_divideComplex, imaginary part

Percentage Accurate: 61.5% → 82.1%

Time: 9.0s

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: 61.5% 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: 82.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot y.im + y.re \cdot y.re\\ \mathbf{if}\;y.im \leq -4 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-161}:\\ \;\;\;\;x.re \cdot \left(\frac{\frac{x.im \cdot y.re}{x.re}}{t\_0} - \frac{y.im}{t\_0}\right)\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{t\_0} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.im y.im) (* y.re y.re))))
   (if (<= y.im -4e+96)
     (/ (- (/ x.im (/ y.im y.re)) x.re) y.im)
     (if (<= y.im -8.2e-161)
       (* x.re (- (/ (/ (* x.im y.re) x.re) t_0) (/ y.im t_0)))
       (if (<= y.im 8.5e-115)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 1.35e+82)
           (* (/ 1.0 t_0) (- (* x.im y.re) (* y.im x.re)))
           (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -4e+96) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -8.2e-161) {
		tmp = x_46_re * ((((x_46_im * y_46_re) / x_46_re) / t_0) - (y_46_im / t_0));
	} else if (y_46_im <= 8.5e-115) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.35e+82) {
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y_46im * y_46im) + (y_46re * y_46re)
    if (y_46im <= (-4d+96)) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) / y_46im
    else if (y_46im <= (-8.2d-161)) then
        tmp = x_46re * ((((x_46im * y_46re) / x_46re) / t_0) - (y_46im / t_0))
    else if (y_46im <= 8.5d-115) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 1.35d+82) then
        tmp = (1.0d0 / t_0) * ((x_46im * y_46re) - (y_46im * x_46re))
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -4e+96) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -8.2e-161) {
		tmp = x_46_re * ((((x_46_im * y_46_re) / x_46_re) / t_0) - (y_46_im / t_0));
	} else if (y_46_im <= 8.5e-115) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.35e+82) {
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re)
	tmp = 0
	if y_46_im <= -4e+96:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im
	elif y_46_im <= -8.2e-161:
		tmp = x_46_re * ((((x_46_im * y_46_re) / x_46_re) / t_0) - (y_46_im / t_0))
	elif y_46_im <= 8.5e-115:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 1.35e+82:
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re))
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -4e+96)
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -8.2e-161)
		tmp = Float64(x_46_re * Float64(Float64(Float64(Float64(x_46_im * y_46_re) / x_46_re) / t_0) - Float64(y_46_im / t_0)));
	elseif (y_46_im <= 8.5e-115)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 1.35e+82)
		tmp = Float64(Float64(1.0 / t_0) * Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)));
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	tmp = 0.0;
	if (y_46_im <= -4e+96)
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -8.2e-161)
		tmp = x_46_re * ((((x_46_im * y_46_re) / x_46_re) / t_0) - (y_46_im / t_0));
	elseif (y_46_im <= 8.5e-115)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 1.35e+82)
		tmp = (1.0 / t_0) * ((x_46_im * y_46_re) - (y_46_im * x_46_re));
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4e+96], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-161], N[(x$46$re * N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision] - N[(y$46$im / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.5e-115], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.35e+82], N[(N[(1.0 / t$95$0), $MachinePrecision] * N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -4.0000000000000002e96

