_divideComplex, real part

Percentage Accurate: 61.1% → 80.0%

Time: 9.5s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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: 61.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (+ (* x.re y.re) (* x.im 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_re * y_46_re) + (x_46_im * 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_46re * y_46re) + (x_46im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + (x_46_im * 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_re * y_46_re) + Float64(x_46_im * 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_re * y_46_re) + (x_46_im * 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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 80.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{y.re \cdot y.re + y.im \cdot y.im} \cdot \left(x.re \cdot y.re + y.im \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -1.2e+83)
     t_0
     (if (<= y.im -1.2e-131)
       (*
        (/ 1.0 (+ (* y.re y.re) (* y.im y.im)))
        (+ (* x.re y.re) (* y.im x.im)))
       (if (<= y.im 9.2e+74) (/ (+ x.re (* x.im (/ y.im y.re))) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e+83) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-131) {
		tmp = (1.0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) * ((x_46_re * y_46_re) + (y_46_im * x_46_im));
	} else if (y_46_im <= 9.2e+74) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-1.2d+83)) then
        tmp = t_0
    else if (y_46im <= (-1.2d-131)) then
        tmp = (1.0d0 / ((y_46re * y_46re) + (y_46im * y_46im))) * ((x_46re * y_46re) + (y_46im * x_46im))
    else if (y_46im <= 9.2d+74) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -1.2e+83) {
		tmp = t_0;
	} else if (y_46_im <= -1.2e-131) {
		tmp = (1.0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) * ((x_46_re * y_46_re) + (y_46_im * x_46_im));
	} else if (y_46_im <= 9.2e+74) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -1.2e+83:
		tmp = t_0
	elif y_46_im <= -1.2e-131:
		tmp = (1.0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) * ((x_46_re * y_46_re) + (y_46_im * x_46_im))
	elif y_46_im <= 9.2e+74:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.2e+83)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-131)
		tmp = Float64(Float64(1.0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) * Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)));
	elseif (y_46_im <= 9.2e+74)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -1.2e+83)
		tmp = t_0;
	elseif (y_46_im <= -1.2e-131)
		tmp = (1.0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) * ((x_46_re * y_46_re) + (y_46_im * x_46_im));
	elseif (y_46_im <= 9.2e+74)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -1.2e+83], t$95$0, If[LessEqual[y$46$im, -1.2e-131], N[(N[(1.0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 9.2e+74], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.19999999999999996e83 or 9.1999999999999994e74 < y.im

   1. Initial program 35.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 84.2%

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

   5. Simplified 84.2%

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

   if -1.19999999999999996e83 < y.im < -1.2e-131

   1. Initial program 88.3%

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

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 88.3%

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

   4. Applied egg-rr 88.3%

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

   if -1.2e-131 < y.im < 9.1999999999999994e74

   1. Initial program 66.1%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 86.1%

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

   5. Simplified 86.1%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 86.2%

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

   7. Applied egg-rr 86.2%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

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

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

      8. /-lowering-/.f64 88.0%

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

   9. Applied egg-rr 88.0%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{1}{y.re \cdot y.re + y.im \cdot y.im} \cdot \left(x.re \cdot y.re + y.im \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq 9.2 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -6 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -6e+36)
     t_0
     (if (<= y.im -1.8e-107)
       (/ (+ (* x.re y.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 7.7e+74) (/ (+ x.re (* x.im (/ y.im y.re))) y.re) t_0)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -6e+36) {
		tmp = t_0;
	} else if (y_46_im <= -1.8e-107) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-6d+36)) then
        tmp = t_0
    else if (y_46im <= (-1.8d-107)) then
        tmp = ((x_46re * y_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 7.7d+74) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -6e+36) {
		tmp = t_0;
	} else if (y_46_im <= -1.8e-107) {
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -6e+36:
		tmp = t_0
	elif y_46_im <= -1.8e-107:
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 7.7e+74:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -6e+36)
		tmp = t_0;
	elseif (y_46_im <= -1.8e-107)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 7.7e+74)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -6e+36)
		tmp = t_0;
	elseif (y_46_im <= -1.8e-107)
		tmp = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 7.7e+74)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -6e+36], t$95$0, If[LessEqual[y$46$im, -1.8e-107], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.7e+74], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6e36 or 7.70000000000000042e74 < y.im

   1. Initial program 38.8%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.5%

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

   5. Simplified 82.5%

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

   if -6e36 < y.im < -1.79999999999999988e-107

   1. Initial program 94.1%

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

   if -1.79999999999999988e-107 < y.im < 7.70000000000000042e74

   1. Initial program 67.0%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 86.5%

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

   5. Simplified 86.5%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 86.5%

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

   7. Applied egg-rr 86.5%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

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

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

      8. /-lowering-/.f64 88.3%

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

   9. Applied egg-rr 88.3%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6 \cdot 10^{+36}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -1.8 \cdot 10^{-107}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -7000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -7000000.0)
     t_0
     (if (<= y.im 7.7e+74) (/ (+ x.re (* x.im (/ y.im y.re))) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -7000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-7000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 7.7d+74) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -7000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -7000000.0:
		tmp = t_0
	elif y_46_im <= 7.7e+74:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7000000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.7e+74)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7000000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.7e+74)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7000000.0], t$95$0, If[LessEqual[y$46$im, 7.7e+74], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7e6 or 7.70000000000000042e74 < y.im

