_divideComplex, real part

Percentage Accurate: 61.6% → 85.1%

Time: 9.1s

Alternatives: 13

Speedup: 2.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.6% 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: 85.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x.re \cdot y.re + x.im \cdot y.im\\ \mathbf{if}\;\frac{t_0}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{t_0}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0 (+ (* x.re y.re) (* x.im y.im))))
   (if (<= (/ t_0 (+ (* y.re y.re) (* y.im y.im))) 2e+289)
     (* (/ 1.0 (hypot y.re y.im)) (/ t_0 (hypot y.re y.im)))
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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))) <= 2e+289) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / 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);
	double tmp;
	if ((t_0 / ((y_46_re * y_46_re) + (y_46_im * y_46_im))) <= 2e+289) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) * (t_0 / Math.hypot(y_46_re, y_46_im));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	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))) <= 2e+289:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) * (t_0 / math.hypot(y_46_re, y_46_im))
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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))) <= 2e+289)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) * Float64(t_0 / hypot(y_46_re, y_46_im)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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))) <= 2e+289)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) * (t_0 / hypot(y_46_re, y_46_im));
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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], 2e+289], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] * N[(t$95$0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $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))) < 2.0000000000000001e289

   1. Initial program 76.8%

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

      1. *-un-lft-identity 76.8%

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

      2. add-sqr-sqrt 76.8%

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

      3. times-frac 76.8%

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

      4. hypot-def 76.8%

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

      5. fma-def 76.8%

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

      6. hypot-def 96.0%

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

   3. Applied egg-rr 96.0%

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

      1. fma-def 96.0%

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

   5. Applied egg-rr 96.0%

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

   if 2.0000000000000001e289 < (/.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 9.7%

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

   2. Taylor expanded in y.re around 0 42.8%

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

      1. +-commutative 42.8%

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

      2. *-commutative 42.8%

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

      3. unpow2 42.8%

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

      4. times-frac 65.7%

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

   4. Simplified 65.7%

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

      1. associate-*r/ 66.9%

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

   6. Applied egg-rr 66.9%

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

4. Final simplification 88.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)} \cdot \frac{x.re \cdot y.re + x.im \cdot y.im}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 2: 79.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re \cdot y.re}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 (- INFINITY))
     (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
     (if (<= t_0 -5e-324)
       t_0
       (if (<= t_0 0.0)
         (/ (/ 1.0 (hypot y.re y.im)) (/ (hypot y.re y.im) (* x.re y.re)))
         (if (<= t_0 2e+289)
           (/ (fma x.re y.re (* x.im y.im)) (fma y.re y.re (* y.im y.im)))
           (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (t_0 <= -5e-324) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) / (hypot(y_46_re, y_46_im) / (x_46_re * y_46_re));
	} else if (t_0 <= 2e+289) {
		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 {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	elseif (t_0 <= -5e-324)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) / Float64(hypot(y_46_re, y_46_im) / Float64(x_46_re * y_46_re)));
	elseif (t_0 <= 2e+289)
		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)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / 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[(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[t$95$0, (-Infinity)], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-324], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+289], 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], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 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))) < -inf.0

   1. Initial program 23.4%

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

   2. Taylor expanded in y.re around inf 26.3%

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

      1. associate-/l* 35.1%

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

      2. associate-/r/ 35.1%

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

      3. unpow2 35.1%

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

      4. associate-/r* 69.5%

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

   4. Simplified 69.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -4.94066e-324

   1. Initial program 98.0%

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

   if -4.94066e-324 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -0.0

   1. Initial program 50.7%

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

      1. *-un-lft-identity 50.7%

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

      2. add-sqr-sqrt 50.7%

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

      3. times-frac 50.7%

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

      4. hypot-def 50.7%

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

      5. fma-def 50.7%

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

      6. hypot-def 95.6%

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

   3. Applied egg-rr 95.6%

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

      1. clear-num 95.6%

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

      2. un-div-inv 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in x.re around inf 50.7%

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

      1. associate-*l/ 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. hypot-def 84.7%

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

      5. *-lft-identity 84.7%

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

   8. Simplified 84.7%

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

   if -0.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.0000000000000001e289

   1. Initial program 99.4%

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

      1. fma-def 99.4%

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

      2. fma-def 99.4%

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

   3. Simplified 99.4%

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

   if 2.0000000000000001e289 < (/.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 9.7%

