_divideComplex, real part

Percentage Accurate: 62.0% → 85.8%

Time: 11.0s

Alternatives: 11

Speedup: 1.1×

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: 62.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      2e+299)
   (*
    (/ 1.0 (hypot y.re y.im))
    (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)))
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+299) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (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))) <= 2e+299)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[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], 2e+299], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.0000000000000001e299

   1. Initial program 77.6%

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

      1. *-un-lft-identity 77.6%

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

      2. associate-*r/ 77.6%

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

      3. fma-define 77.6%

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

      4. add-sqr-sqrt 77.6%

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

      5. times-frac 77.6%

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

      6. fma-define 77.6%

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

      7. hypot-define 77.6%

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

      8. fma-define 77.6%

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

      9. fma-define 77.6%

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

      10. hypot-define 97.7%

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

   4. Applied egg-rr 97.7%

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

   if 2.0000000000000001e299 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 13.9%

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

   3. Taylor expanded in y.re around inf 54.9%

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

      1. associate-/l* 63.5%

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

   5. Simplified 63.5%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+299}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.8% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+65}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{{\left(\sqrt[3]{y.im}\right)}^{2}} \cdot \frac{y.re}{\sqrt[3]{y.im}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.6e+65)
   (* (+ x.re (* x.im (/ y.im y.re))) (/ -1.0 (hypot y.re y.im)))
   (if (<= y.re -2.5e-108)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 8.5e-18)
       (/
        (+ x.im (* (/ x.re (pow (cbrt y.im) 2.0)) (/ y.re (cbrt y.im))))
        y.im)
       (/ (+ x.re (* y.im (/ x.im y.re))) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.6e+65) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -2.5e-108) {
		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));
	} else if (y_46_re <= 8.5e-18) {
		tmp = (x_46_im + ((x_46_re / pow(cbrt(y_46_im), 2.0)) * (y_46_re / cbrt(y_46_im)))) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

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 <= -3.6e+65) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -2.5e-108) {
		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));
	} else if (y_46_re <= 8.5e-18) {
		tmp = (x_46_im + ((x_46_re / Math.pow(Math.cbrt(y_46_im), 2.0)) * (y_46_re / Math.cbrt(y_46_im)))) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.6e+65)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -2.5e-108)
		tmp = 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)));
	elseif (y_46_re <= 8.5e-18)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re / (cbrt(y_46_im) ^ 2.0)) * Float64(y_46_re / cbrt(y_46_im)))) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.6e+65], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -2.5e-108], 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], If[LessEqual[y$46$re, 8.5e-18], N[(N[(x$46$im + N[(N[(x$46$re / N[Power[N[Power[y$46$im, 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(y$46$re / N[Power[y$46$im, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.59999999999999978e65

   1. Initial program 32.2%

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

      1. *-un-lft-identity 32.2%

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

      2. associate-*r/ 32.2%

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

      3. fma-define 32.2%

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

      4. add-sqr-sqrt 32.2%

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

      5. times-frac 32.0%

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

      6. fma-define 32.0%

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

      7. hypot-define 32.0%

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

      8. fma-define 32.0%

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

      9. fma-define 32.0%

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

      10. hypot-define 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in y.re around -inf 77.7%

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

      1. distribute-lft-out 77.7%

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

      2. associate-/l* 80.5%

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

   7. Simplified 80.5%

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

   if -3.59999999999999978e65 < y.re < -2.5e-108

   1. Initial program 94.4%

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

   if -2.5e-108 < y.re < 8.4999999999999995e-18

   1. Initial program 73.4%

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

   3. Taylor expanded in y.im around inf 88.9%

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

      1. *-un-lft-identity 88.9%

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

      2. add-cbrt-cube 71.9%

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

      3. cbrt-prod 81.0%

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

      4. times-frac 81.1%

         \[\leadsto \frac{x.im + \color{blue}{\frac{1}{\sqrt[3]{y.im \cdot y.im}} \cdot \frac{x.re \cdot y.re}{\sqrt[3]{y.im}}}}{y.im} \]