   1. Initial program 42.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.0%

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

   5. Simplified 82.0%

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

   if -7e6 < y.im < 7.70000000000000042e74

   1. Initial program 72.5%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

      2. associate-/r/ N/A

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

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

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

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

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

      5. *-commutative N/A

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

      6. *-lowering-*.f64 81.0%

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

   7. Applied egg-rr 81.0%

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

      1. +-commutative N/A

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

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

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. div-inv N/A

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

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

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

      8. /-lowering-/.f64 82.4%

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

   9. Applied egg-rr 82.4%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7000000:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -4600000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -4600000.0)
     t_0
     (if (<= y.im 7.7e+74) (/ (+ x.re (/ (* y.im x.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -4600000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-4600000.0d0)) then
        tmp = t_0
    else if (y_46im <= 7.7d+74) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -4600000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -4600000.0:
		tmp = t_0
	elif y_46_im <= 7.7e+74:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4600000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.7e+74)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -4600000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.7e+74)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4600000.0], t$95$0, If[LessEqual[y$46$im, 7.7e+74], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.6e6 or 7.70000000000000042e74 < y.im

   1. Initial program 42.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.0%

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

   5. Simplified 82.0%

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

   if -4.6e6 < y.im < 7.70000000000000042e74

   1. Initial program 72.5%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

4. Final simplification 81.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4600000:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -31000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (* x.re (/ y.re y.im))) y.im)))
   (if (<= y.im -31000000.0)
     t_0
     (if (<= y.im 7.7e+74) (/ (+ x.re (* y.im (/ x.im y.re))) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -31000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    if (y_46im <= (-31000000.0d0)) then
        tmp = t_0
    else if (y_46im <= 7.7d+74) then
        tmp = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	double tmp;
	if (y_46_im <= -31000000.0) {
		tmp = t_0;
	} else if (y_46_im <= 7.7e+74) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	tmp = 0
	if y_46_im <= -31000000.0:
		tmp = t_0
	elif y_46_im <= 7.7e+74:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -31000000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.7e+74)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -31000000.0)
		tmp = t_0;
	elseif (y_46_im <= 7.7e+74)
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -31000000.0], t$95$0, If[LessEqual[y$46$im, 7.7e+74], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -3.1e7 or 7.70000000000000042e74 < y.im

   1. Initial program 42.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 82.0%

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

   5. Simplified 82.0%

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

   if -3.1e7 < y.im < 7.70000000000000042e74

   1. Initial program 72.5%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 81.0%

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

   5. Simplified 81.0%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 80.3%

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

   7. Applied egg-rr 80.3%

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

Alternative 6: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.04 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+33}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.04e+80)
   (/ x.re y.re)
   (if (<= y.re 1.9e+33)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.04e+80) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.9e+33) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.04d+80)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.9d+33) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.04e+80) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.9e+33) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.04e+80:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.9e+33:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.04e+80)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.9e+33)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.04e+80)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.9e+33)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.04e+80], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+33], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.04000000000000006e80 or 1.90000000000000001e33 < y.re

   1. Initial program 38.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 75.8%

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

   5. Simplified 75.8%

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

   if -1.04000000000000006e80 < y.re < 1.90000000000000001e33

   1. Initial program 71.1%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 77.5%

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

   5. Simplified 77.5%

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

Alternative 7: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -19500000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 7.7 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -19500000000.0)
   (/ x.im y.im)
   (if (<= y.im 7.7e+74) (/ x.re y.re) (/ x.im 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 <= -19500000000.0) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 7.7e+74) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / 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 <= (-19500000000.0d0)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 7.7d+74) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / 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 <= -19500000000.0) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 7.7e+74) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / 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 <= -19500000000.0:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 7.7e+74:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / 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 <= -19500000000.0)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 7.7e+74)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / 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 <= -19500000000.0)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 7.7e+74)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / 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, -19500000000.0], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 7.7e+74], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.95e10 or 7.70000000000000042e74 < y.im

   1. Initial program 41.9%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 74.4%

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

   5. Simplified 74.4%

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

   if -1.95e10 < y.im < 7.70000000000000042e74

   1. Initial program 72.7%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 66.0%

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

   5. Simplified 66.0%

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

Alternative 8: 42.9% accurate, 5.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im) :precision binary64 (/ x.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_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_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_im;
}

Python

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

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	return Float64(x_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_im;
end

Wolfram

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

Derivation

1. Initial program 58.4%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 45.3%

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

5. Simplified 45.3%

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

Reproduce

herbie shell --seed 2024158 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))