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

   2. Taylor expanded in y.re around 0 42.8%

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

      1. +-commutative 42.8%

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

      2. *-commutative 42.8%

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

      3. unpow2 42.8%

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

      4. times-frac 65.7%

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

   4. Simplified 65.7%

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

      1. associate-*r/ 66.9%

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

   6. Applied egg-rr 66.9%

      \[\leadsto \frac{x.im}{y.im} + \color{blue}{\frac{\frac{y.re}{y.im} \cdot x.re}{y.im}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 0:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re \cdot y.re}}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x.re, y.re, x.im \cdot y.im\right)}{\mathsf{fma}\left(y.re, y.re, y.im \cdot y.im\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 3: 79.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;t_0 \leq -5 \cdot 10^{-324}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 0:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re \cdot y.re}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+289}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (let* ((t_0
         (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))))
   (if (<= t_0 (- INFINITY))
     (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
     (if (<= t_0 -5e-324)
       t_0
       (if (<= t_0 0.0)
         (/ (/ 1.0 (hypot y.re y.im)) (/ (hypot y.re y.im) (* x.re y.re)))
         (if (<= t_0 2e+289)
           t_0
           (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double 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 tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (t_0 <= -5e-324) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / hypot(y_46_re, y_46_im)) / (hypot(y_46_re, y_46_im) / (x_46_re * y_46_re));
	} else if (t_0 <= 2e+289) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / 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 tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (t_0 <= -5e-324) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = (1.0 / Math.hypot(y_46_re, y_46_im)) / (Math.hypot(y_46_re, y_46_im) / (x_46_re * y_46_re));
	} else if (t_0 <= 2e+289) {
		tmp = t_0;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	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))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	elif t_0 <= -5e-324:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = (1.0 / math.hypot(y_46_re, y_46_im)) / (math.hypot(y_46_re, y_46_im) / (x_46_re * y_46_re))
	elif t_0 <= 2e+289:
		tmp = t_0
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	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)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	elseif (t_0 <= -5e-324)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(1.0 / hypot(y_46_re, y_46_im)) / Float64(hypot(y_46_re, y_46_im) / Float64(x_46_re * y_46_re)));
	elseif (t_0 <= 2e+289)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	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));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	elseif (t_0 <= -5e-324)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = (1.0 / hypot(y_46_re, y_46_im)) / (hypot(y_46_re, y_46_im) / (x_46_re * y_46_re));
	elseif (t_0 <= 2e+289)
		tmp = t_0;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := 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]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, -5e-324], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision] / N[(x$46$re * y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+289], t$95$0, N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 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))) < -inf.0

   1. Initial program 23.4%

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

   2. Taylor expanded in y.re around inf 26.3%

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

      1. associate-/l* 35.1%

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

      2. associate-/r/ 35.1%

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

      3. unpow2 35.1%

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

      4. associate-/r* 69.5%

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

   4. Simplified 69.5%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -4.94066e-324 or -0.0 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < 2.0000000000000001e289

   1. Initial program 98.8%

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

   if -4.94066e-324 < (/.f64 (+.f64 (*.f64 x.re y.re) (*.f64 x.im y.im)) (+.f64 (*.f64 y.re y.re) (*.f64 y.im y.im))) < -0.0

   1. Initial program 50.7%

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

      1. *-un-lft-identity 50.7%

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

      2. add-sqr-sqrt 50.7%

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

      3. times-frac 50.7%

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

      4. hypot-def 50.7%

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

      5. fma-def 50.7%

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

      6. hypot-def 95.6%

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

   3. Applied egg-rr 95.6%

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

      1. clear-num 95.6%

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

      2. un-div-inv 95.6%

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

   5. Applied egg-rr 95.6%

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

   6. Taylor expanded in x.re around inf 50.7%

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

      1. associate-*l/ 50.7%

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

      2. unpow2 50.7%

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

      3. unpow2 50.7%

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

      4. hypot-def 84.7%

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

      5. *-lft-identity 84.7%

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

   8. Simplified 84.7%

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

   if 2.0000000000000001e289 < (/.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 9.7%

