_divideComplex, imaginary part

Percentage Accurate: 61.2% → 85.4%

Time: 12.2s

Alternatives: 9

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: 61.2% 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: 85.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im}{\frac{t\_0}{y.re}} - \frac{x.re}{\frac{t\_0}{y.im}}\\ \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+131}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

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.im (/ t_0 y.re)) (/ x.re (/ t_0 y.im)))))
   (if (<= y.re -5.8e+153)
     (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
     (if (<= y.re -5.5e-67)
       t_1
       (if (<= y.re 1.2e-161)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 3.2e+131)
           t_1
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

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_im / (t_0 / y_46_re)) - (x_46_re / (t_0 / y_46_im));
	double tmp;
	if (y_46_re <= -5.8e+153) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= -5.5e-67) {
		tmp = t_1;
	} else if (y_46_re <= 1.2e-161) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.2e+131) {
		tmp = t_1;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46im / (t_0 / y_46re)) - (x_46re / (t_0 / y_46im))
    if (y_46re <= (-5.8d+153)) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46re <= (-5.5d-67)) then
        tmp = t_1
    else if (y_46re <= 1.2d-161) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 3.2d+131) then
        tmp = t_1
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / 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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im / (t_0 / y_46_re)) - (x_46_re / (t_0 / y_46_im));
	double tmp;
	if (y_46_re <= -5.8e+153) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= -5.5e-67) {
		tmp = t_1;
	} else if (y_46_re <= 1.2e-161) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.2e+131) {
		tmp = t_1;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	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_im / (t_0 / y_46_re)) - (x_46_re / (t_0 / y_46_im))
	tmp = 0
	if y_46_re <= -5.8e+153:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_re <= -5.5e-67:
		tmp = t_1
	elif y_46_re <= 1.2e-161:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 3.2e+131:
		tmp = t_1
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im / Float64(t_0 / y_46_re)) - Float64(x_46_re / Float64(t_0 / y_46_im)))
	tmp = 0.0
	if (y_46_re <= -5.8e+153)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_re <= -5.5e-67)
		tmp = t_1;
	elseif (y_46_re <= 1.2e-161)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.2e+131)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	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_im / (t_0 / y_46_re)) - (x_46_re / (t_0 / y_46_im));
	tmp = 0.0;
	if (y_46_re <= -5.8e+153)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_re <= -5.5e-67)
		tmp = t_1;
	elseif (y_46_re <= 1.2e-161)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 3.2e+131)
		tmp = t_1;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	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[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / N[(t$95$0 / y$46$re), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[(t$95$0 / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -5.8e+153], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.5e-67], t$95$1, If[LessEqual[y$46$re, 1.2e-161], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.2e+131], t$95$1, N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -5.80000000000000004e153

   1. Initial program 18.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-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. *-lowering-*.f64 84.7%

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

   5. Simplified 84.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 86.6%

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

   7. Applied egg-rr 86.6%

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

   if -5.80000000000000004e153 < y.re < -5.5000000000000003e-67 or 1.19999999999999999e-161 < y.re < 3.2000000000000002e131

   1. Initial program 76.4%

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

      1. div-sub N/A

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

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-/l* N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

      14. /-lowering-/.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

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

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

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

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

      18. *-lowering-*.f64 86.4%

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

   4. Applied egg-rr 86.4%

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

   if -5.5000000000000003e-67 < y.re < 1.19999999999999999e-161

   1. Initial program 69.6%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 97.5%

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

   5. Simplified 97.5%

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

   if 3.2000000000000002e131 < y.re

   1. Initial program 29.2%

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

   3. 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. /-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. *-lowering-*.f64 77.5%

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

   5. Simplified 77.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 95.1%

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

   7. Applied egg-rr 95.1%

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

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -5.5 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}} - \frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{elif}\;y.re \leq 1.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x.im}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re}} - \frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-67}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -3.5e+99)
     (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
     (if (<= y.re -2.9e-67)
       t_0
       (if (<= y.re 7e-155)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 2.2e+48)
           t_0
           (/ (- x.im (* x.re (/ y.im y.re))) y.re)))))))

