_divideComplex, imaginary part

Percentage Accurate: 60.9% → 81.7%

Time: 8.6s

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: 60.9% 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

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.im \cdot y.im + y.re \cdot y.re\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-11}:\\ \;\;\;\;x.im \cdot \left(\frac{y.re}{t\_0} - \frac{\frac{y.im \cdot x.re}{x.im}}{t\_0}\right)\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.im y.im) (* y.re y.re)))
        (t_1 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.re -2.3e+114)
     t_1
     (if (<= y.re -6.8e-11)
       (* x.im (- (/ y.re t_0) (/ (/ (* y.im x.re) x.im) t_0)))
       (if (<= y.re 7e-140)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 8.6e+54)
           (/ (- (* x.im y.re) (* y.im x.re)) 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_im * y_46_im) + (y_46_re * y_46_re);
	double t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.3e+114) {
		tmp = t_1;
	} else if (y_46_re <= -6.8e-11) {
		tmp = x_46_im * ((y_46_re / t_0) - (((y_46_im * x_46_re) / x_46_im) / t_0));
	} else if (y_46_re <= 7e-140) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 8.6e+54) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / 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_46im * y_46im) + (y_46re * y_46re)
    t_1 = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-2.3d+114)) then
        tmp = t_1
    else if (y_46re <= (-6.8d-11)) then
        tmp = x_46im * ((y_46re / t_0) - (((y_46im * x_46re) / x_46im) / t_0))
    else if (y_46re <= 7d-140) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 8.6d+54) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / 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_im * y_46_im) + (y_46_re * y_46_re);
	double t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.3e+114) {
		tmp = t_1;
	} else if (y_46_re <= -6.8e-11) {
		tmp = x_46_im * ((y_46_re / t_0) - (((y_46_im * x_46_re) / x_46_im) / t_0));
	} else if (y_46_re <= 7e-140) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 8.6e+54) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / 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_im * y_46_im) + (y_46_re * y_46_re)
	t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -2.3e+114:
		tmp = t_1
	elif y_46_re <= -6.8e-11:
		tmp = x_46_im * ((y_46_re / t_0) - (((y_46_im * x_46_re) / x_46_im) / t_0))
	elif y_46_re <= 7e-140:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 8.6e+54:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / 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_im * y_46_im) + Float64(y_46_re * y_46_re))
	t_1 = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.3e+114)
		tmp = t_1;
	elseif (y_46_re <= -6.8e-11)
		tmp = Float64(x_46_im * Float64(Float64(y_46_re / t_0) - Float64(Float64(Float64(y_46_im * x_46_re) / x_46_im) / t_0)));
	elseif (y_46_re <= 7e-140)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 8.6e+54)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / 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_im * y_46_im) + (y_46_re * y_46_re);
	t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.3e+114)
		tmp = t_1;
	elseif (y_46_re <= -6.8e-11)
		tmp = x_46_im * ((y_46_re / t_0) - (((y_46_im * x_46_re) / x_46_im) / t_0));
	elseif (y_46_re <= 7e-140)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 8.6e+54)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / 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$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e+114], t$95$1, If[LessEqual[y$46$re, -6.8e-11], N[(x$46$im * N[(N[(y$46$re / t$95$0), $MachinePrecision] - N[(N[(N[(y$46$im * x$46$re), $MachinePrecision] / x$46$im), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 7e-140], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 8.6e+54], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.3e114 or 8.59999999999999952e54 < y.re

   1. Initial program 41.4%

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

   3. Taylor expanded in y.re around 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 87.4%

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

   5. Simplified 87.4%

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

   if -2.3e114 < y.re < -6.7999999999999998e-11

   1. Initial program 82.5%

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

   3. Taylor expanded in x.im around inf

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-+l- N/A

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

      4. unsub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. neg-sub0 N/A

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

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

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

      9. neg-sub0 N/A

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

   5. Simplified 86.2%

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

   if -6.7999999999999998e-11 < y.re < 6.9999999999999996e-140

   1. Initial program 66.3%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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 87.9%

