_divideComplex, imaginary part

Percentage Accurate: 60.5% → 81.7%

Time: 10.2s

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: 60.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 81.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ t_2 := y.re \cdot x.im - x.re \cdot y.im\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{t\_2}}\\ \mathbf{elif}\;y.re \leq 10^{-67}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im + x.re \cdot \frac{y.re \cdot y.re}{y.im}}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{t\_2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (/ (- x.im (* (/ x.re y.re) y.im)) y.re))
        (t_2 (- (* y.re x.im) (* x.re y.im))))
   (if (<= y.re -2.3e+122)
     t_1
     (if (<= y.re -3.9e-59)
       (/ 1.0 (/ t_0 t_2))
       (if (<= y.re 1e-67)
         (/
          (- (/ (+ (* y.re x.im) (* x.re (/ (* y.re y.re) y.im))) y.im) x.re)
          y.im)
         (if (<= y.re 9.5e+101) (/ t_2 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -2.3e+122) {
		tmp = t_1;
	} else if (y_46_re <= -3.9e-59) {
		tmp = 1.0 / (t_0 / t_2);
	} else if (y_46_re <= 1e-67) {
		tmp = ((((y_46_re * x_46_im) + (x_46_re * ((y_46_re * y_46_re) / y_46_im))) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 9.5e+101) {
		tmp = t_2 / 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) :: t_2
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46im - ((x_46re / y_46re) * y_46im)) / y_46re
    t_2 = (y_46re * x_46im) - (x_46re * y_46im)
    if (y_46re <= (-2.3d+122)) then
        tmp = t_1
    else if (y_46re <= (-3.9d-59)) then
        tmp = 1.0d0 / (t_0 / t_2)
    else if (y_46re <= 1d-67) then
        tmp = ((((y_46re * x_46im) + (x_46re * ((y_46re * y_46re) / y_46im))) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 9.5d+101) then
        tmp = t_2 / 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -2.3e+122) {
		tmp = t_1;
	} else if (y_46_re <= -3.9e-59) {
		tmp = 1.0 / (t_0 / t_2);
	} else if (y_46_re <= 1e-67) {
		tmp = ((((y_46_re * x_46_im) + (x_46_re * ((y_46_re * y_46_re) / y_46_im))) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 9.5e+101) {
		tmp = t_2 / 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im)
	tmp = 0
	if y_46_re <= -2.3e+122:
		tmp = t_1
	elif y_46_re <= -3.9e-59:
		tmp = 1.0 / (t_0 / t_2)
	elif y_46_re <= 1e-67:
		tmp = ((((y_46_re * x_46_im) + (x_46_re * ((y_46_re * y_46_re) / y_46_im))) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 9.5e+101:
		tmp = t_2 / 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(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	t_2 = Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.3e+122)
		tmp = t_1;
	elseif (y_46_re <= -3.9e-59)
		tmp = Float64(1.0 / Float64(t_0 / t_2));
	elseif (y_46_re <= 1e-67)
		tmp = Float64(Float64(Float64(Float64(Float64(y_46_re * x_46_im) + Float64(x_46_re * Float64(Float64(y_46_re * y_46_re) / y_46_im))) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 9.5e+101)
		tmp = Float64(t_2 / 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	tmp = 0.0;
	if (y_46_re <= -2.3e+122)
		tmp = t_1;
	elseif (y_46_re <= -3.9e-59)
		tmp = 1.0 / (t_0 / t_2);
	elseif (y_46_re <= 1e-67)
		tmp = ((((y_46_re * x_46_im) + (x_46_re * ((y_46_re * y_46_re) / y_46_im))) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 9.5e+101)
		tmp = t_2 / 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[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e+122], t$95$1, If[LessEqual[y$46$re, -3.9e-59], N[(1.0 / N[(t$95$0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1e-67], N[(N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 9.5e+101], N[(t$95$2 / t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.3000000000000001e122 or 9.49999999999999947e101 < y.re

   1. Initial program 34.1%

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

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

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

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

      7. /-lowering-/.f64 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. /-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -2.3000000000000001e122 < y.re < -3.90000000000000019e-59