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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3.5e+99) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= -2.9e-67) {
		tmp = t_0;
	} else if (y_46_re <= 7e-155) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.2e+48) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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) :: t_0
    real(8) :: tmp
    t_0 = ((y_46re * x_46im) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-3.5d+99)) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46re <= (-2.9d-67)) then
        tmp = t_0
    else if (y_46re <= 7d-155) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 2.2d+48) then
        tmp = t_0
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / 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 t_0 = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -3.5e+99) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= -2.9e-67) {
		tmp = t_0;
	} else if (y_46_re <= 7e-155) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.2e+48) {
		tmp = t_0;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3.5e+99:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_re <= -2.9e-67:
		tmp = t_0
	elif y_46_re <= 7e-155:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 2.2e+48:
		tmp = t_0
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	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 * x_46_im) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.5e+99)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_re <= -2.9e-67)
		tmp = t_0;
	elseif (y_46_re <= 7e-155)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.2e+48)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	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) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.5e+99)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_re <= -2.9e-67)
		tmp = t_0;
	elseif (y_46_re <= 7e-155)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 2.2e+48)
		tmp = t_0;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	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 * x$46$im), $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]}, If[LessEqual[y$46$re, -3.5e+99], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.9e-67], t$95$0, If[LessEqual[y$46$re, 7e-155], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.2e+48], t$95$0, N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.4999999999999998e99

   1. Initial program 23.5%

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

   3. Taylor expanded in y.re around 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. /-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. *-lowering-*.f64 81.4%

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

   5. Simplified 81.4%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 83.0%

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

   7. Applied egg-rr 83.0%

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

   if -3.4999999999999998e99 < y.re < -2.90000000000000005e-67 or 7.00000000000000031e-155 < y.re < 2.1999999999999999e48

   1. Initial program 82.5%

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

   if -2.90000000000000005e-67 < y.re < 7.00000000000000031e-155

   1. Initial program 69.9%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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 97.6%

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

   5. Simplified 97.6%

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

   if 2.1999999999999999e48 < y.re

   1. Initial program 35.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-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. *-lowering-*.f64 76.7%

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

   5. Simplified 76.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 90.4%

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

   7. Applied egg-rr 90.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -2.9 \cdot 10^{-67}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-155}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{\frac{y.im}{y.re}}{\frac{-1}{x.re}}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.45e-62)
   (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
   (if (<= y.re 4.8e-54)
     (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
     (/ (+ x.im (/ (/ y.im y.re) (/ -1.0 x.re))) 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 <= -1.45e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 4.8e-54) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im + ((y_46_im / y_46_re) / (-1.0 / x_46_re))) / 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 <= (-1.45d-62)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= 4.8d-54) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else
        tmp = (x_46im + ((y_46im / y_46re) / ((-1.0d0) / x_46re))) / 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 <= -1.45e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 4.8e-54) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im + ((y_46_im / y_46_re) / (-1.0 / x_46_re))) / 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 <= -1.45e-62:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= 4.8e-54:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im + ((y_46_im / y_46_re) / (-1.0 / x_46_re))) / 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 <= -1.45e-62)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= 4.8e-54)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_im / y_46_re) / Float64(-1.0 / x_46_re))) / 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 <= -1.45e-62)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= 4.8e-54)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im + ((y_46_im / y_46_re) / (-1.0 / x_46_re))) / 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, -1.45e-62], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.8e-54], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im + N[(N[(y$46$im / y$46$re), $MachinePrecision] / N[(-1.0 / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.44999999999999993e-62

   1. Initial program 50.2%

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

   3. 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. /-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. *-lowering-*.f64 71.4%

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

   5. Simplified 71.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 72.1%

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

   7. Applied egg-rr 72.1%

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

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

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. /-lowering-/.f64 72.2%

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

   9. Applied egg-rr 72.2%

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

   if -1.44999999999999993e-62 < y.re < 4.80000000000000026e-54

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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 94.1%

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

   5. Simplified 94.1%

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

   if 4.80000000000000026e-54 < y.re

   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.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. /-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. *-lowering-*.f64 74.7%

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

   5. Simplified 74.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 85.3%

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

   7. Applied egg-rr 85.3%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. div-inv N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

