_divideComplex, real part

Percentage Accurate: 61.7% → 85.2%

Time: 14.9s

Alternatives: 16

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

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)))
      5e+305)
   (/ (/ (fma x.re y.re (* x.im y.im)) (hypot y.re y.im)) (hypot y.re y.im))
   (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 5e+305) {
		tmp = (fma(x_46_re, y_46_re, (x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))) <= 5e+305)
		tmp = Float64(Float64(fma(x_46_re, y_46_re, Float64(x_46_im * y_46_im)) / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e+305], N[(N[(N[(x$46$re * y$46$re + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $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))) < 5.00000000000000009e305

   1. Initial program 75.3%

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

      1. add-sqr-sqrt 75.3%

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

      2. *-un-lft-identity 75.3%

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

      3. times-frac 75.3%

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

      4. hypot-def 75.3%

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

      5. fma-def 75.3%

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

      6. hypot-def 94.9%

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

   3. Applied egg-rr 94.9%

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

      1. associate-*l/ 95.1%

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

      2. *-un-lft-identity 95.1%

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

   5. Applied egg-rr 95.1%

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

   if 5.00000000000000009e305 < (/.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 9.1%

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

      1. add-sqr-sqrt 9.1%

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

      2. *-un-lft-identity 9.1%

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

      3. times-frac 9.1%

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

      4. hypot-def 9.1%

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

      5. fma-def 9.1%

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

      6. hypot-def 16.1%

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

   3. Applied egg-rr 16.1%

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

   4. Taylor expanded in y.re around 0 25.3%

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

      1. associate-/l* 32.2%

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

   6. Simplified 32.2%

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

   7. Taylor expanded in y.re around 0 54.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 2: 85.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

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))) 5e+305)
     (/ (/ t_0 (hypot y.re y.im)) (hypot y.re y.im))
     (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re)))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (x_46_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))) <= 5e+305) {
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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))) <= 5e+305) {
		tmp = (t_0 / Math.hypot(y_46_re, y_46_im)) / Math.hypot(y_46_re, y_46_im);
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	t_0 = (x_46_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))) <= 5e+305:
		tmp = (t_0 / math.hypot(y_46_re, y_46_im)) / math.hypot(y_46_re, y_46_im)
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(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))) <= 5e+305)
		tmp = Float64(Float64(t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im));
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = (x_46_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))) <= 5e+305)
		tmp = (t_0 / hypot(y_46_re, y_46_im)) / hypot(y_46_re, y_46_im);
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(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], 5e+305], N[(N[(t$95$0 / 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[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $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))) < 5.00000000000000009e305

   1. Initial program 75.3%

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

      1. add-sqr-sqrt 75.3%

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

      2. *-un-lft-identity 75.3%

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

      3. times-frac 75.3%

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

      4. hypot-def 75.3%

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

      5. fma-def 75.3%

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

      6. hypot-def 94.9%

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

   3. Applied egg-rr 94.9%

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

      1. associate-*l/ 95.1%

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

      2. *-un-lft-identity 95.1%

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

   5. Applied egg-rr 95.1%

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

      1. fma-def 95.1%

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

   7. Applied egg-rr 95.1%

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

   if 5.00000000000000009e305 < (/.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 9.1%

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

      1. add-sqr-sqrt 9.1%

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

      2. *-un-lft-identity 9.1%

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

      3. times-frac 9.1%

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

      4. hypot-def 9.1%

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

      5. fma-def 9.1%

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

      6. hypot-def 16.1%

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

   3. Applied egg-rr 16.1%

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

   4. Taylor expanded in y.re around 0 25.3%

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

      1. associate-/l* 32.2%

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

   6. Simplified 32.2%

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

   7. Taylor expanded in y.re around 0 54.4%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 5 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 3: 82.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.im}{\frac{y.re}{y.im}}\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (/ x.im (/ y.re y.im)))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -1.25e+119)
     (/ (- (- x.re) t_0) (hypot y.re y.im))
     (if (<= y.re -1.22e-146)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (if (<= y.re 9.5e-118)
         t_1
         (if (<= y.re 2.5e+46)
           (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
           (if (<= y.re 1.4e+102) t_1 (/ (+ x.re t_0) (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 t_0 = x_46_im / (y_46_re / y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -1.25e+119) {
		tmp = (-x_46_re - t_0) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.22e-146) {
		tmp = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if (y_46_re <= 9.5e-118) {
		tmp = t_1;
	} else if (y_46_re <= 2.5e+46) {
		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_re <= 1.4e+102) {
		tmp = t_1;
	} else {
		tmp = (x_46_re + t_0) / hypot(y_46_re, y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(x_46_im / Float64(y_46_re / y_46_im))
	t_1 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.25e+119)
		tmp = Float64(Float64(Float64(-x_46_re) - t_0) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.22e-146)
		tmp = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im)));
	elseif (y_46_re <= 9.5e-118)
		tmp = t_1;
	elseif (y_46_re <= 2.5e+46)
		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_re <= 1.4e+102)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_re + t_0) / 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_] := Block[{t$95$0 = N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.25e+119], N[(N[((-x$46$re) - t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.22e-146], N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, 9.5e-118], t$95$1, If[LessEqual[y$46$re, 2.5e+46], 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$46$re, 1.4e+102], t$95$1, N[(N[(x$46$re + t$95$0), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -1.25e119

   1. Initial program 43.3%

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

      1. add-sqr-sqrt 43.3%

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

      2. *-un-lft-identity 43.3%

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

      3. times-frac 43.4%

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

      4. hypot-def 43.4%

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

      5. fma-def 43.4%

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

      6. hypot-def 61.5%

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

   3. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   5. Applied egg-rr 61.6%

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

   6. Taylor expanded in y.re around -inf 79.8%

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

      1. distribute-lft-out 79.8%

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

      2. associate-/l* 86.0%

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

   8. Simplified 86.0%

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

   if -1.25e119 < y.re < -1.2200000000000001e-146

   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 -1.2200000000000001e-146 < y.re < 9.49999999999999931e-118 or 2.5000000000000001e46 < y.re < 1.40000000000000009e102

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 9.49999999999999931e-118 < y.re < 2.5000000000000001e46

   1. Initial program 87.0%

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

      1. fma-def 87.1%

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

      2. fma-def 87.1%

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

   3. Simplified 87.1%

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

   if 1.40000000000000009e102 < y.re

   1. Initial program 33.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. *-un-lft-identity 33.4%

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

      3. times-frac 33.4%

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

      4. hypot-def 33.4%

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

      5. fma-def 33.4%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in y.re around inf 85.6%

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

      1. associate-/l* 88.2%

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

   8. Simplified 88.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.25 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.22 \cdot 10^{-146}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.5 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+46}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 4: 80.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ t_2 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+119}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-143}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-118}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

