955e-106 < y.im < 2.05000000000000002e133

   1. Initial program 67.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 85.5%

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

   5. Simplified 85.5%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 85.5%

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

   7. Applied egg-rr 85.5%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.9 \cdot 10^{+85}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \mathbf{elif}\;y.im \leq -4.5 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 2.05 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot \left(\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\right) - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          (- (* y.re (+ (/ x.im y.im) (/ (* y.re x.re) (* y.im y.im)))) x.re)
          y.im)))
   (if (<= y.im -4.8e+138)
     t_0
     (if (<= y.im -4.2e-106)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.im 1.22e+75) (/ (- x.im (/ x.re (/ y.re y.im))) 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 = ((y_46_re * ((x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im)))) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.8e+138) {
		tmp = t_0;
	} else if (y_46_im <= -4.2e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 1.22e+75) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 = ((y_46re * ((x_46im / y_46im) + ((y_46re * x_46re) / (y_46im * y_46im)))) - x_46re) / y_46im
    if (y_46im <= (-4.8d+138)) then
        tmp = t_0
    else if (y_46im <= (-4.2d-106)) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else if (y_46im <= 1.22d+75) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / 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 = ((y_46_re * ((x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im)))) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.8e+138) {
		tmp = t_0;
	} else if (y_46_im <= -4.2e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 1.22e+75) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 = ((y_46_re * ((x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im)))) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -4.8e+138:
		tmp = t_0
	elif y_46_im <= -4.2e-106:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	elif y_46_im <= 1.22e+75:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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(y_46_re * Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re * x_46_re) / Float64(y_46_im * y_46_im)))) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.8e+138)
		tmp = t_0;
	elseif (y_46_im <= -4.2e-106)
		tmp = 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)));
	elseif (y_46_im <= 1.22e+75)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / 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 = ((y_46_re * ((x_46_im / y_46_im) + ((y_46_re * x_46_re) / (y_46_im * y_46_im)))) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -4.8e+138)
		tmp = t_0;
	elseif (y_46_im <= -4.2e-106)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	elseif (y_46_im <= 1.22e+75)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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[(y$46$re * N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re * x$46$re), $MachinePrecision] / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e+138], t$95$0, If[LessEqual[y$46$im, -4.2e-106], 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, 1.22e+75], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -4.8000000000000002e138 or 1.2199999999999999e75 < y.im

   1. Initial program 46.3%

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

      1. clear-num N/A

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

      2. associate-/r/ 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. flip-+ N/A

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

      4. clear-num N/A

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

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

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

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

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

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

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

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

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

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

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

      12. --lowering--.f64 N/A

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

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

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

      14. *-lowering-*.f64 46.3%

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

   4. Applied egg-rr 46.3%

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

   5. Taylor expanded in y.im around inf

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(-1 \cdot x.re + \left(\frac{x.im \cdot y.re}{y.im} + \frac{x.re \cdot {y.re}^{2}}{{y.im}^{2}}\right)\right), \color{blue}{y.im}\right) \]

   7. Simplified 83.6%

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

   if -4.8000000000000002e138 < y.im < -4.20000000000000007e-106

   1. Initial program 88.0%

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

   if -4.20000000000000007e-106 < y.im < 1.2199999999999999e75

   1. Initial program 67.6%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 89.3%

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

   5. Simplified 89.3%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 89.4%

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

   7. Applied egg-rr 89.4%

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

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.8 \cdot 10^{+138}:\\ \;\;\;\;\frac{y.re \cdot \left(\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\right) - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.22 \cdot 10^{+75}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \left(\frac{x.im}{y.im} + \frac{y.re \cdot x.re}{y.im \cdot y.im}\right) - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -1.5e+84)
     t_0
     (if (<= y.im -4.2e-106)
       (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
       (if (<= y.im 1.25e+81) (/ (- x.im (/ x.re (/ y.re y.im))) 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 <= -1.5e+84) {
		tmp = t_0;
	} else if (y_46_im <= -4.2e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 1.25e+81) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 <= (-1.5d+84)) then
        tmp = t_0
    else if (y_46im <= (-4.2d-106)) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    else if (y_46im <= 1.25d+81) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / 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 <= -1.5e+84) {
		tmp = t_0;
	} else if (y_46_im <= -4.2e-106) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} else if (y_46_im <= 1.25e+81) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 <= -1.5e+84:
		tmp = t_0
	elif y_46_im <= -4.2e-106:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	elif y_46_im <= 1.25e+81:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.5e+84)
		tmp = t_0;
	elseif (y_46_im <= -4.2e-106)
		tmp = 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)));
	elseif (y_46_im <= 1.25e+81)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / 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 <= -1.5e+84)
		tmp = t_0;
	elseif (y_46_im <= -4.2e-106)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	elseif (y_46_im <= 1.25e+81)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.5e+84], t$95$0, If[LessEqual[y$46$im, -4.2e-106], 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+81], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.49999999999999998e84 or 1.25e81 < y.im

