_divideComplex, real part

Percentage Accurate: 62.2% → 85.7%

Time: 11.0s

Alternatives: 12

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 62.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.7% accurate, 0.0× speedup

Math

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

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<=
      (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
      1e+299)
   (/ (/ (fma x.im y.im (* x.re y.re)) (hypot y.re y.im)) (hypot y.re y.im))
   (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299) {
		tmp = (fma(x_46_im, y_46_im, (x_46_re * y_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 1e+299)
		tmp = Float64(Float64(fma(x_46_im, y_46_im, Float64(x_46_re * y_46_re)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+299], N[(N[(N[(x$46$im * y$46$im + N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.0000000000000001e299

   1. Initial program 76.4%

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

      1. add-sqr-sqrt 76.4%

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

      2. *-un-lft-identity 76.4%

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

      3. times-frac 76.4%

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

      4. hypot-def 76.4%

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

      5. fma-def 76.4%

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

      6. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. fma-def 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l/ 95.2%

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

      4. div-inv 95.3%

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

      5. +-commutative 95.3%

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

      6. fma-def 95.3%

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

   5. Applied egg-rr 95.3%

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

   if 1.0000000000000001e299 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 13.1%

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

   2. Taylor expanded in y.re around inf 49.4%

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

      1. +-commutative 49.4%

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

      2. associate-/l* 53.4%

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

      3. associate-/r/ 51.9%

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

      4. fma-def 51.9%

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

   4. Simplified 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 51.9%

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

      2. +-commutative 51.9%

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

      3. pow2 51.9%

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

      4. div-inv 51.9%

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

      5. associate-*l* 53.4%

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

      6. pow2 53.4%

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

      7. pow-flip 53.4%

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

      8. metadata-eval 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

      3. metadata-eval 51.9%

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

      4. pow-div 51.9%

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

      5. inv-pow 51.9%

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

      6. pow1 51.9%

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

      7. associate-/r/ 55.0%

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

      8. associate-*l/ 61.0%

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

   8. Applied egg-rr 61.0%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{1}{y.re} \cdot y.im}{\frac{y.re}{x.im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

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

Alternative 2: 85.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+299}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1e+299)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_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_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299) {
		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_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299:
		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_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_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_re * y_46_re) + Float64(x_46_im * 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))) <= 1e+299)
		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(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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], 1e+299], 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[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.0000000000000001e299

   1. Initial program 76.4%

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

      1. add-sqr-sqrt 76.4%

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

      2. *-un-lft-identity 76.4%

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

      3. times-frac 76.4%

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

      4. hypot-def 76.4%

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

      5. fma-def 76.4%

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

      6. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. fma-def 95.2%

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

   5. Applied egg-rr 95.2%

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

   if 1.0000000000000001e299 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 13.1%

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

   2. Taylor expanded in y.re around inf 49.4%

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

      1. +-commutative 49.4%

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

      2. associate-/l* 53.4%

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

      3. associate-/r/ 51.9%

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

      4. fma-def 51.9%

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

   4. Simplified 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 51.9%

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

      2. +-commutative 51.9%

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

      3. pow2 51.9%

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

      4. div-inv 51.9%

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

      5. associate-*l* 53.4%

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

      6. pow2 53.4%

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

      7. pow-flip 53.4%

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

      8. metadata-eval 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

      3. metadata-eval 51.9%

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

      4. pow-div 51.9%

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

      5. inv-pow 51.9%

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

      6. pow1 51.9%

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

      7. associate-/r/ 55.0%

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

      8. associate-*l/ 61.0%

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

   8. Applied egg-rr 61.0%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{1}{y.re} \cdot y.im}{\frac{y.re}{x.im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

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

Alternative 3: 85.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 10^{+299}:\\ \;\;\;\;\frac{t_0 \cdot \frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 1e+299)
     (/ (* t_0 (/ 1.0 (hypot y.re y.im))) (hypot y.re y.im))
     (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299) {
		tmp = (t_0 * (1.0 / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_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_re * y_46_re) + (x_46_im * y_46_im);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299) {
		tmp = (t_0 * (1.0 / Math.hypot(y_46_re, y_46_im))) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im)
	tmp = 0
	if (t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299:
		tmp = (t_0 * (1.0 / math.hypot(y_46_re, y_46_im))) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_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_re * y_46_re) + Float64(x_46_im * 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))) <= 1e+299)
		tmp = Float64(Float64(t_0 * Float64(1.0 / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re * y_46_re) + (x_46_im * y_46_im);
	tmp = 0.0;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 1e+299)
		tmp = (t_0 * (1.0 / hypot(y_46_re, y_46_im))) / hypot(y_46_re, y_46_im);
	else
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * 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], 1e+299], N[(N[(t$95$0 * N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 1.0000000000000001e299