   1. Initial program 78.1%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 78.1%

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

   4. Applied egg-rr 78.1%

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

   if -3.90000000000000019e-59 < y.re < 9.99999999999999943e-68

   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.im around -inf

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

      1. associate-*r/ N/A

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

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

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

   5. Simplified 91.1%

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

   if 9.99999999999999943e-68 < y.re < 9.49999999999999947e101

   1. Initial program 81.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.9 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 10^{-67}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im + x.re \cdot \frac{y.re \cdot y.re}{y.im}}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{+101}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ t_2 := y.re \cdot x.im - x.re \cdot y.im\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{t\_2}}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{t\_2}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (/ (- x.im (* (/ x.re y.re) y.im)) y.re))
        (t_2 (- (* y.re x.im) (* x.re y.im))))
   (if (<= y.re -2.3e+122)
     t_1
     (if (<= y.re -3.3e-59)
       (/ 1.0 (/ t_0 t_2))
       (if (<= y.re 9.2e-69)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 2.4e+102) (/ t_2 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -2.3e+122) {
		tmp = t_1;
	} else if (y_46_re <= -3.3e-59) {
		tmp = 1.0 / (t_0 / t_2);
	} else if (y_46_re <= 9.2e-69) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.4e+102) {
		tmp = t_2 / 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) :: t_2
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46im - ((x_46re / y_46re) * y_46im)) / y_46re
    t_2 = (y_46re * x_46im) - (x_46re * y_46im)
    if (y_46re <= (-2.3d+122)) then
        tmp = t_1
    else if (y_46re <= (-3.3d-59)) then
        tmp = 1.0d0 / (t_0 / t_2)
    else if (y_46re <= 9.2d-69) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 2.4d+102) then
        tmp = t_2 / 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	double tmp;
	if (y_46_re <= -2.3e+122) {
		tmp = t_1;
	} else if (y_46_re <= -3.3e-59) {
		tmp = 1.0 / (t_0 / t_2);
	} else if (y_46_re <= 9.2e-69) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 2.4e+102) {
		tmp = t_2 / 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im)
	tmp = 0
	if y_46_re <= -2.3e+122:
		tmp = t_1
	elif y_46_re <= -3.3e-59:
		tmp = 1.0 / (t_0 / t_2)
	elif y_46_re <= 9.2e-69:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 2.4e+102:
		tmp = t_2 / 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(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	t_2 = Float64(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (y_46_re <= -2.3e+122)
		tmp = t_1;
	elseif (y_46_re <= -3.3e-59)
		tmp = Float64(1.0 / Float64(t_0 / t_2));
	elseif (y_46_re <= 9.2e-69)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 2.4e+102)
		tmp = Float64(t_2 / 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 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	t_2 = (y_46_re * x_46_im) - (x_46_re * y_46_im);
	tmp = 0.0;
	if (y_46_re <= -2.3e+122)
		tmp = t_1;
	elseif (y_46_re <= -3.3e-59)
		tmp = 1.0 / (t_0 / t_2);
	elseif (y_46_re <= 9.2e-69)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 2.4e+102)
		tmp = t_2 / 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[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e+122], t$95$1, If[LessEqual[y$46$re, -3.3e-59], N[(1.0 / N[(t$95$0 / t$95$2), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.2e-69], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 2.4e+102], N[(t$95$2 / t$95$0), $MachinePrecision], t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.3000000000000001e122 or 2.39999999999999994e102 < y.re

   1. Initial program 34.1%

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

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

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

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

      7. /-lowering-/.f64 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. /-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -2.3000000000000001e122 < y.re < -3.29999999999999982e-59

   1. Initial program 78.1%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 78.1%

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

   4. Applied egg-rr 78.1%

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

   if -3.29999999999999982e-59 < y.re < 9.2000000000000003e-69

   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 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 91.1%

