_divideComplex, real part

Percentage Accurate: 60.7% → 79.8%

Time: 8.4s

Alternatives: 8

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.7% 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: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := x.re \cdot y.re + y.im \cdot x.im\\ \mathbf{if}\;y.im \leq -850000000:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -5.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{t\_0} \cdot t\_1\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+115}:\\ \;\;\;\;\frac{t\_1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (+ (* x.re y.re) (* y.im x.im))))
   (if (<= y.im -850000000.0)
     (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
     (if (<= y.im -5.4e-135)
       (* (/ 1.0 t_0) t_1)
       (if (<= y.im 1.7e-70)
         (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
         (if (<= y.im 2.6e+115)
           (/ t_1 t_0)
           (/ (+ x.im (/ (* x.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) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_re * y_46_re) + (y_46_im * x_46_im);
	double tmp;
	if (y_46_im <= -850000000.0) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= -5.4e-135) {
		tmp = (1.0 / t_0) * t_1;
	} else if (y_46_im <= 1.7e-70) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.6e+115) {
		tmp = t_1 / t_0;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (x_46re * y_46re) + (y_46im * x_46im)
    if (y_46im <= (-850000000.0d0)) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else if (y_46im <= (-5.4d-135)) then
        tmp = (1.0d0 / t_0) * t_1
    else if (y_46im <= 1.7d-70) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else if (y_46im <= 2.6d+115) then
        tmp = t_1 / t_0
    else
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (x_46_re * y_46_re) + (y_46_im * x_46_im);
	double tmp;
	if (y_46_im <= -850000000.0) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= -5.4e-135) {
		tmp = (1.0 / t_0) * t_1;
	} else if (y_46_im <= 1.7e-70) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 2.6e+115) {
		tmp = t_1 / t_0;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (x_46_re * y_46_re) + (y_46_im * x_46_im)
	tmp = 0
	if y_46_im <= -850000000.0:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	elif y_46_im <= -5.4e-135:
		tmp = (1.0 / t_0) * t_1
	elif y_46_im <= 1.7e-70:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	elif y_46_im <= 2.6e+115:
		tmp = t_1 / t_0
	else:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / 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))
	t_1 = Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im))
	tmp = 0.0
	if (y_46_im <= -850000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	elseif (y_46_im <= -5.4e-135)
		tmp = Float64(Float64(1.0 / t_0) * t_1);
	elseif (y_46_im <= 1.7e-70)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 2.6e+115)
		tmp = Float64(t_1 / t_0);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (x_46_re * y_46_re) + (y_46_im * x_46_im);
	tmp = 0.0;
	if (y_46_im <= -850000000.0)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	elseif (y_46_im <= -5.4e-135)
		tmp = (1.0 / t_0) * t_1;
	elseif (y_46_im <= 1.7e-70)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 2.6e+115)
		tmp = t_1 / t_0;
	else
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / 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[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -850000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -5.4e-135], N[(N[(1.0 / t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[y$46$im, 1.7e-70], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 2.6e+115], N[(t$95$1 / t$95$0), $MachinePrecision], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -8.5e8

   1. Initial program 39.9%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

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

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

      12. /-lowering-/.f64 76.6%

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

   5. Simplified 76.6%

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

   if -8.5e8 < y.im < -5.39999999999999997e-135

   1. Initial program 90.6%

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

      1. div-inv N/A

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

      2. flip-+ N/A

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

      3. clear-num N/A

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

      4. *-commutative N/A

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

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

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

      6. clear-num N/A

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

      7. flip-+ N/A

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

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

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

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

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

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

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

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

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

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

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

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

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

      14. *-lowering-*.f64 90.7%

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

   4. Applied egg-rr 90.7%

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

   if -5.39999999999999997e-135 < y.im < 1.69999999999999998e-70

   1. Initial program 63.7%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   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 96.2%