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

   2. Taylor expanded in y.re around 0 42.8%

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

      1. +-commutative 42.8%

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

      2. *-commutative 42.8%

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

      3. unpow2 42.8%

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

      4. times-frac 65.7%

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

   4. Simplified 65.7%

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

      1. associate-*r/ 66.9%

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

   6. Applied egg-rr 66.9%

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

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -\infty:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq -5 \cdot 10^{-324}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 0:\\ \;\;\;\;\frac{\frac{1}{\mathsf{hypot}\left(y.re, y.im\right)}}{\frac{\mathsf{hypot}\left(y.re, y.im\right)}{x.re \cdot y.re}}\\ \mathbf{elif}\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im} \leq 2 \cdot 10^{+289}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 4: 79.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{+100}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.6e-11)
   (* (+ x.im (* y.re (/ x.re y.im))) (/ -1.0 (hypot y.re y.im)))
   (if (<= y.im 3.6e-181)
     (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
     (if (<= y.im 2.25e+100)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (+ (/ x.im y.im) (/ 1.0 (* (/ y.im y.re) (/ y.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.6e-11) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 3.6e-181) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (y_46_im <= 2.25e+100) {
		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 {
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	}
	return tmp;
}

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.6e-11) {
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) * (-1.0 / Math.hypot(y_46_re, y_46_im));
	} else if (y_46_im <= 3.6e-181) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (y_46_im <= 2.25e+100) {
		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 {
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.6e-11:
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) * (-1.0 / math.hypot(y_46_re, y_46_im))
	elif y_46_im <= 3.6e-181:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	elif y_46_im <= 2.25e+100:
		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:
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.6e-11)
		tmp = Float64(Float64(x_46_im + Float64(y_46_re * Float64(x_46_re / y_46_im))) * Float64(-1.0 / hypot(y_46_re, y_46_im)));
	elseif (y_46_im <= 3.6e-181)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	elseif (y_46_im <= 2.25e+100)
		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)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(Float64(y_46_im / y_46_re) * Float64(y_46_im / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.6e-11)
		tmp = (x_46_im + (y_46_re * (x_46_re / y_46_im))) * (-1.0 / hypot(y_46_re, y_46_im));
	elseif (y_46_im <= 3.6e-181)
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	elseif (y_46_im <= 2.25e+100)
		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
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.6e-11], N[(N[(x$46$im + N[(y$46$re * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[Sqrt[y$46$re ^ 2 + y$46$im ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.6e-181], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 2.25e+100], 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], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -1.59999999999999997e-11