   1. Initial program 76.4%

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

      1. add-sqr-sqrt 76.4%

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

      2. *-un-lft-identity 76.4%

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

      3. times-frac 76.4%

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

      4. hypot-def 76.4%

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

      5. fma-def 76.4%

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

      6. hypot-def 95.2%

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

   3. Applied egg-rr 95.2%

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

      1. fma-def 95.2%

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

      2. *-commutative 95.2%

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

      3. associate-*l/ 95.2%

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

      4. div-inv 95.3%

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

      5. +-commutative 95.3%

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

      6. fma-def 95.3%

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

   5. Applied egg-rr 95.3%

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

      1. div-inv 95.2%

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

      2. fma-udef 95.2%

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

      3. +-commutative 95.2%

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

      4. fma-def 95.2%

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

   7. Applied egg-rr 95.2%

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

      1. fma-def 95.2%

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

   9. Applied egg-rr 95.2%

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

   if 1.0000000000000001e299 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im)))

   1. Initial program 13.1%

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

   2. Taylor expanded in y.re around inf 49.4%

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

      1. +-commutative 49.4%

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

      2. associate-/l* 53.4%

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

      3. associate-/r/ 51.9%

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

      4. fma-def 51.9%

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

   4. Simplified 51.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 51.9%

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

      2. +-commutative 51.9%

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

      3. pow2 51.9%

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

      4. div-inv 51.9%

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

      5. associate-*l* 53.4%

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

      6. pow2 53.4%

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

      7. pow-flip 53.4%

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

      8. metadata-eval 53.4%

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

   6. Applied egg-rr 53.4%

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

      1. associate-*r* 51.9%

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

      2. *-commutative 51.9%

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

      3. metadata-eval 51.9%

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

      4. pow-div 51.9%

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

      5. inv-pow 51.9%

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

      6. pow1 51.9%

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

      7. associate-/r/ 55.0%

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

      8. associate-*l/ 61.0%

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

   8. Applied egg-rr 61.0%

      \[\leadsto \frac{x.re}{y.re} + \color{blue}{\frac{\frac{1}{y.re} \cdot y.im}{\frac{y.re}{x.im}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

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

Alternative 4: 76.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im \cdot \frac{-1}{y.re}}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -3.3e+127)
   (/ (- x.im) (hypot y.re y.im))
   (if (<= y.im -1.3e+77)
     (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))
     (if (<= y.im -2.65e-129)
       (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
       (if (<= y.im 8e+50)
         (+ (/ x.re y.re) (* x.im (/ (* y.im (/ -1.0 y.re)) (- y.re))))
         (/ (+ x.im (/ x.re (/ y.im y.re))) (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -3.3e+127) {
		tmp = -x_46_im / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -1.3e+77) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	} else if (y_46_im <= -2.65e-129) {
		tmp = fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / fma(y_46_re, y_46_re, (y_46_im * y_46_im));
	} else if (y_46_im <= 8e+50) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -3.3e+127)
		tmp = Float64(Float64(-x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -1.3e+77)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)));
	elseif (y_46_im <= -2.65e-129)
		tmp = Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / fma(y_46_re, y_46_re, Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 8e+50)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im * Float64(-1.0 / y_46_re)) / Float64(-y_46_re))));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -3.3e+127], N[((-x$46$im) / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -1.3e+77], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.65e-129], N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(y$46$re * y$46$re + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8e+50], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -3.29999999999999977e127