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 (/ x.im (pow y.re 2.0)))))
        (t_2 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -2.1e+119)
     t_1
     (if (<= y.re -1.7e-143)
       t_0
       (if (<= y.re 6.4e-118)
         t_2
         (if (<= y.re 2.8e+48) t_0 (if (<= y.re 1.4e+102) t_2 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 * (x_46_im / pow(y_46_re, 2.0)));
	double t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.1e+119) {
		tmp = t_1;
	} else if (y_46_re <= -1.7e-143) {
		tmp = t_0;
	} else if (y_46_re <= 6.4e-118) {
		tmp = t_2;
	} else if (y_46_re <= 2.8e+48) {
		tmp = t_0;
	} else if (y_46_re <= 1.4e+102) {
		tmp = t_2;
	} 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) :: t_2
    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 * (x_46im / (y_46re ** 2.0d0)))
    t_2 = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    if (y_46re <= (-2.1d+119)) then
        tmp = t_1
    else if (y_46re <= (-1.7d-143)) then
        tmp = t_0
    else if (y_46re <= 6.4d-118) then
        tmp = t_2
    else if (y_46re <= 2.8d+48) then
        tmp = t_0
    else if (y_46re <= 1.4d+102) then
        tmp = t_2
    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 * (x_46_im / Math.pow(y_46_re, 2.0)));
	double t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.1e+119) {
		tmp = t_1;
	} else if (y_46_re <= -1.7e-143) {
		tmp = t_0;
	} else if (y_46_re <= 6.4e-118) {
		tmp = t_2;
	} else if (y_46_re <= 2.8e+48) {
		tmp = t_0;
	} else if (y_46_re <= 1.4e+102) {
		tmp = t_2;
	} 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 * (x_46_im / math.pow(y_46_re, 2.0)))
	t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -2.1e+119:
		tmp = t_1
	elif y_46_re <= -1.7e-143:
		tmp = t_0
	elif y_46_re <= 6.4e-118:
		tmp = t_2
	elif y_46_re <= 2.8e+48:
		tmp = t_0
	elif y_46_re <= 1.4e+102:
		tmp = t_2
	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(y_46_im * Float64(x_46_im / (y_46_re ^ 2.0))))
	t_2 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.1e+119)
		tmp = t_1;
	elseif (y_46_re <= -1.7e-143)
		tmp = t_0;
	elseif (y_46_re <= 6.4e-118)
		tmp = t_2;
	elseif (y_46_re <= 2.8e+48)
		tmp = t_0;
	elseif (y_46_re <= 1.4e+102)
		tmp = t_2;
	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 * (x_46_im / (y_46_re ^ 2.0)));
	t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -2.1e+119)
		tmp = t_1;
	elseif (y_46_re <= -1.7e-143)
		tmp = t_0;
	elseif (y_46_re <= 6.4e-118)
		tmp = t_2;
	elseif (y_46_re <= 2.8e+48)
		tmp = t_0;
	elseif (y_46_re <= 1.4e+102)
		tmp = t_2;
	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[(y$46$im * N[(x$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.1e+119], t$95$1, If[LessEqual[y$46$re, -1.7e-143], t$95$0, If[LessEqual[y$46$re, 6.4e-118], t$95$2, If[LessEqual[y$46$re, 2.8e+48], t$95$0, If[LessEqual[y$46$re, 1.4e+102], t$95$2, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -2.09999999999999983e119 or 1.40000000000000009e102 < y.re

   1. Initial program 38.3%

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

   2. Taylor expanded in y.re around inf 77.6%

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

      1. associate-/l* 79.5%

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

      2. associate-/r/ 79.5%

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

   4. Simplified 79.5%

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

   if -2.09999999999999983e119 < y.re < -1.69999999999999992e-143 or 6.40000000000000008e-118 < y.re < 2.80000000000000012e48

   1. Initial program 85.7%

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

   if -1.69999999999999992e-143 < y.re < 6.40000000000000008e-118 or 2.80000000000000012e48 < y.re < 1.40000000000000009e102

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.1 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -1.7 \cdot 10^{-143}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 2.8 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ \end{array} \]

Alternative 5: 80.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-123}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -2.6e+119)
     (+ (/ x.re y.re) (* y.im (/ x.im (pow y.re 2.0))))
     (if (<= y.re -1.9e-144)
       t_0
       (if (<= y.re 9.4e-123)
         t_1
         (if (<= y.re 5.2e+48)
           t_0
           (if (<= y.re 1.55e+103)
             t_1
             (+ (/ x.re y.re) (/ x.im (/ (pow y.re 2.0) 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.6e+119) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_re <= -1.9e-144) {
		tmp = t_0;
	} else if (y_46_re <= 9.4e-123) {
		tmp = t_1;
	} else if (y_46_re <= 5.2e+48) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e+103) {
		tmp = t_1;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (pow(y_46_re, 2.0) / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    t_1 = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    if (y_46re <= (-2.6d+119)) then
        tmp = (x_46re / y_46re) + (y_46im * (x_46im / (y_46re ** 2.0d0)))
    else if (y_46re <= (-1.9d-144)) then
        tmp = t_0
    else if (y_46re <= 9.4d-123) then
        tmp = t_1
    else if (y_46re <= 5.2d+48) then
        tmp = t_0
    else if (y_46re <= 1.55d+103) then
        tmp = t_1
    else
        tmp = (x_46re / y_46re) + (x_46im / ((y_46re ** 2.0d0) / 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.6e+119) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_re <= -1.9e-144) {
		tmp = t_0;
	} else if (y_46_re <= 9.4e-123) {
		tmp = t_1;
	} else if (y_46_re <= 5.2e+48) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e+103) {
		tmp = t_1;
	} else {
		tmp = (x_46_re / y_46_re) + (x_46_im / (Math.pow(y_46_re, 2.0) / 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -2.6e+119:
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_re <= -1.9e-144:
		tmp = t_0
	elif y_46_re <= 9.4e-123:
		tmp = t_1
	elif y_46_re <= 5.2e+48:
		tmp = t_0
	elif y_46_re <= 1.55e+103:
		tmp = t_1
	else:
		tmp = (x_46_re / y_46_re) + (x_46_im / (math.pow(y_46_re, 2.0) / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	t_0 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(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(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.6e+119)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_re <= -1.9e-144)
		tmp = t_0;
	elseif (y_46_re <= 9.4e-123)
		tmp = t_1;
	elseif (y_46_re <= 5.2e+48)
		tmp = t_0;
	elseif (y_46_re <= 1.55e+103)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im / Float64((y_46_re ^ 2.0) / 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) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -2.6e+119)
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_re <= -1.9e-144)
		tmp = t_0;
	elseif (y_46_re <= 9.4e-123)
		tmp = t_1;
	elseif (y_46_re <= 5.2e+48)
		tmp = t_0;
	elseif (y_46_re <= 1.55e+103)
		tmp = t_1;
	else
		tmp = (x_46_re / y_46_re) + (x_46_im / ((y_46_re ^ 2.0) / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := Block[{t$95$0 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(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[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.6e+119], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.9e-144], t$95$0, If[LessEqual[y$46$re, 9.4e-123], t$95$1, If[LessEqual[y$46$re, 5.2e+48], t$95$0, If[LessEqual[y$46$re, 1.55e+103], t$95$1, N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im / N[(N[Power[y$46$re, 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.6e119

   1. Initial program 43.3%

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

   2. Taylor expanded in y.re around inf 74.6%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.3%

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

   4. Simplified 78.3%

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

   if -2.6e119 < y.re < -1.89999999999999996e-144 or 9.4000000000000004e-123 < y.re < 5.1999999999999999e48

   1. Initial program 85.7%

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

   if -1.89999999999999996e-144 < y.re < 9.4000000000000004e-123 or 5.1999999999999999e48 < y.re < 1.5500000000000001e103

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 1.5500000000000001e103 < y.re

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around inf 80.5%

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

      1. associate-/l* 80.8%

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

   4. Simplified 80.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.6 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -1.9 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 9.4 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \end{array} \]