   1. Initial program 30.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

      4. /-lowering-/.f64 81.1%

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

   7. Applied egg-rr 81.1%

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

   if -4.0000000000000002e96 < y.im < -8.1999999999999994e-161

   1. Initial program 86.2%

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

   3. Taylor expanded in x.re around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-+l- N/A

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

      4. unsub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. neg-sub0 N/A

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

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

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

      9. neg-sub0 N/A

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

   5. Simplified 88.0%

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

   if -8.1999999999999994e-161 < y.im < 8.49999999999999953e-115

   1. Initial program 68.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 93.3%

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

   5. Simplified 93.3%

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

   if 8.49999999999999953e-115 < y.im < 1.35e82

   1. Initial program 79.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

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

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 80.0%

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

   4. Applied egg-rr 80.0%

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

   if 1.35e82 < y.im

   1. Initial program 34.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 94.5%

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

   5. Simplified 94.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 95.7%

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

   7. Applied egg-rr 95.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4 \cdot 10^{+96}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-161}:\\ \;\;\;\;x.re \cdot \left(\frac{\frac{x.im \cdot y.re}{x.re}}{y.im \cdot y.im + y.re \cdot y.re} - \frac{y.im}{y.im \cdot y.im + y.re \cdot y.re}\right)\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-115}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - y.im \cdot x.re\\ t_1 := y.im \cdot y.im + y.re \cdot y.re\\ \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{t\_0}{t\_1}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{t\_1} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* y.im x.re)))
        (t_1 (+ (* y.im y.im) (* y.re y.re))))
   (if (<= y.im -2.5e+47)
     (/ (- (/ x.im (/ y.im y.re)) x.re) y.im)
     (if (<= y.im -7.6e-161)
       (/ t_0 t_1)
       (if (<= y.im 5.5e-117)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 4e+80)
           (* (/ 1.0 t_1) t_0)
           (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	double t_1 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -2.5e+47) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -7.6e-161) {
		tmp = t_0 / t_1;
	} else if (y_46_im <= 5.5e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4e+80) {
		tmp = (1.0 / t_1) * t_0;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im * y_46re) - (y_46im * x_46re)
    t_1 = (y_46im * y_46im) + (y_46re * y_46re)
    if (y_46im <= (-2.5d+47)) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) / y_46im
    else if (y_46im <= (-7.6d-161)) then
        tmp = t_0 / t_1
    else if (y_46im <= 5.5d-117) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 4d+80) then
        tmp = (1.0d0 / t_1) * t_0
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	double t_1 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	double tmp;
	if (y_46_im <= -2.5e+47) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -7.6e-161) {
		tmp = t_0 / t_1;
	} else if (y_46_im <= 5.5e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4e+80) {
		tmp = (1.0 / t_1) * t_0;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) - (y_46_im * x_46_re)
	t_1 = (y_46_im * y_46_im) + (y_46_re * y_46_re)
	tmp = 0
	if y_46_im <= -2.5e+47:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im
	elif y_46_im <= -7.6e-161:
		tmp = t_0 / t_1
	elif y_46_im <= 5.5e-117:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 4e+80:
		tmp = (1.0 / t_1) * t_0
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re))
	t_1 = Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re))
	tmp = 0.0
	if (y_46_im <= -2.5e+47)
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -7.6e-161)
		tmp = Float64(t_0 / t_1);
	elseif (y_46_im <= 5.5e-117)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 4e+80)
		tmp = Float64(Float64(1.0 / t_1) * t_0);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) - (y_46_im * x_46_re);
	t_1 = (y_46_im * y_46_im) + (y_46_re * y_46_re);
	tmp = 0.0;
	if (y_46_im <= -2.5e+47)
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -7.6e-161)
		tmp = t_0 / t_1;
	elseif (y_46_im <= 5.5e-117)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 4e+80)
		tmp = (1.0 / t_1) * t_0;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.5e+47], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -7.6e-161], N[(t$95$0 / t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 5.5e-117], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4e+80], N[(N[(1.0 / t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -2.50000000000000011e47