   1. Initial program 45.1%

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

      1. add-sqr-sqrt 45.1%

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

      2. *-un-lft-identity 45.1%

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

      3. times-frac 45.1%

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

      4. hypot-def 45.1%

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

      5. fma-def 45.1%

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

      6. hypot-def 73.0%

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

   3. Applied egg-rr 73.0%

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

      1. fma-def 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-*l/ 73.1%

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

      4. div-inv 73.2%

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

      5. +-commutative 73.2%

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

      6. fma-def 73.2%

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

   5. Applied egg-rr 73.2%

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

   6. Taylor expanded in y.im around -inf 83.9%

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

      1. neg-mul-1 83.9%

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

   8. Simplified 83.9%

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

   if -3.29999999999999977e127 < y.im < -1.3000000000000001e77

   1. Initial program 34.6%

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

   2. Taylor expanded in y.re around inf 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-/l* 79.4%

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

      3. associate-/r/ 79.4%

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

      4. fma-def 79.4%

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

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 79.4%

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

      2. +-commutative 79.4%

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

      3. pow2 79.4%

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

      4. div-inv 79.4%

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

      5. associate-*l* 79.4%

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

      6. pow2 79.4%

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

      7. pow-flip 79.4%

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

      8. metadata-eval 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-*r* 79.4%

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

      2. *-commutative 79.4%

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

      3. metadata-eval 79.4%

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

      4. pow-div 79.4%

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

      5. inv-pow 79.4%

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

      6. pow1 79.4%

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

      7. associate-/r/ 79.4%

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

      8. associate-*l/ 79.4%

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

   8. Applied egg-rr 79.4%

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

   if -1.3000000000000001e77 < y.im < -2.64999999999999987e-129

   1. Initial program 92.5%

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

      1. fma-def 92.5%

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

      2. fma-def 92.5%

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

   3. Simplified 92.5%

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

   if -2.64999999999999987e-129 < y.im < 8.0000000000000006e50

   1. Initial program 67.8%

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

   2. Taylor expanded in y.re around inf 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-/l* 84.6%

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

      3. associate-/r/ 79.9%

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

      4. fma-def 79.9%

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

   4. Simplified 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 79.9%

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

      2. +-commutative 79.9%

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

      3. pow2 79.9%

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

      4. div-inv 79.9%

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

      5. associate-*l* 84.6%

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

      6. pow2 84.6%

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

      7. pow-flip 84.6%

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

      8. metadata-eval 84.6%

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

   6. Applied egg-rr 84.6%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. metadata-eval 79.9%

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

      4. pow-div 79.9%

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

      5. inv-pow 79.9%

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

      6. pow1 79.9%

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

      7. associate-/r/ 80.9%

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

      8. clear-num 80.9%

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

      9. clear-num 80.9%

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

      10. frac-2neg 80.9%

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

      11. associate-*l/ 88.5%

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

      12. distribute-neg-frac 88.5%

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

      13. metadata-eval 88.5%

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

      14. distribute-neg-frac 88.5%

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

   8. Applied egg-rr 88.5%

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

      1. associate-/r/ 89.3%

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

      2. *-commutative 89.3%

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

   10. Simplified 89.3%

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

   if 8.0000000000000006e50 < y.im

   1. Initial program 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. *-un-lft-identity 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. fma-def 42.7%

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

      6. hypot-def 58.1%

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

   3. Applied egg-rr 58.1%

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

      1. fma-def 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*l/ 58.1%

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

      4. div-inv 58.1%

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

      5. +-commutative 58.1%

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

      6. fma-def 58.2%

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

   5. Applied egg-rr 58.2%

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

   6. Taylor expanded in y.im around inf 69.3%

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

      1. associate-/l* 78.5%

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

   8. Simplified 78.5%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -3.3 \cdot 10^{+127}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -1.3 \cdot 10^{+77}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -2.65 \cdot 10^{-129}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.im \leq 8 \cdot 10^{+50}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im \cdot \frac{-1}{y.re}}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 5: 76.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im \cdot \frac{-1}{y.re}}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.4e+125)
   (/ (- x.im) (hypot y.re y.im))
   (if (<= y.im -4.8e+74)
     (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))
     (if (<= y.im -7e-128)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 2.5e+49)
         (+ (/ x.re y.re) (* x.im (/ (* y.im (/ -1.0 y.re)) (- y.re))))
         (/ (+ x.im (/ x.re (/ y.im y.re))) (hypot y.re y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.4e+125) {
		tmp = -x_46_im / hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -4.8e+74) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	} else if (y_46_im <= -7e-128) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.5e+49) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(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 tmp;
	if (y_46_im <= -6.4e+125) {
		tmp = -x_46_im / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_im <= -4.8e+74) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	} else if (y_46_im <= -7e-128) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.5e+49) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	} else {
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / Math.hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -6.4e+125:
		tmp = -x_46_im / math.hypot(y_46_re, y_46_im)
	elif y_46_im <= -4.8e+74:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im))
	elif y_46_im <= -7e-128:
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 2.5e+49:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re))
	else:
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / math.hypot(y_46_re, y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.4e+125)
		tmp = Float64(Float64(-x_46_im) / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= -4.8e+74)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)));
	elseif (y_46_im <= -7e-128)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2.5e+49)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im * Float64(-1.0 / y_46_re)) / Float64(-y_46_re))));
	else
		tmp = Float64(Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))) / hypot(y_46_re, y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -6.4e+125)
		tmp = -x_46_im / hypot(y_46_re, y_46_im);
	elseif (y_46_im <= -4.8e+74)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	elseif (y_46_im <= -7e-128)
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 2.5e+49)
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	else
		tmp = (x_46_im + (x_46_re / (y_46_im / y_46_re))) / hypot(y_46_re, y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -6.4e+125], N[((-x$46$im) / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -4.8e+74], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -7e-128], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.5e+49], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y.im < -6.39999999999999967e125

