_divideComplex, real part

Percentage Accurate: 61.2% → 77.8%

Time: 11.2s

Alternatives: 9

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: 61.2% 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: 77.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ t_1 := \frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+131}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{+80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (/ (* y.re x.re) y.im)) y.im))
        (t_1
         (/ (+ (* y.re x.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -3.6e+131)
     (* (* x.im (+ (/ x.re x.im) (/ y.im y.re))) (/ 1.0 y.re))
     (if (<= y.re -9e+80)
       t_0
       (if (<= y.re -1.75e-65)
         t_1
         (if (<= y.re 1.8e-80)
           t_0
           (if (<= y.re 2.15e+151)
             t_1
             (* (/ 1.0 y.re) (* y.im (+ (/ x.im y.re) (/ x.re y.im)))))))))))

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 * x_46_re) / y_46_im)) / y_46_im;
	double t_1 = ((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 tmp;
	if (y_46_re <= -3.6e+131) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -9e+80) {
		tmp = t_0;
	} else if (y_46_re <= -1.75e-65) {
		tmp = t_1;
	} else if (y_46_re <= 1.8e-80) {
		tmp = t_0;
	} else if (y_46_re <= 2.15e+151) {
		tmp = t_1;
	} else {
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    t_1 = ((y_46re * x_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-3.6d+131)) then
        tmp = (x_46im * ((x_46re / x_46im) + (y_46im / y_46re))) * (1.0d0 / y_46re)
    else if (y_46re <= (-9d+80)) then
        tmp = t_0
    else if (y_46re <= (-1.75d-65)) then
        tmp = t_1
    else if (y_46re <= 1.8d-80) then
        tmp = t_0
    else if (y_46re <= 2.15d+151) then
        tmp = t_1
    else
        tmp = (1.0d0 / y_46re) * (y_46im * ((x_46im / y_46re) + (x_46re / 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 t_0 = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	double t_1 = ((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 tmp;
	if (y_46_re <= -3.6e+131) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -9e+80) {
		tmp = t_0;
	} else if (y_46_re <= -1.75e-65) {
		tmp = t_1;
	} else if (y_46_re <= 1.8e-80) {
		tmp = t_0;
	} else if (y_46_re <= 2.15e+151) {
		tmp = t_1;
	} else {
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)));
	}
	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 * x_46_re) / y_46_im)) / y_46_im
	t_1 = ((y_46_re * x_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3.6e+131:
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re)
	elif y_46_re <= -9e+80:
		tmp = t_0
	elif y_46_re <= -1.75e-65:
		tmp = t_1
	elif y_46_re <= 1.8e-80:
		tmp = t_0
	elif y_46_re <= 2.15e+151:
		tmp = t_1
	else:
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im)
	t_1 = 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)))
	tmp = 0.0
	if (y_46_re <= -3.6e+131)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re / x_46_im) + Float64(y_46_im / y_46_re))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= -9e+80)
		tmp = t_0;
	elseif (y_46_re <= -1.75e-65)
		tmp = t_1;
	elseif (y_46_re <= 1.8e-80)
		tmp = t_0;
	elseif (y_46_re <= 2.15e+151)
		tmp = t_1;
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) + Float64(x_46_re / y_46_im))));
	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 * x_46_re) / y_46_im)) / y_46_im;
	t_1 = ((y_46_re * x_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.6e+131)
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	elseif (y_46_re <= -9e+80)
		tmp = t_0;
	elseif (y_46_re <= -1.75e-65)
		tmp = t_1;
	elseif (y_46_re <= 1.8e-80)
		tmp = t_0;
	elseif (y_46_re <= 2.15e+151)
		tmp = t_1;
	else
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, Block[{t$95$1 = 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]}, If[LessEqual[y$46$re, -3.6e+131], N[(N[(x$46$im * N[(N[(x$46$re / x$46$im), $MachinePrecision] + N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -9e+80], t$95$0, If[LessEqual[y$46$re, -1.75e-65], t$95$1, If[LessEqual[y$46$re, 1.8e-80], t$95$0, If[LessEqual[y$46$re, 2.15e+151], t$95$1, N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.60000000000000031e131

