_divideComplex, imaginary part

Percentage Accurate: 61.8% → 85.3%

Time: 11.1s

Alternatives: 9

Speedup: 1.1×

Specification

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 61.8% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (/ (- (* x.im y.re) (* x.re 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_im * y_46_re) - (x_46_re * 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_46im * y_46re) - (x_46re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - (x_46_re * 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_im * y_46_re) - Float64(x_46_re * 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_im * y_46_re) - (x_46_re * 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$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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: 85.3% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.im \cdot y.re - x.re \cdot y.im\\ \mathbf{if}\;\frac{t\_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t\_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (- (* x.im y.re) (* x.re y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 2e+301)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (/ (- (* x.im (/ y.re y.im)) x.re) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+301) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+301) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+301:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im))
	tmp = 0.0
	if (Float64(t_0 / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 2e+301)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_im * y_46_re) - (x_46_re * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+301)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2e+301], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.00000000000000011e301

   1. Initial program 78.9%

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

      1. *-un-lft-identity 78.9%

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

      2. add-sqr-sqrt 78.9%

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

      3. times-frac 78.9%

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

      4. hypot-def 78.9%

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

      5. hypot-def 95.1%

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

   4. Applied egg-rr 95.1%

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

   if 2.00000000000000011e301 < (/.f64 (-.f64 (*.f64 x.im y.re) (*.f64 x.re y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 12.9%

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

   3. Taylor expanded in y.re around 0 46.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 46.5%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 46.5%

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

      3. unsub-neg 46.5%

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

      4. *-commutative 46.5%

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

   5. Simplified 46.5%

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

      1. *-commutative 46.5%

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

      2. *-un-lft-identity 46.5%

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

      3. pow2 46.5%

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

      4. times-frac 53.8%

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

   7. Applied egg-rr 53.8%

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

   8. Taylor expanded in y.im around 0 46.5%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 46.5%

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

      2. distribute-frac-neg 46.5%

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

      3. +-commutative 46.5%

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

      4. unpow2 46.5%

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

      5. associate-/r* 53.8%

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

      6. rem-square-sqrt 38.9%

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

      7. associate-*r/ 38.9%

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

      8. associate-*l/ 38.9%

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

      9. distribute-frac-neg 38.9%

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

      10. unsub-neg 38.9%

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

      11. associate-*l/ 38.9%

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

      12. associate-*r/ 38.9%

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

      13. rem-square-sqrt 53.8%

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

      14. associate-*r/ 54.4%

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

      15. div-sub 56.7%

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

   10. Simplified 56.7%

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

4. Final simplification 88.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+301}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.im \cdot y.re - x.re \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-105}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.5e+44)
     (/ (- (* y.im (/ x.re y.re)) x.im) (hypot y.re y.im))
     (if (<= y.re -1.75e-105)
       t_0
       (if (<= y.re 3.5e-77)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 1.9e+64)
           t_0
           (* (/ 1.0 (hypot y.re y.im)) (- x.im (/ x.re (/ y.re y.im))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.5e+44) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.75e-105) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e-77) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.9e+64) {
		tmp = t_0;
	} else {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.5e+44) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.75e-105) {
		tmp = t_0;
	} else if (y_46_re <= 3.5e-77) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 1.9e+64) {
		tmp = t_0;
	} else {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (y_46_re / y_46_im)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.5e+44:
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -1.75e-105:
		tmp = t_0
	elif y_46_re <= 3.5e-77:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 1.9e+64:
		tmp = t_0
	else:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (y_46_re / y_46_im)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.5e+44)
		tmp = Float64(Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.75e-105)
		tmp = t_0;
	elseif (y_46_re <= 3.5e-77)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 1.9e+64)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(x_46_im - Float64(x_46_re / Float64(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)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.5e+44)
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -1.75e-105)
		tmp = t_0;
	elseif (y_46_re <= 3.5e-77)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 1.9e+64)
		tmp = t_0;
	else
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (x_46_im - (x_46_re / (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_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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, -2.5e+44], N[(N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.75e-105], t$95$0, If[LessEqual[y$46$re, 3.5e-77], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 1.9e+64], t$95$0, N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(x$46$im - N[(x$46$re / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.4999999999999998e44