   1. Initial program 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. add-sqr-sqrt 56.4%

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

      3. times-frac 56.4%

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

      4. hypot-def 56.4%

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

      5. fma-def 56.4%

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

      6. hypot-def 69.6%

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

   3. Applied egg-rr 69.6%

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

   4. Taylor expanded in y.im around -inf 76.2%

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

      1. neg-mul-1 76.2%

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

      2. +-commutative 76.2%

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

      3. mul-1-neg 76.2%

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

      4. unsub-neg 76.2%

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

      5. *-commutative 76.2%

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

      6. associate-*r/ 83.6%

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

   6. Simplified 83.6%

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

   if -1.59999999999999997e-11 < y.im < 3.5999999999999999e-181

   1. Initial program 60.2%

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

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

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

      1. associate-/l* 70.6%

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

      2. associate-/r/ 71.8%

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

      3. unpow2 71.8%

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

      4. associate-/r* 85.0%

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

   4. Simplified 85.0%

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

   if 3.5999999999999999e-181 < y.im < 2.25000000000000018e100

   1. Initial program 78.3%

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

   if 2.25000000000000018e100 < y.im

   1. Initial program 30.9%

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

   2. Taylor expanded in y.re around 0 68.7%

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

      1. +-commutative 68.7%

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

      2. *-commutative 68.7%

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

      3. unpow2 68.7%

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

   4. Simplified 68.7%

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

      1. clear-num 68.8%

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

      2. inv-pow 68.8%

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

      3. *-commutative 68.8%

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

      4. times-frac 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 83.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.6 \cdot 10^{-11}:\\ \;\;\;\;\left(x.im + y.re \cdot \frac{x.re}{y.im}\right) \cdot \frac{-1}{\mathsf{hypot}\left(y.re, y.im\right)}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 2.25 \cdot 10^{+100}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 5: 78.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.1e-11)
   (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))
   (if (<= y.im 3.6e-181)
     (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
     (if (<= y.im 1.6e+104)
       (/ (+ (* x.re y.re) (* x.im y.im)) (+ (* y.re y.re) (* y.im y.im)))
       (+ (/ x.im y.im) (/ 1.0 (* (/ y.im y.re) (/ y.im x.re))))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.1e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 3.6e-181) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (y_46_im <= 1.6e+104) {
		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 {
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.1d-11)) then
        tmp = (x_46im / y_46im) + (y_46re / (y_46im * (y_46im / x_46re)))
    else if (y_46im <= 3.6d-181) then
        tmp = (x_46re / y_46re) + (x_46im * ((y_46im / y_46re) / y_46re))
    else if (y_46im <= 1.6d+104) then
        tmp = ((x_46re * y_46re) + (x_46im * y_46im)) / ((y_46re * y_46re) + (y_46im * y_46im))
    else
        tmp = (x_46im / y_46im) + (1.0d0 / ((y_46im / y_46re) * (y_46im / x_46re)))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.1e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 3.6e-181) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else if (y_46_im <= 1.6e+104) {
		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 {
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.1e-11:
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)))
	elif y_46_im <= 3.6e-181:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	elif y_46_im <= 1.6e+104:
		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:
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.1e-11)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	elseif (y_46_im <= 3.6e-181)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	elseif (y_46_im <= 1.6e+104)
		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)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(1.0 / Float64(Float64(y_46_im / y_46_re) * Float64(y_46_im / x_46_re))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.1e-11)
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	elseif (y_46_im <= 3.6e-181)
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	elseif (y_46_im <= 1.6e+104)
		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
		tmp = (x_46_im / y_46_im) + (1.0 / ((y_46_im / y_46_re) * (y_46_im / x_46_re)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.1e-11], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 3.6e-181], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.6e+104], 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], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(1.0 / N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y.im < -2.0999999999999999e-11

   1. Initial program 56.4%

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

   2. Taylor expanded in y.re around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.4%

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

      3. frac-times 81.5%

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

      4. *-un-lft-identity 81.5%

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

   6. Applied egg-rr 81.5%

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

   if -2.0999999999999999e-11 < y.im < 3.5999999999999999e-181

   1. Initial program 60.2%

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

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

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

      1. associate-/l* 70.6%

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

      2. associate-/r/ 71.8%

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

      3. unpow2 71.8%

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

      4. associate-/r* 85.0%

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

   4. Simplified 85.0%

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

   if 3.5999999999999999e-181 < y.im < 1.6e104

   1. Initial program 78.3%

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

   if 1.6e104 < y.im

   1. Initial program 30.9%

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

   2. Taylor expanded in y.re around 0 68.7%

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

      1. +-commutative 68.7%

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

      2. *-commutative 68.7%

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

      3. unpow2 68.7%

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

   4. Simplified 68.7%

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

      1. clear-num 68.8%

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

      2. inv-pow 68.8%

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

      3. *-commutative 68.8%

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

      4. times-frac 87.4%

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

   6. Applied egg-rr 87.4%

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

      1. unpow-1 87.4%

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

   8. Simplified 87.4%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 3.6 \cdot 10^{-181}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{elif}\;y.im \leq 1.6 \cdot 10^{+104}:\\ \;\;\;\;\frac{x.re \cdot y.re + x.im \cdot y.im}{y.re \cdot y.re + y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{1}{\frac{y.im}{y.re} \cdot \frac{y.im}{x.re}}\\ \end{array} \]

Alternative 6: 67.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-11} \lor \neg \left(y.im \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -1.9e-11) (not (<= y.im 1.4e-56)))
   (+ (/ x.im y.im) (* y.re (/ x.re (* y.im y.im))))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.9e-11) || !(y_46_im <= 1.4e-56)) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-1.9d-11)) .or. (.not. (y_46im <= 1.4d-56))) then
        tmp = (x_46im / y_46im) + (y_46re * (x_46re / (y_46im * y_46im)))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -1.9e-11) || !(y_46_im <= 1.4e-56)) {
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -1.9e-11) or not (y_46_im <= 1.4e-56):
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -1.9e-11) || !(y_46_im <= 1.4e-56))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re * Float64(x_46_re / Float64(y_46_im * y_46_im))));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -1.9e-11) || ~((y_46_im <= 1.4e-56)))
		tmp = (x_46_im / y_46_im) + (y_46_re * (x_46_re / (y_46_im * y_46_im)));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -1.9e-11], N[Not[LessEqual[y$46$im, 1.4e-56]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re * N[(x$46$re / N[(y$46$im * y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -1.8999999999999999e-11 or 1.39999999999999997e-56 < y.im