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

   5. Simplified 96.2%

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

   if 1.69999999999999998e-70 < y.im < 2.6e115

   1. Initial program 72.0%

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

   if 2.6e115 < y.im

   1. Initial program 35.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.3%

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

   5. Simplified 87.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -850000000:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -5.4 \cdot 10^{-135}:\\ \;\;\;\;\frac{1}{y.re \cdot y.re + y.im \cdot y.im} \cdot \left(x.re \cdot y.re + y.im \cdot x.im\right)\\ \mathbf{elif}\;y.im \leq 1.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.6 \cdot 10^{+115}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.im \leq -850000000:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{-140}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+112}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.im -850000000.0)
     (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
     (if (<= y.im -6.5e-140)
       t_0
       (if (<= y.im 6.5e-71)
         (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
         (if (<= y.im 8.8e+112)
           t_0
           (/ (+ x.im (/ (* x.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) {
	double t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -850000000.0) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= -6.5e-140) {
		tmp = t_0;
	} else if (y_46_im <= 6.5e-71) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.8e+112) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46im <= (-850000000.0d0)) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else if (y_46im <= (-6.5d-140)) then
        tmp = t_0
    else if (y_46im <= 6.5d-71) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else if (y_46im <= 8.8d+112) then
        tmp = t_0
    else
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_im <= -850000000.0) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= -6.5e-140) {
		tmp = t_0;
	} else if (y_46_im <= 6.5e-71) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else if (y_46_im <= 8.8e+112) {
		tmp = t_0;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_im <= -850000000.0:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	elif y_46_im <= -6.5e-140:
		tmp = t_0
	elif y_46_im <= 6.5e-71:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	elif y_46_im <= 8.8e+112:
		tmp = t_0
	else:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_im <= -850000000.0)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	elseif (y_46_im <= -6.5e-140)
		tmp = t_0;
	elseif (y_46_im <= 6.5e-71)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	elseif (y_46_im <= 8.8e+112)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_re * y_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_im <= -850000000.0)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	elseif (y_46_im <= -6.5e-140)
		tmp = t_0;
	elseif (y_46_im <= 6.5e-71)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	elseif (y_46_im <= 8.8e+112)
		tmp = t_0;
	else
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / 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[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * x$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -850000000.0], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -6.5e-140], t$95$0, If[LessEqual[y$46$im, 6.5e-71], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 8.8e+112], t$95$0, N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -8.5e8

   1. Initial program 39.9%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

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

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

      12. /-lowering-/.f64 76.6%

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

   5. Simplified 76.6%

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

   if -8.5e8 < y.im < -6.4999999999999995e-140 or 6.50000000000000005e-71 < y.im < 8.7999999999999997e112

   1. Initial program 78.4%

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

   if -6.4999999999999995e-140 < y.im < 6.50000000000000005e-71

   1. Initial program 63.7%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   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 96.2%

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

   5. Simplified 96.2%

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

   if 8.7999999999999997e112 < y.im

   1. Initial program 35.8%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 87.3%

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

   5. Simplified 87.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -850000000:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq -6.5 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 6.5 \cdot 10^{-71}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 8.8 \cdot 10^{+112}:\\ \;\;\;\;\frac{x.re \cdot y.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -5.7e-49)
   (+ (/ x.im y.im) (* x.re (/ (/ y.re y.im) y.im)))
   (if (<= y.im 7.4e-23)
     (/ (+ x.re (/ (* y.im x.im) y.re)) y.re)
     (/ (+ x.im (/ (* x.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) {
	double tmp;
	if (y_46_im <= -5.7e-49) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 7.4e-23) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-5.7d-49)) then
        tmp = (x_46im / y_46im) + (x_46re * ((y_46re / y_46im) / y_46im))
    else if (y_46im <= 7.4d-23) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -5.7e-49) {
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	} else if (y_46_im <= 7.4e-23) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -5.7e-49:
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im))
	elif y_46_im <= 7.4e-23:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	else:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -5.7e-49)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(x_46_re * Float64(Float64(y_46_re / y_46_im) / y_46_im)));
	elseif (y_46_im <= 7.4e-23)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -5.7e-49)
		tmp = (x_46_im / y_46_im) + (x_46_re * ((y_46_re / y_46_im) / y_46_im));
	elseif (y_46_im <= 7.4e-23)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	else
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -5.7e-49], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(x$46$re * N[(N[(y$46$re / y$46$im), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 7.4e-23], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -5.7000000000000003e-49

   1. Initial program 47.2%

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

   3. Taylor expanded in y.re around 0

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

      1. *-commutative N/A

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

      2. associate-*r/ N/A

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

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

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

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

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

      5. associate-*r/ N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

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

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

      9. unpow2 N/A

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

      10. associate-/r* N/A

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

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

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

      12. /-lowering-/.f64 74.3%

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

   5. Simplified 74.3%

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

   if -5.7000000000000003e-49 < y.im < 7.4000000000000005e-23

   1. Initial program 66.7%

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

   3. Taylor expanded in y.re around inf

      \[\leadsto \color{blue}{\frac{x.re + \frac{x.im \cdot y.im}{y.re}}{y.re}} \]
   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 88.9%