   1. Initial program 51.0%

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

      1. *-un-lft-identity 51.0%

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

      2. add-sqr-sqrt 51.0%

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

      3. times-frac 51.0%

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

      4. hypot-def 51.0%

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

      5. hypot-def 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in y.re around -inf 77.3%

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

      1. +-commutative 77.3%

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

      2. neg-mul-1 77.3%

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

      3. unsub-neg 77.3%

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

      4. associate-/l* 77.5%

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

   7. Simplified 77.5%

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

      1. sub-neg 77.5%

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

      2. distribute-lft-in 77.5%

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

      3. div-inv 77.5%

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

      4. clear-num 78.4%

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

   9. Applied egg-rr 78.4%

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

       1. distribute-lft-out 78.4%

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

       2. sub-neg 78.4%

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

       3. associate-*l/ 78.7%

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

       4. *-lft-identity 78.7%

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

       5. associate-*r/ 77.6%

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

       6. associate-*l/ 79.4%

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

       7. *-commutative 79.4%

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

   11. Simplified 79.4%

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

   if -2.4999999999999998e44 < y.re < -1.75e-105 or 3.50000000000000013e-77 < y.re < 1.9000000000000001e64

   1. Initial program 82.8%

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

   if -1.75e-105 < y.re < 3.50000000000000013e-77

   1. Initial program 75.8%

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

   3. Taylor expanded in y.re around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 81.4%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 81.4%

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

      3. unsub-neg 81.4%

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

      4. *-commutative 81.4%

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

   5. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. *-un-lft-identity 81.4%

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

      3. pow2 81.4%

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

      4. times-frac 92.5%

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in y.im around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 81.4%

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

      2. distribute-frac-neg 81.4%

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

      3. +-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. associate-/r* 92.6%

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

      6. rem-square-sqrt 61.0%

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

      7. associate-*r/ 61.0%

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

      8. associate-*l/ 61.0%

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

      9. distribute-frac-neg 61.0%

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

      10. unsub-neg 61.0%

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

      11. associate-*l/ 61.0%

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

      12. associate-*r/ 61.0%

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

      13. rem-square-sqrt 92.6%

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

      14. associate-*r/ 90.7%

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

      15. div-sub 91.7%

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

   10. Simplified 91.7%

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

   11. Taylor expanded in x.im around 0 93.5%

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

   if 1.9000000000000001e64 < y.re

   1. Initial program 50.9%

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

      1. *-un-lft-identity 50.9%

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

      2. add-sqr-sqrt 50.9%

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

      3. times-frac 50.9%

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

      4. hypot-def 50.9%

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

      5. hypot-def 68.3%

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

   4. Applied egg-rr 68.3%

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

   5. Taylor expanded in y.re around inf 81.4%

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

      1. mul-1-neg 81.4%

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

      2. unsub-neg 81.4%

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

      3. associate-/l* 89.3%

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

   7. Simplified 89.3%

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

4. Final simplification 87.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-105}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 1.9 \cdot 10^{+64}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \left(x.im - \frac{x.re}{\frac{y.re}{y.im}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-96}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))))
   (if (<= y.re -2.5e+44)
     t_1
     (if (<= y.re -4e-96)
       t_0
       (if (<= y.re 1.25e-76)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 3.1e+124) t_0 t_1))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -2.5e+44) {
		tmp = t_1;
	} else if (y_46_re <= -4e-96) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-76) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.1e+124) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46im / y_46re) - (x_46re * (y_46im / (y_46re ** 2.0d0)))
    if (y_46re <= (-2.5d+44)) then
        tmp = t_1
    else if (y_46re <= (-4d-96)) then
        tmp = t_0
    else if (y_46re <= 1.25d-76) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 3.1d+124) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im / Math.pow(y_46_re, 2.0)));
	double tmp;
	if (y_46_re <= -2.5e+44) {
		tmp = t_1;
	} else if (y_46_re <= -4e-96) {
		tmp = t_0;
	} else if (y_46_re <= 1.25e-76) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 3.1e+124) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im / math.pow(y_46_re, 2.0)))
	tmp = 0
	if y_46_re <= -2.5e+44:
		tmp = t_1
	elif y_46_re <= -4e-96:
		tmp = t_0
	elif y_46_re <= 1.25e-76:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 3.1e+124:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))))
	tmp = 0.0
	if (y_46_re <= -2.5e+44)
		tmp = t_1;
	elseif (y_46_re <= -4e-96)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-76)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 3.1e+124)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_im / y_46_re) - (x_46_re * (y_46_im / (y_46_re ^ 2.0)));
	tmp = 0.0;
	if (y_46_re <= -2.5e+44)
		tmp = t_1;
	elseif (y_46_re <= -4e-96)
		tmp = t_0;
	elseif (y_46_re <= 1.25e-76)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 3.1e+124)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.5e+44], t$95$1, If[LessEqual[y$46$re, -4e-96], t$95$0, If[LessEqual[y$46$re, 1.25e-76], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 3.1e+124], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.4999999999999998e44 or 3.1000000000000002e124 < y.re