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

   5. Simplified 91.1%

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

   if 9.2000000000000003e-69 < y.re < 2.39999999999999994e102

   1. Initial program 81.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.3 \cdot 10^{-59}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.im - x.re \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 9.2 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 2.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* y.re x.im) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -1.55e+122)
     t_1
     (if (<= y.re -8.5e-58)
       t_0
       (if (<= y.re 2.6e-67)
         (/ (- (/ (* y.re x.im) y.im) x.re) y.im)
         (if (<= y.re 3.6e+102) 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 = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.55e+122) {
		tmp = t_1;
	} else if (y_46_re <= -8.5e-58) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-67) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.6e+102) {
		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 = ((y_46re * x_46im) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im - ((x_46re / y_46re) * y_46im)) / y_46re
    if (y_46re <= (-1.55d+122)) then
        tmp = t_1
    else if (y_46re <= (-8.5d-58)) then
        tmp = t_0
    else if (y_46re <= 2.6d-67) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 3.6d+102) 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 = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.55e+122) {
		tmp = t_1;
	} else if (y_46_re <= -8.5e-58) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e-67) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.6e+102) {
		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 = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	tmp = 0
	if y_46_re <= -1.55e+122:
		tmp = t_1
	elif y_46_re <= -8.5e-58:
		tmp = t_0
	elif y_46_re <= 2.6e-67:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 3.6e+102:
		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(Float64(y_46_re * x_46_im) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.55e+122)
		tmp = t_1;
	elseif (y_46_re <= -8.5e-58)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-67)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.6e+102)
		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 = ((y_46_re * x_46_im) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.55e+122)
		tmp = t_1;
	elseif (y_46_re <= -8.5e-58)
		tmp = t_0;
	elseif (y_46_re <= 2.6e-67)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 3.6e+102)
		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[(N[(y$46$re * x$46$im), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.55e+122], t$95$1, If[LessEqual[y$46$re, -8.5e-58], t$95$0, If[LessEqual[y$46$re, 2.6e-67], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.6e+102], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.54999999999999999e122 or 3.6000000000000002e102 < y.re

   1. Initial program 34.1%

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

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

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

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

      7. /-lowering-/.f64 81.0%

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

   5. Simplified 81.0%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. /-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 82.1%

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

   7. Applied egg-rr 82.1%

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

   if -1.54999999999999999e122 < y.re < -8.5000000000000004e-58 or 2.5999999999999999e-67 < y.re < 3.6000000000000002e102

   1. Initial program 79.7%

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

   if -8.5000000000000004e-58 < y.re < 2.5999999999999999e-67

   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 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 91.1%

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

   5. Simplified 91.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.55 \cdot 10^{+122}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{elif}\;y.re \leq -8.5 \cdot 10^{-58}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{y.re \cdot x.im - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -4 \cdot 10^{-37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 61:\\ \;\;\;\;\frac{\frac{y.re \cdot x.im}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -4e-37)
     t_0
     (if (<= y.re 61.0) (/ (- (/ (* y.re x.im) y.im) x.re) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -4e-37) {
		tmp = t_0;
	} else if (y_46_re <= 61.0) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - ((x_46re / y_46re) * y_46im)) / y_46re
    if (y_46re <= (-4d-37)) then
        tmp = t_0
    else if (y_46re <= 61.0d0) then
        tmp = (((y_46re * x_46im) / y_46im) - x_46re) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -4e-37) {
		tmp = t_0;
	} else if (y_46_re <= 61.0) {
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	tmp = 0
	if y_46_re <= -4e-37:
		tmp = t_0
	elif y_46_re <= 61.0:
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -4e-37)
		tmp = t_0;
	elseif (y_46_re <= 61.0)
		tmp = Float64(Float64(Float64(Float64(y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -4e-37)
		tmp = t_0;
	elseif (y_46_re <= 61.0)
		tmp = (((y_46_re * x_46_im) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -4e-37], t$95$0, If[LessEqual[y$46$re, 61.0], N[(N[(N[(N[(y$46$re * x$46$im), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.00000000000000027e-37 or 61 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 74.6%

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

   5. Simplified 74.6%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. /-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

   if -4.00000000000000027e-37 < y.re < 61

   1. Initial program 68.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.2%

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

   5. Simplified 87.2%

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

Alternative 5: 77.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im - \frac{x.re}{y.re} \cdot y.im}{y.re}\\ \mathbf{if}\;y.re \leq -5.4 \cdot 10^{-32}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 140:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- x.im (* (/ x.re y.re) y.im)) y.re)))
   (if (<= y.re -5.4e-32)
     t_0
     (if (<= y.re 140.0) (/ (- (* x.im (/ y.re y.im)) x.re) y.im) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -5.4e-32) {
		tmp = t_0;
	} else if (y_46_re <= 140.0) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im - ((x_46re / y_46re) * y_46im)) / y_46re
    if (y_46re <= (-5.4d-32)) then
        tmp = t_0
    else if (y_46re <= 140.0d0) then
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	double tmp;
	if (y_46_re <= -5.4e-32) {
		tmp = t_0;
	} else if (y_46_re <= 140.0) {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re
	tmp = 0
	if y_46_re <= -5.4e-32:
		tmp = t_0
	elif y_46_re <= 140.0:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im - Float64(Float64(x_46_re / y_46_re) * y_46_im)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -5.4e-32)
		tmp = t_0;
	elseif (y_46_re <= 140.0)
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im - ((x_46_re / y_46_re) * y_46_im)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -5.4e-32)
		tmp = t_0;
	elseif (y_46_re <= 140.0)
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im - N[(N[(x$46$re / y$46$re), $MachinePrecision] * y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -5.4e-32], t$95$0, If[LessEqual[y$46$re, 140.0], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.39999999999999962e-32 or 140 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 74.6%