      5. cbrt-prod 88.6%

         \[\leadsto \frac{x.im + \frac{1}{\color{blue}{\sqrt[3]{y.im} \cdot \sqrt[3]{y.im}}} \cdot \frac{x.re \cdot y.re}{\sqrt[3]{y.im}}}{y.im} \]

      6. pow2 88.6%

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

   5. Applied egg-rr 88.6%

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

      1. associate-*l/ 88.7%

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

      2. *-lft-identity 88.7%

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

      3. associate-/l* 89.5%

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

      4. associate-*l/ 89.5%

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

   7. Simplified 89.5%

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

   if 8.4999999999999995e-18 < y.re

   1. Initial program 50.6%

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

   3. Taylor expanded in y.re around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 81.3%

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

   5. Applied egg-rr 81.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+65}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-108}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-18}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{{\left(\sqrt[3]{y.im}\right)}^{2}} \cdot \frac{y.re}{\sqrt[3]{y.im}}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -7.6 \cdot 10^{+68}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -7.6e+68)
   (* (+ x.re (* x.im (/ y.im y.re))) (/ -1.0 (hypot y.re y.im)))
   (if (<= y.re -3e-106)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 3.6e-13)
       (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)
       (/ (+ x.re (* y.im (/ x.im y.re))) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -7.6e+68) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -3e-106) {
		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));
	} else if (y_46_re <= 3.6e-13) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

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 <= -7.6e+68) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_re <= -3e-106) {
		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));
	} else if (y_46_re <= 3.6e-13) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -7.6e+68:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_re <= -3e-106:
		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))
	elif y_46_re <= 3.6e-13:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	else:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -7.6e+68)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_re <= -3e-106)
		tmp = 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)));
	elseif (y_46_re <= 3.6e-13)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -7.6e+68)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -3e-106)
		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));
	elseif (y_46_re <= 3.6e-13)
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	else
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -7.6e+68], N[(N[(x$46$re + N[(x$46$im * N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3e-106], 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], If[LessEqual[y$46$re, 3.6e-13], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -7.6000000000000002e68