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

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

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

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

      8. /-lowering-/.f64 85.3%

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

   9. Applied egg-rr 85.3%

      \[\leadsto \frac{x.im - \color{blue}{\frac{\frac{y.im}{y.re}}{\frac{1}{x.re}}}}{y.re} \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.45 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.8 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{\frac{y.im}{y.re}}{\frac{-1}{x.re}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.05e-62)
   (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
   (if (<= y.re 4.9e-54)
     (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.re))) 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 <= -1.05e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 4.9e-54) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= (-1.05d-62)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= 4.9d-54) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / 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 <= -1.05e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 4.9e-54) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -1.05e-62:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= 4.9e-54:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -1.05e-62)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= 4.9e-54)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / 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 <= -1.05e-62)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= 4.9e-54)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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, -1.05e-62], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.9e-54], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.05e-62

   1. Initial program 50.2%

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

   3. 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. /-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. *-lowering-*.f64 71.4%

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

   5. Simplified 71.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 72.1%

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

   7. Applied egg-rr 72.1%

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

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

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. /-lowering-/.f64 72.2%

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

   9. Applied egg-rr 72.2%

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

   if -1.05e-62 < y.re < 4.90000000000000021e-54

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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 94.1%

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

   5. Simplified 94.1%

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

   if 4.90000000000000021e-54 < y.re

   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.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. /-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. *-lowering-*.f64 74.7%

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

   5. Simplified 74.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 85.3%

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

   7. Applied egg-rr 85.3%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.05 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.5e-62)
   (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
   (if (<= y.re 4.9e-54)
     (/ (- (/ y.re (/ y.im x.im)) x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.re))) 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 <= -1.5e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 4.9e-54) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= (-1.5d-62)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= 4.9d-54) then
        tmp = ((y_46re / (y_46im / x_46im)) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / 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 <= -1.5e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 4.9e-54) {
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -1.5e-62:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= 4.9e-54:
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -1.5e-62)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= 4.9e-54)
		tmp = Float64(Float64(Float64(y_46_re / Float64(y_46_im / x_46_im)) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / 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 <= -1.5e-62)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= 4.9e-54)
		tmp = ((y_46_re / (y_46_im / x_46_im)) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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, -1.5e-62], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.9e-54], N[(N[(N[(y$46$re / N[(y$46$im / x$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.5000000000000001e-62

   1. Initial program 50.2%

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

   3. 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. /-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. *-lowering-*.f64 71.4%

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

   5. Simplified 71.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 72.1%

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

   7. Applied egg-rr 72.1%

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

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

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. /-lowering-/.f64 72.2%

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

   9. Applied egg-rr 72.2%

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

   if -1.5000000000000001e-62 < y.re < 4.90000000000000021e-54

   1. Initial program 73.0%

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

   3. Taylor expanded in y.re around 0

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 69.0%

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

   5. Simplified 69.0%

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

      1. clear-num N/A

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

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

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

      3. clear-num N/A

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

      4. associate-/r* N/A

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

      5. clear-num N/A

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

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

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      10. *-rgt-identity N/A

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

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

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

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

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

      13. *-lowering-*.f64 93.9%

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

   7. Applied egg-rr 93.9%

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

      1. clear-num N/A

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

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

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

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

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

      11. /-lowering-/.f64 93.1%

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

   9. Applied egg-rr 93.1%

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

   if 4.90000000000000021e-54 < y.re

   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.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. /-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. *-lowering-*.f64 74.7%

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

   5. Simplified 74.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 85.3%

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

   7. Applied egg-rr 85.3%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 4.9 \cdot 10^{-54}:\\ \;\;\;\;\frac{\frac{y.re}{\frac{y.im}{x.im}} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.3e-62)
   (/ (- x.im (/ y.im (/ y.re x.re))) y.re)
   (if (<= y.re 1.05e-72)
     (- 0.0 (/ x.re y.im))
     (/ (- x.im (* x.re (/ y.im y.re))) 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 <= -1.3e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 1.05e-72) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= (-1.3d-62)) then
        tmp = (x_46im - (y_46im / (y_46re / x_46re))) / y_46re
    else if (y_46re <= 1.05d-72) then
        tmp = 0.0d0 - (x_46re / y_46im)
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / 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 <= -1.3e-62) {
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	} else if (y_46_re <= 1.05e-72) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -1.3e-62:
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re
	elif y_46_re <= 1.05e-72:
		tmp = 0.0 - (x_46_re / y_46_im)
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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 <= -1.3e-62)
		tmp = Float64(Float64(x_46_im - Float64(y_46_im / Float64(y_46_re / x_46_re))) / y_46_re);
	elseif (y_46_re <= 1.05e-72)
		tmp = Float64(0.0 - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / 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 <= -1.3e-62)
		tmp = (x_46_im - (y_46_im / (y_46_re / x_46_re))) / y_46_re;
	elseif (y_46_re <= 1.05e-72)
		tmp = 0.0 - (x_46_re / y_46_im);
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / 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, -1.3e-62], N[(N[(x$46$im - N[(y$46$im / N[(y$46$re / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.05e-72], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.3e-62