   1. Initial program 36.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 78.9%

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

   5. Simplified 78.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

      4. /-lowering-/.f64 78.9%

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

   7. Applied egg-rr 78.9%

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

   if -2.50000000000000011e47 < y.im < -7.6000000000000003e-161

   1. Initial program 89.9%

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

   if -7.6000000000000003e-161 < y.im < 5.50000000000000025e-117

   1. Initial program 68.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 93.3%

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

   5. Simplified 93.3%

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

   if 5.50000000000000025e-117 < y.im < 4e80

   1. Initial program 79.9%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

      3. flip-+ N/A

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

      4. clear-num N/A

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

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

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 80.0%

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

   4. Applied egg-rr 80.0%

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

   if 4e80 < y.im

   1. Initial program 34.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 94.5%

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

   5. Simplified 94.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 95.7%

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

   7. Applied egg-rr 95.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -7.6 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\frac{1}{y.im \cdot y.im + y.re \cdot y.re} \cdot \left(x.im \cdot y.re - y.im \cdot x.re\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 83.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{if}\;y.im \leq -6.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-161}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))))
   (if (<= y.im -6.3e+47)
     (/ (- (/ x.im (/ y.im y.re)) x.re) y.im)
     (if (<= y.im -8.2e-161)
       t_0
       (if (<= y.im 5.5e-117)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 4e+80) t_0 (/ (- (* y.re (/ x.im y.im)) x.re) y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -6.3e+47) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -8.2e-161) {
		tmp = t_0;
	} else if (y_46_im <= 5.5e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4e+80) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    if (y_46im <= (-6.3d+47)) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) / y_46im
    else if (y_46im <= (-8.2d-161)) then
        tmp = t_0
    else if (y_46im <= 5.5d-117) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 4d+80) then
        tmp = t_0
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double tmp;
	if (y_46_im <= -6.3e+47) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= -8.2e-161) {
		tmp = t_0;
	} else if (y_46_im <= 5.5e-117) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 4e+80) {
		tmp = t_0;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	tmp = 0
	if y_46_im <= -6.3e+47:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im
	elif y_46_im <= -8.2e-161:
		tmp = t_0
	elif y_46_im <= 5.5e-117:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 4e+80:
		tmp = t_0
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	tmp = 0.0
	if (y_46_im <= -6.3e+47)
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) / y_46_im);
	elseif (y_46_im <= -8.2e-161)
		tmp = t_0;
	elseif (y_46_im <= 5.5e-117)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 4e+80)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	tmp = 0.0;
	if (y_46_im <= -6.3e+47)
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_im <= -8.2e-161)
		tmp = t_0;
	elseif (y_46_im <= 5.5e-117)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 4e+80)
		tmp = t_0;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.3e+47], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -8.2e-161], t$95$0, If[LessEqual[y$46$im, 5.5e-117], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 4e+80], t$95$0, N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -6.30000000000000003e47

   1. Initial program 36.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 78.9%

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

   5. Simplified 78.9%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

      4. /-lowering-/.f64 78.9%

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

   7. Applied egg-rr 78.9%

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

   if -6.30000000000000003e47 < y.im < -8.1999999999999994e-161 or 5.50000000000000025e-117 < y.im < 4e80

   1. Initial program 85.3%

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

   if -8.1999999999999994e-161 < y.im < 5.50000000000000025e-117

   1. Initial program 68.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 93.3%

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

   5. Simplified 93.3%

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

   if 4e80 < y.im

   1. Initial program 34.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

      9. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 94.5%

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

   5. Simplified 94.5%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 95.7%

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

   7. Applied egg-rr 95.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.3 \cdot 10^{+47}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -8.2 \cdot 10^{-161}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.im \leq 5.5 \cdot 10^{-117}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 4 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1550:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.7e-64)
   (/ (- (/ x.im (/ y.im y.re)) x.re) y.im)
   (if (<= y.im 1550.0)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.7e-64) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1550.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-3.7d-64)) then
        tmp = ((x_46im / (y_46im / y_46re)) - x_46re) / y_46im
    else if (y_46im <= 1550.0d0) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.7e-64) {
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 1550.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -3.7e-64:
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im
	elif y_46_im <= 1550.0:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.7e-64)
		tmp = Float64(Float64(Float64(x_46_im / Float64(y_46_im / y_46_re)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 1550.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -3.7e-64)
		tmp = ((x_46_im / (y_46_im / y_46_re)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 1550.0)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.7e-64], N[(N[(N[(x$46$im / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1550.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.69999999999999999e-64

   1. Initial program 52.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 76.5%

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

   5. Simplified 76.5%

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

      1. clear-num N/A

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

      2. un-div-inv N/A

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

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

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

      4. /-lowering-/.f64 76.6%

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

   7. Applied egg-rr 76.6%

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

   if -3.69999999999999999e-64 < y.im < 1550

   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 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 84.0%

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

   5. Simplified 84.0%

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

   if 1550 < y.im

   1. Initial program 47.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 83.1%

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

   5. Simplified 83.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{x.im}{\frac{y.im}{y.re}} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1550:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.3e-68)
   (/ (- (* x.im (/ y.re y.im)) x.re) y.im)
   (if (<= y.im 7.0)
     (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
     (/ (- (* y.re (/ x.im y.im)) x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.3e-68) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 7.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-3.3d-68)) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else if (y_46im <= 7.0d0) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = ((y_46re * (x_46im / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.3e-68) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else if (y_46_im <= 7.0) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -3.3e-68:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	elif y_46_im <= 7.0:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.3e-68)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	elseif (y_46_im <= 7.0)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(Float64(y_46_re * Float64(x_46_im / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -3.3e-68)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	elseif (y_46_im <= 7.0)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = ((y_46_re * (x_46_im / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.3e-68], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7.0], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(N[(y$46$re * N[(x$46$im / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -3.2999999999999998e-68