   1. Initial program 46.9%

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

      1. *-un-lft-identity 46.9%

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

      2. add-sqr-sqrt 46.9%

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

      3. times-frac 46.9%

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

      4. hypot-def 46.9%

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

      5. hypot-def 69.0%

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

   4. Applied egg-rr 69.0%

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

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

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

      1. +-commutative 73.8%

         \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg 73.8%

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

      3. unsub-neg 73.8%

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

      4. *-lft-identity 73.8%

         \[\leadsto \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{\color{blue}{1 \cdot {y.re}^{2}}} \]

      5. times-frac 76.3%

         \[\leadsto \frac{x.im}{y.re} - \color{blue}{\frac{x.re}{1} \cdot \frac{y.im}{{y.re}^{2}}} \]

      6. /-rgt-identity 76.3%

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

   7. Simplified 76.3%

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

   if -2.4999999999999998e44 < y.re < -3.9999999999999996e-96 or 1.2499999999999999e-76 < y.re < 3.1000000000000002e124

   1. Initial program 82.8%

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

   if -3.9999999999999996e-96 < y.re < 1.2499999999999999e-76

   1. Initial program 75.8%

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

   3. Taylor expanded in y.re around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 81.4%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 81.4%

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

      3. unsub-neg 81.4%

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

      4. *-commutative 81.4%

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

   5. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. *-un-lft-identity 81.4%

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

      3. pow2 81.4%

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

      4. times-frac 92.5%

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in y.im around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 81.4%

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

      2. distribute-frac-neg 81.4%

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

      3. +-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. associate-/r* 92.6%

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

      6. rem-square-sqrt 61.0%

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

      7. associate-*r/ 61.0%

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

      8. associate-*l/ 61.0%

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

      9. distribute-frac-neg 61.0%

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

      10. unsub-neg 61.0%

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

      11. associate-*l/ 61.0%

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

      12. associate-*r/ 61.0%

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

      13. rem-square-sqrt 92.6%

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

      14. associate-*r/ 90.7%

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

      15. div-sub 91.7%

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

   10. Simplified 91.7%

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

   11. Taylor expanded in x.im around 0 93.5%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.5 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{-96}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{-76}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 3.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-102}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.3e+44)
     (/ (- (* y.im (/ x.re y.re)) x.im) (hypot y.re y.im))
     (if (<= y.re -1.75e-102)
       t_0
       (if (<= y.re 3.6e-77)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 9e+124)
           t_0
           (- (/ x.im y.re) (* x.re (/ y.im (pow y.re 2.0))))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.3e+44) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.75e-102) {
		tmp = t_0;
	} else if (y_46_re <= 3.6e-77) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 9e+124) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / pow(y_46_re, 2.0)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.3e+44) {
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.75e-102) {
		tmp = t_0;
	} else if (y_46_re <= 3.6e-77) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 9e+124) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / Math.pow(y_46_re, 2.0)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.3e+44:
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -1.75e-102:
		tmp = t_0
	elif y_46_re <= 3.6e-77:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 9e+124:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / math.pow(y_46_re, 2.0)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.3e+44)
		tmp = Float64(Float64(Float64(y_46_im * Float64(x_46_re / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.75e-102)
		tmp = t_0;
	elseif (y_46_re <= 3.6e-77)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 9e+124)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_re) - Float64(x_46_re * Float64(y_46_im / (y_46_re ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.3e+44)
		tmp = ((y_46_im * (x_46_re / y_46_re)) - x_46_im) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -1.75e-102)
		tmp = t_0;
	elseif (y_46_re <= 3.6e-77)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 9e+124)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_re) - (x_46_re * (y_46_im / (y_46_re ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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, -2.3e+44], N[(N[(N[(y$46$im * N[(x$46$re / y$46$re), $MachinePrecision]), $MachinePrecision] - x$46$im), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.75e-102], t$95$0, If[LessEqual[y$46$re, 3.6e-77], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 9e+124], t$95$0, N[(N[(x$46$im / y$46$re), $MachinePrecision] - N[(x$46$re * N[(y$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.30000000000000004e44