   1. Initial program 48.3%

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

   3. Taylor expanded in y.re around 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 80.9%

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

   5. Simplified 80.9%

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

   if -1.49999999999999998e84 < y.im < -4.20000000000000007e-106

   1. Initial program 93.5%

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

   if -4.20000000000000007e-106 < y.im < 1.25e81

   1. Initial program 67.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 88.6%

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

   5. Simplified 88.6%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 88.6%

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

   7. Applied egg-rr 88.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.5 \cdot 10^{+84}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -4.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

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 -3.9e-19)
     t_0
     (if (<= y.im 1.25e+81) (/ (- x.im (/ x.re (/ y.re y.im))) 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 <= -3.9e-19) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e+81) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 <= (-3.9d-19)) then
        tmp = t_0
    else if (y_46im <= 1.25d+81) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / 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 <= -3.9e-19) {
		tmp = t_0;
	} else if (y_46_im <= 1.25e+81) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 <= -3.9e-19:
		tmp = t_0
	elif y_46_im <= 1.25e+81:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.9e-19)
		tmp = t_0;
	elseif (y_46_im <= 1.25e+81)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / 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 <= -3.9e-19)
		tmp = t_0;
	elseif (y_46_im <= 1.25e+81)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -3.9e-19], t$95$0, If[LessEqual[y$46$im, 1.25e+81], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.89999999999999995e-19 or 1.25e81 < y.im

   1. Initial program 57.3%

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

   3. Taylor expanded in y.re around 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 77.5%

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

   5. Simplified 77.5%

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

   if -3.89999999999999995e-19 < y.im < 1.25e81

   1. Initial program 72.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 84.5%

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

   5. Simplified 84.5%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 84.5%

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

   7. Applied egg-rr 84.5%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.9 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.25 \cdot 10^{+81}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -6800000.0)
     t_0
     (if (<= y.im 8e+128) (/ (- x.im (/ x.re (/ y.re y.im))) 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 <= -6800000.0) {
		tmp = t_0;
	} else if (y_46_im <= 8e+128) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 <= (-6800000.0d0)) then
        tmp = t_0
    else if (y_46im <= 8d+128) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / 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 <= -6800000.0) {
		tmp = t_0;
	} else if (y_46_im <= 8e+128) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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 <= -6800000.0:
		tmp = t_0
	elif y_46_im <= 8e+128:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -6800000.0)
		tmp = t_0;
	elseif (y_46_im <= 8e+128)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / 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 <= -6800000.0)
		tmp = t_0;
	elseif (y_46_im <= 8e+128)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / 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[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6800000.0], t$95$0, If[LessEqual[y$46$im, 8e+128], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.8e6 or 8.0000000000000006e128 < y.im

   1. Initial program 54.0%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{x.re}{y.im}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{x.re}{y.im}\right)}\right) \]

      4. /-lowering-/.f64 72.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(x.re, \color{blue}{y.im}\right)\right) \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{0 - \frac{x.re}{y.im}} \]

   if -6.8e6 < y.im < 8.0000000000000006e128

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 80.4%

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

   5. Simplified 80.4%

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

      1. /-lowering-/.f64 N/A

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

      2. --lowering--.f64 N/A

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

      3. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x.im, \left(x.re \cdot \frac{1}{\frac{y.re}{y.im}}\right)\right), y.re\right) \]

      4. un-div-inv N/A

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

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

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

      6. /-lowering-/.f64 80.4%

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

   7. Applied egg-rr 80.4%

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

Alternative 7: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -10800000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7.5 \cdot 10^{+128}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -10800000.0)
     t_0
     (if (<= y.im 7.5e+128) (/ (- 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 <= -10800000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.5e+128) {
		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 <= (-10800000.0d0)) then
        tmp = t_0
    else if (y_46im <= 7.5d+128) 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 <= -10800000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.5e+128) {
		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 <= -10800000.0:
		tmp = t_0
	elif y_46_im <= 7.5e+128:
		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(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -10800000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.5e+128)
		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 <= -10800000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.5e+128)
		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[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -10800000.0], t$95$0, If[LessEqual[y$46$im, 7.5e+128], 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 < -1.08e7 or 7.50000000000000076e128 < y.im

   1. Initial program 54.0%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{x.re}{y.im}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{x.re}{y.im}\right)}\right) \]

      4. /-lowering-/.f64 72.0%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(x.re, \color{blue}{y.im}\right)\right) \]

   5. Simplified 72.0%

      \[\leadsto \color{blue}{0 - \frac{x.re}{y.im}} \]

   if -1.08e7 < y.im < 7.50000000000000076e128

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      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. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 80.4%