   1. Initial program 56.4%

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

      1. *-un-lft-identity 56.4%

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

      2. add-sqr-sqrt 56.4%

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

      3. times-frac 56.4%

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

      4. hypot-def 56.4%

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

      5. fma-def 56.4%

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

      6. hypot-def 69.8%

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

   3. Applied egg-rr 69.8%

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

   4. Taylor expanded in y.re around 0 70.0%

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

      1. +-commutative 70.0%

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

      2. *-commutative 70.0%

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

      3. unpow2 70.0%

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

      4. associate-*r/ 72.6%

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

   6. Simplified 72.6%

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

   if -1.8999999999999999e-11 < y.im < 1.39999999999999997e-56

   1. Initial program 63.7%

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

   2. Taylor expanded in y.re around inf 64.5%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.9 \cdot 10^{-11} \lor \neg \left(y.im \leq 1.4 \cdot 10^{-56}\right):\\ \;\;\;\;\frac{x.im}{y.im} + y.re \cdot \frac{x.re}{y.im \cdot y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 7: 68.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-11} \lor \neg \left(y.im \leq 3.5 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.im -2.05e-11) (not (<= y.im 3.5e-127)))
   (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im)))
   (/ x.re y.re)))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.05e-11) || !(y_46_im <= 3.5e-127)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46im <= (-2.05d-11)) .or. (.not. (y_46im <= 3.5d-127))) then
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    else
        tmp = x_46re / y_46re
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_im <= -2.05e-11) || !(y_46_im <= 3.5e-127)) {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	} else {
		tmp = x_46_re / y_46_re;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_im <= -2.05e-11) or not (y_46_im <= 3.5e-127):
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	else:
		tmp = x_46_re / y_46_re
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_im <= -2.05e-11) || !(y_46_im <= 3.5e-127))
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	else
		tmp = Float64(x_46_re / y_46_re);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_im <= -2.05e-11) || ~((y_46_im <= 3.5e-127)))
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	else
		tmp = x_46_re / y_46_re;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$im, -2.05e-11], N[Not[LessEqual[y$46$im, 3.5e-127]], $MachinePrecision]], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$46$re / y$46$re), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.05e-11 or 3.49999999999999989e-127 < y.im

   1. Initial program 57.3%

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

   2. Taylor expanded in y.re around 0 68.5%

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

      1. +-commutative 68.5%

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

      2. *-commutative 68.5%

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

      3. unpow2 68.5%

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

      4. times-frac 76.7%

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

   4. Simplified 76.7%

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

   if -2.05e-11 < y.im < 3.49999999999999989e-127

   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. Taylor expanded in y.re around inf 67.0%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.05 \cdot 10^{-11} \lor \neg \left(y.im \leq 3.5 \cdot 10^{-127}\right):\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \end{array} \]

Alternative 8: 70.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+107} \lor \neg \left(y.re \leq 3 \cdot 10^{+177}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (or (<= y.re -1.65e+107) (not (<= y.re 3e+177)))
   (/ x.re y.re)
   (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.65e+107) || !(y_46_re <= 3e+177)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if ((y_46re <= (-1.65d+107)) .or. (.not. (y_46re <= 3d+177))) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if ((y_46_re <= -1.65e+107) || !(y_46_re <= 3e+177)) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if (y_46_re <= -1.65e+107) or not (y_46_re <= 3e+177):
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if ((y_46_re <= -1.65e+107) || !(y_46_re <= 3e+177))
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if ((y_46_re <= -1.65e+107) || ~((y_46_re <= 3e+177)))
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[Or[LessEqual[y$46$re, -1.65e+107], N[Not[LessEqual[y$46$re, 3e+177]], $MachinePrecision]], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y.re < -1.65000000000000016e107 or 3e177 < y.re