   1. Initial program 45.1%

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

      1. add-sqr-sqrt 45.1%

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

      2. *-un-lft-identity 45.1%

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

      3. times-frac 45.1%

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

      4. hypot-def 45.1%

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

      5. fma-def 45.1%

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

      6. hypot-def 73.0%

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

   3. Applied egg-rr 73.0%

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

      1. fma-def 73.0%

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

      2. *-commutative 73.0%

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

      3. associate-*l/ 73.1%

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

      4. div-inv 73.2%

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

      5. +-commutative 73.2%

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

      6. fma-def 73.2%

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

   5. Applied egg-rr 73.2%

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

   6. Taylor expanded in y.im around -inf 83.9%

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

      1. neg-mul-1 83.9%

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

   8. Simplified 83.9%

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

   if -6.39999999999999967e125 < y.im < -4.80000000000000017e74

   1. Initial program 34.6%

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

   2. Taylor expanded in y.re around inf 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-/l* 79.4%

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

      3. associate-/r/ 79.4%

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

      4. fma-def 79.4%

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

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 79.4%

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

      2. +-commutative 79.4%

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

      3. pow2 79.4%

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

      4. div-inv 79.4%

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

      5. associate-*l* 79.4%

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

      6. pow2 79.4%

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

      7. pow-flip 79.4%

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

      8. metadata-eval 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-*r* 79.4%

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

      2. *-commutative 79.4%

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

      3. metadata-eval 79.4%

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

      4. pow-div 79.4%

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

      5. inv-pow 79.4%

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

      6. pow1 79.4%

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

      7. associate-/r/ 79.4%

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

      8. associate-*l/ 79.4%

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

   8. Applied egg-rr 79.4%

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

   if -4.80000000000000017e74 < y.im < -6.99999999999999999e-128

   1. Initial program 92.5%

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

   if -6.99999999999999999e-128 < y.im < 2.5000000000000002e49

   1. Initial program 67.8%

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

   2. Taylor expanded in y.re around inf 83.6%

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

      1. +-commutative 83.6%

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

      2. associate-/l* 84.6%

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

      3. associate-/r/ 79.9%

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

      4. fma-def 79.9%

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

   4. Simplified 79.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 79.9%

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

      2. +-commutative 79.9%

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

      3. pow2 79.9%

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

      4. div-inv 79.9%

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

      5. associate-*l* 84.6%

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

      6. pow2 84.6%

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

      7. pow-flip 84.6%

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

      8. metadata-eval 84.6%

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

   6. Applied egg-rr 84.6%

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

      1. associate-*r* 79.9%

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

      2. *-commutative 79.9%

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

      3. metadata-eval 79.9%

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

      4. pow-div 79.9%

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

      5. inv-pow 79.9%

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

      6. pow1 79.9%

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

      7. associate-/r/ 80.9%

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

      8. clear-num 80.9%

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

      9. clear-num 80.9%

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

      10. frac-2neg 80.9%

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

      11. associate-*l/ 88.5%

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

      12. distribute-neg-frac 88.5%

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

      13. metadata-eval 88.5%

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

      14. distribute-neg-frac 88.5%

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

   8. Applied egg-rr 88.5%

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

      1. associate-/r/ 89.3%

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

      2. *-commutative 89.3%

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

   10. Simplified 89.3%

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

   if 2.5000000000000002e49 < y.im

   1. Initial program 42.7%

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

      1. add-sqr-sqrt 42.7%

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

      2. *-un-lft-identity 42.7%

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

      3. times-frac 42.7%

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

      4. hypot-def 42.7%

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

      5. fma-def 42.7%

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

      6. hypot-def 58.1%

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

   3. Applied egg-rr 58.1%

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

      1. fma-def 58.1%

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

      2. *-commutative 58.1%

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

      3. associate-*l/ 58.1%

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

      4. div-inv 58.1%

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

      5. +-commutative 58.1%

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

      6. fma-def 58.2%

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

   5. Applied egg-rr 58.2%

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

   6. Taylor expanded in y.im around inf 69.3%

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

      1. associate-/l* 78.5%

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

   8. Simplified 78.5%

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

4. Final simplification 85.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{-x.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -7 \cdot 10^{-128}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.5 \cdot 10^{+49}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im \cdot \frac{-1}{y.re}}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im + \frac{x.re}{\frac{y.im}{y.re}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 6: 79.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -9.6 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))))
   (if (<= y.re -9.6e+137)
     t_1
     (if (<= y.re -3.1e-112)
       t_0
       (if (<= y.re 5.8e-140)
         (+ (/ x.im y.im) (* y.re (/ x.re (pow y.im 2.0))))
         (if (<= y.re 1.65e+86) 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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -9.6e+137) {
		tmp = t_1;
	} else if (y_46_re <= -3.1e-112) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-140) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.65e+86) {
		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_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re / y_46re) + ((y_46im * (1.0d0 / y_46re)) / (y_46re / x_46im))
    if (y_46re <= (-9.6d+137)) then
        tmp = t_1
    else if (y_46re <= (-3.1d-112)) then
        tmp = t_0
    else if (y_46re <= 5.8d-140) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im ** 2.0d0)))
    else if (y_46re <= 1.65d+86) 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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -9.6e+137) {
		tmp = t_1;
	} else if (y_46_re <= -3.1e-112) {
		tmp = t_0;
	} else if (y_46_re <= 5.8e-140) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / Math.pow(y_46_im, 2.0)));
	} else if (y_46_re <= 1.65e+86) {
		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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im))
	tmp = 0
	if y_46_re <= -9.6e+137:
		tmp = t_1
	elif y_46_re <= -3.1e-112:
		tmp = t_0
	elif y_46_re <= 5.8e-140:
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / math.pow(y_46_im, 2.0)))
	elif y_46_re <= 1.65e+86:
		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_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -9.6e+137)
		tmp = t_1;
	elseif (y_46_re <= -3.1e-112)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-140)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / (y_46_im ^ 2.0))));
	elseif (y_46_re <= 1.65e+86)
		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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	tmp = 0.0;
	if (y_46_re <= -9.6e+137)
		tmp = t_1;
	elseif (y_46_re <= -3.1e-112)
		tmp = t_0;
	elseif (y_46_re <= 5.8e-140)
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im ^ 2.0)));
	elseif (y_46_re <= 1.65e+86)
		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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -9.6e+137], t$95$1, If[LessEqual[y$46$re, -3.1e-112], t$95$0, If[LessEqual[y$46$re, 5.8e-140], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[Power[y$46$im, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 1.65e+86], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -9.59999999999999932e137 or 1.65e86 < y.re