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

   5. Simplified 87.9%

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

   if 6.9999999999999996e-140 < y.re < 8.59999999999999952e54

   1. Initial program 83.7%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+114}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-11}:\\ \;\;\;\;x.im \cdot \left(\frac{y.re}{y.im \cdot y.im + y.re \cdot y.re} - \frac{\frac{y.im \cdot x.re}{x.im}}{y.im \cdot y.im + y.re \cdot y.re}\right)\\ \mathbf{elif}\;y.re \leq 7 \cdot 10^{-140}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 8.6 \cdot 10^{+54}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot \frac{y.re}{y.im \cdot y.im}\\ t_1 := \frac{\frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot t\_0\right) + x.re \cdot \left(t\_0 + -1\right)}{y.im}\\ \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 y.im) (- y.re (* y.re t_0))) (* x.re (+ t_0 -1.0)))
          y.im)))
   (if (<= y.im -2.05e+37)
     t_1
     (if (<= y.im 7e-81)
       (/ (- x.im (* x.re (/ y.im y.re))) y.re)
       (if (<= y.im 3.1e+129)
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re)))
         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 / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.05e+37) {
		tmp = t_1;
	} else if (y_46_im <= 7e-81) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.1e+129) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} 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 * (y_46re / (y_46im * y_46im))
    t_1 = (((x_46im / y_46im) * (y_46re - (y_46re * t_0))) + (x_46re * (t_0 + (-1.0d0)))) / y_46im
    if (y_46im <= (-2.05d+37)) then
        tmp = t_1
    else if (y_46im <= 7d-81) then
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 3.1d+129) then
        tmp = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    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 / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im;
	double tmp;
	if (y_46_im <= -2.05e+37) {
		tmp = t_1;
	} else if (y_46_im <= 7e-81) {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 3.1e+129) {
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	} 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 / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im
	tmp = 0
	if y_46_im <= -2.05e+37:
		tmp = t_1
	elif y_46_im <= 7e-81:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 3.1e+129:
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(y_46_re * Float64(y_46_re / Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(Float64(Float64(x_46_im / y_46_im) * Float64(y_46_re - Float64(y_46_re * t_0))) + Float64(x_46_re * Float64(t_0 + -1.0))) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.05e+37)
		tmp = t_1;
	elseif (y_46_im <= 7e-81)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 3.1e+129)
		tmp = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)));
	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 / y_46_im) * (y_46_re - (y_46_re * t_0))) + (x_46_re * (t_0 + -1.0))) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -2.05e+37)
		tmp = t_1;
	elseif (y_46_im <= 7e-81)
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 3.1e+129)
		tmp = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	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[(y$46$re * N[(y$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x$46$im / y$46$im), $MachinePrecision] * N[(y$46$re - N[(y$46$re * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x$46$re * N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -2.05e+37], t$95$1, If[LessEqual[y$46$im, 7e-81], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+129], N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -2.0499999999999999e37 or 3.1e129 < y.im

   1. Initial program 41.5%

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

   3. Taylor expanded in y.im around inf

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

   5. Simplified 91.7%

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

   if -2.0499999999999999e37 < y.im < 6.99999999999999973e-81

   1. Initial program 71.0%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.im + -1 \cdot \frac{x.re \cdot y.im}{y.re}}{y.re}} \]
   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 86.0%

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

   5. Simplified 86.0%

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

   if 6.99999999999999973e-81 < y.im < 3.1e129

   1. Initial program 79.6%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{+37}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im}\right)\right) + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \mathbf{elif}\;y.im \leq 7 \cdot 10^{-81}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+129}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im}{y.im} \cdot \left(y.re - y.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im}\right)\right) + x.re \cdot \left(y.re \cdot \frac{y.re}{y.im \cdot y.im} + -1\right)}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ t_1 := \frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -9 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+55}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* y.im x.re)) (+ (* y.im y.im) (* y.re y.re))))
        (t_1 (/ (- x.im (* x.re (/ y.im y.re))) y.re)))
   (if (<= y.re -9e+111)
     t_1
     (if (<= y.re -3.4e-43)
       t_0
       (if (<= y.re 1.55e-137)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.05e+55) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -9e+111) {
		tmp = t_1;
	} else if (y_46_re <= -3.4e-43) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-137) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.05e+55) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (y_46im * x_46re)) / ((y_46im * y_46im) + (y_46re * y_46re))
    t_1 = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    if (y_46re <= (-9d+111)) then
        tmp = t_1
    else if (y_46re <= (-3.4d-43)) then
        tmp = t_0
    else if (y_46re <= 1.55d-137) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 1.05d+55) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	double t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	double tmp;
	if (y_46_re <= -9e+111) {
		tmp = t_1;
	} else if (y_46_re <= -3.4e-43) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e-137) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.05e+55) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re))
	t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	tmp = 0
	if y_46_re <= -9e+111:
		tmp = t_1
	elif y_46_re <= -3.4e-43:
		tmp = t_0
	elif y_46_re <= 1.55e-137:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 1.05e+55:
		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(x_46_im * y_46_re) - Float64(y_46_im * x_46_re)) / Float64(Float64(y_46_im * y_46_im) + Float64(y_46_re * y_46_re)))
	t_1 = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -9e+111)
		tmp = t_1;
	elseif (y_46_re <= -3.4e-43)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-137)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.05e+55)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (y_46_im * x_46_re)) / ((y_46_im * y_46_im) + (y_46_re * y_46_re));
	t_1 = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -9e+111)
		tmp = t_1;
	elseif (y_46_re <= -3.4e-43)
		tmp = t_0;
	elseif (y_46_re <= 1.55e-137)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.05e+55)
		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[(x$46$im * y$46$re), $MachinePrecision] - N[(y$46$im * x$46$re), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$im * y$46$im), $MachinePrecision] + N[(y$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -9e+111], t$95$1, If[LessEqual[y$46$re, -3.4e-43], t$95$0, If[LessEqual[y$46$re, 1.55e-137], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.05e+55], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.00000000000000001e111 or 1.05e55 < y.re