   1. Initial program 29.4%

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

   2. Taylor expanded in y.re around inf 79.8%

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

   if -1.65000000000000016e107 < y.re < 3e177

   1. Initial program 69.0%

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

   2. Taylor expanded in y.re around 0 62.5%

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

      1. +-commutative 62.5%

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

      2. *-commutative 62.5%

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

      3. unpow2 62.5%

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

      4. times-frac 70.7%

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

   4. Simplified 70.7%

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

      1. associate-*r/ 72.1%

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

   6. Applied egg-rr 72.1%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.re \leq -1.65 \cdot 10^{+107} \lor \neg \left(y.re \leq 3 \cdot 10^{+177}\right):\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 9: 68.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.85e-11)
   (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))
   (if (<= y.im 5e-127)
     (/ x.re y.re)
     (+ (/ x.im y.im) (* (/ y.re y.im) (/ x.re y.im))))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.85e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 5e-127) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.85d-11)) then
        tmp = (x_46im / y_46im) + (y_46re / (y_46im * (y_46im / x_46re)))
    else if (y_46im <= 5d-127) then
        tmp = x_46re / y_46re
    else
        tmp = (x_46im / y_46im) + ((y_46re / y_46im) * (x_46re / y_46im))
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.85e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 5e-127) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.85e-11:
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)))
	elif y_46_im <= 5e-127:
		tmp = x_46_re / y_46_re
	else:
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im))
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.85e-11)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	elseif (y_46_im <= 5e-127)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(y_46_re / y_46_im) * Float64(x_46_re / y_46_im)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.85e-11)
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	elseif (y_46_im <= 5e-127)
		tmp = x_46_re / y_46_re;
	else
		tmp = (x_46_im / y_46_im) + ((y_46_re / y_46_im) * (x_46_re / y_46_im));
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.85e-11], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 5e-127], N[(x$46$re / y$46$re), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(y$46$re / y$46$im), $MachinePrecision] * N[(x$46$re / y$46$im), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.8500000000000001e-11

   1. Initial program 56.4%

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

   2. Taylor expanded in y.re around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.4%

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

      3. frac-times 81.5%

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

      4. *-un-lft-identity 81.5%

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

   6. Applied egg-rr 81.5%

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

   if -1.8500000000000001e-11 < y.im < 4.9999999999999997e-127

   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. Taylor expanded in y.re around inf 67.0%

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

   if 4.9999999999999997e-127 < y.im

   1. Initial program 58.1%

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

   2. Taylor expanded in y.re around 0 65.3%

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

      1. +-commutative 65.3%

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

      2. *-commutative 65.3%

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

      3. unpow2 65.3%

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

      4. times-frac 72.6%

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

   4. Simplified 72.6%

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

4. Final simplification 73.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.85 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 5 \cdot 10^{-127}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im} \cdot \frac{x.re}{y.im}\\ \end{array} \]

Alternative 10: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.8e-11)
   (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))
   (if (<= y.im 1.02e-119)
     (+ (/ x.re y.re) (* (/ y.im y.re) (/ x.im y.re)))
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.8e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 1.02e-119) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.8d-11)) then
        tmp = (x_46im / y_46im) + (y_46re / (y_46im * (y_46im / x_46re)))
    else if (y_46im <= 1.02d-119) then
        tmp = (x_46re / y_46re) + ((y_46im / y_46re) * (x_46im / y_46re))
    else
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.8e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 1.02e-119) {
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.8e-11:
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)))
	elif y_46_im <= 1.02e-119:
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re))
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.8e-11)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	elseif (y_46_im <= 1.02e-119)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(Float64(y_46_im / y_46_re) * Float64(x_46_im / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.8e-11)
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	elseif (y_46_im <= 1.02e-119)
		tmp = (x_46_re / y_46_re) + ((y_46_im / y_46_re) * (x_46_im / y_46_re));
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.8e-11], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 1.02e-119], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(N[(y$46$im / y$46$re), $MachinePrecision] * N[(x$46$im / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.79999999999999992e-11