Alternative 6: 81.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -1.15e+119)
     (+ (/ x.re y.re) (* y.im (/ x.im (pow y.re 2.0))))
     (if (<= y.re -7.8e-144)
       t_0
       (if (<= y.re 3e-120)
         t_1
         (if (<= y.re 4.6e+48)
           t_0
           (if (<= y.re 2.6e+102)
             t_1
             (/ (+ x.re (/ x.im (/ y.re y.im))) (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 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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -1.15e+119) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / pow(y_46_re, 2.0)));
	} else if (y_46_re <= -7.8e-144) {
		tmp = t_0;
	} else if (y_46_re <= 3e-120) {
		tmp = t_1;
	} else if (y_46_re <= 4.6e+48) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e+102) {
		tmp = t_1;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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 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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -1.15e+119) {
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / Math.pow(y_46_re, 2.0)));
	} else if (y_46_re <= -7.8e-144) {
		tmp = t_0;
	} else if (y_46_re <= 3e-120) {
		tmp = t_1;
	} else if (y_46_re <= 4.6e+48) {
		tmp = t_0;
	} else if (y_46_re <= 2.6e+102) {
		tmp = t_1;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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):
	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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -1.15e+119:
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / math.pow(y_46_re, 2.0)))
	elif y_46_re <= -7.8e-144:
		tmp = t_0
	elif y_46_re <= 3e-120:
		tmp = t_1
	elif y_46_re <= 4.6e+48:
		tmp = t_0
	elif y_46_re <= 2.6e+102:
		tmp = t_1
	else:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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)
	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(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -1.15e+119)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(y_46_im * Float64(x_46_im / (y_46_re ^ 2.0))));
	elseif (y_46_re <= -7.8e-144)
		tmp = t_0;
	elseif (y_46_re <= 3e-120)
		tmp = t_1;
	elseif (y_46_re <= 4.6e+48)
		tmp = t_0;
	elseif (y_46_re <= 2.6e+102)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / 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)
	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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -1.15e+119)
		tmp = (x_46_re / y_46_re) + (y_46_im * (x_46_im / (y_46_re ^ 2.0)));
	elseif (y_46_re <= -7.8e-144)
		tmp = t_0;
	elseif (y_46_re <= 3e-120)
		tmp = t_1;
	elseif (y_46_re <= 4.6e+48)
		tmp = t_0;
	elseif (y_46_re <= 2.6e+102)
		tmp = t_1;
	else
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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_] := 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[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -1.15e+119], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(y$46$im * N[(x$46$im / N[Power[y$46$re, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -7.8e-144], t$95$0, If[LessEqual[y$46$re, 3e-120], t$95$1, If[LessEqual[y$46$re, 4.6e+48], t$95$0, If[LessEqual[y$46$re, 2.6e+102], t$95$1, N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -1.15e119

   1. Initial program 43.3%

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

   2. Taylor expanded in y.re around inf 74.6%

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

      1. associate-/l* 78.3%

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

      2. associate-/r/ 78.3%

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

   4. Simplified 78.3%

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

   if -1.15e119 < y.re < -7.8000000000000003e-144 or 3.00000000000000011e-120 < y.re < 4.6e48

   1. Initial program 85.7%

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

   if -7.8000000000000003e-144 < y.re < 3.00000000000000011e-120 or 4.6e48 < y.re < 2.60000000000000006e102

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 2.60000000000000006e102 < y.re

   1. Initial program 33.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. *-un-lft-identity 33.4%

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

      3. times-frac 33.4%

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

      4. hypot-def 33.4%

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

      5. fma-def 33.4%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in y.re around inf 85.6%

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

      1. associate-/l* 88.2%

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

   8. Simplified 88.2%

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

4. Final simplification 83.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.15 \cdot 10^{+119}:\\ \;\;\;\;\frac{x.re}{y.re} + y.im \cdot \frac{x.im}{{y.re}^{2}}\\ \mathbf{elif}\;y.re \leq -7.8 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 3 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 4.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 7: 81.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x.im \cdot \left(-y.im\right)}{y.re} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+103}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -2.25e+120)
     (/ (- (/ (* x.im (- y.im)) y.re) x.re) (hypot y.re y.im))
     (if (<= y.re -3e-142)
       t_0
       (if (<= y.re 2.05e-120)
         t_1
         (if (<= y.re 4e+44)
           t_0
           (if (<= y.re 1.55e+103)
             t_1
             (/ (+ x.re (/ x.im (/ y.re y.im))) (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 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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.25e+120) {
		tmp = (((x_46_im * -y_46_im) / y_46_re) - x_46_re) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -3e-142) {
		tmp = t_0;
	} else if (y_46_re <= 2.05e-120) {
		tmp = t_1;
	} else if (y_46_re <= 4e+44) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e+103) {
		tmp = t_1;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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 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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.25e+120) {
		tmp = (((x_46_im * -y_46_im) / y_46_re) - x_46_re) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -3e-142) {
		tmp = t_0;
	} else if (y_46_re <= 2.05e-120) {
		tmp = t_1;
	} else if (y_46_re <= 4e+44) {
		tmp = t_0;
	} else if (y_46_re <= 1.55e+103) {
		tmp = t_1;
	} else {
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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):
	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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -2.25e+120:
		tmp = (((x_46_im * -y_46_im) / y_46_re) - x_46_re) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -3e-142:
		tmp = t_0
	elif y_46_re <= 2.05e-120:
		tmp = t_1
	elif y_46_re <= 4e+44:
		tmp = t_0
	elif y_46_re <= 1.55e+103:
		tmp = t_1
	else:
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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)
	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(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.25e+120)
		tmp = Float64(Float64(Float64(Float64(x_46_im * Float64(-y_46_im)) / y_46_re) - x_46_re) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -3e-142)
		tmp = t_0;
	elseif (y_46_re <= 2.05e-120)
		tmp = t_1;
	elseif (y_46_re <= 4e+44)
		tmp = t_0;
	elseif (y_46_re <= 1.55e+103)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_46_re + Float64(x_46_im / Float64(y_46_re / y_46_im))) / 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)
	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 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -2.25e+120)
		tmp = (((x_46_im * -y_46_im) / y_46_re) - x_46_re) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -3e-142)
		tmp = t_0;
	elseif (y_46_re <= 2.05e-120)
		tmp = t_1;
	elseif (y_46_re <= 4e+44)
		tmp = t_0;
	elseif (y_46_re <= 1.55e+103)
		tmp = t_1;
	else
		tmp = (x_46_re + (x_46_im / (y_46_re / y_46_im))) / 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_] := 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[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.25e+120], N[(N[(N[(N[(x$46$im * (-y$46$im)), $MachinePrecision] / y$46$re), $MachinePrecision] - x$46$re), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -3e-142], t$95$0, If[LessEqual[y$46$re, 2.05e-120], t$95$1, If[LessEqual[y$46$re, 4e+44], t$95$0, If[LessEqual[y$46$re, 1.55e+103], t$95$1, N[(N[(x$46$re + N[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.24999999999999988e120

   1. Initial program 44.4%

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

      1. add-sqr-sqrt 44.4%

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

      2. *-un-lft-identity 44.4%

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

      3. times-frac 44.4%

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

      4. hypot-def 44.4%

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

      5. fma-def 44.4%

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

      6. hypot-def 63.0%

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

   3. Applied egg-rr 63.0%

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

      1. associate-*l/ 63.2%

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

      2. *-un-lft-identity 63.2%

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

   5. Applied egg-rr 63.2%

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

   6. Taylor expanded in y.re around -inf 81.8%

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

      1. neg-mul-1 81.8%

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

      2. +-commutative 81.8%

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

      3. unsub-neg 81.8%

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

      4. associate-*r/ 81.8%

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

      5. neg-mul-1 81.8%

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

      6. distribute-rgt-neg-in 81.8%

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

   8. Simplified 81.8%

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

   if -2.24999999999999988e120 < y.re < -3.0000000000000001e-142 or 2.05000000000000017e-120 < y.re < 4.0000000000000004e44

   1. Initial program 84.7%

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

   if -3.0000000000000001e-142 < y.re < 2.05000000000000017e-120 or 4.0000000000000004e44 < y.re < 1.5500000000000001e103

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 1.5500000000000001e103 < y.re

