_divideComplex, imaginary part

Percentage Accurate: 61.3% → 90.9%

Time: 11.3s

Alternatives: 8

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    code = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	return ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re + y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ t_1 := \frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{-160}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (-
          (/ x.im (+ y.re (* y.im (/ y.im y.re))))
          (/ x.re (/ (+ (* y.re y.re) (* y.im y.im)) y.im))))
        (t_1 (/ (- (* (/ x.im y.im) y.re) x.re) y.im)))
   (if (<= y.im -2.4e+187)
     t_1
     (if (<= y.im -1.5e-160)
       t_0
       (if (<= y.im 2e-194)
         (/ (- x.im (/ (* y.im x.re) y.re)) y.re)
         (if (<= y.im 1.35e+154) t_0 t_1))))))

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 * (y_46_im / y_46_re)))) - (x_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / y_46_im));
	double t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.4e+187) {
		tmp = t_1;
	} else if (y_46_im <= -1.5e-160) {
		tmp = t_0;
	} else if (y_46_im <= 2e-194) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 * (y_46im / y_46re)))) - (x_46re / (((y_46re * y_46re) + (y_46im * y_46im)) / y_46im))
    t_1 = (((x_46im / y_46im) * y_46re) - x_46re) / y_46im
    if (y_46im <= (-2.4d+187)) then
        tmp = t_1
    else if (y_46im <= (-1.5d-160)) then
        tmp = t_0
    else if (y_46im <= 2d-194) then
        tmp = (x_46im - ((y_46im * x_46re) / y_46re)) / y_46re
    else if (y_46im <= 1.35d+154) then
        tmp = t_0
    else
        tmp = t_1
    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 * (y_46_im / y_46_re)))) - (x_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / y_46_im));
	double t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -2.4e+187) {
		tmp = t_1;
	} else if (y_46_im <= -1.5e-160) {
		tmp = t_0;
	} else if (y_46_im <= 2e-194) {
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 1.35e+154) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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 * (y_46_im / y_46_re)))) - (x_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / y_46_im))
	t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -2.4e+187:
		tmp = t_1
	elif y_46_im <= -1.5e-160:
		tmp = t_0
	elif y_46_im <= 2e-194:
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re
	elif y_46_im <= 1.35e+154:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im / Float64(y_46_re + Float64(y_46_im * Float64(y_46_im / y_46_re)))) - Float64(x_46_re / Float64(Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)) / y_46_im)))
	t_1 = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.4e+187)
		tmp = t_1;
	elseif (y_46_im <= -1.5e-160)
		tmp = t_0;
	elseif (y_46_im <= 2e-194)
		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+154)
		tmp = t_0;
	else
		tmp = t_1;
	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 * (y_46_im / y_46_re)))) - (x_46_re / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / y_46_im));
	t_1 = (((x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.4e+187)
		tmp = t_1;
	elseif (y_46_im <= -1.5e-160)
		tmp = t_0;
	elseif (y_46_im <= 2e-194)
		tmp = (x_46_im - ((y_46_im * x_46_re) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 1.35e+154)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / N[(N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.4e+187], t$95$1, If[LessEqual[y$46$im, -1.5e-160], t$95$0, If[LessEqual[y$46$im, 2e-194], 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+154], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.39999999999999985e187 or 1.35000000000000003e154 < y.im

   1. Initial program 34.4%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 87.0%

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

   5. Simplified 87.0%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 97.3%

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

   7. Applied egg-rr 97.3%

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

   if -2.39999999999999985e187 < y.im < -1.49999999999999998e-160 or 2.00000000000000004e-194 < y.im < 1.35000000000000003e154

   1. Initial program 69.7%

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

      1. div-sub N/A

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

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-/l* N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      18. *-lowering-*.f64 73.3%