   1. Initial program 29.3%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 29.3%

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

   5. Simplified 29.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 40.4%

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

   7. Applied egg-rr 40.4%

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

   8. Taylor expanded in x.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 84.8%

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

   10. Simplified 84.8%

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

   if -3.60000000000000031e131 < y.re < -9.00000000000000013e80 or -1.75000000000000002e-65 < y.re < 1.8e-80

   1. Initial program 65.9%

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

   3. Taylor expanded in y.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 95.2%

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

   5. Simplified 95.2%

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

   if -9.00000000000000013e80 < y.re < -1.75000000000000002e-65 or 1.8e-80 < y.re < 2.14999999999999991e151

   1. Initial program 87.9%

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

   if 2.14999999999999991e151 < y.re

   1. Initial program 21.3%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 21.3%

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

   5. Simplified 21.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 51.2%

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

   7. Applied egg-rr 51.2%

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

   8. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 88.4%

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

   10. Simplified 88.4%

       \[\leadsto \color{blue}{\left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)} \cdot \frac{1}{y.re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+131}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -9 \cdot 10^{+80}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-65}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.8 \cdot 10^{-80}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 2.15 \cdot 10^{+151}:\\ \;\;\;\;\frac{y.re \cdot x.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+126}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{t\_0}, y.im, \frac{y.re \cdot x.re}{t\_0}\right)\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\frac{t\_0}{y.re \cdot x.re + x.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im))))
   (if (<= y.re -4.4e+126)
     (* (* x.im (+ (/ x.re x.im) (/ y.im y.re))) (/ 1.0 y.re))
     (if (<= y.re -1.25e-54)
       (fma (/ x.im t_0) y.im (/ (* y.re x.re) t_0))
       (if (<= y.re 3.6e-85)
         (/ (+ x.im (/ (* y.re x.re) y.im)) y.im)
         (if (<= y.re 1.85e+151)
           (/ 1.0 (/ t_0 (+ (* y.re x.re) (* x.im y.im))))
           (* (/ 1.0 y.re) (* y.im (+ (/ x.im y.re) (/ x.re y.im))))))))))

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 tmp;
	if (y_46_re <= -4.4e+126) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -1.25e-54) {
		tmp = fma((x_46_im / t_0), y_46_im, ((y_46_re * x_46_re) / t_0));
	} else if (y_46_re <= 3.6e-85) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 1.85e+151) {
		tmp = 1.0 / (t_0 / ((y_46_re * x_46_re) + (x_46_im * y_46_im)));
	} else {
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	tmp = 0.0
	if (y_46_re <= -4.4e+126)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re / x_46_im) + Float64(y_46_im / y_46_re))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= -1.25e-54)
		tmp = fma(Float64(x_46_im / t_0), y_46_im, Float64(Float64(y_46_re * x_46_re) / t_0));
	elseif (y_46_re <= 3.6e-85)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 1.85e+151)
		tmp = Float64(1.0 / Float64(t_0 / Float64(Float64(y_46_re * x_46_re) + Float64(x_46_im * y_46_im))));
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) + Float64(x_46_re / y_46_im))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -4.4e+126], N[(N[(x$46$im * N[(N[(x$46$re / x$46$im), $MachinePrecision] + N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.25e-54], N[(N[(x$46$im / t$95$0), $MachinePrecision] * y$46$im + N[(N[(y$46$re * x$46$re), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 3.6e-85], 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, 1.85e+151], N[(1.0 / N[(t$95$0 / N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -4.39999999999999997e126