   1. Initial program 51.0%

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

      1. *-un-lft-identity 51.0%

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

      2. add-sqr-sqrt 51.0%

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

      3. times-frac 51.0%

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

      4. hypot-def 51.0%

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

      5. hypot-def 71.9%

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

   4. Applied egg-rr 71.9%

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

   5. Taylor expanded in y.re around -inf 77.3%

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

      1. +-commutative 77.3%

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

      2. neg-mul-1 77.3%

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

      3. unsub-neg 77.3%

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

      4. associate-/l* 77.5%

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

   7. Simplified 77.5%

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

      1. sub-neg 77.5%

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

      2. distribute-lft-in 77.5%

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

      3. div-inv 77.5%

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

      4. clear-num 78.4%

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

   9. Applied egg-rr 78.4%

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

       1. distribute-lft-out 78.4%

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

       2. sub-neg 78.4%

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

       3. associate-*l/ 78.7%

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

       4. *-lft-identity 78.7%

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

       5. associate-*r/ 77.6%

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

       6. associate-*l/ 79.4%

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

       7. *-commutative 79.4%

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

   11. Simplified 79.4%

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

   if -2.30000000000000004e44 < y.re < -1.74999999999999993e-102 or 3.6e-77 < y.re < 9.0000000000000008e124

   1. Initial program 82.8%

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

   if -1.74999999999999993e-102 < y.re < 3.6e-77

   1. Initial program 75.8%

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

   3. Taylor expanded in y.re around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 81.4%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 81.4%

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

      3. unsub-neg 81.4%

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

      4. *-commutative 81.4%

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

   5. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. *-un-lft-identity 81.4%

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

      3. pow2 81.4%

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

      4. times-frac 92.5%

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in y.im around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 81.4%

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

      2. distribute-frac-neg 81.4%

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

      3. +-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. associate-/r* 92.6%

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

      6. rem-square-sqrt 61.0%

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

      7. associate-*r/ 61.0%

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

      8. associate-*l/ 61.0%

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

      9. distribute-frac-neg 61.0%

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

      10. unsub-neg 61.0%

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

      11. associate-*l/ 61.0%

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

      12. associate-*r/ 61.0%

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

      13. rem-square-sqrt 92.6%

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

      14. associate-*r/ 90.7%

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

      15. div-sub 91.7%

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

   10. Simplified 91.7%

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

   11. Taylor expanded in x.im around 0 93.5%

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

   if 9.0000000000000008e124 < y.re

   1. Initial program 37.6%

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

      1. *-un-lft-identity 37.6%

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

      2. add-sqr-sqrt 37.5%

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

      3. times-frac 37.6%

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

      4. hypot-def 37.6%

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

      5. hypot-def 62.3%

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

   4. Applied egg-rr 62.3%

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

   5. Taylor expanded in y.re around inf 76.6%

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

      1. +-commutative 76.6%

         \[\leadsto \color{blue}{\frac{x.im}{y.re} + -1 \cdot \frac{x.re \cdot y.im}{{y.re}^{2}}} \]

      2. mul-1-neg 76.6%

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

      3. unsub-neg 76.6%

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

      4. *-lft-identity 76.6%

         \[\leadsto \frac{x.im}{y.re} - \frac{x.re \cdot y.im}{\color{blue}{1 \cdot {y.re}^{2}}} \]

      5. times-frac 80.8%

         \[\leadsto \frac{x.im}{y.re} - \color{blue}{\frac{x.re}{1} \cdot \frac{y.im}{{y.re}^{2}}} \]