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

   5. Simplified 74.6%

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

      1. clear-num N/A

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

      2. associate-*r/ N/A

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

      3. div-inv N/A

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

      4. times-frac N/A

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

      5. clear-num N/A

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

      6. /-rgt-identity N/A

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

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

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

      8. /-lowering-/.f64 75.4%

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

   7. Applied egg-rr 75.4%

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

   if -5.39999999999999962e-32 < y.re < 140

   1. Initial program 68.7%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.2%

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

   5. Simplified 87.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 86.4%

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

   7. Applied egg-rr 86.4%

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

Alternative 6: 73.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0 - x.re}{y.im}\\ \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+53}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (- 0.0 x.re) y.im)))
   (if (<= y.im -4.5e+53)
     t_0
     (if (<= y.im 2.6e+92) (/ (- x.im (* x.re (/ y.im y.re))) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (0.0 - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.5e+53) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e+92) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.0d0 - x_46re) / y_46im
    if (y_46im <= (-4.5d+53)) then
        tmp = t_0
    else if (y_46im <= 2.6d+92) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (0.0 - x_46_re) / y_46_im;
	double tmp;
	if (y_46_im <= -4.5e+53) {
		tmp = t_0;
	} else if (y_46_im <= 2.6e+92) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (0.0 - x_46_re) / y_46_im
	tmp = 0
	if y_46_im <= -4.5e+53:
		tmp = t_0
	elif y_46_im <= 2.6e+92:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(0.0 - x_46_re) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -4.5e+53)
		tmp = t_0;
	elseif (y_46_im <= 2.6e+92)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (0.0 - x_46_re) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -4.5e+53)
		tmp = t_0;
	elseif (y_46_im <= 2.6e+92)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(0.0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -4.5e+53], t$95$0, If[LessEqual[y$46$im, 2.6e+92], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -4.5000000000000002e53 or 2.5999999999999999e92 < y.im

   1. Initial program 42.2%

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

   3. Taylor expanded in y.re around 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 78.7%

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

   5. Simplified 78.7%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 74.3%

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

   8. Simplified 74.3%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 74.3%

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

   10. Applied egg-rr 74.3%

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

   if -4.5000000000000002e53 < y.im < 2.5999999999999999e92

   1. Initial program 71.3%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. associate-/l* N/A

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

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

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

      7. /-lowering-/.f64 71.8%

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

   5. Simplified 71.8%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -4.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+92}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 96:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.7e-49)
   (/ x.im y.re)
   (if (<= y.re 96.0) (/ (- 0.0 x.re) y.im) (/ x.im y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.7e-49) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 96.0) {
		tmp = (0.0 - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.7d-49)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 96.0d0) then
        tmp = (0.0d0 - x_46re) / y_46im
    else
        tmp = x_46im / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.7e-49) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 96.0) {
		tmp = (0.0 - x_46_re) / y_46_im;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.7e-49:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 96.0:
		tmp = (0.0 - x_46_re) / y_46_im
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.7e-49)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 96.0)
		tmp = Float64(Float64(0.0 - x_46_re) / y_46_im);
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.7e-49)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 96.0)
		tmp = (0.0 - x_46_re) / y_46_im;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.7e-49], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 96.0], N[(N[(0.0 - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -4.70000000000000021e-49 or 96 < y.re

   1. Initial program 53.2%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 61.0%

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

   5. Simplified 61.0%

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

   if -4.70000000000000021e-49 < y.re < 96

   1. Initial program 68.2%

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

   3. Taylor expanded in y.re around 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 87.8%

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

   5. Simplified 87.8%

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

   6. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. --lowering--.f64 68.5%

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

   8. Simplified 68.5%

      \[\leadsto \frac{\color{blue}{0 - x.re}}{y.im} \]
   9. Step-by-step derivation

      1. sub0-neg N/A

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

      2. neg-lowering-neg.f64 68.5%

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

   10. Applied egg-rr 68.5%

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

4. Final simplification 64.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 96:\\ \;\;\;\;\frac{0 - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.2% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.0%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 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 2024160 
(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))))