   1. Initial program 27.6%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 27.6%

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

   5. Simplified 27.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 37.9%

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

   7. Applied egg-rr 37.9%

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

   8. Taylor expanded in x.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 81.5%

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

   10. Simplified 81.5%

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

   if -4.39999999999999997e126 < y.re < -1.25000000000000004e-54

   1. Initial program 75.7%

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

      1. div-inv N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-in N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. associate-*l* N/A

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

      7. fma-define N/A

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

      8. un-div-inv N/A

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

      9. fma-undefine N/A

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

      10. *-commutative N/A

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

      11. fma-undefine N/A

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

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

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

   4. Applied egg-rr 80.1%

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

   if -1.25000000000000004e-54 < y.re < 3.5999999999999998e-85

   1. Initial program 71.1%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. /-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 96.8%

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

   5. Simplified 96.8%

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

   if 3.5999999999999998e-85 < y.re < 1.8499999999999999e151

   1. Initial program 92.8%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 92.8%

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

   4. Applied egg-rr 92.8%

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

   if 1.8499999999999999e151 < y.re

   1. Initial program 21.3%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 21.3%

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

   5. Simplified 21.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 51.2%

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

   7. Applied egg-rr 51.2%

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

   8. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 88.4%

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

   10. Simplified 88.4%

       \[\leadsto \color{blue}{\left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)} \cdot \frac{1}{y.re} \]
3. Recombined 5 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.4 \cdot 10^{+126}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.25 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x.im}{y.re \cdot y.re + y.im \cdot y.im}, y.im, \frac{y.re \cdot x.re}{y.re \cdot y.re + y.im \cdot y.im}\right)\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 1.85 \cdot 10^{+151}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}\\ \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+123}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{+152}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/
          1.0
          (/
           (+ (* y.re y.re) (* y.im y.im))
           (+ (* y.re x.re) (* x.im y.im))))))
   (if (<= y.re -3.2e+123)
     (* (* x.im (+ (/ x.re x.im) (/ y.im y.re))) (/ 1.0 y.re))
     (if (<= y.re -6.8e-65)
       t_0
       (if (<= y.re 6e-85)
         (/ (+ x.im (/ (* y.re x.re) y.im)) y.im)
         (if (<= y.re 5.1e+152)
           t_0
           (* (/ 1.0 y.re) (* y.im (+ (/ x.im y.re) (/ x.re y.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_re) + (x_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -3.2e+123) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -6.8e-65) {
		tmp = t_0;
	} else if (y_46_re <= 6e-85) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 5.1e+152) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / 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) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 / (((y_46re * y_46re) + (y_46im * y_46im)) / ((y_46re * x_46re) + (x_46im * y_46im)))
    if (y_46re <= (-3.2d+123)) then
        tmp = (x_46im * ((x_46re / x_46im) + (y_46im / y_46re))) * (1.0d0 / y_46re)
    else if (y_46re <= (-6.8d-65)) then
        tmp = t_0
    else if (y_46re <= 6d-85) then
        tmp = (x_46im + ((y_46re * x_46re) / y_46im)) / y_46im
    else if (y_46re <= 5.1d+152) then
        tmp = t_0
    else
        tmp = (1.0d0 / y_46re) * (y_46im * ((x_46im / y_46re) + (x_46re / 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 t_0 = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_re) + (x_46_im * y_46_im)));
	double tmp;
	if (y_46_re <= -3.2e+123) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -6.8e-65) {
		tmp = t_0;
	} else if (y_46_re <= 6e-85) {
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	} else if (y_46_re <= 5.1e+152) {
		tmp = t_0;
	} else {
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_re) + (x_46_im * y_46_im)))
	tmp = 0
	if y_46_re <= -3.2e+123:
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re)
	elif y_46_re <= -6.8e-65:
		tmp = t_0
	elif y_46_re <= 6e-85:
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im
	elif y_46_re <= 5.1e+152:
		tmp = t_0
	else:
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(1.0 / Float64(Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)) / Float64(Float64(y_46_re * x_46_re) + Float64(x_46_im * y_46_im))))
	tmp = 0.0
	if (y_46_re <= -3.2e+123)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re / x_46_im) + Float64(y_46_im / y_46_re))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= -6.8e-65)
		tmp = t_0;
	elseif (y_46_re <= 6e-85)
		tmp = Float64(Float64(x_46_im + Float64(Float64(y_46_re * x_46_re) / y_46_im)) / y_46_im);
	elseif (y_46_re <= 5.1e+152)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / y_46_re) * Float64(y_46_im * Float64(Float64(x_46_im / y_46_re) + Float64(x_46_re / y_46_im))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = 1.0 / (((y_46_re * y_46_re) + (y_46_im * y_46_im)) / ((y_46_re * x_46_re) + (x_46_im * y_46_im)));
	tmp = 0.0;
	if (y_46_re <= -3.2e+123)
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	elseif (y_46_re <= -6.8e-65)
		tmp = t_0;
	elseif (y_46_re <= 6e-85)
		tmp = (x_46_im + ((y_46_re * x_46_re) / y_46_im)) / y_46_im;
	elseif (y_46_re <= 5.1e+152)
		tmp = t_0;
	else
		tmp = (1.0 / y_46_re) * (y_46_im * ((x_46_im / y_46_re) + (x_46_re / y_46_im)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(1.0 / N[(N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * x$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.2e+123], N[(N[(x$46$im * N[(N[(x$46$re / x$46$im), $MachinePrecision] + N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -6.8e-65], t$95$0, If[LessEqual[y$46$re, 6e-85], 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, 5.1e+152], t$95$0, N[(N[(1.0 / y$46$re), $MachinePrecision] * N[(y$46$im * N[(N[(x$46$im / y$46$re), $MachinePrecision] + N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.20000000000000005e123

