_divideComplex, real part

Percentage Accurate: 61.0% → 82.3%

Time: 8.2s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+136}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-81}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+83}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* y.re x.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ (+ x.re (/ x.im (/ y.re y.im))) y.re)))
   (if (<= y.re -2.6e+136)
     t_1
     (if (<= y.re -1.9e-81)
       t_0
       (if (<= y.re 2.5e-130)
         (/ (+ x.im (/ (* y.re x.re) y.im)) y.im)
         (if (<= y.re 6.2e+83) 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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.6e+136) {
		tmp = t_1;
	} else if (y_46_re <= -1.9e-81) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-130) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 6.2e+83) {
		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_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re + (x_46im / (y_46re / y_46im))) / y_46re
    if (y_46re <= (-2.6d+136)) then
        tmp = t_1
    else if (y_46re <= (-1.9d-81)) then
        tmp = t_0
    else if (y_46re <= 2.5d-130) then
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    else if (y_46re <= 6.2d+83) 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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.6e+136) {
		tmp = t_1;
	} else if (y_46_re <= -1.9e-81) {
		tmp = t_0;
	} else if (y_46_re <= 2.5e-130) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 6.2e+83) {
		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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re
	tmp = 0
	if y_46_re <= -2.6e+136:
		tmp = t_1
	elif y_46_re <= -1.9e-81:
		tmp = t_0
	elif y_46_re <= 2.5e-130:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im
	elif y_46_re <= 6.2e+83:
		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_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.6e+136)
		tmp = t_1;
	elseif (y_46_re <= -1.9e-81)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-130)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 6.2e+83)
		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_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.6e+136)
		tmp = t_1;
	elseif (y_46_re <= -1.9e-81)
		tmp = t_0;
	elseif (y_46_re <= 2.5e-130)
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	elseif (y_46_re <= 6.2e+83)
		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$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+136], t$95$1, If[LessEqual[y$46$re, -1.9e-81], t$95$0, If[LessEqual[y$46$re, 2.5e-130], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 6.2e+83], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.6000000000000001e136 or 6.19999999999999984e83 < y.re

   1. Initial program 44.8%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 86.0%

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

   5. Simplified 86.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 90.3%

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

   7. Applied egg-rr 90.3%

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

   if -2.6000000000000001e136 < y.re < -1.8999999999999999e-81 or 2.4999999999999998e-130 < y.re < 6.19999999999999984e83

   1. Initial program 81.3%

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

   if -1.8999999999999999e-81 < y.re < 2.4999999999999998e-130

   1. Initial program 65.6%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 92.7%

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

   5. Simplified 92.7%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+136}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-81}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{-130}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 6.2 \cdot 10^{+83}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 76.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{-71}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 5.4 \cdot 10^{+17}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (/ x.im (/ y.re y.im))) y.re)))
   (if (<= y.re -2.1e-71)
     t_0
     (if (<= y.re 5.4e+17) (/ (+ x.im (/ (* y.re x.re) y.im)) 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_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.1e-71) {
		tmp = t_0;
	} else if (y_46_re <= 5.4e+17) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / 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_46re + (x_46im / (y_46re / y_46im))) / y_46re
    if (y_46re <= (-2.1d-71)) then
        tmp = t_0
    else if (y_46re <= 5.4d+17) then
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / 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_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	double tmp;
	if (y_46_re <= -2.1e-71) {
		tmp = t_0;
	} else if (y_46_re <= 5.4e+17) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / 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_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re
	tmp = 0
	if y_46_re <= -2.1e-71:
		tmp = t_0
	elif y_46_re <= 5.4e+17:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / 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_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -2.1e-71)
		tmp = t_0;
	elseif (y_46_re <= 5.4e+17)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / 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_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -2.1e-71)
		tmp = t_0;
	elseif (y_46_re <= 5.4e+17)
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / 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$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e-71], t$95$0, If[LessEqual[y$46$re, 5.4e+17], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.1000000000000001e-71 or 5.4e17 < y.re

   1. Initial program 62.6%

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

   3. Taylor expanded in y.re around inf

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

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

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

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

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

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

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

      4. *-lowering-*.f64 78.0%

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

   5. Simplified 78.0%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 80.4%

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

   7. Applied egg-rr 80.4%

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

   if -2.1000000000000001e-71 < y.re < 5.4e17

   1. Initial program 68.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 84.0%

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

   5. Simplified 84.0%

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

Alternative 3: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.15 \cdot 10^{-32}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.15e-32)
   (/ x.re y.re)
   (if (<= y.re 3.1e+18)
     (/ (+ x.im (/ (* y.re x.re) y.im)) y.im)
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.15e-32) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.1e+18) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.15d-32)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.1d+18) then
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.15e-32) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.1e+18) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.15e-32:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.1e+18:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.15e-32)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.1e+18)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.15e-32)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.1e+18)
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.15e-32], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.1e+18], N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.14999999999999995e-32 or 3.1e18 < y.re

   1. Initial program 61.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 72.2%

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

   5. Simplified 72.2%

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

   if -2.14999999999999995e-32 < y.re < 3.1e18

   1. Initial program 69.3%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 81.2%

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

   5. Simplified 81.2%

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

Alternative 4: 71.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.22 \cdot 10^{-30}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.22e-30)
   (/ x.re y.re)
   (if (<= y.re 2.7e+18)
     (/ (+ x.im (* x.re (/ y.re y.im))) y.im)
     (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.22e-30) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.7e+18) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.22d-30)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.7d+18) then
        tmp = (x_46im + (x_46re * (y_46re / y_46im))) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.22e-30) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.7e+18) {
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.22e-30:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.7e+18:
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.22e-30)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.7e+18)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(y_46_re / y_46_im))) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.22e-30)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.7e+18)
		tmp = (x_46_im + (x_46_re * (y_46_re / y_46_im))) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.22e-30], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.7e+18], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.22e-30 or 2.7e18 < y.re

   1. Initial program 61.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 72.2%

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

   5. Simplified 72.2%

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

   if -1.22e-30 < y.re < 2.7e18

   1. Initial program 69.3%

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

   3. Taylor expanded in x.im around inf

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

      9. associate-/r* N/A

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

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

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

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

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

      12. *-commutative N/A

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

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

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

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

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

      15. unpow2 N/A

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

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

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

      17. unpow2 N/A

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

      18. *-lowering-*.f64 67.6%

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

   5. Simplified 67.6%

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

   6. Taylor expanded in y.im around inf

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

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

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

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

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

      3. associate-/l* N/A

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

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

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

      5. /-lowering-/.f64 80.6%

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

   8. Simplified 80.6%

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

Alternative 5: 64.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -8.2 \cdot 10^{-67}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 1.3 \cdot 10^{+18}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -8.2e-67)
   (/ x.im y.im)
   (if (<= y.im 1.3e+18) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -8.2e-67) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.3e+18) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-8.2d-67)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 1.3d+18) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -8.2e-67) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 1.3e+18) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -8.2e-67:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 1.3e+18:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -8.2e-67)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 1.3e+18)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -8.2e-67)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 1.3e+18)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -8.2e-67], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.3e+18], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -8.1999999999999994e-67 or 1.3e18 < y.im

   1. Initial program 55.0%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 64.0%

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

   5. Simplified 64.0%

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

   if -8.1999999999999994e-67 < y.im < 1.3e18

   1. Initial program 77.5%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 69.3%

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

   5. Simplified 69.3%

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

Alternative 6: 43.1% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.6%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 41.4%

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

5. Simplified 41.4%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))