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

   4. Applied egg-rr 73.3%

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

   5. Taylor expanded in y.im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 91.4%

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

   7. Simplified 91.4%

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

   if -1.49999999999999998e-160 < y.im < 2.00000000000000004e-194

   1. Initial program 68.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 91.8%

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

   5. Simplified 91.8%

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

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.4 \cdot 10^{+187}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq -1.5 \cdot 10^{-160}:\\ \;\;\;\;\frac{x.im}{y.re + y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{elif}\;y.im \leq 2 \cdot 10^{-194}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{x.im}{y.re + y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{y.re + y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -7 \cdot 10^{-26}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (/ x.im (+ y.re (* y.im (/ y.im y.re)))) (/ x.re y.im))))
   (if (<= y.im -7e-26)
     t_0
     (if (<= y.im 1.55e-12) (/ (- 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 * (y_46_im / y_46_re)))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -7e-26) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e-12) {
		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 * (y_46im / y_46re)))) - (x_46re / y_46im)
    if (y_46im <= (-7d-26)) then
        tmp = t_0
    else if (y_46im <= 1.55d-12) 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 * (y_46_im / y_46_re)))) - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -7e-26) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e-12) {
		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 * (y_46_im / y_46_re)))) - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -7e-26:
		tmp = t_0
	elif y_46_im <= 1.55e-12:
		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(x_46_im / Float64(y_46_re + Float64(y_46_im * Float64(y_46_im / y_46_re)))) - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -7e-26)
		tmp = t_0;
	elseif (y_46_im <= 1.55e-12)
		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 * (y_46_im / y_46_re)))) - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -7e-26)
		tmp = t_0;
	elseif (y_46_im <= 1.55e-12)
		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[(x$46$im / N[(y$46$re + N[(y$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -7e-26], t$95$0, If[LessEqual[y$46$im, 1.55e-12], 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 < -6.9999999999999997e-26 or 1.5500000000000001e-12 < y.im

   1. Initial program 50.2%

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

      1. div-sub N/A

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

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-/l* N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      18. *-lowering-*.f64 54.2%

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

   4. Applied egg-rr 54.2%

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

   5. Taylor expanded in y.im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 64.5%

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

   7. Simplified 64.5%

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

   8. Taylor expanded in y.re around 0

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

   10. Simplified 81.7%

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

   if -6.9999999999999997e-26 < y.im < 1.5500000000000001e-12

   1. Initial program 72.8%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 85.6%

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

   5. Simplified 85.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7 \cdot 10^{-26}:\\ \;\;\;\;\frac{x.im}{y.re + y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re + y.im \cdot \frac{y.im}{y.re}} - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -6.3 \cdot 10^{-25}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- (* (/ x.im y.im) y.re) x.re) y.im)))
   (if (<= y.im -6.3e-25)
     t_0
     (if (<= y.im 1.8e+41) (/ (- 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_im) * y_46_re) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -6.3e-25) {
		tmp = t_0;
	} else if (y_46_im <= 1.8e+41) {
		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_46im) * y_46re) - x_46re) / y_46im
    if (y_46im <= (-6.3d-25)) then
        tmp = t_0
    else if (y_46im <= 1.8d+41) 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_im) * y_46_re) - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -6.3e-25) {
		tmp = t_0;
	} else if (y_46_im <= 1.8e+41) {
		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_im) * y_46_re) - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -6.3e-25:
		tmp = t_0
	elif y_46_im <= 1.8e+41:
		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(Float64(x_46_im / y_46_im) * y_46_re) - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.3e-25)
		tmp = t_0;
	elseif (y_46_im <= 1.8e+41)
		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_im) * y_46_re) - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -6.3e-25)
		tmp = t_0;
	elseif (y_46_im <= 1.8e+41)
		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[(N[(x$46$im / y$46$im), $MachinePrecision] * y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6.3e-25], t$95$0, If[LessEqual[y$46$im, 1.8e+41], 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 < -6.29999999999999961e-25 or 1.80000000000000013e41 < y.im

   1. Initial program 47.7%

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

   3. Taylor expanded in y.re around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. unpow2 N/A

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

      5. associate-/r* N/A

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

      6. div-sub N/A

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

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

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

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

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

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

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

      10. *-commutative N/A

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

      11. *-lowering-*.f64 75.7%

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

   5. Simplified 75.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. /-lowering-/.f64 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -6.29999999999999961e-25 < y.im < 1.80000000000000013e41

