_divideComplex, real part

Percentage Accurate: 61.7% → 80.1%

Time: 9.5s

Alternatives: 6

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% 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: 80.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.5e-24)
   (/ (+ x.im (/ y.re (/ y.im x.re))) y.im)
   (if (<= y.im 3.15e-68)
     (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
     (if (<= y.im 1.15e+63)
       (/ (+ (* y.re x.re) (* y.im x.im)) (+ (* y.re y.re) (* y.im y.im)))
       (/ (+ x.im (/ x.re (/ y.im y.re))) 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 <= -3.5e-24) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else if (y_46_im <= 3.15e-68) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.15e+63) {
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_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) :: tmp
    if (y_46im <= (-3.5d-24)) then
        tmp = (x_46im + (y_46re / (y_46im / x_46re))) / y_46im
    else if (y_46im <= 3.15d-68) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46im <= 1.15d+63) then
        tmp = ((y_46re * x_46re) + (y_46im * x_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46im + (x_46re / (y_46im / y_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 tmp;
	if (y_46_im <= -3.5e-24) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else if (y_46_im <= 3.15e-68) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_im <= 1.15e+63) {
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -3.5e-24:
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im
	elif y_46_im <= 3.15e-68:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_im <= 1.15e+63:
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -3.5e-24)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_46_re))) / y_46_im);
	elseif (y_46_im <= 3.15e-68)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_im <= 1.15e+63)
		tmp = Float64(Float64(Float64(y_46_re * x_46_re) + Float64(y_46_im * x_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_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)
	tmp = 0.0;
	if (y_46_im <= -3.5e-24)
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	elseif (y_46_im <= 3.15e-68)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_im <= 1.15e+63)
		tmp = ((y_46_re * x_46_re) + (y_46_im * x_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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, -3.5e-24], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 3.15e-68], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, 1.15e+63], N[(N[(N[(y$46$re * x$46$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], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -3.4999999999999996e-24

   1. Initial program 48.7%

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

   3. Taylor expanded in y.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 82.1%

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

   5. Simplified 82.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 82.2%

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

   7. Applied egg-rr 82.2%

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

   if -3.4999999999999996e-24 < y.im < 3.1499999999999999e-68

   1. Initial program 71.3%

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

   3. Taylor expanded in x.re around inf

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 66.8%

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

   5. Simplified 66.8%

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

   6. Taylor expanded in y.re around inf

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

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

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

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

      5. /-lowering-/.f64 89.9%

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

   8. Simplified 89.9%

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

   if 3.1499999999999999e-68 < y.im < 1.14999999999999997e63

   1. Initial program 80.2%

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

   if 1.14999999999999997e63 < y.im

   1. Initial program 37.5%

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

   3. Taylor expanded in y.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 82.8%

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

   7. Applied egg-rr 82.8%

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

4. Final simplification 85.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 3.15 \cdot 10^{-68}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.15 \cdot 10^{+63}:\\ \;\;\;\;\frac{y.re \cdot x.re + y.im \cdot x.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -7.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -7.5e-25)
   (/ (+ x.im (/ y.re (/ y.im x.re))) y.im)
   (if (<= y.im 1.8e+41)
     (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
     (/ (+ x.im (/ x.re (/ y.im y.re))) 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 <= -7.5e-25) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else if (y_46_im <= 1.8e+41) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_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) :: tmp
    if (y_46im <= (-7.5d-25)) then
        tmp = (x_46im + (y_46re / (y_46im / x_46re))) / y_46im
    else if (y_46im <= 1.8d+41) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else
        tmp = (x_46im + (x_46re / (y_46im / y_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 tmp;
	if (y_46_im <= -7.5e-25) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	} else if (y_46_im <= 1.8e+41) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -7.5e-25:
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im
	elif y_46_im <= 1.8e+41:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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 <= -7.5e-25)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_46_re))) / y_46_im);
	elseif (y_46_im <= 1.8e+41)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_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)
	tmp = 0.0;
	if (y_46_im <= -7.5e-25)
		tmp = (x_46_im + (y_46_re / (y_46_im / x_46_re))) / y_46_im;
	elseif (y_46_im <= 1.8e+41)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / 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, -7.5e-25], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+41], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -7.49999999999999989e-25