   1. Initial program 56.4%

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

   2. Taylor expanded in y.re around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.4%

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

      3. frac-times 81.5%

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

      4. *-un-lft-identity 81.5%

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

   6. Applied egg-rr 81.5%

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

   if -1.79999999999999992e-11 < y.im < 1.02e-119

   1. Initial program 63.3%

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

   2. Taylor expanded in y.re around inf 70.8%

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

      1. unpow2 70.8%

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

      2. times-frac 82.1%

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

   4. Simplified 82.1%

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

   if 1.02e-119 < y.im

   1. Initial program 57.6%

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

   2. Taylor expanded in y.re around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 73.4%

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

   4. Simplified 73.4%

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

      1. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.8 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 1.02 \cdot 10^{-119}:\\ \;\;\;\;\frac{x.re}{y.re} + \frac{y.im}{y.re} \cdot \frac{x.im}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 11: 74.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -1.7e-11)
   (+ (/ x.im y.im) (/ y.re (* y.im (/ y.im x.re))))
   (if (<= y.im 8.5e-120)
     (+ (/ x.re y.re) (* x.im (/ (/ y.im y.re) y.re)))
     (+ (/ x.im y.im) (/ (* x.re (/ y.re y.im)) y.im)))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.7e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 8.5e-120) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-1.7d-11)) then
        tmp = (x_46im / y_46im) + (y_46re / (y_46im * (y_46im / x_46re)))
    else if (y_46im <= 8.5d-120) then
        tmp = (x_46re / y_46re) + (x_46im * ((y_46im / y_46re) / y_46re))
    else
        tmp = (x_46im / y_46im) + ((x_46re * (y_46re / y_46im)) / y_46im)
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -1.7e-11) {
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	} else if (y_46_im <= 8.5e-120) {
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	} else {
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -1.7e-11:
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)))
	elif y_46_im <= 8.5e-120:
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re))
	else:
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im)
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -1.7e-11)
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(y_46_re / Float64(y_46_im * Float64(y_46_im / x_46_re))));
	elseif (y_46_im <= 8.5e-120)
		tmp = Float64(Float64(x_46_re / y_46_re) + Float64(x_46_im * Float64(Float64(y_46_im / y_46_re) / y_46_re)));
	else
		tmp = Float64(Float64(x_46_im / y_46_im) + Float64(Float64(x_46_re * Float64(y_46_re / y_46_im)) / y_46_im));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -1.7e-11)
		tmp = (x_46_im / y_46_im) + (y_46_re / (y_46_im * (y_46_im / x_46_re)));
	elseif (y_46_im <= 8.5e-120)
		tmp = (x_46_re / y_46_re) + (x_46_im * ((y_46_im / y_46_re) / y_46_re));
	else
		tmp = (x_46_im / y_46_im) + ((x_46_re * (y_46_re / y_46_im)) / y_46_im);
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -1.7e-11], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(y$46$re / N[(y$46$im * N[(y$46$im / x$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y$46$im, 8.5e-120], N[(N[(x$46$re / y$46$re), $MachinePrecision] + N[(x$46$im * N[(N[(y$46$im / y$46$re), $MachinePrecision] / y$46$re), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$46$im / y$46$im), $MachinePrecision] + N[(N[(x$46$re * N[(y$46$re / y$46$im), $MachinePrecision]), $MachinePrecision] / y$46$im), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y.im < -1.6999999999999999e-11