      6. /-rgt-identity 80.8%

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

   7. Simplified 80.8%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{y.im \cdot \frac{x.re}{y.re} - x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.75 \cdot 10^{-102}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3.6 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 9 \cdot 10^{+124}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re} - x.re \cdot \frac{y.im}{{y.re}^{2}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+127}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-106}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -2.9e+127)
     (/ x.im y.re)
     (if (<= y.re -3.2e-106)
       t_0
       (if (<= y.re 6.6e-78)
         (/ (- (/ (* x.im y.re) y.im) x.re) y.im)
         (if (<= y.re 7.8e+153) t_0 (/ x.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.9e+127) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-106) {
		tmp = t_0;
	} else if (y_46_re <= 6.6e-78) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7.8e+153) {
		tmp = t_0;
	} else {
		tmp = x_46_im / 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) :: t_0
    real(8) :: tmp
    t_0 = ((x_46im * y_46re) - (x_46re * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    if (y_46re <= (-2.9d+127)) then
        tmp = x_46im / y_46re
    else if (y_46re <= (-3.2d-106)) then
        tmp = t_0
    else if (y_46re <= 6.6d-78) then
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    else if (y_46re <= 7.8d+153) then
        tmp = t_0
    else
        tmp = x_46im / 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 t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double tmp;
	if (y_46_re <= -2.9e+127) {
		tmp = x_46_im / y_46_re;
	} else if (y_46_re <= -3.2e-106) {
		tmp = t_0;
	} else if (y_46_re <= 6.6e-78) {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	} else if (y_46_re <= 7.8e+153) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -2.9e+127:
		tmp = x_46_im / y_46_re
	elif y_46_re <= -3.2e-106:
		tmp = t_0
	elif y_46_re <= 6.6e-78:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im
	elif y_46_re <= 7.8e+153:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_im * y_46_re) - Float64(x_46_re * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	tmp = 0.0
	if (y_46_re <= -2.9e+127)
		tmp = Float64(x_46_im / y_46_re);
	elseif (y_46_re <= -3.2e-106)
		tmp = t_0;
	elseif (y_46_re <= 6.6e-78)
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	elseif (y_46_re <= 7.8e+153)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = ((x_46_im * y_46_re) - (x_46_re * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -2.9e+127)
		tmp = x_46_im / y_46_re;
	elseif (y_46_re <= -3.2e-106)
		tmp = t_0;
	elseif (y_46_re <= 6.6e-78)
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	elseif (y_46_re <= 7.8e+153)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] - N[(x$46$re * 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, -2.9e+127], N[(x$46$im / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -3.2e-106], t$95$0, If[LessEqual[y$46$re, 6.6e-78], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 7.8e+153], t$95$0, N[(x$46$im / y$46$re), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.9000000000000002e127 or 7.79999999999999966e153 < y.re

   1. Initial program 33.3%

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

   3. Taylor expanded in y.re around inf 74.8%

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

   if -2.9000000000000002e127 < y.re < -3.2e-106 or 6.59999999999999963e-78 < y.re < 7.79999999999999966e153

   1. Initial program 79.9%

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

   if -3.2e-106 < y.re < 6.59999999999999963e-78

   1. Initial program 75.8%

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

   3. Taylor expanded in y.re around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 81.4%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 81.4%