   1. Initial program 33.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. *-un-lft-identity 33.4%

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

      3. times-frac 33.4%

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

      4. hypot-def 33.4%

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

      5. fma-def 33.4%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in y.re around inf 85.6%

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

      1. associate-/l* 88.2%

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

   8. Simplified 88.2%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.25 \cdot 10^{+120}:\\ \;\;\;\;\frac{\frac{x.im \cdot \left(-y.im\right)}{y.re} - x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -3 \cdot 10^{-142}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.05 \cdot 10^{-120}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.55 \cdot 10^{+103}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 8: 82.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ t_1 := \frac{x.im}{\frac{y.re}{y.im}}\\ t_2 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-x.re\right) - t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{-144}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-123}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + t_1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im))))
        (t_1 (/ x.im (/ y.re y.im)))
        (t_2 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -3.3e+119)
     (/ (- (- x.re) t_1) (hypot y.re y.im))
     (if (<= y.re -1.35e-144)
       t_0
       (if (<= y.re 1.1e-123)
         t_2
         (if (<= y.re 5e+48)
           t_0
           (if (<= y.re 1.4e+102) t_2 (/ (+ x.re t_1) (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 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_im / (y_46_re / y_46_im);
	double t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -3.3e+119) {
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.35e-144) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-123) {
		tmp = t_2;
	} else if (y_46_re <= 5e+48) {
		tmp = t_0;
	} else if (y_46_re <= 1.4e+102) {
		tmp = t_2;
	} else {
		tmp = (x_46_re + t_1) / 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 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_im / (y_46_re / y_46_im);
	double t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -3.3e+119) {
		tmp = (-x_46_re - t_1) / Math.hypot(y_46_re, y_46_im);
	} else if (y_46_re <= -1.35e-144) {
		tmp = t_0;
	} else if (y_46_re <= 1.1e-123) {
		tmp = t_2;
	} else if (y_46_re <= 5e+48) {
		tmp = t_0;
	} else if (y_46_re <= 1.4e+102) {
		tmp = t_2;
	} else {
		tmp = (x_46_re + t_1) / 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):
	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_im / (y_46_re / y_46_im)
	t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -3.3e+119:
		tmp = (-x_46_re - t_1) / math.hypot(y_46_re, y_46_im)
	elif y_46_re <= -1.35e-144:
		tmp = t_0
	elif y_46_re <= 1.1e-123:
		tmp = t_2
	elif y_46_re <= 5e+48:
		tmp = t_0
	elif y_46_re <= 1.4e+102:
		tmp = t_2
	else:
		tmp = (x_46_re + t_1) / 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)
	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(x_46_im / Float64(y_46_re / y_46_im))
	t_2 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -3.3e+119)
		tmp = Float64(Float64(Float64(-x_46_re) - t_1) / hypot(y_46_re, y_46_im));
	elseif (y_46_re <= -1.35e-144)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-123)
		tmp = t_2;
	elseif (y_46_re <= 5e+48)
		tmp = t_0;
	elseif (y_46_re <= 1.4e+102)
		tmp = t_2;
	else
		tmp = Float64(Float64(x_46_re + t_1) / 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)
	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_im / (y_46_re / y_46_im);
	t_2 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -3.3e+119)
		tmp = (-x_46_re - t_1) / hypot(y_46_re, y_46_im);
	elseif (y_46_re <= -1.35e-144)
		tmp = t_0;
	elseif (y_46_re <= 1.1e-123)
		tmp = t_2;
	elseif (y_46_re <= 5e+48)
		tmp = t_0;
	elseif (y_46_re <= 1.4e+102)
		tmp = t_2;
	else
		tmp = (x_46_re + t_1) / 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_] := 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[(x$46$im / N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.3e+119], N[(N[((-x$46$re) - t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$re, -1.35e-144], t$95$0, If[LessEqual[y$46$re, 1.1e-123], t$95$2, If[LessEqual[y$46$re, 5e+48], t$95$0, If[LessEqual[y$46$re, 1.4e+102], t$95$2, N[(N[(x$46$re + t$95$1), $MachinePrecision] / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.3000000000000002e119

   1. Initial program 43.3%

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

      1. add-sqr-sqrt 43.3%

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

      2. *-un-lft-identity 43.3%

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

      3. times-frac 43.4%

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

      4. hypot-def 43.4%

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

      5. fma-def 43.4%

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

      6. hypot-def 61.5%

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

   3. Applied egg-rr 61.5%

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

      1. associate-*l/ 61.6%

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

      2. *-un-lft-identity 61.6%

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

   5. Applied egg-rr 61.6%

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

   6. Taylor expanded in y.re around -inf 79.8%

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

      1. distribute-lft-out 79.8%

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

      2. associate-/l* 86.0%

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

   8. Simplified 86.0%

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

   if -3.3000000000000002e119 < y.re < -1.34999999999999988e-144 or 1.10000000000000003e-123 < y.re < 4.99999999999999973e48

   1. Initial program 85.7%

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

   if -1.34999999999999988e-144 < y.re < 1.10000000000000003e-123 or 4.99999999999999973e48 < y.re < 1.40000000000000009e102

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 1.40000000000000009e102 < y.re

   1. Initial program 33.4%

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

      1. add-sqr-sqrt 33.4%

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

      2. *-un-lft-identity 33.4%

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

      3. times-frac 33.4%

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

      4. hypot-def 33.4%

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

      5. fma-def 33.4%

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

      6. hypot-def 57.4%

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

   3. Applied egg-rr 57.4%

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

      1. associate-*l/ 57.6%

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

      2. *-un-lft-identity 57.6%

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

   5. Applied egg-rr 57.6%

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

   6. Taylor expanded in y.re around inf 85.6%

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

      1. associate-/l* 88.2%

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

   8. Simplified 88.2%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.3 \cdot 10^{+119}:\\ \;\;\;\;\frac{\left(-x.re\right) - \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.re \leq -1.35 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-123}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 5 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.4 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re + \frac{x.im}{\frac{y.re}{y.im}}}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 9: 79.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ t_1 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-143}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-122}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+102}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re)))))
        (t_1
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= y.re -3.4e+161)
     (/ x.re y.re)
     (if (<= y.re -2.5e-143)
       t_1
       (if (<= y.re 1.45e-122)
         t_0
         (if (<= y.re 1.6e+48)
           t_1
           (if (<= y.re 8e+102)
             t_0
             (if (<= y.re 1.7e+104)
               (/ x.im (/ (pow y.re 2.0) y.im))
               (/ x.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 t_0 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double t_1 = ((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 tmp;
	if (y_46_re <= -3.4e+161) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -2.5e-143) {
		tmp = t_1;
	} else if (y_46_re <= 1.45e-122) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e+48) {
		tmp = t_1;
	} else if (y_46_re <= 8e+102) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e+104) {
		tmp = x_46_im / (pow(y_46_re, 2.0) / y_46_im);
	} else {
		tmp = x_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 t_0 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double t_1 = ((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 tmp;
	if (y_46_re <= -3.4e+161) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -2.5e-143) {
		tmp = t_1;
	} else if (y_46_re <= 1.45e-122) {
		tmp = t_0;
	} else if (y_46_re <= 1.6e+48) {
		tmp = t_1;
	} else if (y_46_re <= 8e+102) {
		tmp = t_0;
	} else if (y_46_re <= 1.7e+104) {
		tmp = x_46_im / (Math.pow(y_46_re, 2.0) / y_46_im);
	} else {
		tmp = x_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):
	t_0 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	t_1 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	tmp = 0
	if y_46_re <= -3.4e+161:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -2.5e-143:
		tmp = t_1
	elif y_46_re <= 1.45e-122:
		tmp = t_0
	elif y_46_re <= 1.6e+48:
		tmp = t_1
	elif y_46_re <= 8e+102:
		tmp = t_0
	elif y_46_re <= 1.7e+104:
		tmp = x_46_im / (math.pow(y_46_re, 2.0) / y_46_im)
	else:
		tmp = x_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)
	t_0 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	t_1 = 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)))
	tmp = 0.0
	if (y_46_re <= -3.4e+161)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -2.5e-143)
		tmp = t_1;
	elseif (y_46_re <= 1.45e-122)
		tmp = t_0;
	elseif (y_46_re <= 1.6e+48)
		tmp = t_1;
	elseif (y_46_re <= 8e+102)
		tmp = t_0;
	elseif (y_46_re <= 1.7e+104)
		tmp = Float64(x_46_im / Float64((y_46_re ^ 2.0) / y_46_im));
	else
		tmp = Float64(x_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)
	t_0 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	t_1 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	tmp = 0.0;
	if (y_46_re <= -3.4e+161)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -2.5e-143)
		tmp = t_1;
	elseif (y_46_re <= 1.45e-122)
		tmp = t_0;
	elseif (y_46_re <= 1.6e+48)
		tmp = t_1;
	elseif (y_46_re <= 8e+102)
		tmp = t_0;
	elseif (y_46_re <= 1.7e+104)
		tmp = x_46_im / ((y_46_re ^ 2.0) / y_46_im);
	else
		tmp = x_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_] := Block[{t$95$0 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -3.4e+161], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -2.5e-143], t$95$1, If[LessEqual[y$46$re, 1.45e-122], t$95$0, If[LessEqual[y$46$re, 1.6e+48], t$95$1, If[LessEqual[y$46$re, 8e+102], t$95$0, If[LessEqual[y$46$re, 1.7e+104], N[(x$46$im / N[(N[Power[y$46$re, 2.0], $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision], N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.39999999999999993e161