   1. Initial program 32.2%

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

      1. *-un-lft-identity 32.2%

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

      2. associate-*r/ 32.2%

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

      3. fma-define 32.2%

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

      4. add-sqr-sqrt 32.2%

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

      5. times-frac 32.0%

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

      6. fma-define 32.0%

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

      7. hypot-define 32.0%

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

      8. fma-define 32.0%

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

      9. fma-define 32.0%

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

      10. hypot-define 57.6%

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

   4. Applied egg-rr 57.6%

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

   5. Taylor expanded in y.re around -inf 77.7%

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

      1. distribute-lft-out 77.7%

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

      2. associate-/l* 80.5%

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

   7. Simplified 80.5%

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

   if -7.6000000000000002e68 < y.re < -3.00000000000000019e-106

   1. Initial program 94.4%

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

   if -3.00000000000000019e-106 < y.re < 3.5999999999999998e-13

   1. Initial program 73.4%

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

   3. Taylor expanded in y.im around inf 88.9%

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

   if 3.5999999999999998e-13 < y.re

   1. Initial program 50.6%

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

   3. Taylor expanded in y.re around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 81.3%

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

   5. Applied egg-rr 81.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -7.6 \cdot 10^{+68}:\\ \;\;\;\;\left(x.re + x.im \cdot \frac{y.im}{y.re}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -3.1e+62)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
   (if (<= y.re -3.2e-102)
     (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
     (if (<= y.re 5.5e-13)
       (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)
       (/ (+ x.re (* y.im (/ x.im y.re))) y.re)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.1e+62) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= -3.2e-102) {
		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));
	} else if (y_46_re <= 5.5e-13) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-3.1d+62)) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46re <= (-3.2d-102)) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46re <= 5.5d-13) then
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    else
        tmp = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -3.1e+62) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= -3.2e-102) {
		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));
	} else if (y_46_re <= 5.5e-13) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -3.1e+62:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= -3.2e-102:
		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))
	elif y_46_re <= 5.5e-13:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	else:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -3.1e+62)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= -3.2e-102)
		tmp = 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)));
	elseif (y_46_re <= 5.5e-13)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -3.1e+62)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= -3.2e-102)
		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));
	elseif (y_46_re <= 5.5e-13)
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	else
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -3.1e+62], 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$46$re, -3.2e-102], 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], If[LessEqual[y$46$re, 5.5e-13], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.10000000000000014e62

   1. Initial program 32.2%

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

   3. Taylor expanded in y.re around inf 77.7%

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

      1. associate-/l* 80.4%

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

   5. Simplified 80.4%

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

   if -3.10000000000000014e62 < y.re < -3.19999999999999986e-102

   1. Initial program 94.4%

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

   if -3.19999999999999986e-102 < y.re < 5.49999999999999979e-13

   1. Initial program 73.4%

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

   3. Taylor expanded in y.im around inf 88.9%

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

   if 5.49999999999999979e-13 < y.re

   1. Initial program 50.6%

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

   3. Taylor expanded in y.re around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 81.3%

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

   5. Applied egg-rr 81.3%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.1 \cdot 10^{+62}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.5 \cdot 10^{-13}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 72.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-42} \lor \neg \left(y.re \leq 1.28 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.8e-42) (not (<= y.re 1.28e-14)))
   (/ x.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_re <= -5.8e-42) || !(y_46_re <= 1.28e-14)) {
		tmp = x_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_46re <= (-5.8d-42)) .or. (.not. (y_46re <= 1.28d-14))) then
        tmp = x_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_re <= -5.8e-42) || !(y_46_re <= 1.28e-14)) {
		tmp = x_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_re <= -5.8e-42) or not (y_46_re <= 1.28e-14):
		tmp = x_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_re <= -5.8e-42) || !(y_46_re <= 1.28e-14))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re * Float64(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_re <= -5.8e-42) || ~((y_46_re <= 1.28e-14)))
		tmp = x_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[Or[LessEqual[y$46$re, -5.8e-42], N[Not[LessEqual[y$46$re, 1.28e-14]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -5.8000000000000006e-42 or 1.28e-14 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf 65.8%

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

   if -5.8000000000000006e-42 < y.re < 1.28e-14

   1. Initial program 76.1%

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

   3. Taylor expanded in y.im around inf 86.0%

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

      1. associate-/l* 84.7%

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

   5. Simplified 84.7%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-42} \lor \neg \left(y.re \leq 1.28 \cdot 10^{-14}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-42} \lor \neg \left(y.re \leq 1.75 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.8e-42) (not (<= y.re 1.75e-13)))
   (/ x.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_re <= -5.8e-42) || !(y_46_re <= 1.75e-13)) {
		tmp = x_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_46re <= (-5.8d-42)) .or. (.not. (y_46re <= 1.75d-13))) then
        tmp = x_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_re <= -5.8e-42) || !(y_46_re <= 1.75e-13)) {
		tmp = x_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_re <= -5.8e-42) or not (y_46_re <= 1.75e-13):
		tmp = x_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_re <= -5.8e-42) || !(y_46_re <= 1.75e-13))
		tmp = Float64(x_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_re <= -5.8e-42) || ~((y_46_re <= 1.75e-13)))
		tmp = x_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[Or[LessEqual[y$46$re, -5.8e-42], N[Not[LessEqual[y$46$re, 1.75e-13]], $MachinePrecision]], N[(x$46$re / 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 2 regimes

2. if y.re < -5.8000000000000006e-42 or 1.7500000000000001e-13 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf 65.8%