   1. Initial program 33.6%

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

   2. Taylor expanded in y.re around inf 74.4%

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

      1. +-commutative 74.4%

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

      2. associate-/l* 75.0%

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

      3. associate-/r/ 76.0%

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

      4. fma-def 76.0%

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

   4. Simplified 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 76.0%

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

      2. +-commutative 76.0%

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

      3. pow2 76.0%

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

      4. div-inv 76.1%

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

      5. associate-*l* 75.0%

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

      6. pow2 75.0%

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

      7. pow-flip 74.0%

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

      8. metadata-eval 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. associate-*r* 76.1%

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

      2. *-commutative 76.1%

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

      3. metadata-eval 76.1%

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

      4. pow-div 76.0%

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

      5. inv-pow 76.0%

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

      6. pow1 76.0%

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

      7. associate-/r/ 79.2%

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

      8. associate-*l/ 84.2%

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

   8. Applied egg-rr 84.2%

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

   if -9.59999999999999932e137 < y.re < -3.0999999999999998e-112 or 5.79999999999999995e-140 < y.re < 1.65e86

   1. Initial program 84.0%

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

   if -3.0999999999999998e-112 < y.re < 5.79999999999999995e-140

   1. Initial program 63.9%

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

   2. Taylor expanded in y.re around 0 86.4%

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

      1. associate-/l* 88.0%

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

      2. associate-/r/ 85.2%

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

   4. Simplified 85.2%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -9.6 \cdot 10^{+137}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{-112}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 5.8 \cdot 10^{-140}:\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{{y.im}^{2}}\\ \mathbf{elif}\;y.re \leq 1.65 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 7: 79.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+133}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.01 \cdot 10^{-162}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-207}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))))
   (if (<= y.re -3.6e+133)
     t_1
     (if (<= y.re -1.01e-162)
       t_0
       (if (<= y.re 1.6e-207) (/ x.im y.im) (if (<= y.re 4.1e+86) 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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -3.6e+133) {
		tmp = t_1;
	} else if (y_46_re <= -1.01e-162) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-207) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 4.1e+86) {
		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_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (x_46re / y_46re) + ((y_46im * (1.0d0 / y_46re)) / (y_46re / x_46im))
    if (y_46re <= (-3.6d+133)) then
        tmp = t_1
    else if (y_46re <= (-1.01d-162)) then
        tmp = t_0
    else if (y_46re <= 1.6d-207) then
        tmp = x_46im / y_46im
    else if (y_46re <= 4.1d+86) 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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_re <= -3.6e+133) {
		tmp = t_1;
	} else if (y_46_re <= -1.01e-162) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e-207) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_re <= 4.1e+86) {
		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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im))
	tmp = 0
	if y_46_re <= -3.6e+133:
		tmp = t_1
	elif y_46_re <= -1.01e-162:
		tmp = t_0
	elif y_46_re <= 1.6e-207:
		tmp = x_46_im / y_46_im
	elif y_46_re <= 4.1e+86:
		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_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)))
	t_1 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_re <= -3.6e+133)
		tmp = t_1;
	elseif (y_46_re <= -1.01e-162)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-207)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_re <= 4.1e+86)
		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_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.6e+133)
		tmp = t_1;
	elseif (y_46_re <= -1.01e-162)
		tmp = t_0;
	elseif (y_46_re <= 1.6e-207)
		tmp = x_46_im / y_46_im;
	elseif (y_46_re <= 4.1e+86)
		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$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.6e+133], t$95$1, If[LessEqual[y$46$re, -1.01e-162], t$95$0, If[LessEqual[y$46$re, 1.6e-207], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$re, 4.1e+86], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -3.59999999999999978e133 or 4.0999999999999999e86 < y.re