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 80.2%

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

   if -3.39999999999999993e161 < y.re < -2.5000000000000001e-143 or 1.4500000000000001e-122 < y.re < 1.6000000000000001e48

   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 -2.5000000000000001e-143 < y.re < 1.4500000000000001e-122 or 1.6000000000000001e48 < y.re < 7.99999999999999982e102

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 7.99999999999999982e102 < y.re < 1.6999999999999998e104

   1. Initial program 98.4%

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

   2. Taylor expanded in x.re around 0 98.4%

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

      1. associate-/l* 100.0%

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

      2. unpow2 100.0%

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

      3. fma-udef 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y.im around 0 100.0%

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

   if 1.6999999999999998e104 < y.re

   1. Initial program 31.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. *-un-lft-identity 31.7%

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

      3. times-frac 31.7%

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

      4. hypot-def 31.7%

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

      5. fma-def 31.7%

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

      6. hypot-def 56.3%

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

   3. Applied egg-rr 56.3%

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

   4. Taylor expanded in y.re around inf 80.2%

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

      1. expm1-log1p-u 79.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\right)} \]

      2. expm1-udef 32.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)} - 1} \]

      3. associate-*l/ 32.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 79.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

      2. expm1-log1p 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -2.5 \cdot 10^{-143}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.45 \cdot 10^{-122}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 8 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.7 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im}{\frac{{y.re}^{2}}{y.im}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 10: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ t_2 := \frac{x.re \cdot y.re + x.im \cdot y.im}{t_0}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.22 \cdot 10^{+104}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re)))))
        (t_2 (/ (+ (* x.re y.re) (* x.im y.im)) t_0)))
   (if (<= y.re -3.4e+161)
     (/ x.re y.re)
     (if (<= y.re -1.85e-144)
       t_2
       (if (<= y.re 6e-118)
         t_1
         (if (<= y.re 1.6e+48)
           t_2
           (if (<= y.re 1.22e+104)
             t_1
             (if (<= y.re 2.1e+104)
               (/ (* x.im y.im) t_0)
               (/ x.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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	double tmp;
	if (y_46_re <= -3.4e+161) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.85e-144) {
		tmp = t_2;
	} else if (y_46_re <= 6e-118) {
		tmp = t_1;
	} else if (y_46_re <= 1.6e+48) {
		tmp = t_2;
	} else if (y_46_re <= 1.22e+104) {
		tmp = t_1;
	} else if (y_46_re <= 2.1e+104) {
		tmp = (x_46_im * y_46_im) / t_0;
	} else {
		tmp = x_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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	double tmp;
	if (y_46_re <= -3.4e+161) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -1.85e-144) {
		tmp = t_2;
	} else if (y_46_re <= 6e-118) {
		tmp = t_1;
	} else if (y_46_re <= 1.6e+48) {
		tmp = t_2;
	} else if (y_46_re <= 1.22e+104) {
		tmp = t_1;
	} else if (y_46_re <= 2.1e+104) {
		tmp = (x_46_im * y_46_im) / t_0;
	} else {
		tmp = x_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):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0
	tmp = 0
	if y_46_re <= -3.4e+161:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -1.85e-144:
		tmp = t_2
	elif y_46_re <= 6e-118:
		tmp = t_1
	elif y_46_re <= 1.6e+48:
		tmp = t_2
	elif y_46_re <= 1.22e+104:
		tmp = t_1
	elif y_46_re <= 2.1e+104:
		tmp = (x_46_im * y_46_im) / t_0
	else:
		tmp = x_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)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	t_2 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / t_0)
	tmp = 0.0
	if (y_46_re <= -3.4e+161)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -1.85e-144)
		tmp = t_2;
	elseif (y_46_re <= 6e-118)
		tmp = t_1;
	elseif (y_46_re <= 1.6e+48)
		tmp = t_2;
	elseif (y_46_re <= 1.22e+104)
		tmp = t_1;
	elseif (y_46_re <= 2.1e+104)
		tmp = Float64(Float64(x_46_im * y_46_im) / t_0);
	else
		tmp = Float64(x_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)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	tmp = 0.0;
	if (y_46_re <= -3.4e+161)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -1.85e-144)
		tmp = t_2;
	elseif (y_46_re <= 6e-118)
		tmp = t_1;
	elseif (y_46_re <= 1.6e+48)
		tmp = t_2;
	elseif (y_46_re <= 1.22e+104)
		tmp = t_1;
	elseif (y_46_re <= 2.1e+104)
		tmp = (x_46_im * y_46_im) / t_0;
	else
		tmp = x_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_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.4e+161], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -1.85e-144], t$95$2, If[LessEqual[y$46$re, 6e-118], t$95$1, If[LessEqual[y$46$re, 1.6e+48], t$95$2, If[LessEqual[y$46$re, 1.22e+104], t$95$1, If[LessEqual[y$46$re, 2.1e+104], N[(N[(x$46$im * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x$46$re / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y.re < -3.39999999999999993e161

   1. Initial program 35.6%

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

   2. Taylor expanded in y.re around inf 80.2%

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

   if -3.39999999999999993e161 < y.re < -1.8500000000000001e-144 or 6.00000000000000035e-118 < y.re < 1.6000000000000001e48

   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 -1.8500000000000001e-144 < y.re < 6.00000000000000035e-118 or 1.6000000000000001e48 < y.re < 1.22e104

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 1.22e104 < y.re < 2.0999999999999998e104

   1. Initial program 98.4%

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

   2. Taylor expanded in x.re around 0 98.4%

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

   if 2.0999999999999998e104 < y.re

   1. Initial program 31.7%

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

      1. add-sqr-sqrt 31.7%

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

      2. *-un-lft-identity 31.7%

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

      3. times-frac 31.7%

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

      4. hypot-def 31.7%

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

      5. fma-def 31.7%

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

      6. hypot-def 56.3%

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

   3. Applied egg-rr 56.3%

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

   4. Taylor expanded in y.re around inf 80.2%

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

      1. expm1-log1p-u 79.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)\right)} \]