   1. Initial program 27.6%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 27.6%

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

   5. Simplified 27.6%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 37.9%

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

   7. Applied egg-rr 37.9%

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

   8. Taylor expanded in x.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 81.5%

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

   10. Simplified 81.5%

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

   if -3.20000000000000005e123 < y.re < -6.79999999999999973e-65 or 6.00000000000000044e-85 < y.re < 5.0999999999999996e152

   1. Initial program 83.1%

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

      1. clear-num N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

      9. *-lowering-*.f64 83.2%

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

   4. Applied egg-rr 83.2%

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

   if -6.79999999999999973e-65 < y.re < 6.00000000000000044e-85

   1. Initial program 70.8%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. /-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 97.8%

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

   5. Simplified 97.8%

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

   if 5.0999999999999996e152 < y.re

   1. Initial program 21.3%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 21.3%

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

   5. Simplified 21.3%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 51.2%

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

   7. Applied egg-rr 51.2%

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

   8. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 88.4%

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

   10. Simplified 88.4%

       \[\leadsto \color{blue}{\left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)} \cdot \frac{1}{y.re} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.2 \cdot 10^{+123}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -6.8 \cdot 10^{-65}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-85}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.re \leq 5.1 \cdot 10^{+152}:\\ \;\;\;\;\frac{1}{\frac{y.re \cdot y.re + y.im \cdot y.im}{y.re \cdot x.re + x.im \cdot y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.re} \cdot \left(y.im \cdot \left(\frac{x.im}{y.re} + \frac{x.re}{y.im}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.05 \cdot 10^{+176}:\\ \;\;\;\;\left(x.im \cdot \left(\frac{x.re}{x.im} + \frac{y.im}{y.re}\right)\right) \cdot \frac{1}{y.re}\\ \mathbf{elif}\;y.re \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 2.7 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)))
   (if (<= y.re -1.05e+176)
     (* (* x.im (+ (/ x.re x.im) (/ y.im y.re))) (/ 1.0 y.re))
     (if (<= y.re -1.5e-62)
       t_0
       (if (<= y.re 2.7e-18) (/ (+ 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_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.05e+176) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -1.5e-62) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-18) {
		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_46im) / y_46re)) / y_46re
    if (y_46re <= (-1.05d+176)) then
        tmp = (x_46im * ((x_46re / x_46im) + (y_46im / y_46re))) * (1.0d0 / y_46re)
    else if (y_46re <= (-1.5d-62)) then
        tmp = t_0
    else if (y_46re <= 2.7d-18) 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_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.05e+176) {
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	} else if (y_46_re <= -1.5e-62) {
		tmp = t_0;
	} else if (y_46_re <= 2.7e-18) {
		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_im) / y_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -1.05e+176:
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re)
	elif y_46_re <= -1.5e-62:
		tmp = t_0
	elif y_46_re <= 2.7e-18:
		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(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.05e+176)
		tmp = Float64(Float64(x_46_im * Float64(Float64(x_46_re / x_46_im) + Float64(y_46_im / y_46_re))) * Float64(1.0 / y_46_re));
	elseif (y_46_re <= -1.5e-62)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-18)
		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_im) / y_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.05e+176)
		tmp = (x_46_im * ((x_46_re / x_46_im) + (y_46_im / y_46_re))) * (1.0 / y_46_re);
	elseif (y_46_re <= -1.5e-62)