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

      3. unsub-neg 81.4%

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

      4. *-commutative 81.4%

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

   5. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. *-un-lft-identity 81.4%

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

      3. pow2 81.4%

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

      4. times-frac 92.5%

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

   7. Applied egg-rr 92.5%

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

   8. Taylor expanded in y.im around 0 81.4%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 81.4%

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

      2. distribute-frac-neg 81.4%

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

      3. +-commutative 81.4%

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

      4. unpow2 81.4%

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

      5. associate-/r* 92.6%

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

      6. rem-square-sqrt 61.0%

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

      7. associate-*r/ 61.0%

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

      8. associate-*l/ 61.0%

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

      9. distribute-frac-neg 61.0%

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

      10. unsub-neg 61.0%

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

      11. associate-*l/ 61.0%

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

      12. associate-*r/ 61.0%

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

      13. rem-square-sqrt 92.6%

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

      14. associate-*r/ 90.7%

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

      15. div-sub 91.7%

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

   10. Simplified 91.7%

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

   11. Taylor expanded in x.im around 0 93.5%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.9 \cdot 10^{+127}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{elif}\;y.re \leq -3.2 \cdot 10^{-106}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \mathbf{elif}\;y.re \leq 7.8 \cdot 10^{+153}:\\ \;\;\;\;\frac{x.im \cdot y.re - x.re \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.re}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+51} \lor \neg \left(y.re \leq 4.8 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.35e+51) (not (<= y.re 4.8e+63)))
   (/ x.im y.re)
   (/ (- (* x.im (/ y.re y.im)) x.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_re <= -2.35e+51) || !(y_46_re <= 4.8e+63)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.35d+51)) .or. (.not. (y_46re <= 4.8d+63))) then
        tmp = x_46im / y_46re
    else
        tmp = ((x_46im * (y_46re / y_46im)) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.35e+51) || !(y_46_re <= 4.8e+63)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_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_re <= -2.35e+51) or not (y_46_re <= 4.8e+63):
		tmp = x_46_im / y_46_re
	else:
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_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_re <= -2.35e+51) || !(y_46_re <= 4.8e+63))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(x_46_im * Float64(y_46_re / y_46_im)) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.35e+51) || ~((y_46_re <= 4.8e+63)))
		tmp = x_46_im / y_46_re;
	else
		tmp = ((x_46_im * (y_46_re / y_46_im)) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.35e+51], N[Not[LessEqual[y$46$re, 4.8e+63]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(x$46$im * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.3500000000000001e51 or 4.8e63 < y.re

   1. Initial program 52.0%

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

   3. Taylor expanded in y.re around inf 67.3%

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

   if -2.3500000000000001e51 < y.re < 4.8e63

   1. Initial program 77.1%

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

   3. Taylor expanded in y.re around 0 71.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 71.7%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. *-commutative 71.7%

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

   5. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. *-un-lft-identity 71.7%

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

      3. pow2 71.7%

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

      4. times-frac 78.7%

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

   7. Applied egg-rr 78.7%

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

   8. Taylor expanded in y.im around 0 71.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 71.7%

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

      2. distribute-frac-neg 71.7%

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

      3. +-commutative 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/r* 78.7%

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

      6. rem-square-sqrt 50.6%

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

      7. associate-*r/ 50.6%

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

      8. associate-*l/ 50.6%

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

      9. distribute-frac-neg 50.6%

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

      10. unsub-neg 50.6%

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

      11. associate-*l/ 50.6%

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

      12. associate-*r/ 50.6%

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

      13. rem-square-sqrt 78.7%

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

      14. associate-*r/ 77.6%

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

      15. div-sub 78.9%

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

   10. Simplified 78.9%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.35 \cdot 10^{+51} \lor \neg \left(y.re \leq 4.8 \cdot 10^{+63}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im \cdot \frac{y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+51} \lor \neg \left(y.re \leq 1.5 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -3.4e+51) (not (<= y.re 1.5e+61)))
   (/ x.im y.re)
   (/ (- (/ (* x.im y.re) y.im) x.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_re <= -3.4e+51) || !(y_46_re <= 1.5e+61)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-3.4d+51)) .or. (.not. (y_46re <= 1.5d+61))) then
        tmp = x_46im / y_46re
    else
        tmp = (((x_46im * y_46re) / y_46im) - x_46re) / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -3.4e+51) || !(y_46_re <= 1.5e+61)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_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_re <= -3.4e+51) or not (y_46_re <= 1.5e+61):
		tmp = x_46_im / y_46_re
	else:
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_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_re <= -3.4e+51) || !(y_46_re <= 1.5e+61))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(Float64(Float64(x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -3.4e+51) || ~((y_46_re <= 1.5e+61)))
		tmp = x_46_im / y_46_re;
	else
		tmp = (((x_46_im * y_46_re) / y_46_im) - x_46_re) / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -3.4e+51], N[Not[LessEqual[y$46$re, 1.5e+61]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[(N[(N[(N[(x$46$im * y$46$re), $MachinePrecision] / y$46$im), $MachinePrecision] - x$46$re), $MachinePrecision] / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -3.39999999999999984e51 or 1.5e61 < y.re