      2. expm1-udef 32.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot x.re\right)} - 1} \]

      3. associate-*l/ 32.9%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot x.re}{\mathsf{hypot}\left(y.re, y.im\right)}}\right)} - 1 \]

      4. *-un-lft-identity 32.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x.re}}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1 \]

   6. Applied egg-rr 32.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 79.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\right)\right)} \]

      2. expm1-log1p 80.4%

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

   8. Simplified 80.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -1.85 \cdot 10^{-144}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.6 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.22 \cdot 10^{+104}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 2.1 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \end{array} \]

Alternative 11: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{+15} \lor \neg \left(y.re \leq 1300000000\right) \land \left(y.re \leq 5 \cdot 10^{+41} \lor \neg \left(y.re \leq 1.5 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.re -8.5e+56)
   (/ x.re y.re)
   (if (<= y.re -6e+16)
     (/ (* x.im y.im) (+ (* y.re y.re) (* y.im y.im)))
     (if (or (<= y.re -3.1e+15)
             (and (not (<= y.re 1300000000.0))
                  (or (<= y.re 5e+41) (not (<= y.re 1.5e+102)))))
       (/ x.re y.re)
       (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+56) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -6e+16) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if ((y_46_re <= -3.1e+15) || (!(y_46_re <= 1300000000.0) && ((y_46_re <= 5e+41) || !(y_46_re <= 1.5e+102)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46re <= (-8.5d+56)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-6d+16)) then
        tmp = (x_46im * y_46im) / ((y_46re * y_46re) + (y_46im * y_46im))
    else if ((y_46re <= (-3.1d+15)) .or. (.not. (y_46re <= 1300000000.0d0)) .and. (y_46re <= 5d+41) .or. (.not. (y_46re <= 1.5d+102))) then
        tmp = x_46re / y_46re
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_re <= -8.5e+56) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -6e+16) {
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	} else if ((y_46_re <= -3.1e+15) || (!(y_46_re <= 1300000000.0) && ((y_46_re <= 5e+41) || !(y_46_re <= 1.5e+102)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_re <= -8.5e+56:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -6e+16:
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im))
	elif (y_46_re <= -3.1e+15) or (not (y_46_re <= 1300000000.0) and ((y_46_re <= 5e+41) or not (y_46_re <= 1.5e+102))):
		tmp = x_46_re / y_46_re
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_re <= -8.5e+56)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -6e+16)
		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_re <= -3.1e+15) || (!(y_46_re <= 1300000000.0) && ((y_46_re <= 5e+41) || !(y_46_re <= 1.5e+102))))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_re <= -8.5e+56)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -6e+16)
		tmp = (x_46_im * y_46_im) / ((y_46_re * y_46_re) + (y_46_im * y_46_im));
	elseif ((y_46_re <= -3.1e+15) || (~((y_46_re <= 1300000000.0)) && ((y_46_re <= 5e+41) || ~((y_46_re <= 1.5e+102)))))
		tmp = x_46_re / y_46_re;
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$re, -8.5e+56], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -6e+16], 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[Or[LessEqual[y$46$re, -3.1e+15], And[N[Not[LessEqual[y$46$re, 1300000000.0]], $MachinePrecision], Or[LessEqual[y$46$re, 5e+41], N[Not[LessEqual[y$46$re, 1.5e+102]], $MachinePrecision]]]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y.re < -8.4999999999999998e56 or -6e16 < y.re < -3.1e15 or 1.3e9 < y.re < 5.00000000000000022e41 or 1.4999999999999999e102 < y.re

   1. Initial program 50.0%

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

   2. Taylor expanded in y.re around inf 73.6%

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

   if -8.4999999999999998e56 < y.re < -6e16

   1. Initial program 99.7%

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

   2. Taylor expanded in x.re around 0 99.7%

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

   if -3.1e15 < y.re < 1.3e9 or 5.00000000000000022e41 < y.re < 1.4999999999999999e102

   1. Initial program 65.3%

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

      1. add-sqr-sqrt 65.3%

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

      2. *-un-lft-identity 65.3%

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

      3. times-frac 65.3%

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

      4. hypot-def 65.4%

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

      5. fma-def 65.4%

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

      6. hypot-def 82.1%

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

   3. Applied egg-rr 82.1%

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

   4. Taylor expanded in y.re around 0 39.2%

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

      1. associate-/l* 39.8%

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

   6. Simplified 39.8%

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

   7. Taylor expanded in y.re around 0 76.4%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -8.5 \cdot 10^{+56}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -6 \cdot 10^{+16}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -3.1 \cdot 10^{+15} \lor \neg \left(y.re \leq 1300000000\right) \land \left(y.re \leq 5 \cdot 10^{+41} \lor \neg \left(y.re \leq 1.5 \cdot 10^{+102}\right)\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 12: 73.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im \cdot y.im}{t_0}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re \cdot y.re}{t_0}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))
   (if (<= y.re -2.3e+59)
     (/ x.re y.re)
     (if (<= y.re -5.2e+15)
       (/ (* x.im y.im) t_0)
       (if (<= y.re -4e+15)
         (/ x.re y.re)
         (if (<= y.re 3.8e-26)
           t_1
           (if (<= y.re 6.4e+46)
             (/ (* x.re y.re) t_0)
             (if (<= y.re 2.5e+102) t_1 (/ x.re y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.3e+59) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.2e+15) {
		tmp = (x_46_im * y_46_im) / t_0;
	} else if (y_46_re <= -4e+15) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.8e-26) {
		tmp = t_1;
	} else if (y_46_re <= 6.4e+46) {
		tmp = (x_46_re * y_46_re) / t_0;
	} else if (y_46_re <= 2.5e+102) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    if (y_46re <= (-2.3d+59)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-5.2d+15)) then
        tmp = (x_46im * y_46im) / t_0
    else if (y_46re <= (-4d+15)) then
        tmp = x_46re / y_46re
    else if (y_46re <= 3.8d-26) then
        tmp = t_1
    else if (y_46re <= 6.4d+46) then
        tmp = (x_46re * y_46re) / t_0
    else if (y_46re <= 2.5d+102) then
        tmp = t_1
    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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double tmp;
	if (y_46_re <= -2.3e+59) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -5.2e+15) {
		tmp = (x_46_im * y_46_im) / t_0;
	} else if (y_46_re <= -4e+15) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= 3.8e-26) {
		tmp = t_1;
	} else if (y_46_re <= 6.4e+46) {
		tmp = (x_46_re * y_46_re) / t_0;
	} else if (y_46_re <= 2.5e+102) {
		tmp = t_1;
	} 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):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	tmp = 0
	if y_46_re <= -2.3e+59:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -5.2e+15:
		tmp = (x_46_im * y_46_im) / t_0
	elif y_46_re <= -4e+15:
		tmp = x_46_re / y_46_re
	elif y_46_re <= 3.8e-26:
		tmp = t_1
	elif y_46_re <= 6.4e+46:
		tmp = (x_46_re * y_46_re) / t_0
	elif y_46_re <= 2.5e+102:
		tmp = t_1
	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)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	tmp = 0.0
	if (y_46_re <= -2.3e+59)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -5.2e+15)
		tmp = Float64(Float64(x_46_im * y_46_im) / t_0);
	elseif (y_46_re <= -4e+15)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= 3.8e-26)
		tmp = t_1;
	elseif (y_46_re <= 6.4e+46)
		tmp = Float64(Float64(x_46_re * y_46_re) / t_0);
	elseif (y_46_re <= 2.5e+102)
		tmp = t_1;
	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)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	tmp = 0.0;
	if (y_46_re <= -2.3e+59)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -5.2e+15)
		tmp = (x_46_im * y_46_im) / t_0;
	elseif (y_46_re <= -4e+15)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= 3.8e-26)
		tmp = t_1;
	elseif (y_46_re <= 6.4e+46)
		tmp = (x_46_re * y_46_re) / t_0;
	elseif (y_46_re <= 2.5e+102)
		tmp = t_1;
	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_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y$46$re, -2.3e+59], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -5.2e+15], N[(N[(x$46$im * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, -4e+15], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, 3.8e-26], t$95$1, If[LessEqual[y$46$re, 6.4e+46], N[(N[(x$46$re * y$46$re), $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[y$46$re, 2.5e+102], t$95$1, N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -2.30000000000000008e59 or -5.2e15 < y.re < -4e15 or 2.5e102 < y.re