		tmp = t_0;
	elseif (y_46_re <= 2.7e-18)
		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[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.05e+176], N[(N[(x$46$im * N[(N[(x$46$re / x$46$im), $MachinePrecision] + N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.5e-62], t$95$0, If[LessEqual[y$46$re, 2.7e-18], 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 3 regimes

2. if y.re < -1.05e176

   1. Initial program 21.7%

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

   3. Taylor expanded in y.re around inf

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

      1. unpow2 N/A

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

      2. *-lowering-*.f64 21.7%

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

   5. Simplified 21.7%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

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

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

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

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

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

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

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

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

      9. *-commutative N/A

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

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

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

      11. /-lowering-/.f64 29.8%

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

   7. Applied egg-rr 29.8%

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

   8. Taylor expanded in x.im around inf

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

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

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

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

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

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

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

      4. /-lowering-/.f64 83.7%

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

   10. Simplified 83.7%

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

   if -1.05e176 < y.re < -1.5000000000000001e-62 or 2.69999999999999989e-18 < y.re

   1. Initial program 61.4%

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

   3. Taylor expanded in y.re around 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 72.6%

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

   5. Simplified 72.6%

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

   if -1.5000000000000001e-62 < y.re < 2.69999999999999989e-18

   1. Initial program 72.8%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. /-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 95.2%

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

   5. Simplified 95.2%

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

Alternative 5: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}\\ \mathbf{if}\;y.re \leq -1.5 \cdot 10^{-62}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.3 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.re (/ (* x.im y.im) y.re)) y.re)))
   (if (<= y.re -1.5e-62)
     t_0
     (if (<= y.re 1.3e-25) (/ (+ 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_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.5e-62) {
		tmp = t_0;
	} else if (y_46_re <= 1.3e-25) {
		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_46im) / y_46re)) / y_46re
    if (y_46re <= (-1.5d-62)) then
        tmp = t_0
    else if (y_46re <= 1.3d-25) 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_im) / y_46_re)) / y_46_re;
	double tmp;
	if (y_46_re <= -1.5e-62) {
		tmp = t_0;
	} else if (y_46_re <= 1.3e-25) {
		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_im) / y_46_re)) / y_46_re
	tmp = 0
	if y_46_re <= -1.5e-62:
		tmp = t_0
	elif y_46_re <= 1.3e-25:
		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(Float64(x_46_im * y_46_im) / y_46_re)) / y_46_re)
	tmp = 0.0
	if (y_46_re <= -1.5e-62)
		tmp = t_0;
	elseif (y_46_re <= 1.3e-25)
		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_im) / y_46_re)) / y_46_re;
	tmp = 0.0;
	if (y_46_re <= -1.5e-62)
		tmp = t_0;
	elseif (y_46_re <= 1.3e-25)
		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[(N[(x$46$im * y$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]}, If[LessEqual[y$46$re, -1.5e-62], t$95$0, If[LessEqual[y$46$re, 1.3e-25], 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 < -1.5000000000000001e-62 or 1.3e-25 < y.re