   1. Initial program 48.7%

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

   3. Taylor expanded in y.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 82.1%

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

   5. Simplified 82.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 82.2%

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

   7. Applied egg-rr 82.2%

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

   if -7.49999999999999989e-25 < y.im < 1.80000000000000013e41

   1. Initial program 73.8%

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

   3. Taylor expanded in x.re around inf

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

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

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

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

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

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

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

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

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

      5. unpow2 N/A

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

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

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

      7. unpow2 N/A

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

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

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

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

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

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

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

      11. *-commutative N/A

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

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

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

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

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

      14. unpow2 N/A

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

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

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

      16. unpow2 N/A

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

      17. *-lowering-*.f64 69.3%

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

   5. Simplified 69.3%

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

   6. Taylor expanded in y.re around inf

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

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

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

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

      5. /-lowering-/.f64 83.2%

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

   8. Simplified 83.2%

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

   if 1.80000000000000013e41 < y.im

   1. Initial program 39.1%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 78.2%

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

   5. Simplified 78.2%

      \[\leadsto \color{blue}{\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}} \]
   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 82.0%

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

   7. Applied egg-rr 82.0%

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

Alternative 3: 72.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.4 \cdot 10^{+25}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.im + \frac{y.re}{\frac{y.im}{x.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.4e+25)
   (/ x.re y.re)
   (if (<= y.re 2.8e+41)
     (/ (+ x.im (/ y.re (/ y.im x.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 <= -1.4e+25) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.8e+41) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_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 <= (-1.4d+25)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 2.8d+41) then
        tmp = (x_46im + (y_46re / (y_46im / x_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 <= -1.4e+25) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 2.8e+41) {
		tmp = (x_46_im + (y_46_re / (y_46_im / x_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 <= -1.4e+25:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 2.8e+41:
		tmp = (x_46_im + (y_46_re / (y_46_im / x_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 <= -1.4e+25)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 2.8e+41)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re / Float64(y_46_im / x_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 <= -1.4e+25)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 2.8e+41)
		tmp = (x_46_im + (y_46_re / (y_46_im / x_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, -1.4e+25], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.8e+41], N[(N[(x$46$im + N[(y$46$re / N[(y$46$im / x$46$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 < -1.4000000000000001e25 or 2.7999999999999999e41 < y.re

   1. Initial program 47.4%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 67.0%

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

   5. Simplified 67.0%

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

   if -1.4000000000000001e25 < y.re < 2.7999999999999999e41

   1. Initial program 69.3%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.6%

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

   5. Simplified 79.6%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

      5. /-lowering-/.f64 79.7%

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

   7. Applied egg-rr 79.7%

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

Alternative 4: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.86 \cdot 10^{+27}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.4 \cdot 10^{+39}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.86e+27)
   (/ x.re y.re)
   (if (<= y.re 3.4e+39)
     (/ (+ 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 <= -1.86e+27) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.4e+39) {
		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 <= (-1.86d+27)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.4d+39) 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 <= -1.86e+27) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.4e+39) {
		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 <= -1.86e+27:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.4e+39:
		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 <= -1.86e+27)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.4e+39)
		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 <= -1.86e+27)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.4e+39)
		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, -1.86e+27], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.4e+39], 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 < -1.86e27 or 3.3999999999999999e39 < y.re

   1. Initial program 47.4%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 67.0%

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

   5. Simplified 67.0%

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

   if -1.86e27 < y.re < 3.3999999999999999e39

   1. Initial program 69.3%

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

   3. Taylor expanded in y.im around inf

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

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

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

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

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

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

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

      4. *-commutative N/A

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

      5. *-lowering-*.f64 79.6%

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

   5. Simplified 79.6%

      \[\leadsto \color{blue}{\frac{x.im + \frac{y.re \cdot x.re}{y.im}}{y.im}} \]
   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 79.0%

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

   7. Applied egg-rr 79.0%

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

Alternative 5: 63.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.45 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 3.1 \cdot 10^{+41}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.45e-13)
   (/ x.im y.im)
   (if (<= y.im 3.1e+41) (/ 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 <= -1.45e-13) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 3.1e+41) {
		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 <= (-1.45d-13)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 3.1d+41) 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 <= -1.45e-13) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 3.1e+41) {
		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 <= -1.45e-13:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 3.1e+41:
		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 <= -1.45e-13)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 3.1e+41)
		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 <= -1.45e-13)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 3.1e+41)
		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, -1.45e-13], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 3.1e+41], 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 < -1.4499999999999999e-13 or 3.1e41 < y.im

   1. Initial program 43.7%

      \[\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \]
   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 73.7%

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

   5. Simplified 73.7%

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

   if -1.4499999999999999e-13 < y.im < 3.1e41

   1. Initial program 74.0%

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

   3. Taylor expanded in y.re around inf

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

      1. /-lowering-/.f64 61.2%

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

   5. Simplified 61.2%

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

Alternative 6: 42.5% 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 59.5%

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

3. Taylor expanded in y.re around 0

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

   1. /-lowering-/.f64 45.5%

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

5. Simplified 45.5%

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

Reproduce

herbie shell --seed 2024164 
(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))))