   1. Initial program 33.6%

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

   2. Taylor expanded in y.re around inf 74.4%

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

      1. +-commutative 74.4%

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

      2. associate-/l* 75.0%

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

      3. associate-/r/ 76.0%

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

      4. fma-def 76.0%

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

   4. Simplified 76.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 76.0%

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

      2. +-commutative 76.0%

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

      3. pow2 76.0%

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

      4. div-inv 76.1%

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

      5. associate-*l* 75.0%

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

      6. pow2 75.0%

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

      7. pow-flip 74.0%

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

      8. metadata-eval 74.0%

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

   6. Applied egg-rr 74.0%

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

      1. associate-*r* 76.1%

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

      2. *-commutative 76.1%

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

      3. metadata-eval 76.1%

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

      4. pow-div 76.0%

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

      5. inv-pow 76.0%

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

      6. pow1 76.0%

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

      7. associate-/r/ 79.2%

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

      8. associate-*l/ 84.2%

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

   8. Applied egg-rr 84.2%

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

   if -3.59999999999999978e133 < y.re < -1.01e-162 or 1.6000000000000002e-207 < y.re < 4.0999999999999999e86

   1. Initial program 82.0%

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

   if -1.01e-162 < y.re < 1.6000000000000002e-207

   1. Initial program 62.0%

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

   2. Taylor expanded in y.re around 0 84.3%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.6 \cdot 10^{+133}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.re \leq -1.01 \cdot 10^{-162}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{-207}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.re \leq 4.1 \cdot 10^{+86}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \end{array} \]

Alternative 8: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im \cdot \frac{-1}{y.re}}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.1e+127)
   (/ x.im y.im)
   (if (<= y.im -6.8e+76)
     (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))
     (if (<= y.im -2.3e-23)
       (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 1.8e+65)
         (+ (/ x.re y.re) (* x.im (/ (* y.im (/ -1.0 y.re)) (- y.re))))
         (/ x.im y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.1e+127) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -6.8e+76) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	} else if (y_46_im <= -2.3e-23) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.8e+65) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.1d+127)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-6.8d+76)) then
        tmp = (x_46re / y_46re) + ((y_46im * (1.0d0 / y_46re)) / (y_46re / x_46im))
    else if (y_46im <= (-2.3d-23)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 1.8d+65) then
        tmp = (x_46re / y_46re) + (x_46im * ((y_46im * ((-1.0d0) / y_46re)) / -y_46re))
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.1e+127) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -6.8e+76) {
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	} else if (y_46_im <= -2.3e-23) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.8e+65) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.1e+127:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -6.8e+76:
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im))
	elif y_46_im <= -2.3e-23:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.8e+65:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re))
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.1e+127)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -6.8e+76)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)));
	elseif (y_46_im <= -2.3e-23)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.8e+65)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im * Float64(-1.0 / y_46_re)) / Float64(-y_46_re))));
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.1e+127)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -6.8e+76)
		tmp = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	elseif (y_46_im <= -2.3e-23)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.8e+65)
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im * (-1.0 / y_46_re)) / -y_46_re));
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.1e+127], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -6.8e+76], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, -2.3e-23], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.8e+65], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im * N[(-1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / (-y$46$re)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.09999999999999992e127 or 1.79999999999999989e65 < y.im