   1. Initial program 52.0%

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

   3. Taylor expanded in y.re around inf 67.3%

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

   if -3.39999999999999984e51 < y.re < 1.5e61

   1. Initial program 77.1%

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

   3. Taylor expanded in y.re around 0 71.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   4. Step-by-step derivation

      1. +-commutative 71.7%

         \[\leadsto \color{blue}{\frac{x.im \cdot y.re}{{y.im}^{2}} + -1 \cdot \frac{x.re}{y.im}} \]

      2. mul-1-neg 71.7%

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

      3. unsub-neg 71.7%

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

      4. *-commutative 71.7%

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

   5. Simplified 71.7%

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

      1. *-commutative 71.7%

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

      2. *-un-lft-identity 71.7%

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

      3. pow2 71.7%

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

      4. times-frac 78.7%

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

   7. Applied egg-rr 78.7%

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

   8. Taylor expanded in y.im around 0 71.7%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im} + \frac{x.im \cdot y.re}{{y.im}^{2}}} \]
   9. Step-by-step derivation

      1. mul-1-neg 71.7%

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

      2. distribute-frac-neg 71.7%

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

      3. +-commutative 71.7%

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

      4. unpow2 71.7%

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

      5. associate-/r* 78.7%

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

      6. rem-square-sqrt 50.6%

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

      7. associate-*r/ 50.6%

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

      8. associate-*l/ 50.6%

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

      9. distribute-frac-neg 50.6%

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

      10. unsub-neg 50.6%

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

      11. associate-*l/ 50.6%

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

      12. associate-*r/ 50.6%

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

      13. rem-square-sqrt 78.7%

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

      14. associate-*r/ 77.6%

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

      15. div-sub 78.9%

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

   10. Simplified 78.9%

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

   11. Taylor expanded in x.im around 0 80.0%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+51} \lor \neg \left(y.re \leq 1.5 \cdot 10^{+61}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x.im \cdot y.re}{y.im} - x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -0.55 \lor \neg \left(y.re \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -0.55) (not (<= y.re 9e+43)))
   (/ x.im y.re)
   (/ (- x.re) y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -0.55) || !(y_46_re <= 9e+43)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_46_re / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-0.55d0)) .or. (.not. (y_46re <= 9d+43))) then
        tmp = x_46im / y_46re
    else
        tmp = -x_46re / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -0.55) || !(y_46_re <= 9e+43)) {
		tmp = x_46_im / y_46_re;
	} else {
		tmp = -x_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_re <= -0.55) or not (y_46_re <= 9e+43):
		tmp = x_46_im / y_46_re
	else:
		tmp = -x_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_re <= -0.55) || !(y_46_re <= 9e+43))
		tmp = Float64(x_46_im / y_46_re);
	else
		tmp = Float64(Float64(-x_46_re) / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -0.55) || ~((y_46_re <= 9e+43)))
		tmp = x_46_im / y_46_re;
	else
		tmp = -x_46_re / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -0.55], N[Not[LessEqual[y$46$re, 9e+43]], $MachinePrecision]], N[(x$46$im / y$46$re), $MachinePrecision], N[((-x$46$re) / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -0.55000000000000004 or 9e43 < y.re

   1. Initial program 54.2%

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

   3. Taylor expanded in y.re around inf 63.8%

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

   if -0.55000000000000004 < y.re < 9e43

   1. Initial program 78.0%

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

   3. Taylor expanded in y.re around 0 63.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x.re}{y.im}} \]
   4. Step-by-step derivation

      1. associate-*r/ 63.2%

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

      2. neg-mul-1 63.2%

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

   5. Simplified 63.2%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -0.55 \lor \neg \left(y.re \leq 9 \cdot 10^{+43}\right):\\ \;\;\;\;\frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x.re}{y.im}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 43.6% accurate, 5.0× speedup

Math

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

FPCore

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

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_re;
}

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_46re
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_re;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.1%

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

3. Taylor expanded in y.re around inf 39.5%

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

4. Final simplification 39.5%

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

Reproduce

herbie shell --seed 2024040 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, imaginary part"
  :precision binary64
  (/ (- (* x.im y.re) (* x.re y.im)) (+ (* y.re y.re) (* y.im y.im))))