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

   if -5.8000000000000006e-42 < y.re < 1.7500000000000001e-13

   1. Initial program 76.1%

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

   3. Taylor expanded in y.im around inf 86.0%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.8 \cdot 10^{-42} \lor \neg \left(y.re \leq 1.75 \cdot 10^{-13}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{-45} \lor \neg \left(y.re \leq 5.2 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -9.2e-45) (not (<= y.re 5.2e-18)))
   (/ (+ x.re (* x.im (/ y.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_re <= -9.2e-45) || !(y_46_re <= 5.2e-18)) {
		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_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_46re <= (-9.2d-45)) .or. (.not. (y_46re <= 5.2d-18))) then
        tmp = (x_46re + (x_46im * (y_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_re <= -9.2e-45) || !(y_46_re <= 5.2e-18)) {
		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_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_re <= -9.2e-45) or not (y_46_re <= 5.2e-18):
		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_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_re <= -9.2e-45) || !(y_46_re <= 5.2e-18))
		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(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_re <= -9.2e-45) || ~((y_46_re <= 5.2e-18)))
		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_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[Or[LessEqual[y$46$re, -9.2e-45], N[Not[LessEqual[y$46$re, 5.2e-18]], $MachinePrecision]], 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[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -9.19999999999999967e-45 or 5.2000000000000001e-18 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf 74.5%

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

      1. associate-/l* 77.4%

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

   5. Simplified 77.4%

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

   if -9.19999999999999967e-45 < y.re < 5.2000000000000001e-18

   1. Initial program 76.1%

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

   3. Taylor expanded in y.im around inf 86.0%

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

4. Final simplification 81.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.2 \cdot 10^{-45} \lor \neg \left(y.re \leq 5.2 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -4.2e-42)
   (/ (+ x.re (* x.im (/ y.im y.re))) y.re)
   (if (<= y.re 8.5e-12)
     (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)
     (/ (+ x.re (* y.im (/ x.im y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e-42) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 8.5e-12) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-4.2d-42)) then
        tmp = (x_46re + (x_46im * (y_46im / y_46re))) / y_46re
    else if (y_46re <= 8.5d-12) then
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    else
        tmp = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -4.2e-42) {
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	} else if (y_46_re <= 8.5e-12) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -4.2e-42:
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re
	elif y_46_re <= 8.5e-12:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	else:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -4.2e-42)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im * Float64(y_46_im / y_46_re))) / y_46_re);
	elseif (y_46_re <= 8.5e-12)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -4.2e-42)
		tmp = (x_46_re + (x_46_im * (y_46_im / y_46_re))) / y_46_re;
	elseif (y_46_re <= 8.5e-12)
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	else
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -4.2e-42], 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$46$re, 8.5e-12], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -4.20000000000000013e-42