   1. Initial program 41.4%

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

   3. Taylor expanded in y.re around 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 87.4%

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

   5. Simplified 87.4%

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

   if -9.00000000000000001e111 < y.re < -3.4000000000000001e-43 or 1.54999999999999989e-137 < y.re < 1.05e55

   1. Initial program 83.2%

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

   if -3.4000000000000001e-43 < y.re < 1.54999999999999989e-137

   1. Initial program 65.5%

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

   3. Taylor expanded in y.re around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   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 89.3%

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

   5. Simplified 89.3%

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

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9 \cdot 10^{+111}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.4 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{-137}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.05 \cdot 10^{+55}:\\ \;\;\;\;\frac{x.im \cdot y.re - y.im \cdot x.re}{y.im \cdot y.im + y.re \cdot y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1500:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1500.0)
   (/ (- x.im (/ x.re (/ y.re y.im))) y.re)
   (if (<= y.re 1.9e-45)
     (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
     (/ (- x.im (* x.re (/ y.im y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1500.0) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= 1.9e-45) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1500.0d0)) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else if (y_46re <= 1.9d-45) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else
        tmp = (x_46im - (x_46re * (y_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1500.0) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= 1.9e-45) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else {
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1500.0:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	elif y_46_re <= 1.9e-45:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	else:
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1500.0)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_re <= 1.9e-45)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	else
		tmp = Float64(Float64(x_46_im - Float64(x_46_re * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1500.0)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_re <= 1.9e-45)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	else
		tmp = (x_46_im - (x_46_re * (y_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1500.0], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.9e-45], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$im - N[(x$46$re * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1500

   1. Initial program 64.8%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. 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 80.2%

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

   5. Simplified 80.2%

      \[\leadsto \color{blue}{\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}} \]
   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. un-div-inv N/A

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

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

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

      4. /-lowering-/.f64 80.3%

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

   7. Applied egg-rr 80.3%

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

   if -1500 < y.re < 1.89999999999999999e-45

   1. Initial program 70.9%

      \[\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 84.9%

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

   5. Simplified 84.9%

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

   if 1.89999999999999999e-45 < y.re

   1. Initial program 50.8%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. 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 81.8%

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

   5. Simplified 81.8%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1500:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{-45}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+36}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -2.2e+36)
     t_0
     (if (<= y.im 9e+29) (/ (- x.im (/ x.re (/ y.re y.im))) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.2e+36) {
		tmp = t_0;
	} else if (y_46_im <= 9e+29) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.0d0 - (x_46re / y_46im)
    if (y_46im <= (-2.2d+36)) then
        tmp = t_0
    else if (y_46im <= 9d+29) then
        tmp = (x_46im - (x_46re / (y_46re / y_46im))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 0.0 - (x_46_re / y_46_im);
	double tmp;
	if (y_46_im <= -2.2e+36) {
		tmp = t_0;
	} else if (y_46_im <= 9e+29) {
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 0.0 - (x_46_re / y_46_im)
	tmp = 0
	if y_46_im <= -2.2e+36:
		tmp = t_0
	elif y_46_im <= 9e+29:
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -2.2e+36)
		tmp = t_0;
	elseif (y_46_im <= 9e+29)
		tmp = Float64(Float64(x_46_im - Float64(x_46_re / Float64(y_46_re / y_46_im))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 0.0 - (x_46_re / y_46_im);
	tmp = 0.0;
	if (y_46_im <= -2.2e+36)
		tmp = t_0;
	elseif (y_46_im <= 9e+29)
		tmp = (x_46_im - (x_46_re / (y_46_re / y_46_im))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -2.2e+36], t$95$0, If[LessEqual[y$46$im, 9e+29], N[(N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.2e36 or 9.0000000000000005e29 < y.im