   1. Initial program 43.1%

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

   2. Taylor expanded in y.re around 0 71.1%

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

   if -2.09999999999999992e127 < y.im < -6.7999999999999994e76

   1. Initial program 34.6%

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

   2. Taylor expanded in y.re around inf 68.3%

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

      1. +-commutative 68.3%

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

      2. associate-/l* 79.4%

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

      3. associate-/r/ 79.4%

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

      4. fma-def 79.4%

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

   4. Simplified 79.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 79.4%

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

      2. +-commutative 79.4%

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

      3. pow2 79.4%

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

      4. div-inv 79.4%

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

      5. associate-*l* 79.4%

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

      6. pow2 79.4%

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

      7. pow-flip 79.4%

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

      8. metadata-eval 79.4%

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

   6. Applied egg-rr 79.4%

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

      1. associate-*r* 79.4%

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

      2. *-commutative 79.4%

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

      3. metadata-eval 79.4%

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

      4. pow-div 79.4%

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

      5. inv-pow 79.4%

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

      6. pow1 79.4%

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

      7. associate-/r/ 79.4%

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

      8. associate-*l/ 79.4%

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

   8. Applied egg-rr 79.4%

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

   if -6.7999999999999994e76 < y.im < -2.3000000000000001e-23

   1. Initial program 91.0%

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

   2. Taylor expanded in x.re around 0 59.7%

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

   if -2.3000000000000001e-23 < y.im < 1.79999999999999989e65

   1. Initial program 72.2%

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

   2. Taylor expanded in y.re around inf 80.9%

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

      1. +-commutative 80.9%

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

      2. associate-/l* 81.7%

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

      3. associate-/r/ 77.8%

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

      4. fma-def 77.8%

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

   4. Simplified 77.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 77.8%

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

      2. +-commutative 77.8%

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

      3. pow2 77.8%

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

      4. div-inv 77.8%

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

      5. associate-*l* 81.7%

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

      6. pow2 81.7%

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

      7. pow-flip 81.7%

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

      8. metadata-eval 81.7%

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

   6. Applied egg-rr 81.7%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

      3. metadata-eval 77.8%

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

      4. pow-div 77.8%

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

      5. inv-pow 77.8%

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

      6. pow1 77.8%

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

      7. associate-/r/ 78.6%

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

      8. clear-num 78.6%

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

      9. clear-num 78.6%

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

      10. frac-2neg 78.6%

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

      11. associate-*l/ 85.0%

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

      12. distribute-neg-frac 85.0%

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

      13. metadata-eval 85.0%

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

      14. distribute-neg-frac 85.0%

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

   8. Applied egg-rr 85.0%

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

      1. associate-/r/ 85.7%

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

      2. *-commutative 85.7%

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

   10. Simplified 85.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{+127}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -6.8 \cdot 10^{+76}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.8 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{y.im \cdot \frac{-1}{y.re}}{-y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 9: 71.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+65}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (/ x.re y.re) (/ (* y.im (/ 1.0 y.re)) (/ y.re x.im)))))
   (if (<= y.im -6.4e+125)
     (/ x.im y.im)
     (if (<= y.im -4.8e+74)
       t_0
       (if (<= y.im -9e-23)
         (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
         (if (<= y.im 2.7e+65) t_0 (/ x.im y.im)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_im <= -6.4e+125) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4.8e+74) {
		tmp = t_0;
	} else if (y_46_im <= -9e-23) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.7e+65) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x_46re / y_46re) + ((y_46im * (1.0d0 / y_46re)) / (y_46re / x_46im))
    if (y_46im <= (-6.4d+125)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-4.8d+74)) then
        tmp = t_0
    else if (y_46im <= (-9d-23)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 2.7d+65) then
        tmp = t_0
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	double tmp;
	if (y_46_im <= -6.4e+125) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -4.8e+74) {
		tmp = t_0;
	} else if (y_46_im <= -9e-23) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 2.7e+65) {
		tmp = t_0;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im))
	tmp = 0
	if y_46_im <= -6.4e+125:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -4.8e+74:
		tmp = t_0
	elif y_46_im <= -9e-23:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 2.7e+65:
		tmp = t_0
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im * Float64(1.0 / y_46_re)) / Float64(y_46_re / x_46_im)))
	tmp = 0.0
	if (y_46_im <= -6.4e+125)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -4.8e+74)
		tmp = t_0;
	elseif (y_46_im <= -9e-23)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 2.7e+65)
		tmp = t_0;
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_re / y_46_re) + ((y_46_im * (1.0 / y_46_re)) / (y_46_re / x_46_im));
	tmp = 0.0;
	if (y_46_im <= -6.4e+125)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -4.8e+74)
		tmp = t_0;
	elseif (y_46_im <= -9e-23)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 2.7e+65)
		tmp = t_0;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im * N[(1.0 / y$46$re), $MachinePrecision]), $MachinePrecision] / N[(y$46$re / x$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$im, -6.4e+125], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -4.8e+74], t$95$0, If[LessEqual[y$46$im, -9e-23], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.7e+65], t$95$0, N[(x$46$im / y$46$im), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.39999999999999967e125 or 2.70000000000000019e65 < y.im

   1. Initial program 43.1%

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

   2. Taylor expanded in y.re around 0 71.1%

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

   if -6.39999999999999967e125 < y.im < -4.80000000000000017e74 or -8.9999999999999995e-23 < y.im < 2.70000000000000019e65

   1. Initial program 69.6%

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

   2. Taylor expanded in y.re around inf 80.0%

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

      1. +-commutative 80.0%

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

      2. associate-/l* 81.6%

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

      3. associate-/r/ 77.9%

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

      4. fma-def 77.9%

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

   4. Simplified 77.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x.im}{{y.re}^{2}}, y.im, \frac{x.re}{y.re}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 77.9%