   1. Initial program 72.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.1%

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

   5. Simplified 81.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot y.re - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -8.5 \cdot 10^{-31}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.26 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -8.5e-31)
     t_0
     (if (<= y.im 1.26e+42) (/ (- 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 <= -8.5e-31) {
		tmp = t_0;
	} else if (y_46_im <= 1.26e+42) {
		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 <= (-8.5d-31)) then
        tmp = t_0
    else if (y_46im <= 1.26d+42) 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 <= -8.5e-31) {
		tmp = t_0;
	} else if (y_46_im <= 1.26e+42) {
		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 <= -8.5e-31:
		tmp = t_0
	elif y_46_im <= 1.26e+42:
		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 <= -8.5e-31)
		tmp = t_0;
	elseif (y_46_im <= 1.26e+42)
		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 <= -8.5e-31)
		tmp = t_0;
	elseif (y_46_im <= 1.26e+42)
		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, -8.5e-31], t$95$0, If[LessEqual[y$46$im, 1.26e+42], 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 < -8.5000000000000007e-31 or 1.26e42 < y.im

   1. Initial program 47.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 69.0%

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

   5. Simplified 69.0%

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

   if -8.5000000000000007e-31 < y.im < 1.26e42

   1. Initial program 72.1%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 81.1%

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

   5. Simplified 81.1%

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

4. Final simplification 75.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -8.5 \cdot 10^{-31}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.26 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.im - \frac{y.im \cdot x.re}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.55 \cdot 10^{-23}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.4 \cdot 10^{+42}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -1.55e-23)
     t_0
     (if (<= y.im 1.4e+42) (/ (- x.im (/ x.re (/ y.re y.im))) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.55e-23) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e+42) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-1.55d-23)) then
        tmp = t_0
    else if (y_46im <= 1.4d+42) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.55e-23) {
		tmp = t_0;
	} else if (y_46_im <= 1.4e+42) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.55e-23:
		tmp = t_0
	elif y_46_im <= 1.4e+42:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.55e-23)
		tmp = t_0;
	elseif (y_46_im <= 1.4e+42)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.55e-23)
		tmp = t_0;
	elseif (y_46_im <= 1.4e+42)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.55e-23], t$95$0, If[LessEqual[y$46$im, 1.4e+42], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.5499999999999999e-23 or 1.4e42 < y.im

   1. Initial program 47.7%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 69.0%

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

   5. Simplified 69.0%

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

   if -1.5499999999999999e-23 < y.im < 1.4e42

   1. Initial program 72.1%

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

   3. Taylor expanded in y.im around 0

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

      1. +-commutative N/A

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

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x.im}{y.re}\right), \color{blue}{\left(y.im \cdot \left(-1 \cdot \frac{x.re}{{y.re}^{2}} + -1 \cdot \frac{x.im \cdot y.im}{{y.re}^{3}}\right)\right)}\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \left(\color{blue}{y.im} \cdot \left(-1 \cdot \frac{x.re}{{y.re}^{2}} + -1 \cdot \frac{x.im \cdot y.im}{{y.re}^{3}}\right)\right)\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(y.im, \color{blue}{\left(-1 \cdot \frac{x.re}{{y.re}^{2}} + -1 \cdot \frac{x.im \cdot y.im}{{y.re}^{3}}\right)}\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x.im, y.re\right), \mathsf{*.f64}\left(y.im, \left(\frac{-1 \cdot x.re}{{y.re}^{2}} + \color{blue}{-1} \cdot \frac{x.im \cdot y.im}{{y.re}^{3}}\right)\right)\right) \]