   1. Initial program 50.2%

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

   3. 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. /-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. *-lowering-*.f64 71.4%

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

   5. Simplified 71.4%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 72.1%

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

   7. Applied egg-rr 72.1%

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

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

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

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. clear-num N/A

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

      6. un-div-inv N/A

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

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

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

      8. /-lowering-/.f64 72.2%

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

   9. Applied egg-rr 72.2%

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

   if -1.3e-62 < y.re < 1.05e-72

   1. Initial program 72.4%

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

   3. Taylor expanded in y.re around 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 81.9%

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

   5. Simplified 81.9%

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

   if 1.05e-72 < y.re

   1. Initial program 50.3%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. /-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. *-lowering-*.f64 74.0%

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

   5. Simplified 74.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 84.3%

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

   7. Applied egg-rr 84.3%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.3 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im - \frac{y.im}{\frac{y.re}{x.re}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{-72}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-73}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.re -3.4e-66)
     t_0
     (if (<= y.re 9.5e-73) (- 0.0 (/ x.re y.im)) 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 - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -3.4e-66) {
		tmp = t_0;
	} else if (y_46_re <= 9.5e-73) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} 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 - (x_46re * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-3.4d-66)) then
        tmp = t_0
    else if (y_46re <= 9.5d-73) then
        tmp = 0.0d0 - (x_46re / y_46im)
    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 - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -3.4e-66) {
		tmp = t_0;
	} else if (y_46_re <= 9.5e-73) {
		tmp = 0.0 - (x_46_re / y_46_im);
	} 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 - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -3.4e-66:
		tmp = t_0
	elif y_46_re <= 9.5e-73:
		tmp = 0.0 - (x_46_re / y_46_im)
	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(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -3.4e-66)
		tmp = t_0;
	elseif (y_46_re <= 9.5e-73)
		tmp = Float64(0.0 - Float64(x_46_re / y_46_im));
	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 - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -3.4e-66)
		tmp = t_0;
	elseif (y_46_re <= 9.5e-73)
		tmp = 0.0 - (x_46_re / y_46_im);
	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[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -3.4e-66], t$95$0, If[LessEqual[y$46$re, 9.5e-73], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.39999999999999997e-66 or 9.50000000000000005e-73 < y.re

   1. Initial program 50.2%

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

   3. 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. /-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. *-lowering-*.f64 72.5%

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

   5. Simplified 72.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 77.2%

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

   7. Applied egg-rr 77.2%

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

   if -3.39999999999999997e-66 < y.re < 9.50000000000000005e-73

   1. Initial program 72.4%

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

   3. Taylor expanded in y.re around 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 81.9%

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

   5. Simplified 81.9%

      \[\leadsto \color{blue}{0 - \frac{x.re}{y.im}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{-66}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-73}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 2.9 \cdot 10^{-16}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.5e-62)
   (/ x.im y.re)
   (if (<= y.re 2.9e-16) (- 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 <= -1.5e-62) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.9e-16) {
		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 <= (-1.5d-62)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 2.9d-16) 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 <= -1.5e-62) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 2.9e-16) {
		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 <= -1.5e-62:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 2.9e-16:
		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 <= -1.5e-62)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 2.9e-16)
		tmp = Float64(0.0 - Float64(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 <= -1.5e-62)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 2.9e-16)
		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, -1.5e-62], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.9e-16], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.5000000000000001e-62 or 2.8999999999999998e-16 < y.re

   1. Initial program 48.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 61.0%

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

   5. Simplified 61.0%

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

   if -1.5000000000000001e-62 < y.re < 2.8999999999999998e-16

   1. Initial program 74.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 78.4%

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

   5. Simplified 78.4%

      \[\leadsto \color{blue}{0 - \frac{x.re}{y.im}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 42.2% 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 58.5%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 40.9%

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

5. Simplified 40.9%

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

Reproduce

herbie shell --seed 2024159 
(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))))