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

   5. Simplified 88.9%

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

   if 7.4000000000000005e-23 < y.im

   1. Initial program 54.4%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 76.5%

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

   5. Simplified 76.5%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;\frac{x.im}{y.im} + x.re \cdot \frac{\frac{y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 7.4 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -7.7 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)))
   (if (<= y.im -7.7e-28)
     t_0
     (if (<= y.im 8e-23) (/ (+ x.re (/ (* y.im x.im) y.re)) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	double tmp;
	if (y_46_im <= -7.7e-28) {
		tmp = t_0;
	} else if (y_46_im <= 8e-23) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    if (y_46im <= (-7.7d-28)) then
        tmp = t_0
    else if (y_46im <= 8d-23) then
        tmp = (x_46re + ((y_46im * x_46im) / y_46re)) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	double tmp;
	if (y_46_im <= -7.7e-28) {
		tmp = t_0;
	} else if (y_46_im <= 8e-23) {
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	tmp = 0
	if y_46_im <= -7.7e-28:
		tmp = t_0
	elif y_46_im <= 8e-23:
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.7e-28)
		tmp = t_0;
	elseif (y_46_im <= 8e-23)
		tmp = Float64(Float64(x_46_re + Float64(Float64(y_46_im * x_46_im) / y_46_re)) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7.7e-28)
		tmp = t_0;
	elseif (y_46_im <= 8e-23)
		tmp = (x_46_re + ((y_46_im * x_46_im) / y_46_re)) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.7e-28], t$95$0, If[LessEqual[y$46$im, 8e-23], N[(N[(x$46$re + N[(N[(y$46$im * x$46$im), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.7000000000000001e-28 or 7.99999999999999968e-23 < y.im

   1. Initial program 49.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.9%

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

   5. Simplified 75.9%

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

   if -7.7000000000000001e-28 < y.im < 7.99999999999999968e-23

   1. Initial program 66.9%

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

   3. Taylor expanded in y.re around 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 87.8%

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

   5. Simplified 87.8%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.7 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.re + \frac{y.im \cdot x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{if}\;y.im \leq -7.2 \cdot 10^{-28}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)))
   (if (<= y.im -7.2e-28)
     t_0
     (if (<= y.im 3.3e-24) (/ (+ x.re (* y.im (/ x.im y.re))) y.re) t_0))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	double tmp;
	if (y_46_im <= -7.2e-28) {
		tmp = t_0;
	} else if (y_46_im <= 3.3e-24) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    if (y_46im <= (-7.2d-28)) then
        tmp = t_0
    else if (y_46im <= 3.3d-24) then
        tmp = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	double tmp;
	if (y_46_im <= -7.2e-28) {
		tmp = t_0;
	} else if (y_46_im <= 3.3e-24) {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	tmp = 0
	if y_46_im <= -7.2e-28:
		tmp = t_0
	elif y_46_im <= 3.3e-24:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	else:
		tmp = t_0
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im)
	tmp = 0.0
	if (y_46_im <= -7.2e-28)
		tmp = t_0;
	elseif (y_46_im <= 3.3e-24)
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	tmp = 0.0;
	if (y_46_im <= -7.2e-28)
		tmp = t_0;
	elseif (y_46_im <= 3.3e-24)
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]}, If[LessEqual[y$46$im, -7.2e-28], t$95$0, If[LessEqual[y$46$im, 3.3e-24], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -7.1999999999999997e-28 or 3.29999999999999984e-24 < y.im

   1. Initial program 49.9%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 75.9%

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

   5. Simplified 75.9%

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

   if -7.1999999999999997e-28 < y.im < 3.29999999999999984e-24

   1. Initial program 66.9%

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

   3. Taylor expanded in y.re around 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 87.8%

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

   5. Simplified 87.8%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

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

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

      4. /-lowering-/.f64 87.1%

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

   7. Applied egg-rr 87.1%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -7.2 \cdot 10^{-28}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{elif}\;y.im \leq 3.3 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -2.3e+60)
   (/ x.re y.re)
   (if (<= y.re 2e-6) (/ (+ x.im (/ (* x.re y.re) y.im)) y.im) (/ x.re y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.3e+60) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2e-6) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-2.3d+60)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2d-6) then
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -2.3e+60) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2e-6) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -2.3e+60:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2e-6:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -2.3e+60)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2e-6)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -2.3e+60)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2e-6)
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -2.3e+60], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2e-6], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.30000000000000017e60 or 1.99999999999999991e-6 < y.re

   1. Initial program 45.2%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 70.1%

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

   5. Simplified 70.1%

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

   if -2.30000000000000017e60 < y.re < 1.99999999999999991e-6

   1. Initial program 70.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 80.3%

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

   5. Simplified 80.3%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+60}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -66000000000:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 2.1 \cdot 10^{-54}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -66000000000.0)
   (/ x.im y.im)
   (if (<= y.im 2.1e-54) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -66000000000.0) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.1e-54) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-66000000000.0d0)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 2.1d-54) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -66000000000.0) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 2.1e-54) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -66000000000.0:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 2.1e-54:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -66000000000.0)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 2.1e-54)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -66000000000.0)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 2.1e-54)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -66000000000.0], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 2.1e-54], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.6e10 or 2.1e-54 < y.im

   1. Initial program 49.0%

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

   3. Taylor expanded in y.re around 0

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

      1. /-lowering-/.f64 65.0%

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

   5. Simplified 65.0%

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

   if -6.6e10 < y.im < 2.1e-54

   1. Initial program 68.5%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 69.9%

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

   5. Simplified 69.9%

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

Alternative 8: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.7%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 39.4%

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

5. Simplified 39.4%

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

Reproduce

herbie shell --seed 2024156 
(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))))