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

      2. +-commutative 77.9%

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

      3. pow2 77.9%

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

      4. div-inv 77.9%

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

      5. associate-*l* 81.6%

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

      6. pow2 81.6%

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

      7. pow-flip 81.6%

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

      8. metadata-eval 81.6%

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

   6. Applied egg-rr 81.6%

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

      1. associate-*r* 77.9%

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

      2. *-commutative 77.9%

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

      3. metadata-eval 77.9%

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

      4. pow-div 77.9%

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

      5. inv-pow 77.9%

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

      6. pow1 77.9%

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

      7. associate-/r/ 78.7%

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

      8. associate-*l/ 84.6%

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

   8. Applied egg-rr 84.6%

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

   if -4.80000000000000017e74 < y.im < -8.9999999999999995e-23

   1. Initial program 91.0%

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

   2. Taylor expanded in x.re around 0 59.7%

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

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -4.8 \cdot 10^{+74}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{elif}\;y.im \leq -9 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 2.7 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im \cdot \frac{1}{y.re}}{\frac{y.re}{x.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 10: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -6.4e+125)
   (/ x.im y.im)
   (if (<= y.im -9.8e+73)
     (/ x.re y.re)
     (if (<= y.im -3.5e-23)
       (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.im 1.85e+65) (/ x.re y.re) (/ x.im y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.4e+125) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -9.8e+73) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -3.5e-23) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.85e+65) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-6.4d+125)) then
        tmp = x_46im / y_46im
    else if (y_46im <= (-9.8d+73)) then
        tmp = x_46re / y_46re
    else if (y_46im <= (-3.5d-23)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if (y_46im <= 1.85d+65) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -6.4e+125) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= -9.8e+73) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_im <= -3.5e-23) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_im <= 1.85e+65) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -6.4e+125:
		tmp = x_46_im / y_46_im
	elif y_46_im <= -9.8e+73:
		tmp = x_46_re / y_46_re
	elif y_46_im <= -3.5e-23:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif y_46_im <= 1.85e+65:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -6.4e+125)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= -9.8e+73)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_im <= -3.5e-23)
		tmp = Float64(Float64(x_46_im * y_46_im) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_im <= 1.85e+65)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -6.4e+125)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= -9.8e+73)
		tmp = x_46_re / y_46_re;
	elseif (y_46_im <= -3.5e-23)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif (y_46_im <= 1.85e+65)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -6.4e+125], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, -9.8e+73], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$im, -3.5e-23], N[(N[(x$46$im * y$46$im), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.85e+65], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -6.39999999999999967e125 or 1.84999999999999997e65 < y.im

   1. Initial program 43.1%

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

   2. Taylor expanded in y.re around 0 71.1%

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

   if -6.39999999999999967e125 < y.im < -9.7999999999999998e73 or -3.49999999999999993e-23 < y.im < 1.84999999999999997e65

   1. Initial program 69.6%

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

   2. Taylor expanded in y.re around inf 67.8%

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

   if -9.7999999999999998e73 < y.im < -3.49999999999999993e-23

   1. Initial program 91.0%

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

   2. Taylor expanded in x.re around 0 59.7%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq -9.8 \cdot 10^{+73}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.im \leq -3.5 \cdot 10^{-23}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.im \leq 1.85 \cdot 10^{+65}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 11: 61.9% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125} \lor \neg \left(y.im \leq -2.05 \cdot 10^{+69}\right) \land \left(y.im \leq -8 \cdot 10^{-24} \lor \neg \left(y.im \leq 3.8 \cdot 10^{+65}\right)\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -6.4e+125)
         (and (not (<= y.im -2.05e+69))
              (or (<= y.im -8e-24) (not (<= y.im 3.8e+65)))))
   (/ x.im y.im)
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.4e+125) || (!(y_46_im <= -2.05e+69) && ((y_46_im <= -8e-24) || !(y_46_im <= 3.8e+65)))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-6.4d+125)) .or. (.not. (y_46im <= (-2.05d+69))) .and. (y_46im <= (-8d-24)) .or. (.not. (y_46im <= 3.8d+65))) then
        tmp = x_46im / y_46im
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -6.4e+125) || (!(y_46_im <= -2.05e+69) && ((y_46_im <= -8e-24) || !(y_46_im <= 3.8e+65)))) {
		tmp = x_46_im / y_46_im;
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -6.4e+125) or (not (y_46_im <= -2.05e+69) and ((y_46_im <= -8e-24) or not (y_46_im <= 3.8e+65))):
		tmp = x_46_im / y_46_im
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -6.4e+125) || (!(y_46_im <= -2.05e+69) && ((y_46_im <= -8e-24) || !(y_46_im <= 3.8e+65))))
		tmp = Float64(x_46_im / y_46_im);
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -6.4e+125) || (~((y_46_im <= -2.05e+69)) && ((y_46_im <= -8e-24) || ~((y_46_im <= 3.8e+65)))))
		tmp = x_46_im / y_46_im;
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -6.4e+125], And[N[Not[LessEqual[y$46$im, -2.05e+69]], $MachinePrecision], Or[LessEqual[y$46$im, -8e-24], N[Not[LessEqual[y$46$im, 3.8e+65]], $MachinePrecision]]]], N[(x$46$im / y$46$im), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -6.39999999999999967e125 or -2.05e69 < y.im < -7.99999999999999939e-24 or 3.80000000000000011e65 < y.im

   1. Initial program 51.1%

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

   2. Taylor expanded in y.re around 0 67.6%

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

   if -6.39999999999999967e125 < y.im < -2.05e69 or -7.99999999999999939e-24 < y.im < 3.80000000000000011e65

   1. Initial program 69.6%

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

   2. Taylor expanded in y.re around inf 67.8%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -6.4 \cdot 10^{+125} \lor \neg \left(y.im \leq -2.05 \cdot 10^{+69}\right) \land \left(y.im \leq -8 \cdot 10^{-24} \lor \neg \left(y.im \leq 3.8 \cdot 10^{+65}\right)\right):\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 12: 42.6% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.6%

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

2. Taylor expanded in y.re around 0 41.3%

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

3. Final simplification 41.3%

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

Reproduce

herbie shell --seed 2023314 
(FPCore (x.re x.im y.re y.im)
  :name "_divideComplex, real part"
  :precision binary64
  (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))