   1. Initial program 53.2%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 71.4%

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

   5. Simplified 71.4%

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

   if -1.5000000000000001e-62 < y.re < 1.3e-25

   1. Initial program 72.8%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

      3. /-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 95.2%

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

   5. Simplified 95.2%

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

Alternative 6: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5 \cdot 10^{+110}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 1.15 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -5e+110)
   (/ x.re y.re)
   (if (<= y.re 1.15e-25)
     (/ (+ 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 <= -5e+110) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.15e-25) {
		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 <= (-5d+110)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 1.15d-25) 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 <= -5e+110) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 1.15e-25) {
		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 <= -5e+110:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 1.15e-25:
		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 <= -5e+110)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 1.15e-25)
		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 <= -5e+110)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 1.15e-25)
		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, -5e+110], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 1.15e-25], 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 < -4.99999999999999978e110 or 1.15e-25 < y.re

   1. Initial program 43.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 70.0%

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

   5. Simplified 70.0%

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

   if -4.99999999999999978e110 < y.re < 1.15e-25

   1. Initial program 73.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 78.8%

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

   5. Simplified 78.8%

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

Alternative 7: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.4 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.25 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.4e+112)
   (/ x.re y.re)
   (if (<= y.re 3.25e-18)
     (/ (+ x.im (/ x.re (/ y.im y.re))) 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 <= -2.4e+112) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.25e-18) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= (-2.4d+112)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.25d-18) then
        tmp = (x_46im + (x_46re / (y_46im / y_46re))) / 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 <= -2.4e+112) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.25e-18) {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -2.4e+112:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.25e-18:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -2.4e+112)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.25e-18)
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / 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 <= -2.4e+112)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.25e-18)
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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, -2.4e+112], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.25e-18], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $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 < -2.4e112 or 3.25000000000000004e-18 < y.re

   1. Initial program 43.3%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 70.0%

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

   5. Simplified 70.0%

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

   if -2.4e112 < y.re < 3.25000000000000004e-18

   1. Initial program 73.6%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 78.8%

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

   5. Simplified 78.8%

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

      1. clear-num N/A

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

      2. associate-/r* N/A

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

      3. clear-num N/A

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

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

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

      5. /-lowering-/.f64 77.6%

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

   7. Applied egg-rr 77.6%

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

Alternative 8: 63.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.5 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 4.7 \cdot 10^{-33}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -9.5e-42)
   (/ x.re y.re)
   (if (<= y.re 4.7e-33) (/ x.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 <= -9.5e-42) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 4.7e-33) {
		tmp = x_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 <= (-9.5d-42)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 4.7d-33) then
        tmp = x_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 <= -9.5e-42) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 4.7e-33) {
		tmp = x_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 <= -9.5e-42:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 4.7e-33:
		tmp = x_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 <= -9.5e-42)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 4.7e-33)
		tmp = Float64(x_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 <= -9.5e-42)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 4.7e-33)
		tmp = x_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, -9.5e-42], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 4.7e-33], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.49999999999999948e-42 or 4.7000000000000002e-33 < y.re

   1. Initial program 52.6%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 61.8%

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

   5. Simplified 61.8%

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

   if -9.49999999999999948e-42 < y.re < 4.7000000000000002e-33

   1. Initial program 72.9%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 71.9%

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

   5. Simplified 71.9%

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

Alternative 9: 42.8% 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 60.9%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 40.5%

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

5. Simplified 40.5%

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

Reproduce

herbie shell --seed 2024159 
(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))))