   1. Initial program 54.8%

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

   3. Taylor expanded in y.re around inf 73.8%

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

      1. associate-/l* 75.5%

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

   5. Simplified 75.5%

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

   if -4.20000000000000013e-42 < y.re < 8.4999999999999997e-12

   1. Initial program 76.1%

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

   3. Taylor expanded in y.im around inf 86.0%

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

   if 8.4999999999999997e-12 < y.re

   1. Initial program 50.6%

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

   3. Taylor expanded in y.re around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 81.3%

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

   5. Applied egg-rr 81.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -4.2 \cdot 10^{-42}:\\ \;\;\;\;\frac{x.re + x.im \cdot \frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.re \leq 8.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -1.15e-43)
   (/ (+ x.re (/ x.im (/ y.re y.im))) y.re)
   (if (<= y.re 2.02e-16)
     (/ (+ x.im (/ (* x.re y.re) y.im)) y.im)
     (/ (+ x.re (* y.im (/ x.im y.re))) y.re))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e-43) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= 2.02e-16) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-1.15d-43)) then
        tmp = (x_46re + (x_46im / (y_46re / y_46im))) / y_46re
    else if (y_46re <= 2.02d-16) then
        tmp = (x_46im + ((x_46re * y_46re) / y_46im)) / y_46im
    else
        tmp = (x_46re + (y_46im * (x_46im / y_46re))) / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -1.15e-43) {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	} else if (y_46_re <= 2.02e-16) {
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	} else {
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -1.15e-43:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re
	elif y_46_re <= 2.02e-16:
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im
	else:
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -1.15e-43)
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / y_46_re);
	elseif (y_46_re <= 2.02e-16)
		tmp = Float64(Float64(x_46_im + Float64(Float64(x_46_re * y_46_re) / y_46_im)) / y_46_im);
	else
		tmp = Float64(Float64(x_46_re + Float64(y_46_im * Float64(x_46_im / y_46_re))) / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -1.15e-43)
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / y_46_re;
	elseif (y_46_re <= 2.02e-16)
		tmp = (x_46_im + ((x_46_re * y_46_re) / y_46_im)) / y_46_im;
	else
		tmp = (x_46_re + (y_46_im * (x_46_im / y_46_re))) / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -1.15e-43], N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 2.02e-16], N[(N[(x$46$im + N[(N[(x$46$re * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision], N[(N[(x$46$re + N[(y$46$im * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y$46$re), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -1.1499999999999999e-43

   1. Initial program 54.8%

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

   3. Taylor expanded in y.re around inf 73.8%

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

      1. associate-/l* 75.5%

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

   5. Simplified 75.5%

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

   6. Taylor expanded in x.im around 0 73.8%

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

      1. associate-*l/ 75.5%

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

      2. associate-/r/ 75.5%

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

   8. Simplified 75.5%

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

   if -1.1499999999999999e-43 < y.re < 2.02000000000000004e-16

   1. Initial program 76.1%

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

   3. Taylor expanded in y.im around inf 86.0%

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

   if 2.02000000000000004e-16 < y.re

   1. Initial program 50.6%

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

   3. Taylor expanded in y.re around inf 75.0%

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

      1. *-commutative 75.0%

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

      2. associate-/l* 81.3%

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

   5. Applied egg-rr 81.3%

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

4. Final simplification 82.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{-43}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{y.re}\\ \mathbf{elif}\;y.re \leq 2.02 \cdot 10^{-16}:\\ \;\;\;\;\frac{x.im + \frac{x.re \cdot y.re}{y.im}}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + y.im \cdot \frac{x.im}{y.re}}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -5.4 \cdot 10^{-45} \lor \neg \left(y.re \leq 2 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -5.4e-45) (not (<= y.re 2e-20)))
   (/ 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_re <= -5.4e-45) || !(y_46_re <= 2e-20)) {
		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_46re <= (-5.4d-45)) .or. (.not. (y_46re <= 2d-20))) 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_re <= -5.4e-45) || !(y_46_re <= 2e-20)) {
		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_re <= -5.4e-45) or not (y_46_re <= 2e-20):
		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_re <= -5.4e-45) || !(y_46_re <= 2e-20))
		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_re <= -5.4e-45) || ~((y_46_re <= 2e-20)))
		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[Or[LessEqual[y$46$re, -5.4e-45], N[Not[LessEqual[y$46$re, 2e-20]], $MachinePrecision]], 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.re < -5.3999999999999997e-45 or 1.99999999999999989e-20 < y.re

   1. Initial program 52.5%

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

   3. Taylor expanded in y.re around inf 65.8%

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

   if -5.3999999999999997e-45 < y.re < 1.99999999999999989e-20

   1. Initial program 76.1%

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

   3. Taylor expanded in y.re around 0 67.1%

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

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -5.4 \cdot 10^{-45} \lor \neg \left(y.re \leq 2 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 42.4% 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 63.2%

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

3. Taylor expanded in y.re around 0 41.1%

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

4. Final simplification 41.1%

   \[\leadsto \frac{x.im}{y.im} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024071 
(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))))