   1. Initial program 43.4%

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

   2. Taylor expanded in y.re around inf 76.4%

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

   if -2.30000000000000008e59 < y.re < -5.2e15

   1. Initial program 99.7%

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

   2. Taylor expanded in x.re around 0 99.7%

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

   if -4e15 < y.re < 3.80000000000000015e-26 or 6.3999999999999996e46 < y.re < 2.5e102

   1. Initial program 62.9%

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

      1. add-sqr-sqrt 62.9%

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

      2. *-un-lft-identity 62.9%

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

      3. times-frac 62.9%

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

      4. hypot-def 62.9%

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

      5. fma-def 62.9%

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

      6. hypot-def 81.2%

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

   3. Applied egg-rr 81.2%

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

   4. Taylor expanded in y.re around 0 38.3%

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

      1. associate-/l* 39.0%

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

   6. Simplified 39.0%

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

   7. Taylor expanded in y.re around 0 78.3%

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

   if 3.80000000000000015e-26 < y.re < 6.3999999999999996e46

   1. Initial program 89.1%

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

   2. Taylor expanded in x.re around inf 69.0%

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

      1. *-commutative 69.0%

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

   4. Simplified 69.0%

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

4. Final simplification 77.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq -4 \cdot 10^{+15}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq 3.8 \cdot 10^{-26}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 6.4 \cdot 10^{+46}:\\ \;\;\;\;\frac{x.re \cdot y.re}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 2.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 13: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y.re \cdot y.re + y.im \cdot y.im\\ t_1 := \frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ t_2 := \frac{x.re \cdot y.re + x.im \cdot y.im}{t_0}\\ \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-147}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-118}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im \cdot y.im}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* y.re y.re) (* y.im y.im)))
        (t_1 (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re)))))
        (t_2 (/ (+ (* x.re y.re) (* x.im y.im)) t_0)))
   (if (<= y.re -3.4e+161)
     (/ x.re y.re)
     (if (<= y.re -4.8e-147)
       t_2
       (if (<= y.re 1.1e-118)
         t_1
         (if (<= y.re 5.2e+48)
           t_2
           (if (<= y.re 1.75e+102)
             t_1
             (if (<= y.re 1.25e+107)
               (/ (* x.im y.im) t_0)
               (/ x.re y.re)))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	double tmp;
	if (y_46_re <= -3.4e+161) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -4.8e-147) {
		tmp = t_2;
	} else if (y_46_re <= 1.1e-118) {
		tmp = t_1;
	} else if (y_46_re <= 5.2e+48) {
		tmp = t_2;
	} else if (y_46_re <= 1.75e+102) {
		tmp = t_1;
	} else if (y_46_re <= 1.25e+107) {
		tmp = (x_46_im * y_46_im) / t_0;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = (y_46re * y_46re) + (y_46im * y_46im)
    t_1 = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    t_2 = ((x_46re * y_46re) + (x_46im * y_46im)) / t_0
    if (y_46re <= (-3.4d+161)) then
        tmp = x_46re / y_46re
    else if (y_46re <= (-4.8d-147)) then
        tmp = t_2
    else if (y_46re <= 1.1d-118) then
        tmp = t_1
    else if (y_46re <= 5.2d+48) then
        tmp = t_2
    else if (y_46re <= 1.75d+102) then
        tmp = t_1
    else if (y_46re <= 1.25d+107) then
        tmp = (x_46im * y_46im) / t_0
    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 t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	double t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	double t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	double tmp;
	if (y_46_re <= -3.4e+161) {
		tmp = x_46_re / y_46_re;
	} else if (y_46_re <= -4.8e-147) {
		tmp = t_2;
	} else if (y_46_re <= 1.1e-118) {
		tmp = t_1;
	} else if (y_46_re <= 5.2e+48) {
		tmp = t_2;
	} else if (y_46_re <= 1.75e+102) {
		tmp = t_1;
	} else if (y_46_re <= 1.25e+107) {
		tmp = (x_46_im * y_46_im) / t_0;
	} 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):
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im)
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0
	tmp = 0
	if y_46_re <= -3.4e+161:
		tmp = x_46_re / y_46_re
	elif y_46_re <= -4.8e-147:
		tmp = t_2
	elif y_46_re <= 1.1e-118:
		tmp = t_1
	elif y_46_re <= 5.2e+48:
		tmp = t_2
	elif y_46_re <= 1.75e+102:
		tmp = t_1
	elif y_46_re <= 1.25e+107:
		tmp = (x_46_im * y_46_im) / t_0
	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)
	t_0 = Float64(Float64(y_46_re * y_46_re) + Float64(y_46_im * y_46_im))
	t_1 = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))))
	t_2 = Float64(Float64(Float64(x_46_re * y_46_re) + Float64(x_46_im * y_46_im)) / t_0)
	tmp = 0.0
	if (y_46_re <= -3.4e+161)
		tmp = Float64(x_46_re / y_46_re);
	elseif (y_46_re <= -4.8e-147)
		tmp = t_2;
	elseif (y_46_re <= 1.1e-118)
		tmp = t_1;
	elseif (y_46_re <= 5.2e+48)
		tmp = t_2;
	elseif (y_46_re <= 1.75e+102)
		tmp = t_1;
	elseif (y_46_re <= 1.25e+107)
		tmp = Float64(Float64(x_46_im * y_46_im) / t_0);
	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)
	t_0 = (y_46_re * y_46_re) + (y_46_im * y_46_im);
	t_1 = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	t_2 = ((x_46_re * y_46_re) + (x_46_im * y_46_im)) / t_0;
	tmp = 0.0;
	if (y_46_re <= -3.4e+161)
		tmp = x_46_re / y_46_re;
	elseif (y_46_re <= -4.8e-147)
		tmp = t_2;
	elseif (y_46_re <= 1.1e-118)
		tmp = t_1;
	elseif (y_46_re <= 5.2e+48)
		tmp = t_2;
	elseif (y_46_re <= 1.75e+102)
		tmp = t_1;
	elseif (y_46_re <= 1.25e+107)
		tmp = (x_46_im * y_46_im) / t_0;
	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_] := Block[{t$95$0 = N[(N[(y$46$re * y$46$re), $MachinePrecision] + N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x$46$re * y$46$re), $MachinePrecision] + N[(x$46$im * y$46$im), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[y$46$re, -3.4e+161], N[(x$46$re / y$46$re), $MachinePrecision], If[LessEqual[y$46$re, -4.8e-147], t$95$2, If[LessEqual[y$46$re, 1.1e-118], t$95$1, If[LessEqual[y$46$re, 5.2e+48], t$95$2, If[LessEqual[y$46$re, 1.75e+102], t$95$1, If[LessEqual[y$46$re, 1.25e+107], N[(N[(x$46$im * y$46$im), $MachinePrecision] / t$95$0), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y.re < -3.39999999999999993e161 or 1.25e107 < y.re