      6. unpow2 N/A

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

      7. times-frac N/A

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

      8. associate-*r/ N/A

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

      9. cube-mult N/A

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

      10. unpow2 N/A

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

      11. times-frac N/A

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

      12. distribute-lft-out N/A

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

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

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

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

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

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

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

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

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

      17. unpow2 N/A

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

      18. associate-/r* N/A

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

      19. /-lowering-/.f64 N/A

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

   5. Simplified 68.0%

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

   6. Taylor expanded in x.re around inf

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

      1. /-lowering-/.f64 73.8%

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

   8. Simplified 73.8%

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

      1. div-inv N/A

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

      2. fma-define N/A

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

      3. associate-*r/ N/A

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

      4. associate-*r/ N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l/ N/A

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

      7. associate-*r/ N/A

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

      8. mul-1-neg N/A

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

      9. distribute-rgt-neg-out N/A

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

      10. fmm-undef N/A

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

      11. div-inv N/A

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

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

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

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

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

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

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

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

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

      16. /-lowering-/.f64 76.6%

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

   10. Applied egg-rr 76.6%

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

       1. associate-*r/ N/A

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

       2. sub-div N/A

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

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

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

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

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

       5. *-commutative N/A

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

       6. clear-num N/A

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

       7. un-div-inv N/A

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

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

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

       9. /-lowering-/.f64 80.4%

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

   12. Applied egg-rr 80.4%

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

Alternative 6: 65.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.55 \cdot 10^{+149}:\\ \;\;\;\;\frac{x.im}{y.re + \frac{y.im \cdot y.im}{y.re}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -1.9e-34)
     t_0
     (if (<= y.im 1.55e+149) (/ x.im (+ y.re (/ (* y.im y.im) y.re))) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.9e-34) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e+149) {
		tmp = x_46_im / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-1.9d-34)) then
        tmp = t_0
    else if (y_46im <= 1.55d+149) then
        tmp = x_46im / (y_46re + ((y_46im * y_46im) / y_46re))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -1.9e-34) {
		tmp = t_0;
	} else if (y_46_im <= 1.55e+149) {
		tmp = x_46_im / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -1.9e-34:
		tmp = t_0
	elif y_46_im <= 1.55e+149:
		tmp = x_46_im / (y_46_re + ((y_46_im * y_46_im) / y_46_re))
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.9e-34)
		tmp = t_0;
	elseif (y_46_im <= 1.55e+149)
		tmp = Float64(x_46_im / Float64(y_46_re + Float64(Float64(y_46_im * y_46_im) / y_46_re)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -1.9e-34)
		tmp = t_0;
	elseif (y_46_im <= 1.55e+149)
		tmp = x_46_im / (y_46_re + ((y_46_im * y_46_im) / y_46_re));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.9e-34], t$95$0, If[LessEqual[y$46$im, 1.55e+149], N[(x$46$im / N[(y$46$re + N[(N[(y$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.9000000000000001e-34 or 1.54999999999999993e149 < y.im

   1. Initial program 45.8%

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

   3. Taylor expanded in y.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 74.0%

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

   5. Simplified 74.0%

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

   if -1.9000000000000001e-34 < y.im < 1.54999999999999993e149

   1. Initial program 70.0%

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

      1. div-sub N/A

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

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

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

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

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

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

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

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

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

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

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

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

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

      11. associate-/l* N/A

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

      12. clear-num N/A

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

      13. un-div-inv N/A

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

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

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

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

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

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

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

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

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

      18. *-lowering-*.f64 69.3%

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

   4. Applied egg-rr 69.3%

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

   5. Taylor expanded in y.im around 0

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

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

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

      2. unpow2 N/A

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 87.9%

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

   7. Simplified 87.9%

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

   8. Taylor expanded in x.im around inf

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

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

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

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

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

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

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

      4. unpow2 N/A

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

      5. *-lowering-*.f64 67.5%

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

   10. Simplified 67.5%

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

Alternative 7: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-34}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 105000:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -1.7e-34) t_0 (if (<= y.im 105000.0) (/ 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 <= -1.7e-34) {
		tmp = t_0;
	} else if (y_46_im <= 105000.0) {
		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 <= (-1.7d-34)) then
        tmp = t_0
    else if (y_46im <= 105000.0d0) 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 <= -1.7e-34) {
		tmp = t_0;
	} else if (y_46_im <= 105000.0) {
		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 <= -1.7e-34:
		tmp = t_0
	elif y_46_im <= 105000.0:
		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 <= -1.7e-34)
		tmp = t_0;
	elseif (y_46_im <= 105000.0)
		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 <= -1.7e-34)
		tmp = t_0;
	elseif (y_46_im <= 105000.0)
		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, -1.7e-34], t$95$0, If[LessEqual[y$46$im, 105000.0], N[(x$46$im / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.7e-34 or 105000 < y.im

   1. Initial program 49.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   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 66.8%

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

   5. Simplified 66.8%

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

   if -1.7e-34 < y.im < 105000

   1. Initial program 73.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 66.6%

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

   5. Simplified 66.6%

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

Alternative 8: 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 60.3%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 39.9%

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

5. Simplified 39.9%

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

Reproduce

herbie shell --seed 2024164 
(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))))