   1. Initial program 56.4%

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

   2. Taylor expanded in y.re around 0 72.2%

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

      1. +-commutative 72.2%

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

      2. *-commutative 72.2%

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

      3. unpow2 72.2%

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

      4. times-frac 81.4%

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

   4. Simplified 81.4%

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

      1. *-commutative 81.4%

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

      2. clear-num 81.4%

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

      3. frac-times 81.5%

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

      4. *-un-lft-identity 81.5%

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

   6. Applied egg-rr 81.5%

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

   if -1.6999999999999999e-11 < y.im < 8.50000000000000059e-120

   1. Initial program 63.3%

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

   2. Taylor expanded in y.re around inf 70.8%

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

      1. associate-/l* 70.2%

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

      2. associate-/r/ 71.2%

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

      3. unpow2 71.2%

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

      4. associate-/r* 82.7%

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

   4. Simplified 82.7%

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

   if 8.50000000000000059e-120 < y.im

   1. Initial program 57.6%

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

   2. Taylor expanded in y.re around 0 66.0%

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

      1. +-commutative 66.0%

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

      2. *-commutative 66.0%

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

      3. unpow2 66.0%

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

      4. times-frac 73.4%

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

   4. Simplified 73.4%

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

      1. associate-*r/ 74.4%

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

   6. Applied egg-rr 74.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -1.7 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{y.re}{y.im \cdot \frac{y.im}{x.re}}\\ \mathbf{elif}\;y.im \leq 8.5 \cdot 10^{-120}:\\ \;\;\;\;\frac{x.re}{y.re} + x.im \cdot \frac{\frac{y.im}{y.re}}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im} + \frac{x.re \cdot \frac{y.re}{y.im}}{y.im}\\ \end{array} \]

Alternative 12: 62.9% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \end{array} \]

FPCore

(FPCore (x.re x.im y.re y.im)
 :precision binary64
 (if (<= y.im -2.2e-11)
   (/ x.im y.im)
   (if (<= y.im 6.8e-52) (/ x.re y.re) (/ x.im y.im))))

C

double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.2e-11) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 6.8e-52) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Fortran

real(8) function code(x_46re, x_46im, y_46re, y_46im)
    real(8), intent (in) :: x_46re
    real(8), intent (in) :: x_46im
    real(8), intent (in) :: y_46re
    real(8), intent (in) :: y_46im
    real(8) :: tmp
    if (y_46im <= (-2.2d-11)) then
        tmp = x_46im / y_46im
    else if (y_46im <= 6.8d-52) then
        tmp = x_46re / y_46re
    else
        tmp = x_46im / y_46im
    end if
    code = tmp
end function

Java

public static double code(double x_46_re, double x_46_im, double y_46_re, double y_46_im) {
	double tmp;
	if (y_46_im <= -2.2e-11) {
		tmp = x_46_im / y_46_im;
	} else if (y_46_im <= 6.8e-52) {
		tmp = x_46_re / y_46_re;
	} else {
		tmp = x_46_im / y_46_im;
	}
	return tmp;
}

Python

def code(x_46_re, x_46_im, y_46_re, y_46_im):
	tmp = 0
	if y_46_im <= -2.2e-11:
		tmp = x_46_im / y_46_im
	elif y_46_im <= 6.8e-52:
		tmp = x_46_re / y_46_re
	else:
		tmp = x_46_im / y_46_im
	return tmp

Julia

function code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0
	if (y_46_im <= -2.2e-11)
		tmp = Float64(x_46_im / y_46_im);
	elseif (y_46_im <= 6.8e-52)
		tmp = Float64(x_46_re / y_46_re);
	else
		tmp = Float64(x_46_im / y_46_im);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x_46_re, x_46_im, y_46_re, y_46_im)
	tmp = 0.0;
	if (y_46_im <= -2.2e-11)
		tmp = x_46_im / y_46_im;
	elseif (y_46_im <= 6.8e-52)
		tmp = x_46_re / y_46_re;
	else
		tmp = x_46_im / y_46_im;
	end
	tmp_2 = tmp;
end

Wolfram

code[x$46$re_, x$46$im_, y$46$re_, y$46$im_] := If[LessEqual[y$46$im, -2.2e-11], N[(x$46$im / y$46$im), $MachinePrecision], If[LessEqual[y$46$im, 6.8e-52], N[(x$46$re / y$46$re), $MachinePrecision], N[(x$46$im / y$46$im), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y.im < -2.2000000000000002e-11 or 6.80000000000000035e-52 < y.im

   1. Initial program 56.1%

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

   2. Taylor expanded in y.re around 0 61.8%

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

   if -2.2000000000000002e-11 < y.im < 6.80000000000000035e-52

   1. Initial program 64.1%

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

   2. Taylor expanded in y.re around inf 64.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y.im \leq -2.2 \cdot 10^{-11}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \mathbf{elif}\;y.im \leq 6.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{x.re}{y.re}\\ \mathbf{else}:\\ \;\;\;\;\frac{x.im}{y.im}\\ \end{array} \]

Alternative 13: 42.7% accurate, 5.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.3%

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

2. Taylor expanded in y.re around 0 43.6%

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

3. Final simplification 43.6%

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

Reproduce

herbie shell --seed 2023201 
(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))))