   1. Initial program 52.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 76.5%

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

   5. Simplified 76.5%

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

   if -3.2999999999999998e-68 < y.im < 7

   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 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 84.0%

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

   5. Simplified 84.0%

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

   if 7 < y.im

   1. Initial program 47.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 83.1%

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

   5. Simplified 83.1%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 85.6%

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

   7. Applied egg-rr 85.6%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 7:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{y.re \cdot \frac{x.im}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-63}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* x.im (/ y.re y.im)) x.re) y.im)))
   (if (<= y.im -1.8e-63)
     t_0
     (if (<= y.im 1.16e+66) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.8e-63) {
		tmp = t_0;
	} else if (y_46_im <= 1.16e+66) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    if (y_46im <= (-1.8d-63)) then
        tmp = t_0
    else if (y_46im <= 1.16d+66) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -1.8e-63) {
		tmp = t_0;
	} else if (y_46_im <= 1.16e+66) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -1.8e-63:
		tmp = t_0
	elif y_46_im <= 1.16e+66:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.8e-63)
		tmp = t_0;
	elseif (y_46_im <= 1.16e+66)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.8e-63)
		tmp = t_0;
	elseif (y_46_im <= 1.16e+66)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.8e-63], t$95$0, If[LessEqual[y$46$im, 1.16e+66], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.80000000000000004e-63 or 1.16e66 < y.im

   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 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. associate-/l* N/A

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

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

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

      11. /-lowering-/.f64 82.1%

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

   5. Simplified 82.1%

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

   if -1.80000000000000004e-63 < y.im < 1.16e66

   1. Initial program 74.9%

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

   3. Taylor expanded in y.re around 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%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-63}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.16 \cdot 10^{+66}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 72.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -1.3e-22)
     t_0
     (if (<= y.im 2.95e+79) (/ (- x.im (/ (* y.im x.re) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.3e-22) {
		tmp = t_0;
	} else if (y_46_im <= 2.95e+79) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-1.3d-22)) then
        tmp = t_0
    else if (y_46im <= 2.95d+79) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.3e-22) {
		tmp = t_0;
	} else if (y_46_im <= 2.95e+79) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.3e-22:
		tmp = t_0
	elif y_46_im <= 2.95e+79:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.3e-22)
		tmp = t_0;
	elseif (y_46_im <= 2.95e+79)
		tmp = Float64(Float64(x_46_im - Float64(Float64(y_46_im * x_46_re) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.3e-22)
		tmp = t_0;
	elseif (y_46_im <= 2.95e+79)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.3e-22], t$95$0, If[LessEqual[y$46$im, 2.95e+79], N[(N[(x$46$im - N[(N[(y$46$im * x$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.3e-22 or 2.95e79 < y.im

   1. Initial program 43.8%

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

   3. Taylor expanded in y.re around 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 73.7%

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

   5. Simplified 73.7%

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

   if -1.3e-22 < y.im < 2.95e79

   1. Initial program 76.2%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 78.9%

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

   5. Simplified 78.9%

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

4. Final simplification 76.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.3 \cdot 10^{-22}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.8 \cdot 10^{-125}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 29000000000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -4.8e-125)
     t_0
     (if (<= y.im 29000000000.0) (/ x.im y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4.8e-125) {
		tmp = t_0;
	} else if (y_46_im <= 29000000000.0) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-4.8d-125)) then
        tmp = t_0
    else if (y_46im <= 29000000000.0d0) then
        tmp = x_46im / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -4.8e-125) {
		tmp = t_0;
	} else if (y_46_im <= 29000000000.0) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -4.8e-125:
		tmp = t_0
	elif y_46_im <= 29000000000.0:
		tmp = x_46_im / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -4.8e-125)
		tmp = t_0;
	elseif (y_46_im <= 29000000000.0)
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -4.8e-125)
		tmp = t_0;
	elseif (y_46_im <= 29000000000.0)
		tmp = x_46_im / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -4.8e-125], t$95$0, If[LessEqual[y$46$im, 29000000000.0], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.8000000000000003e-125 or 2.9e10 < y.im

   1. Initial program 54.9%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 65.6%

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

   5. Simplified 65.6%

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

   if -4.8000000000000003e-125 < y.im < 2.9e10

   1. Initial program 71.6%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 68.8%

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

   5. Simplified 68.8%

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

Alternative 9: 42.8% 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 61.8%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 40.7%

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

5. Simplified 40.7%

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

Reproduce

herbie shell --seed 2024155 
(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))))