   1. Initial program 33.4%

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

   2. Taylor expanded in y.re around inf 80.2%

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

   if -3.39999999999999993e161 < y.re < -4.79999999999999997e-147 or 1.09999999999999992e-118 < y.re < 5.1999999999999999e48

   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 -4.79999999999999997e-147 < y.re < 1.09999999999999992e-118 or 5.1999999999999999e48 < y.re < 1.75000000000000005e102

   1. Initial program 53.8%

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

      1. add-sqr-sqrt 53.7%

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

      2. *-un-lft-identity 53.7%

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

      3. times-frac 53.8%

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

      4. hypot-def 53.9%

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

      5. fma-def 53.9%

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

      6. hypot-def 77.3%

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

   3. Applied egg-rr 77.3%

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

   4. Taylor expanded in y.re around 0 39.1%

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

      1. associate-/l* 40.1%

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

   6. Simplified 40.1%

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

   7. Taylor expanded in y.re around 0 82.1%

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

   if 1.75000000000000005e102 < y.re < 1.25e107

   1. Initial program 98.4%

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

   2. Taylor expanded in x.re around 0 98.4%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -3.4 \cdot 10^{+161}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{elif}\;y.re \leq -4.8 \cdot 10^{-147}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.1 \cdot 10^{-118}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 5.2 \cdot 10^{+48}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;y.re \leq 1.75 \cdot 10^{+102}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \mathbf{elif}\;y.re \leq 1.25 \cdot 10^{+107}:\\ \;\;\;\;\frac{x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 14: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+53} \lor \neg \left(y.re \leq 1200000000 \lor \neg \left(y.re \leq 3.1 \cdot 10^{+43}\right) \land y.re \leq 1.4 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -6.5e+53)
         (not
          (or (<= y.re 1200000000.0)
              (and (not (<= y.re 3.1e+43)) (<= y.re 1.4e+102)))))
   (/ x.re y.re)
   (* (/ 1.0 y.im) (+ x.im (/ x.re (/ y.im y.re))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.5e+53) || !((y_46_re <= 1200000000.0) || (!(y_46_re <= 3.1e+43) && (y_46_re <= 1.4e+102)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-6.5d+53)) .or. (.not. (y_46re <= 1200000000.0d0) .or. (.not. (y_46re <= 3.1d+43)) .and. (y_46re <= 1.4d+102))) then
        tmp = x_46re / y_46re
    else
        tmp = (1.0d0 / y_46im) * (x_46im + (x_46re / (y_46im / y_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -6.5e+53) || !((y_46_re <= 1200000000.0) || (!(y_46_re <= 3.1e+43) && (y_46_re <= 1.4e+102)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -6.5e+53) or not ((y_46_re <= 1200000000.0) or (not (y_46_re <= 3.1e+43) and (y_46_re <= 1.4e+102))):
		tmp = x_46_re / y_46_re
	else:
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / y_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -6.5e+53) || !((y_46_re <= 1200000000.0) || (!(y_46_re <= 3.1e+43) && (y_46_re <= 1.4e+102))))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(1.0 / y_46_im) * Float64(x_46_im + Float64(x_46_re / Float64(y_46_im / y_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -6.5e+53) || ~(((y_46_re <= 1200000000.0) || (~((y_46_re <= 3.1e+43)) && (y_46_re <= 1.4e+102)))))
		tmp = x_46_re / y_46_re;
	else
		tmp = (1.0 / y_46_im) * (x_46_im + (x_46_re / (y_46_im / 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$46$re, -6.5e+53], N[Not[Or[LessEqual[y$46$re, 1200000000.0], And[N[Not[LessEqual[y$46$re, 3.1e+43]], $MachinePrecision], LessEqual[y$46$re, 1.4e+102]]]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(1.0 / y$46$im), $MachinePrecision] * N[(x$46$im + N[(x$46$re / N[(y$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -6.50000000000000017e53 or 1.2e9 < y.re < 3.1000000000000002e43 or 1.40000000000000009e102 < y.re

   1. Initial program 50.0%

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

   2. Taylor expanded in y.re around inf 72.7%

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

   if -6.50000000000000017e53 < y.re < 1.2e9 or 3.1000000000000002e43 < y.re < 1.40000000000000009e102

   1. Initial program 66.4%

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

      1. add-sqr-sqrt 66.4%

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

      2. *-un-lft-identity 66.4%

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

      3. times-frac 66.4%

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

      4. hypot-def 66.5%

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

      5. fma-def 66.5%

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

      6. hypot-def 82.7%

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

   3. Applied egg-rr 82.7%

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

   4. Taylor expanded in y.re around 0 38.7%

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

      1. associate-/l* 39.3%

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

   6. Simplified 39.3%

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

   7. Taylor expanded in y.re around 0 74.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -6.5 \cdot 10^{+53} \lor \neg \left(y.re \leq 1200000000 \lor \neg \left(y.re \leq 3.1 \cdot 10^{+43}\right) \land y.re \leq 1.4 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y.im} \cdot \left(x.im + \frac{x.re}{\frac{y.im}{y.re}}\right)\\ \end{array} \]

Alternative 15: 63.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+46} \lor \neg \left(y.re \leq 7.5 \cdot 10^{-23} \lor \neg \left(y.re \leq 2.3 \cdot 10^{+45}\right) \land y.re \leq 1.4 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -2.7e+46)
         (not
          (or (<= y.re 7.5e-23)
              (and (not (<= y.re 2.3e+45)) (<= y.re 1.4e+102)))))
   (/ x.re y.re)
   (/ x.im y.im)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.7e+46) || !((y_46_re <= 7.5e-23) || (!(y_46_re <= 2.3e+45) && (y_46_re <= 1.4e+102)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-2.7d+46)) .or. (.not. (y_46re <= 7.5d-23) .or. (.not. (y_46re <= 2.3d+45)) .and. (y_46re <= 1.4d+102))) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -2.7e+46) || !((y_46_re <= 7.5e-23) || (!(y_46_re <= 2.3e+45) && (y_46_re <= 1.4e+102)))) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -2.7e+46) or not ((y_46_re <= 7.5e-23) or (not (y_46_re <= 2.3e+45) and (y_46_re <= 1.4e+102))):
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -2.7e+46) || !((y_46_re <= 7.5e-23) || (!(y_46_re <= 2.3e+45) && (y_46_re <= 1.4e+102))))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -2.7e+46) || ~(((y_46_re <= 7.5e-23) || (~((y_46_re <= 2.3e+45)) && (y_46_re <= 1.4e+102)))))
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -2.7e+46], N[Not[Or[LessEqual[y$46$re, 7.5e-23], And[N[Not[LessEqual[y$46$re, 2.3e+45]], $MachinePrecision], LessEqual[y$46$re, 1.4e+102]]]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -2.7000000000000002e46 or 7.4999999999999998e-23 < y.re < 2.30000000000000012e45 or 1.40000000000000009e102 < y.re

   1. Initial program 54.4%

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

   2. Taylor expanded in y.re around inf 69.3%

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

   if -2.7000000000000002e46 < y.re < 7.4999999999999998e-23 or 2.30000000000000012e45 < y.re < 1.40000000000000009e102

   1. Initial program 64.4%

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

   2. Taylor expanded in y.re around 0 61.4%

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

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -2.7 \cdot 10^{+46} \lor \neg \left(y.re \leq 7.5 \cdot 10^{-23} \lor \neg \left(y.re \leq 2.3 \cdot 10^{+45}\right) \land y.re \leq 1.4 \cdot 10^{+102}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 16: 42.4% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 60.1%

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

2. Taylor expanded in y.re around 0 41.7%

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

3. Final simplification 41.7%

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

Reproduce

herbie shell --seed 2024024 
(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))))