   1. Initial program 48.7%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 77.4%

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

   5. Simplified 77.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 77.4%

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

   7. Applied egg-rr 77.4%

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

   if -2.2e36 < y.im < 9.0000000000000005e29

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. 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 80.6%

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

   5. Simplified 80.6%

      \[\leadsto \color{blue}{\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}} \]
   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. un-div-inv N/A

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

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

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

      4. /-lowering-/.f64 80.6%

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

   7. Applied egg-rr 80.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{+36}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 9 \cdot 10^{+29}:\\ \;\;\;\;\frac{x.im - \frac{x.re}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0 - \frac{x.re}{y.im}\\ \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+37}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 1.52 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- 0.0 (/ x.re y.im))))
   (if (<= y.im -1.95e+37)
     t_0
     (if (<= y.im 1.52e+32) (/ (- 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 <= -1.95e+37) {
		tmp = t_0;
	} else if (y_46_im <= 1.52e+32) {
		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 <= (-1.95d+37)) then
        tmp = t_0
    else if (y_46im <= 1.52d+32) 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 <= -1.95e+37) {
		tmp = t_0;
	} else if (y_46_im <= 1.52e+32) {
		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 <= -1.95e+37:
		tmp = t_0
	elif y_46_im <= 1.52e+32:
		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(0.0 - Float64(x_46_re / y_46_im))
	tmp = 0.0
	if (y_46_im <= -1.95e+37)
		tmp = t_0;
	elseif (y_46_im <= 1.52e+32)
		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 <= -1.95e+37)
		tmp = t_0;
	elseif (y_46_im <= 1.52e+32)
		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[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -1.95e+37], t$95$0, If[LessEqual[y$46$im, 1.52e+32], 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 < -1.9499999999999999e37 or 1.5200000000000001e32 < y.im

   1. Initial program 48.7%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 77.4%

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

   5. Simplified 77.4%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 77.4%

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

   7. Applied egg-rr 77.4%

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

   if -1.9499999999999999e37 < y.im < 1.5200000000000001e32

   1. Initial program 73.2%

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

   3. Taylor expanded in y.re around inf

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

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

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

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

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

      5. 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 80.6%

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

   5. Simplified 80.6%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.95 \cdot 10^{+37}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{elif}\;y.im \leq 1.52 \cdot 10^{+32}:\\ \;\;\;\;\frac{x.im - x.re \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -6e+65)
   (/ x.im y.re)
   (if (<= y.re 9.5e-7) (- 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 <= -6e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9.5e-7) {
		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 <= (-6d+65)) then
        tmp = x_46im / y_46re
    else if (y_46re <= 9.5d-7) 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 <= -6e+65) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= 9.5e-7) {
		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 <= -6e+65:
		tmp = x_46_im / y_46_re
	elif y_46_re <= 9.5e-7:
		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 <= -6e+65)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= 9.5e-7)
		tmp = Float64(0.0 - Float64(x_46_re / y_46_im));
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -6e+65)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= 9.5e-7)
		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, -6e+65], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-7], N[(0.0 - N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.0000000000000004e65 or 9.5000000000000001e-7 < y.re

   1. Initial program 53.4%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 69.8%

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

   5. Simplified 69.8%

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

   if -6.0000000000000004e65 < y.re < 9.5000000000000001e-7

   1. Initial program 72.6%

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

   3. Taylor expanded in y.re around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

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

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

      4. /-lowering-/.f64 69.8%

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

   5. Simplified 69.8%

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

      1. sub0-neg N/A

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

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

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

      3. /-lowering-/.f64 69.8%

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

   7. Applied egg-rr 69.8%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-7}:\\ \;\;\;\;0 - \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.7% 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 63.3%

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

3. Taylor expanded in y.re around inf

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

   1. /-lowering-/.f64 40.9%

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

5. Simplified 40.9%

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

Reproduce

herbie shell --seed 2024154 
(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))))