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

   5. Simplified 80.4%

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

Alternative 8: 64.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.8 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -6.8e-37) t_0 (if (<= y.im 8e-8) (/ x.im 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 <= -6.8e-37) {
		tmp = t_0;
	} else if (y_46_im <= 8e-8) {
		tmp = x_46_im / 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 <= (-6.8d-37)) then
        tmp = t_0
    else if (y_46im <= 8d-8) then
        tmp = x_46im / 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 <= -6.8e-37) {
		tmp = t_0;
	} else if (y_46_im <= 8e-8) {
		tmp = x_46_im / 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 <= -6.8e-37:
		tmp = t_0
	elif y_46_im <= 8e-8:
		tmp = x_46_im / 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(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -6.8e-37)
		tmp = t_0;
	elseif (y_46_im <= 8e-8)
		tmp = Float64(x_46_im / 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 <= -6.8e-37)
		tmp = t_0;
	elseif (y_46_im <= 8e-8)
		tmp = x_46_im / 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[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.8e-37], t$95$0, If[LessEqual[y$46$im, 8e-8], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.80000000000000037e-37 or 8.0000000000000002e-8 < y.im

   1. Initial program 57.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto 0 - \color{blue}{\frac{x.re}{y.im}} \]

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

         \[\leadsto \mathsf{\_.f64}\left(0, \color{blue}{\left(\frac{x.re}{y.im}\right)}\right) \]

      4. /-lowering-/.f64 63.8%

         \[\leadsto \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(x.re, \color{blue}{y.im}\right)\right) \]

   5. Simplified 63.8%

      \[\leadsto \color{blue}{0 - \frac{x.re}{y.im}} \]

   if -6.80000000000000037e-37 < y.im < 8.0000000000000002e-8

   1. Initial program 73.6%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 68.6%

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

   5. Simplified 68.6%

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

Alternative 9: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot \frac{-1}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{-38}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-8}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* x.re (/ -1.0 y.im))))
   (if (<= y.im -6e-38) t_0 (if (<= y.im 8e-8) (/ x.im 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_re * (-1.0 / y_46_im);
	double tmp;
	if (y_46_im <= -6e-38) {
		tmp = t_0;
	} else if (y_46_im <= 8e-8) {
		tmp = x_46_im / 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_46re * ((-1.0d0) / y_46im)
    if (y_46im <= (-6d-38)) then
        tmp = t_0
    else if (y_46im <= 8d-8) then
        tmp = x_46im / 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_re * (-1.0 / y_46_im);
	double tmp;
	if (y_46_im <= -6e-38) {
		tmp = t_0;
	} else if (y_46_im <= 8e-8) {
		tmp = x_46_im / 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_re * (-1.0 / y_46_im)
	tmp = 0
	if y_46_im <= -6e-38:
		tmp = t_0
	elif y_46_im <= 8e-8:
		tmp = x_46_im / 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(x_46_re * Float64(-1.0 / y_46_im))
	tmp = 0.0
	if (y_46_im <= -6e-38)
		tmp = t_0;
	elseif (y_46_im <= 8e-8)
		tmp = Float64(x_46_im / 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_re * (-1.0 / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -6e-38)
		tmp = t_0;
	elseif (y_46_im <= 8e-8)
		tmp = x_46_im / 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[(x$46$re * N[(-1.0 / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6e-38], t$95$0, If[LessEqual[y$46$im, 8e-8], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -5.99999999999999977e-38 or 8.0000000000000002e-8 < y.im

   1. Initial program 57.9%

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

   3. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

         \[\leadsto x.re \cdot \left(\left(0 - \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right) + \frac{\color{blue}{x.im \cdot y.re}}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right) \]

      3. associate-+l- N/A

         \[\leadsto x.re \cdot \left(0 - \color{blue}{\left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} - \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right)}\right) \]

      4. unsub-neg N/A

         \[\leadsto x.re \cdot \left(0 - \left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} + \color{blue}{\left(\mathsf{neg}\left(\frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}\right)\right)}\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto x.re \cdot \left(0 - \left(\frac{y.im}{{y.im}^{2} + {y.re}^{2}} + -1 \cdot \color{blue}{\frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)}}\right)\right) \]

      6. +-commutative N/A

         \[\leadsto x.re \cdot \left(0 - \left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \color{blue}{\frac{y.im}{{y.im}^{2} + {y.re}^{2}}}\right)\right) \]

      7. neg-sub0 N/A

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

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

         \[\leadsto \mathsf{*.f64}\left(x.re, \color{blue}{\left(\mathsf{neg}\left(\left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right)\right)\right)}\right) \]

      9. neg-sub0 N/A

         \[\leadsto \mathsf{*.f64}\left(x.re, \left(0 - \color{blue}{\left(-1 \cdot \frac{x.im \cdot y.re}{x.re \cdot \left({y.im}^{2} + {y.re}^{2}\right)} + \frac{y.im}{{y.im}^{2} + {y.re}^{2}}\right)}\right)\right) \]

   5. Simplified 53.8%

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

   6. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 63.5%

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

   8. Simplified 63.5%

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

   if -5.99999999999999977e-38 < y.im < 8.0000000000000002e-8

   1. Initial program 73.6%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 68.6%

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

   5. Simplified 68.6%

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

Alternative 10: 42.6% 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 64.7%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 39.4%

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

5. Simplified 39.4%

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

Reproduce

herbie shell --seed 2024158 
(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))))